# Properties

 Label 1050.2.a.h.1.1 Level $1050$ Weight $2$ Character 1050.1 Self dual yes Analytic conductor $8.384$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1050.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{21} -4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +2.00000 q^{41} +1.00000 q^{42} +12.0000 q^{43} +4.00000 q^{44} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} +4.00000 q^{59} -2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} +8.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} -4.00000 q^{76} -4.00000 q^{77} -2.00000 q^{78} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} -1.00000 q^{84} -12.0000 q^{86} +6.00000 q^{87} -4.00000 q^{88} +2.00000 q^{89} -2.00000 q^{91} +8.00000 q^{92} -8.00000 q^{93} -8.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ −4.00000 −0.852803
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.00000 0.696311
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 2.00000 0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −4.00000 −0.529813
$$58$$ −6.00000 −0.787839
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 8.00000 1.01600
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ −4.00000 −0.455842
$$78$$ −2.00000 −0.226455
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ −12.0000 −1.29399
$$87$$ 6.00000 0.643268
$$88$$ −4.00000 −0.426401
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 8.00000 0.834058
$$93$$ −8.00000 −0.829561
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 2.00000 0.198030
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 20.0000 1.93347 0.966736 0.255774i $$-0.0823304\pi$$
0.966736 + 0.255774i $$0.0823304\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ −1.00000 −0.0944911
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 2.00000 0.184900
$$118$$ −4.00000 −0.368230
$$119$$ 2.00000 0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 2.00000 0.181071
$$123$$ 2.00000 0.180334
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 4.00000 0.346844
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ −8.00000 −0.681005
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ −8.00000 −0.671345
$$143$$ 8.00000 0.668994
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −14.0000 −1.15865
$$147$$ 1.00000 0.0824786
$$148$$ 2.00000 0.164399
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 4.00000 0.324443
$$153$$ −2.00000 −0.161690
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 12.0000 0.914991
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 4.00000 0.300658
$$178$$ −2.00000 −0.149906
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 2.00000 0.148250
$$183$$ −2.00000 −0.147844
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ −8.00000 −0.585018
$$188$$ 8.00000 0.583460
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ −6.00000 −0.422159
$$203$$ −6.00000 −0.421117
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 8.00000 0.556038
$$208$$ 2.00000 0.138675
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 8.00000 0.548151
$$214$$ −20.0000 −1.36717
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 8.00000 0.543075
$$218$$ 2.00000 0.135457
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ −2.00000 −0.134231
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ −6.00000 −0.393919
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ −2.00000 −0.129641
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ −8.00000 −0.509028
$$248$$ 8.00000 0.508001
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 32.0000 2.01182
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ −12.0000 −0.747087
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ −12.0000 −0.741362
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ 2.00000 0.122398
$$268$$ −12.0000 −0.733017
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ −2.00000 −0.121046
$$274$$ 10.0000 0.604122
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ −14.0000 −0.841178 −0.420589 0.907251i $$-0.638177\pi$$
−0.420589 + 0.907251i $$0.638177\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ −2.00000 −0.118056
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 14.0000 0.819288
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 4.00000 0.232104
$$298$$ 18.0000 1.04271
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ −8.00000 −0.460348
$$303$$ 6.00000 0.344691
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −14.0000 −0.786318 −0.393159 0.919470i $$-0.628618\pi$$
−0.393159 + 0.919470i $$0.628618\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 20.0000 1.11629
$$322$$ 8.00000 0.445823
$$323$$ 8.00000 0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ −2.00000 −0.110600
$$328$$ −2.00000 −0.110432
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 2.00000 0.109599
$$334$$ −16.0000 −0.875481
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ 4.00000 0.216295
$$343$$ −1.00000 −0.0539949
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 6.00000 0.321634
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ −4.00000 −0.213201
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 2.00000 0.105851
$$358$$ 20.0000 1.05703
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 10.0000 0.525588
$$363$$ 5.00000 0.262432
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ −8.00000 −0.414781
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ 12.0000 0.618031
$$378$$ 1.00000 0.0514344
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.0000 0.818631
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 12.0000 0.609994
$$388$$ −10.0000 −0.507673
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ −1.00000 −0.0505076
$$393$$ 12.0000 0.605320
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 12.0000 0.598506
$$403$$ −16.0000 −0.797017
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 8.00000 0.396545
$$408$$ 2.00000 0.0990148
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ −8.00000 −0.394132
$$413$$ −4.00000 −0.196827
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 20.0000 0.979404
$$418$$ 16.0000 0.782586
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 8.00000 0.388973
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 2.00000 0.0967868
$$428$$ 20.0000 0.966736
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 6.00000 0.288342 0.144171 0.989553i $$-0.453949\pi$$
0.144171 + 0.989553i $$0.453949\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ −32.0000 −1.53077
$$438$$ −14.0000 −0.668946
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 4.00000 0.190261
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −18.0000 −0.851371
$$448$$ −1.00000 −0.0472456
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 14.0000 0.658505
$$453$$ 8.00000 0.375873
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 38.0000 1.77757 0.888783 0.458329i $$-0.151552\pi$$
0.888783 + 0.458329i $$0.151552\pi$$
$$458$$ 26.0000 1.21490
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −34.0000 −1.58354 −0.791769 0.610821i $$-0.790840\pi$$
−0.791769 + 0.610821i $$0.790840\pi$$
$$462$$ 4.00000 0.186097
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ 20.0000 0.925490 0.462745 0.886492i $$-0.346865\pi$$
0.462745 + 0.886492i $$0.346865\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 12.0000 0.554109
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ −4.00000 −0.184115
$$473$$ 48.0000 2.20704
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2.00000 0.0916698
$$477$$ −6.00000 −0.274721
$$478$$ 16.0000 0.731823
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ −18.0000 −0.819878
$$483$$ −8.00000 −0.364013
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 24.0000 1.08754 0.543772 0.839233i $$-0.316996\pi$$
0.543772 + 0.839233i $$0.316996\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ −12.0000 −0.540453
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ −8.00000 −0.358849
$$498$$ 12.0000 0.537733
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ −4.00000 −0.178529
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ −32.0000 −1.42257
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ −14.0000 −0.619324
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 2.00000 0.0882162
$$515$$ 0 0
$$516$$ 12.0000 0.528271
$$517$$ 32.0000 1.40736
$$518$$ 2.00000 0.0878750
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 16.0000 0.696971
$$528$$ 4.00000 0.174078
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 4.00000 0.173422
$$533$$ 4.00000 0.173259
$$534$$ −2.00000 −0.0865485
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ −20.0000 −0.863064
$$538$$ −30.0000 −1.29339
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 24.0000 1.03089
$$543$$ −10.0000 −0.429141
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ −8.00000 −0.340503
$$553$$ 0 0
$$554$$ 14.0000 0.594803
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 8.00000 0.338667
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ −10.0000 −0.421825
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ −1.00000 −0.0419961
$$568$$ −8.00000 −0.335673
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 8.00000 0.334497
$$573$$ −16.0000 −0.668410
$$574$$ 2.00000 0.0834784
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −26.0000 −1.08239 −0.541197 0.840896i $$-0.682029\pi$$
−0.541197 + 0.840896i $$0.682029\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 10.0000 0.414513
$$583$$ −24.0000 −0.993978
$$584$$ −14.0000 −0.579324
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 2.00000 0.0821995
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ −16.0000 −0.654836
$$598$$ −16.0000 −0.654289
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 12.0000 0.489083
$$603$$ −12.0000 −0.488678
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 4.00000 0.162221
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ 16.0000 0.647291
$$612$$ −2.00000 −0.0808452
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ −10.0000 −0.402585 −0.201292 0.979531i $$-0.564514\pi$$
−0.201292 + 0.979531i $$0.564514\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ −8.00000 −0.320771
$$623$$ −2.00000 −0.0801283
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ 34.0000 1.35891
$$627$$ −16.0000 −0.638978
$$628$$ −14.0000 −0.558661
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 24.0000 0.955425 0.477712 0.878516i $$-0.341466\pi$$
0.477712 + 0.878516i $$0.341466\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 14.0000 0.556011
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 2.00000 0.0792429
$$638$$ −24.0000 −0.950169
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ −20.0000 −0.789337
$$643$$ 20.0000 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 8.00000 0.313545
$$652$$ −12.0000 −0.469956
$$653$$ 34.0000 1.33052 0.665261 0.746611i $$-0.268320\pi$$
0.665261 + 0.746611i $$0.268320\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 14.0000 0.546192
$$658$$ 8.00000 0.311872
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ −4.00000 −0.155347
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 48.0000 1.85857
$$668$$ 16.0000 0.619059
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 1.00000 0.0385758
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 32.0000 1.22534
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ −26.0000 −0.991962
$$688$$ 12.0000 0.457496
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ −4.00000 −0.151947
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ −4.00000 −0.151511
$$698$$ 34.0000 1.28692
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −8.00000 −0.301726
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ −6.00000 −0.225653
$$708$$ 4.00000 0.150329
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −2.00000 −0.0749532
$$713$$ −64.0000 −2.39682
$$714$$ −2.00000 −0.0748481
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ −16.0000 −0.597531
$$718$$ 8.00000 0.298557
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 3.00000 0.111648
$$723$$ 18.0000 0.669427
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ 24.0000 0.890111 0.445055 0.895503i $$-0.353184\pi$$
0.445055 + 0.895503i $$0.353184\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ −2.00000 −0.0739221
$$733$$ −30.0000 −1.10808 −0.554038 0.832492i $$-0.686914\pi$$
−0.554038 + 0.832492i $$0.686914\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ −48.0000 −1.76810
$$738$$ −2.00000 −0.0736210
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ −6.00000 −0.220267
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ 14.0000 0.512576
$$747$$ −12.0000 −0.439057
$$748$$ −8.00000 −0.292509
$$749$$ −20.0000 −0.730784
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 4.00000 0.145768
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ 34.0000 1.23575 0.617876 0.786276i $$-0.287994\pi$$
0.617876 + 0.786276i $$0.287994\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ 2.00000 0.0724999 0.0362500 0.999343i $$-0.488459\pi$$
0.0362500 + 0.999343i $$0.488459\pi$$
$$762$$ 0 0
$$763$$ 2.00000 0.0724049
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 8.00000 0.288863
$$768$$ 1.00000 0.0360844
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ 14.0000 0.503871
$$773$$ 26.0000 0.935155 0.467578 0.883952i $$-0.345127\pi$$
0.467578 + 0.883952i $$0.345127\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ −2.00000 −0.0717496
$$778$$ 18.0000 0.645331
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 16.0000 0.572159
$$783$$ 6.00000 0.214423
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ −14.0000 −0.497783
$$792$$ −4.00000 −0.142134
$$793$$ −4.00000 −0.142044
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ 2.00000 0.0706665
$$802$$ −18.0000 −0.635602
$$803$$ 56.0000 1.97620
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ 16.0000 0.563576
$$807$$ 30.0000 1.05605
$$808$$ −6.00000 −0.211079
$$809$$ −38.0000 −1.33601 −0.668004 0.744157i $$-0.732851\pi$$
−0.668004 + 0.744157i $$0.732851\pi$$
$$810$$ 0 0
$$811$$ 36.0000 1.26413 0.632065 0.774915i $$-0.282207\pi$$
0.632065 + 0.774915i $$0.282207\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ −24.0000 −0.841717
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ −48.0000 −1.67931
$$818$$ −10.0000 −0.349642
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ 46.0000 1.60541 0.802706 0.596376i $$-0.203393\pi$$
0.802706 + 0.596376i $$0.203393\pi$$
$$822$$ 10.0000 0.348790
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 8.00000 0.278019
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ −14.0000 −0.485655
$$832$$ 2.00000 0.0693375
$$833$$ −2.00000 −0.0692959
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ −8.00000 −0.276520
$$838$$ −12.0000 −0.414533
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 26.0000 0.896019
$$843$$ 10.0000 0.344418
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ −8.00000 −0.275046
$$847$$ −5.00000 −0.171802
$$848$$ −6.00000 −0.206041
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 16.0000 0.548473
$$852$$ 8.00000 0.274075
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ −2.00000 −0.0684386
$$855$$ 0 0
$$856$$ −20.0000 −0.683586
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ −2.00000 −0.0681598
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −6.00000 −0.203888
$$867$$ −13.0000 −0.441503
$$868$$ 8.00000 0.271538
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 2.00000 0.0677285
$$873$$ −10.0000 −0.338449
$$874$$ 32.0000 1.08242
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ 10.0000 0.337676 0.168838 0.985644i $$-0.445999\pi$$
0.168838 + 0.985644i $$0.445999\pi$$
$$878$$ −32.0000 −1.07995
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 0 0
$$893$$ −32.0000 −1.07084
$$894$$ 18.0000 0.602010
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 16.0000 0.534224
$$898$$ 14.0000 0.467186
$$899$$ −48.0000 −1.60089
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ −8.00000 −0.266371
$$903$$ −12.0000 −0.399335
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ −48.0000 −1.58857
$$914$$ −38.0000 −1.25693
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ −12.0000 −0.396275
$$918$$ 2.00000 0.0660098
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ 34.0000 1.11973
$$923$$ 16.0000 0.526646
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ −8.00000 −0.262754
$$928$$ −6.00000 −0.196960
$$929$$ 10.0000 0.328089 0.164045 0.986453i $$-0.447546\pi$$
0.164045 + 0.986453i $$0.447546\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ −10.0000 −0.327561
$$933$$ 8.00000 0.261908
$$934$$ −20.0000 −0.654420
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ −18.0000 −0.588034 −0.294017 0.955800i $$-0.594992\pi$$
−0.294017 + 0.955800i $$0.594992\pi$$
$$938$$ −12.0000 −0.391814
$$939$$ −34.0000 −1.10955
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 14.0000 0.456145
$$943$$ 16.0000 0.521032
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ 28.0000 0.908918
$$950$$ 0 0
$$951$$ −14.0000 −0.453981
$$952$$ −2.00000 −0.0648204
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ 24.0000 0.775810
$$958$$ 0 0
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −4.00000 −0.128965
$$963$$ 20.0000 0.644491
$$964$$ 18.0000 0.579741
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −20.0000 −0.641171
$$974$$ −24.0000 −0.769010
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 12.0000 0.383718
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ −36.0000 −1.14881
$$983$$ −32.0000 −1.02064 −0.510321 0.859984i $$-0.670473\pi$$
−0.510321 + 0.859984i $$0.670473\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ −8.00000 −0.254643
$$988$$ −8.00000 −0.254514
$$989$$ 96.0000 3.05262
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 28.0000 0.888553
$$994$$ 8.00000 0.253745
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ −38.0000 −1.20347 −0.601736 0.798695i $$-0.705524\pi$$
−0.601736 + 0.798695i $$0.705524\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ 2.00000 0.0632772
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 1050.2.a.h.1.1 1
3.2 odd 2 3150.2.a.w.1.1 1
4.3 odd 2 8400.2.a.p.1.1 1
5.2 odd 4 1050.2.g.d.799.1 2
5.3 odd 4 1050.2.g.d.799.2 2
5.4 even 2 210.2.a.c.1.1 1
7.6 odd 2 7350.2.a.p.1.1 1
15.2 even 4 3150.2.g.e.2899.2 2
15.8 even 4 3150.2.g.e.2899.1 2
15.14 odd 2 630.2.a.b.1.1 1
20.19 odd 2 1680.2.a.q.1.1 1
35.4 even 6 1470.2.i.f.961.1 2
35.9 even 6 1470.2.i.f.361.1 2
35.19 odd 6 1470.2.i.b.361.1 2
35.24 odd 6 1470.2.i.b.961.1 2
35.34 odd 2 1470.2.a.q.1.1 1
40.19 odd 2 6720.2.a.k.1.1 1
40.29 even 2 6720.2.a.bp.1.1 1
60.59 even 2 5040.2.a.i.1.1 1
105.104 even 2 4410.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.c.1.1 1 5.4 even 2
630.2.a.b.1.1 1 15.14 odd 2
1050.2.a.h.1.1 1 1.1 even 1 trivial
1050.2.g.d.799.1 2 5.2 odd 4
1050.2.g.d.799.2 2 5.3 odd 4
1470.2.a.q.1.1 1 35.34 odd 2
1470.2.i.b.361.1 2 35.19 odd 6
1470.2.i.b.961.1 2 35.24 odd 6
1470.2.i.f.361.1 2 35.9 even 6
1470.2.i.f.961.1 2 35.4 even 6
1680.2.a.q.1.1 1 20.19 odd 2
3150.2.a.w.1.1 1 3.2 odd 2
3150.2.g.e.2899.1 2 15.8 even 4
3150.2.g.e.2899.2 2 15.2 even 4
4410.2.a.l.1.1 1 105.104 even 2
5040.2.a.i.1.1 1 60.59 even 2
6720.2.a.k.1.1 1 40.19 odd 2
6720.2.a.bp.1.1 1 40.29 even 2
7350.2.a.p.1.1 1 7.6 odd 2
8400.2.a.p.1.1 1 4.3 odd 2