# Properties

 Label 7098.2.a.o.1.1 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.6778153547$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 7098.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -3.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +7.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +3.00000 q^{38} -1.00000 q^{40} -4.00000 q^{41} +1.00000 q^{42} +5.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} -7.00000 q^{51} -6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{55} +1.00000 q^{56} -3.00000 q^{57} +1.00000 q^{58} +10.0000 q^{59} +1.00000 q^{60} -13.0000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +8.00000 q^{67} -7.00000 q^{68} -1.00000 q^{69} +1.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} -13.0000 q^{73} +1.00000 q^{74} -4.00000 q^{75} -3.00000 q^{76} -3.00000 q^{77} -12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} -7.00000 q^{85} -5.00000 q^{86} -1.00000 q^{87} -3.00000 q^{88} +12.0000 q^{89} -1.00000 q^{90} -1.00000 q^{92} +8.00000 q^{93} -3.00000 q^{95} -1.00000 q^{96} -6.00000 q^{97} -1.00000 q^{98} +3.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 1.00000 0.267261
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −3.00000 −0.688247 −0.344124 0.938924i $$-0.611824\pi$$
−0.344124 + 0.938924i $$0.611824\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −1.00000 −0.218218
$$22$$ −3.00000 −0.639602
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 3.00000 0.522233
$$34$$ 7.00000 1.20049
$$35$$ −1.00000 −0.169031
$$36$$ 1.00000 0.166667
$$37$$ −1.00000 −0.164399 −0.0821995 0.996616i $$-0.526194\pi$$
−0.0821995 + 0.996616i $$0.526194\pi$$
$$38$$ 3.00000 0.486664
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 1.00000 0.149071
$$46$$ 1.00000 0.147442
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ 4.00000 0.565685
$$51$$ −7.00000 −0.980196
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 3.00000 0.404520
$$56$$ 1.00000 0.133631
$$57$$ −3.00000 −0.397360
$$58$$ 1.00000 0.131306
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −3.00000 −0.369274
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ −7.00000 −0.848875
$$69$$ −1.00000 −0.120386
$$70$$ 1.00000 0.119523
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −13.0000 −1.52153 −0.760767 0.649025i $$-0.775177\pi$$
−0.760767 + 0.649025i $$0.775177\pi$$
$$74$$ 1.00000 0.116248
$$75$$ −4.00000 −0.461880
$$76$$ −3.00000 −0.344124
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 4.00000 0.441726
$$83$$ −2.00000 −0.219529 −0.109764 0.993958i $$-0.535010\pi$$
−0.109764 + 0.993958i $$0.535010\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ −7.00000 −0.759257
$$86$$ −5.00000 −0.539164
$$87$$ −1.00000 −0.107211
$$88$$ −3.00000 −0.319801
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −1.00000 −0.104257
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ −3.00000 −0.307794
$$96$$ −1.00000 −0.102062
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 3.00000 0.301511
$$100$$ −4.00000 −0.400000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 7.00000 0.693103
$$103$$ 1.00000 0.0985329 0.0492665 0.998786i $$-0.484312\pi$$
0.0492665 + 0.998786i $$0.484312\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 6.00000 0.582772
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −7.00000 −0.670478 −0.335239 0.942133i $$-0.608817\pi$$
−0.335239 + 0.942133i $$0.608817\pi$$
$$110$$ −3.00000 −0.286039
$$111$$ −1.00000 −0.0949158
$$112$$ −1.00000 −0.0944911
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 3.00000 0.280976
$$115$$ −1.00000 −0.0932505
$$116$$ −1.00000 −0.0928477
$$117$$ 0 0
$$118$$ −10.0000 −0.920575
$$119$$ 7.00000 0.641689
$$120$$ −1.00000 −0.0912871
$$121$$ −2.00000 −0.181818
$$122$$ 13.0000 1.17696
$$123$$ −4.00000 −0.360668
$$124$$ 8.00000 0.718421
$$125$$ −9.00000 −0.804984
$$126$$ 1.00000 0.0890871
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 5.00000 0.440225
$$130$$ 0 0
$$131$$ −17.0000 −1.48530 −0.742648 0.669681i $$-0.766431\pi$$
−0.742648 + 0.669681i $$0.766431\pi$$
$$132$$ 3.00000 0.261116
$$133$$ 3.00000 0.260133
$$134$$ −8.00000 −0.691095
$$135$$ 1.00000 0.0860663
$$136$$ 7.00000 0.600245
$$137$$ −17.0000 −1.45241 −0.726204 0.687479i $$-0.758717\pi$$
−0.726204 + 0.687479i $$0.758717\pi$$
$$138$$ 1.00000 0.0851257
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −1.00000 −0.0830455
$$146$$ 13.0000 1.07589
$$147$$ 1.00000 0.0824786
$$148$$ −1.00000 −0.0821995
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ 4.00000 0.326599
$$151$$ −11.0000 −0.895167 −0.447584 0.894242i $$-0.647715\pi$$
−0.447584 + 0.894242i $$0.647715\pi$$
$$152$$ 3.00000 0.243332
$$153$$ −7.00000 −0.565916
$$154$$ 3.00000 0.241747
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ 19.0000 1.51637 0.758183 0.652042i $$-0.226088\pi$$
0.758183 + 0.652042i $$0.226088\pi$$
$$158$$ 12.0000 0.954669
$$159$$ −6.00000 −0.475831
$$160$$ −1.00000 −0.0790569
$$161$$ 1.00000 0.0788110
$$162$$ −1.00000 −0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ 3.00000 0.233550
$$166$$ 2.00000 0.155230
$$167$$ 15.0000 1.16073 0.580367 0.814355i $$-0.302909\pi$$
0.580367 + 0.814355i $$0.302909\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 0 0
$$170$$ 7.00000 0.536875
$$171$$ −3.00000 −0.229416
$$172$$ 5.00000 0.381246
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 1.00000 0.0758098
$$175$$ 4.00000 0.302372
$$176$$ 3.00000 0.226134
$$177$$ 10.0000 0.751646
$$178$$ −12.0000 −0.899438
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 26.0000 1.93256 0.966282 0.257485i $$-0.0828937\pi$$
0.966282 + 0.257485i $$0.0828937\pi$$
$$182$$ 0 0
$$183$$ −13.0000 −0.960988
$$184$$ 1.00000 0.0737210
$$185$$ −1.00000 −0.0735215
$$186$$ −8.00000 −0.586588
$$187$$ −21.0000 −1.53567
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 3.00000 0.217643
$$191$$ −9.00000 −0.651217 −0.325609 0.945505i $$-0.605569\pi$$
−0.325609 + 0.945505i $$0.605569\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −8.00000 −0.575853 −0.287926 0.957653i $$-0.592966\pi$$
−0.287926 + 0.957653i $$0.592966\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −14.0000 −0.997459 −0.498729 0.866758i $$-0.666200\pi$$
−0.498729 + 0.866758i $$0.666200\pi$$
$$198$$ −3.00000 −0.213201
$$199$$ −9.00000 −0.637993 −0.318997 0.947756i $$-0.603346\pi$$
−0.318997 + 0.947756i $$0.603346\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 8.00000 0.564276
$$202$$ 14.0000 0.985037
$$203$$ 1.00000 0.0701862
$$204$$ −7.00000 −0.490098
$$205$$ −4.00000 −0.279372
$$206$$ −1.00000 −0.0696733
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ −9.00000 −0.622543
$$210$$ 1.00000 0.0690066
$$211$$ 15.0000 1.03264 0.516321 0.856395i $$-0.327301\pi$$
0.516321 + 0.856395i $$0.327301\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ −6.00000 −0.411113
$$214$$ −2.00000 −0.136717
$$215$$ 5.00000 0.340997
$$216$$ −1.00000 −0.0680414
$$217$$ −8.00000 −0.543075
$$218$$ 7.00000 0.474100
$$219$$ −13.0000 −0.878459
$$220$$ 3.00000 0.202260
$$221$$ 0 0
$$222$$ 1.00000 0.0671156
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −4.00000 −0.266667
$$226$$ 10.0000 0.665190
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ −3.00000 −0.198680
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ −3.00000 −0.197386
$$232$$ 1.00000 0.0656532
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 10.0000 0.650945
$$237$$ −12.0000 −0.779484
$$238$$ −7.00000 −0.453743
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 1.00000 0.0641500
$$244$$ −13.0000 −0.832240
$$245$$ 1.00000 0.0638877
$$246$$ 4.00000 0.255031
$$247$$ 0 0
$$248$$ −8.00000 −0.508001
$$249$$ −2.00000 −0.126745
$$250$$ 9.00000 0.569210
$$251$$ −17.0000 −1.07303 −0.536515 0.843891i $$-0.680260\pi$$
−0.536515 + 0.843891i $$0.680260\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ −3.00000 −0.188608
$$254$$ −4.00000 −0.250982
$$255$$ −7.00000 −0.438357
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ −5.00000 −0.311286
$$259$$ 1.00000 0.0621370
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ 17.0000 1.05026
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ −6.00000 −0.368577
$$266$$ −3.00000 −0.183942
$$267$$ 12.0000 0.734388
$$268$$ 8.00000 0.488678
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ −7.00000 −0.424437
$$273$$ 0 0
$$274$$ 17.0000 1.02701
$$275$$ −12.0000 −0.723627
$$276$$ −1.00000 −0.0601929
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 8.00000 0.479808
$$279$$ 8.00000 0.478947
$$280$$ 1.00000 0.0597614
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ −3.00000 −0.177705
$$286$$ 0 0
$$287$$ 4.00000 0.236113
$$288$$ −1.00000 −0.0589256
$$289$$ 32.0000 1.88235
$$290$$ 1.00000 0.0587220
$$291$$ −6.00000 −0.351726
$$292$$ −13.0000 −0.760767
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 10.0000 0.582223
$$296$$ 1.00000 0.0581238
$$297$$ 3.00000 0.174078
$$298$$ −22.0000 −1.27443
$$299$$ 0 0
$$300$$ −4.00000 −0.230940
$$301$$ −5.00000 −0.288195
$$302$$ 11.0000 0.632979
$$303$$ −14.0000 −0.804279
$$304$$ −3.00000 −0.172062
$$305$$ −13.0000 −0.744378
$$306$$ 7.00000 0.400163
$$307$$ 16.0000 0.913168 0.456584 0.889680i $$-0.349073\pi$$
0.456584 + 0.889680i $$0.349073\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 1.00000 0.0568880
$$310$$ −8.00000 −0.454369
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ −19.0000 −1.07223
$$315$$ −1.00000 −0.0563436
$$316$$ −12.0000 −0.675053
$$317$$ −24.0000 −1.34797 −0.673987 0.738743i $$-0.735420\pi$$
−0.673987 + 0.738743i $$0.735420\pi$$
$$318$$ 6.00000 0.336463
$$319$$ −3.00000 −0.167968
$$320$$ 1.00000 0.0559017
$$321$$ 2.00000 0.111629
$$322$$ −1.00000 −0.0557278
$$323$$ 21.0000 1.16847
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ −7.00000 −0.387101
$$328$$ 4.00000 0.220863
$$329$$ 0 0
$$330$$ −3.00000 −0.165145
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ −2.00000 −0.109764
$$333$$ −1.00000 −0.0547997
$$334$$ −15.0000 −0.820763
$$335$$ 8.00000 0.437087
$$336$$ −1.00000 −0.0545545
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 0 0
$$339$$ −10.0000 −0.543125
$$340$$ −7.00000 −0.379628
$$341$$ 24.0000 1.29967
$$342$$ 3.00000 0.162221
$$343$$ −1.00000 −0.0539949
$$344$$ −5.00000 −0.269582
$$345$$ −1.00000 −0.0538382
$$346$$ −6.00000 −0.322562
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ −1.00000 −0.0536056
$$349$$ −16.0000 −0.856460 −0.428230 0.903670i $$-0.640863\pi$$
−0.428230 + 0.903670i $$0.640863\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ −3.00000 −0.159901
$$353$$ 16.0000 0.851594 0.425797 0.904819i $$-0.359994\pi$$
0.425797 + 0.904819i $$0.359994\pi$$
$$354$$ −10.0000 −0.531494
$$355$$ −6.00000 −0.318447
$$356$$ 12.0000 0.635999
$$357$$ 7.00000 0.370479
$$358$$ −6.00000 −0.317110
$$359$$ −26.0000 −1.37223 −0.686114 0.727494i $$-0.740685\pi$$
−0.686114 + 0.727494i $$0.740685\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −10.0000 −0.526316
$$362$$ −26.0000 −1.36653
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ −13.0000 −0.680451
$$366$$ 13.0000 0.679521
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −4.00000 −0.208232
$$370$$ 1.00000 0.0519875
$$371$$ 6.00000 0.311504
$$372$$ 8.00000 0.414781
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 21.0000 1.08588
$$375$$ −9.00000 −0.464758
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 1.00000 0.0514344
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ −3.00000 −0.153897
$$381$$ 4.00000 0.204926
$$382$$ 9.00000 0.460480
$$383$$ −17.0000 −0.868659 −0.434330 0.900754i $$-0.643015\pi$$
−0.434330 + 0.900754i $$0.643015\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −3.00000 −0.152894
$$386$$ 8.00000 0.407189
$$387$$ 5.00000 0.254164
$$388$$ −6.00000 −0.304604
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 7.00000 0.354005
$$392$$ −1.00000 −0.0505076
$$393$$ −17.0000 −0.857537
$$394$$ 14.0000 0.705310
$$395$$ −12.0000 −0.603786
$$396$$ 3.00000 0.150756
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ 9.00000 0.451129
$$399$$ 3.00000 0.150188
$$400$$ −4.00000 −0.200000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 0 0
$$404$$ −14.0000 −0.696526
$$405$$ 1.00000 0.0496904
$$406$$ −1.00000 −0.0496292
$$407$$ −3.00000 −0.148704
$$408$$ 7.00000 0.346552
$$409$$ 29.0000 1.43396 0.716979 0.697095i $$-0.245524\pi$$
0.716979 + 0.697095i $$0.245524\pi$$
$$410$$ 4.00000 0.197546
$$411$$ −17.0000 −0.838548
$$412$$ 1.00000 0.0492665
$$413$$ −10.0000 −0.492068
$$414$$ 1.00000 0.0491473
$$415$$ −2.00000 −0.0981761
$$416$$ 0 0
$$417$$ −8.00000 −0.391762
$$418$$ 9.00000 0.440204
$$419$$ −11.0000 −0.537385 −0.268693 0.963226i $$-0.586592\pi$$
−0.268693 + 0.963226i $$0.586592\pi$$
$$420$$ −1.00000 −0.0487950
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ −15.0000 −0.730189
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 28.0000 1.35820
$$426$$ 6.00000 0.290701
$$427$$ 13.0000 0.629114
$$428$$ 2.00000 0.0966736
$$429$$ 0 0
$$430$$ −5.00000 −0.241121
$$431$$ 14.0000 0.674356 0.337178 0.941441i $$-0.390528\pi$$
0.337178 + 0.941441i $$0.390528\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 8.00000 0.384455 0.192228 0.981350i $$-0.438429\pi$$
0.192228 + 0.981350i $$0.438429\pi$$
$$434$$ 8.00000 0.384012
$$435$$ −1.00000 −0.0479463
$$436$$ −7.00000 −0.335239
$$437$$ 3.00000 0.143509
$$438$$ 13.0000 0.621164
$$439$$ −27.0000 −1.28864 −0.644320 0.764756i $$-0.722859\pi$$
−0.644320 + 0.764756i $$0.722859\pi$$
$$440$$ −3.00000 −0.143019
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −34.0000 −1.61539 −0.807694 0.589601i $$-0.799285\pi$$
−0.807694 + 0.589601i $$0.799285\pi$$
$$444$$ −1.00000 −0.0474579
$$445$$ 12.0000 0.568855
$$446$$ 4.00000 0.189405
$$447$$ 22.0000 1.04056
$$448$$ −1.00000 −0.0472456
$$449$$ −9.00000 −0.424736 −0.212368 0.977190i $$-0.568118\pi$$
−0.212368 + 0.977190i $$0.568118\pi$$
$$450$$ 4.00000 0.188562
$$451$$ −12.0000 −0.565058
$$452$$ −10.0000 −0.470360
$$453$$ −11.0000 −0.516825
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ −28.0000 −1.30978 −0.654892 0.755722i $$-0.727286\pi$$
−0.654892 + 0.755722i $$0.727286\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ −7.00000 −0.326732
$$460$$ −1.00000 −0.0466252
$$461$$ 27.0000 1.25752 0.628758 0.777601i $$-0.283564\pi$$
0.628758 + 0.777601i $$0.283564\pi$$
$$462$$ 3.00000 0.139573
$$463$$ −15.0000 −0.697109 −0.348555 0.937288i $$-0.613327\pi$$
−0.348555 + 0.937288i $$0.613327\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 8.00000 0.370991
$$466$$ 18.0000 0.833834
$$467$$ −9.00000 −0.416470 −0.208235 0.978079i $$-0.566772\pi$$
−0.208235 + 0.978079i $$0.566772\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 19.0000 0.875474
$$472$$ −10.0000 −0.460287
$$473$$ 15.0000 0.689701
$$474$$ 12.0000 0.551178
$$475$$ 12.0000 0.550598
$$476$$ 7.00000 0.320844
$$477$$ −6.00000 −0.274721
$$478$$ 8.00000 0.365911
$$479$$ −3.00000 −0.137073 −0.0685367 0.997649i $$-0.521833\pi$$
−0.0685367 + 0.997649i $$0.521833\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ −26.0000 −1.18427
$$483$$ 1.00000 0.0455016
$$484$$ −2.00000 −0.0909091
$$485$$ −6.00000 −0.272446
$$486$$ −1.00000 −0.0453609
$$487$$ 24.0000 1.08754 0.543772 0.839233i $$-0.316996\pi$$
0.543772 + 0.839233i $$0.316996\pi$$
$$488$$ 13.0000 0.588482
$$489$$ 4.00000 0.180886
$$490$$ −1.00000 −0.0451754
$$491$$ 40.0000 1.80517 0.902587 0.430507i $$-0.141665\pi$$
0.902587 + 0.430507i $$0.141665\pi$$
$$492$$ −4.00000 −0.180334
$$493$$ 7.00000 0.315264
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 8.00000 0.359211
$$497$$ 6.00000 0.269137
$$498$$ 2.00000 0.0896221
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ −9.00000 −0.402492
$$501$$ 15.0000 0.670151
$$502$$ 17.0000 0.758747
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ −14.0000 −0.622992
$$506$$ 3.00000 0.133366
$$507$$ 0 0
$$508$$ 4.00000 0.177471
$$509$$ 27.0000 1.19675 0.598377 0.801215i $$-0.295813\pi$$
0.598377 + 0.801215i $$0.295813\pi$$
$$510$$ 7.00000 0.309965
$$511$$ 13.0000 0.575086
$$512$$ −1.00000 −0.0441942
$$513$$ −3.00000 −0.132453
$$514$$ −6.00000 −0.264649
$$515$$ 1.00000 0.0440653
$$516$$ 5.00000 0.220113
$$517$$ 0 0
$$518$$ −1.00000 −0.0439375
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 15.0000 0.657162 0.328581 0.944476i $$-0.393430\pi$$
0.328581 + 0.944476i $$0.393430\pi$$
$$522$$ 1.00000 0.0437688
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ −17.0000 −0.742648
$$525$$ 4.00000 0.174574
$$526$$ 24.0000 1.04645
$$527$$ −56.0000 −2.43940
$$528$$ 3.00000 0.130558
$$529$$ −22.0000 −0.956522
$$530$$ 6.00000 0.260623
$$531$$ 10.0000 0.433963
$$532$$ 3.00000 0.130066
$$533$$ 0 0
$$534$$ −12.0000 −0.519291
$$535$$ 2.00000 0.0864675
$$536$$ −8.00000 −0.345547
$$537$$ 6.00000 0.258919
$$538$$ 4.00000 0.172452
$$539$$ 3.00000 0.129219
$$540$$ 1.00000 0.0430331
$$541$$ 25.0000 1.07483 0.537417 0.843317i $$-0.319400\pi$$
0.537417 + 0.843317i $$0.319400\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 26.0000 1.11577
$$544$$ 7.00000 0.300123
$$545$$ −7.00000 −0.299847
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ −17.0000 −0.726204
$$549$$ −13.0000 −0.554826
$$550$$ 12.0000 0.511682
$$551$$ 3.00000 0.127804
$$552$$ 1.00000 0.0425628
$$553$$ 12.0000 0.510292
$$554$$ 28.0000 1.18961
$$555$$ −1.00000 −0.0424476
$$556$$ −8.00000 −0.339276
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 0 0
$$560$$ −1.00000 −0.0422577
$$561$$ −21.0000 −0.886621
$$562$$ 10.0000 0.421825
$$563$$ −41.0000 −1.72794 −0.863972 0.503540i $$-0.832031\pi$$
−0.863972 + 0.503540i $$0.832031\pi$$
$$564$$ 0 0
$$565$$ −10.0000 −0.420703
$$566$$ −14.0000 −0.588464
$$567$$ −1.00000 −0.0419961
$$568$$ 6.00000 0.251754
$$569$$ 2.00000 0.0838444 0.0419222 0.999121i $$-0.486652\pi$$
0.0419222 + 0.999121i $$0.486652\pi$$
$$570$$ 3.00000 0.125656
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 0 0
$$573$$ −9.00000 −0.375980
$$574$$ −4.00000 −0.166957
$$575$$ 4.00000 0.166812
$$576$$ 1.00000 0.0416667
$$577$$ −38.0000 −1.58196 −0.790980 0.611842i $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ −32.0000 −1.33102
$$579$$ −8.00000 −0.332469
$$580$$ −1.00000 −0.0415227
$$581$$ 2.00000 0.0829740
$$582$$ 6.00000 0.248708
$$583$$ −18.0000 −0.745484
$$584$$ 13.0000 0.537944
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ −16.0000 −0.660391 −0.330195 0.943913i $$-0.607115\pi$$
−0.330195 + 0.943913i $$0.607115\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ −24.0000 −0.988903
$$590$$ −10.0000 −0.411693
$$591$$ −14.0000 −0.575883
$$592$$ −1.00000 −0.0410997
$$593$$ −16.0000 −0.657041 −0.328521 0.944497i $$-0.606550\pi$$
−0.328521 + 0.944497i $$0.606550\pi$$
$$594$$ −3.00000 −0.123091
$$595$$ 7.00000 0.286972
$$596$$ 22.0000 0.901155
$$597$$ −9.00000 −0.368345
$$598$$ 0 0
$$599$$ −3.00000 −0.122577 −0.0612883 0.998120i $$-0.519521\pi$$
−0.0612883 + 0.998120i $$0.519521\pi$$
$$600$$ 4.00000 0.163299
$$601$$ 4.00000 0.163163 0.0815817 0.996667i $$-0.474003\pi$$
0.0815817 + 0.996667i $$0.474003\pi$$
$$602$$ 5.00000 0.203785
$$603$$ 8.00000 0.325785
$$604$$ −11.0000 −0.447584
$$605$$ −2.00000 −0.0813116
$$606$$ 14.0000 0.568711
$$607$$ −11.0000 −0.446476 −0.223238 0.974764i $$-0.571663\pi$$
−0.223238 + 0.974764i $$0.571663\pi$$
$$608$$ 3.00000 0.121666
$$609$$ 1.00000 0.0405220
$$610$$ 13.0000 0.526355
$$611$$ 0 0
$$612$$ −7.00000 −0.282958
$$613$$ −31.0000 −1.25208 −0.626039 0.779792i $$-0.715325\pi$$
−0.626039 + 0.779792i $$0.715325\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ −4.00000 −0.161296
$$616$$ 3.00000 0.120873
$$617$$ 37.0000 1.48956 0.744782 0.667308i $$-0.232553\pi$$
0.744782 + 0.667308i $$0.232553\pi$$
$$618$$ −1.00000 −0.0402259
$$619$$ −23.0000 −0.924448 −0.462224 0.886763i $$-0.652948\pi$$
−0.462224 + 0.886763i $$0.652948\pi$$
$$620$$ 8.00000 0.321288
$$621$$ −1.00000 −0.0401286
$$622$$ −4.00000 −0.160385
$$623$$ −12.0000 −0.480770
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 26.0000 1.03917
$$627$$ −9.00000 −0.359425
$$628$$ 19.0000 0.758183
$$629$$ 7.00000 0.279108
$$630$$ 1.00000 0.0398410
$$631$$ −47.0000 −1.87104 −0.935520 0.353273i $$-0.885069\pi$$
−0.935520 + 0.353273i $$0.885069\pi$$
$$632$$ 12.0000 0.477334
$$633$$ 15.0000 0.596196
$$634$$ 24.0000 0.953162
$$635$$ 4.00000 0.158735
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 3.00000 0.118771
$$639$$ −6.00000 −0.237356
$$640$$ −1.00000 −0.0395285
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ −2.00000 −0.0789337
$$643$$ 39.0000 1.53801 0.769005 0.639243i $$-0.220752\pi$$
0.769005 + 0.639243i $$0.220752\pi$$
$$644$$ 1.00000 0.0394055
$$645$$ 5.00000 0.196875
$$646$$ −21.0000 −0.826234
$$647$$ −22.0000 −0.864909 −0.432455 0.901656i $$-0.642352\pi$$
−0.432455 + 0.901656i $$0.642352\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 30.0000 1.17760
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ 4.00000 0.156652
$$653$$ −35.0000 −1.36966 −0.684828 0.728705i $$-0.740123\pi$$
−0.684828 + 0.728705i $$0.740123\pi$$
$$654$$ 7.00000 0.273722
$$655$$ −17.0000 −0.664245
$$656$$ −4.00000 −0.156174
$$657$$ −13.0000 −0.507178
$$658$$ 0 0
$$659$$ 26.0000 1.01282 0.506408 0.862294i $$-0.330973\pi$$
0.506408 + 0.862294i $$0.330973\pi$$
$$660$$ 3.00000 0.116775
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 8.00000 0.310929
$$663$$ 0 0
$$664$$ 2.00000 0.0776151
$$665$$ 3.00000 0.116335
$$666$$ 1.00000 0.0387492
$$667$$ 1.00000 0.0387202
$$668$$ 15.0000 0.580367
$$669$$ −4.00000 −0.154649
$$670$$ −8.00000 −0.309067
$$671$$ −39.0000 −1.50558
$$672$$ 1.00000 0.0385758
$$673$$ 3.00000 0.115642 0.0578208 0.998327i $$-0.481585\pi$$
0.0578208 + 0.998327i $$0.481585\pi$$
$$674$$ −23.0000 −0.885927
$$675$$ −4.00000 −0.153960
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 10.0000 0.384048
$$679$$ 6.00000 0.230259
$$680$$ 7.00000 0.268438
$$681$$ −20.0000 −0.766402
$$682$$ −24.0000 −0.919007
$$683$$ 9.00000 0.344375 0.172188 0.985064i $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$684$$ −3.00000 −0.114708
$$685$$ −17.0000 −0.649537
$$686$$ 1.00000 0.0381802
$$687$$ 10.0000 0.381524
$$688$$ 5.00000 0.190623
$$689$$ 0 0
$$690$$ 1.00000 0.0380693
$$691$$ 16.0000 0.608669 0.304334 0.952565i $$-0.401566\pi$$
0.304334 + 0.952565i $$0.401566\pi$$
$$692$$ 6.00000 0.228086
$$693$$ −3.00000 −0.113961
$$694$$ 24.0000 0.911028
$$695$$ −8.00000 −0.303457
$$696$$ 1.00000 0.0379049
$$697$$ 28.0000 1.06058
$$698$$ 16.0000 0.605609
$$699$$ −18.0000 −0.680823
$$700$$ 4.00000 0.151186
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ 3.00000 0.113147
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −16.0000 −0.602168
$$707$$ 14.0000 0.526524
$$708$$ 10.0000 0.375823
$$709$$ −30.0000 −1.12667 −0.563337 0.826227i $$-0.690483\pi$$
−0.563337 + 0.826227i $$0.690483\pi$$
$$710$$ 6.00000 0.225176
$$711$$ −12.0000 −0.450035
$$712$$ −12.0000 −0.449719
$$713$$ −8.00000 −0.299602
$$714$$ −7.00000 −0.261968
$$715$$ 0 0
$$716$$ 6.00000 0.224231
$$717$$ −8.00000 −0.298765
$$718$$ 26.0000 0.970311
$$719$$ −20.0000 −0.745874 −0.372937 0.927857i $$-0.621649\pi$$
−0.372937 + 0.927857i $$0.621649\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ −1.00000 −0.0372419
$$722$$ 10.0000 0.372161
$$723$$ 26.0000 0.966950
$$724$$ 26.0000 0.966282
$$725$$ 4.00000 0.148556
$$726$$ 2.00000 0.0742270
$$727$$ −35.0000 −1.29808 −0.649039 0.760755i $$-0.724829\pi$$
−0.649039 + 0.760755i $$0.724829\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 13.0000 0.481152
$$731$$ −35.0000 −1.29452
$$732$$ −13.0000 −0.480494
$$733$$ 8.00000 0.295487 0.147743 0.989026i $$-0.452799\pi$$
0.147743 + 0.989026i $$0.452799\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 1.00000 0.0368856
$$736$$ 1.00000 0.0368605
$$737$$ 24.0000 0.884051
$$738$$ 4.00000 0.147242
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ −1.00000 −0.0367607
$$741$$ 0 0
$$742$$ −6.00000 −0.220267
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 22.0000 0.806018
$$746$$ −6.00000 −0.219676
$$747$$ −2.00000 −0.0731762
$$748$$ −21.0000 −0.767836
$$749$$ −2.00000 −0.0730784
$$750$$ 9.00000 0.328634
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ −17.0000 −0.619514
$$754$$ 0 0
$$755$$ −11.0000 −0.400331
$$756$$ −1.00000 −0.0363696
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 4.00000 0.145287
$$759$$ −3.00000 −0.108893
$$760$$ 3.00000 0.108821
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ 7.00000 0.253417
$$764$$ −9.00000 −0.325609
$$765$$ −7.00000 −0.253086
$$766$$ 17.0000 0.614235
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −39.0000 −1.40638 −0.703188 0.711004i $$-0.748241\pi$$
−0.703188 + 0.711004i $$0.748241\pi$$
$$770$$ 3.00000 0.108112
$$771$$ 6.00000 0.216085
$$772$$ −8.00000 −0.287926
$$773$$ −21.0000 −0.755318 −0.377659 0.925945i $$-0.623271\pi$$
−0.377659 + 0.925945i $$0.623271\pi$$
$$774$$ −5.00000 −0.179721
$$775$$ −32.0000 −1.14947
$$776$$ 6.00000 0.215387
$$777$$ 1.00000 0.0358748
$$778$$ −18.0000 −0.645331
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ −18.0000 −0.644091
$$782$$ −7.00000 −0.250319
$$783$$ −1.00000 −0.0357371
$$784$$ 1.00000 0.0357143
$$785$$ 19.0000 0.678139
$$786$$ 17.0000 0.606370
$$787$$ 17.0000 0.605985 0.302992 0.952993i $$-0.402014\pi$$
0.302992 + 0.952993i $$0.402014\pi$$
$$788$$ −14.0000 −0.498729
$$789$$ −24.0000 −0.854423
$$790$$ 12.0000 0.426941
$$791$$ 10.0000 0.355559
$$792$$ −3.00000 −0.106600
$$793$$ 0 0
$$794$$ −8.00000 −0.283909
$$795$$ −6.00000 −0.212798
$$796$$ −9.00000 −0.318997
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ −3.00000 −0.106199
$$799$$ 0 0
$$800$$ 4.00000 0.141421
$$801$$ 12.0000 0.423999
$$802$$ 6.00000 0.211867
$$803$$ −39.0000 −1.37628
$$804$$ 8.00000 0.282138
$$805$$ 1.00000 0.0352454
$$806$$ 0 0
$$807$$ −4.00000 −0.140807
$$808$$ 14.0000 0.492518
$$809$$ 52.0000 1.82822 0.914111 0.405463i $$-0.132890\pi$$
0.914111 + 0.405463i $$0.132890\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ −7.00000 −0.245803 −0.122902 0.992419i $$-0.539220\pi$$
−0.122902 + 0.992419i $$0.539220\pi$$
$$812$$ 1.00000 0.0350931
$$813$$ 20.0000 0.701431
$$814$$ 3.00000 0.105150
$$815$$ 4.00000 0.140114
$$816$$ −7.00000 −0.245049
$$817$$ −15.0000 −0.524784
$$818$$ −29.0000 −1.01396
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ −36.0000 −1.25641 −0.628204 0.778048i $$-0.716210\pi$$
−0.628204 + 0.778048i $$0.716210\pi$$
$$822$$ 17.0000 0.592943
$$823$$ −12.0000 −0.418294 −0.209147 0.977884i $$-0.567069\pi$$
−0.209147 + 0.977884i $$0.567069\pi$$
$$824$$ −1.00000 −0.0348367
$$825$$ −12.0000 −0.417786
$$826$$ 10.0000 0.347945
$$827$$ −9.00000 −0.312961 −0.156480 0.987681i $$-0.550015\pi$$
−0.156480 + 0.987681i $$0.550015\pi$$
$$828$$ −1.00000 −0.0347524
$$829$$ 15.0000 0.520972 0.260486 0.965478i $$-0.416117\pi$$
0.260486 + 0.965478i $$0.416117\pi$$
$$830$$ 2.00000 0.0694210
$$831$$ −28.0000 −0.971309
$$832$$ 0 0
$$833$$ −7.00000 −0.242536
$$834$$ 8.00000 0.277017
$$835$$ 15.0000 0.519096
$$836$$ −9.00000 −0.311272
$$837$$ 8.00000 0.276520
$$838$$ 11.0000 0.379989
$$839$$ −28.0000 −0.966667 −0.483334 0.875436i $$-0.660574\pi$$
−0.483334 + 0.875436i $$0.660574\pi$$
$$840$$ 1.00000 0.0345033
$$841$$ −28.0000 −0.965517
$$842$$ −34.0000 −1.17172
$$843$$ −10.0000 −0.344418
$$844$$ 15.0000 0.516321
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ −6.00000 −0.206041
$$849$$ 14.0000 0.480479
$$850$$ −28.0000 −0.960392
$$851$$ 1.00000 0.0342796
$$852$$ −6.00000 −0.205557
$$853$$ 32.0000 1.09566 0.547830 0.836590i $$-0.315454\pi$$
0.547830 + 0.836590i $$0.315454\pi$$
$$854$$ −13.0000 −0.444851
$$855$$ −3.00000 −0.102598
$$856$$ −2.00000 −0.0683586
$$857$$ −38.0000 −1.29806 −0.649028 0.760765i $$-0.724824\pi$$
−0.649028 + 0.760765i $$0.724824\pi$$
$$858$$ 0 0
$$859$$ 22.0000 0.750630 0.375315 0.926897i $$-0.377534\pi$$
0.375315 + 0.926897i $$0.377534\pi$$
$$860$$ 5.00000 0.170499
$$861$$ 4.00000 0.136320
$$862$$ −14.0000 −0.476842
$$863$$ −28.0000 −0.953131 −0.476566 0.879139i $$-0.658119\pi$$
−0.476566 + 0.879139i $$0.658119\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 6.00000 0.204006
$$866$$ −8.00000 −0.271851
$$867$$ 32.0000 1.08678
$$868$$ −8.00000 −0.271538
$$869$$ −36.0000 −1.22122
$$870$$ 1.00000 0.0339032
$$871$$ 0 0
$$872$$ 7.00000 0.237050
$$873$$ −6.00000 −0.203069
$$874$$ −3.00000 −0.101477
$$875$$ 9.00000 0.304256
$$876$$ −13.0000 −0.439229
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 27.0000 0.911206
$$879$$ −14.0000 −0.472208
$$880$$ 3.00000 0.101130
$$881$$ −51.0000 −1.71823 −0.859117 0.511780i $$-0.828986\pi$$
−0.859117 + 0.511780i $$0.828986\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ −3.00000 −0.100958 −0.0504790 0.998725i $$-0.516075\pi$$
−0.0504790 + 0.998725i $$0.516075\pi$$
$$884$$ 0 0
$$885$$ 10.0000 0.336146
$$886$$ 34.0000 1.14225
$$887$$ 42.0000 1.41022 0.705111 0.709097i $$-0.250897\pi$$
0.705111 + 0.709097i $$0.250897\pi$$
$$888$$ 1.00000 0.0335578
$$889$$ −4.00000 −0.134156
$$890$$ −12.0000 −0.402241
$$891$$ 3.00000 0.100504
$$892$$ −4.00000 −0.133930
$$893$$ 0 0
$$894$$ −22.0000 −0.735790
$$895$$ 6.00000 0.200558
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 9.00000 0.300334
$$899$$ −8.00000 −0.266815
$$900$$ −4.00000 −0.133333
$$901$$ 42.0000 1.39922
$$902$$ 12.0000 0.399556
$$903$$ −5.00000 −0.166390
$$904$$ 10.0000 0.332595
$$905$$ 26.0000 0.864269
$$906$$ 11.0000 0.365451
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ 57.0000 1.88849 0.944247 0.329238i $$-0.106792\pi$$
0.944247 + 0.329238i $$0.106792\pi$$
$$912$$ −3.00000 −0.0993399
$$913$$ −6.00000 −0.198571
$$914$$ 28.0000 0.926158
$$915$$ −13.0000 −0.429767
$$916$$ 10.0000 0.330409
$$917$$ 17.0000 0.561389
$$918$$ 7.00000 0.231034
$$919$$ −30.0000 −0.989609 −0.494804 0.869004i $$-0.664760\pi$$
−0.494804 + 0.869004i $$0.664760\pi$$
$$920$$ 1.00000 0.0329690
$$921$$ 16.0000 0.527218
$$922$$ −27.0000 −0.889198
$$923$$ 0 0
$$924$$ −3.00000 −0.0986928
$$925$$ 4.00000 0.131519
$$926$$ 15.0000 0.492931
$$927$$ 1.00000 0.0328443
$$928$$ 1.00000 0.0328266
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ −8.00000 −0.262330
$$931$$ −3.00000 −0.0983210
$$932$$ −18.0000 −0.589610
$$933$$ 4.00000 0.130954
$$934$$ 9.00000 0.294489
$$935$$ −21.0000 −0.686773
$$936$$ 0 0
$$937$$ 4.00000 0.130674 0.0653372 0.997863i $$-0.479188\pi$$
0.0653372 + 0.997863i $$0.479188\pi$$
$$938$$ 8.00000 0.261209
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ −19.0000 −0.619053
$$943$$ 4.00000 0.130258
$$944$$ 10.0000 0.325472
$$945$$ −1.00000 −0.0325300
$$946$$ −15.0000 −0.487692
$$947$$ −25.0000 −0.812391 −0.406195 0.913786i $$-0.633145\pi$$
−0.406195 + 0.913786i $$0.633145\pi$$
$$948$$ −12.0000 −0.389742
$$949$$ 0 0
$$950$$ −12.0000 −0.389331
$$951$$ −24.0000 −0.778253
$$952$$ −7.00000 −0.226871
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 6.00000 0.194257
$$955$$ −9.00000 −0.291233
$$956$$ −8.00000 −0.258738
$$957$$ −3.00000 −0.0969762
$$958$$ 3.00000 0.0969256
$$959$$ 17.0000 0.548959
$$960$$ 1.00000 0.0322749
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 2.00000 0.0644491
$$964$$ 26.0000 0.837404
$$965$$ −8.00000 −0.257529
$$966$$ −1.00000 −0.0321745
$$967$$ 43.0000 1.38279 0.691393 0.722478i $$-0.256997\pi$$
0.691393 + 0.722478i $$0.256997\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 21.0000 0.674617
$$970$$ 6.00000 0.192648
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 8.00000 0.256468
$$974$$ −24.0000 −0.769010
$$975$$ 0 0
$$976$$ −13.0000 −0.416120
$$977$$ −15.0000 −0.479893 −0.239946 0.970786i $$-0.577130\pi$$
−0.239946 + 0.970786i $$0.577130\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 36.0000 1.15056
$$980$$ 1.00000 0.0319438
$$981$$ −7.00000 −0.223493
$$982$$ −40.0000 −1.27645
$$983$$ −51.0000 −1.62665 −0.813324 0.581811i $$-0.802344\pi$$
−0.813324 + 0.581811i $$0.802344\pi$$
$$984$$ 4.00000 0.127515
$$985$$ −14.0000 −0.446077
$$986$$ −7.00000 −0.222925
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −5.00000 −0.158991
$$990$$ −3.00000 −0.0953463
$$991$$ −20.0000 −0.635321 −0.317660 0.948205i $$-0.602897\pi$$
−0.317660 + 0.948205i $$0.602897\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ −8.00000 −0.253872
$$994$$ −6.00000 −0.190308
$$995$$ −9.00000 −0.285319
$$996$$ −2.00000 −0.0633724
$$997$$ 18.0000 0.570066 0.285033 0.958518i $$-0.407995\pi$$
0.285033 + 0.958518i $$0.407995\pi$$
$$998$$ 14.0000 0.443162
$$999$$ −1.00000 −0.0316386
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 7098.2.a.o.1.1 1
13.5 odd 4 546.2.c.c.337.2 yes 2
13.8 odd 4 546.2.c.c.337.1 2
13.12 even 2 7098.2.a.y.1.1 1
39.5 even 4 1638.2.c.e.883.1 2
39.8 even 4 1638.2.c.e.883.2 2
52.31 even 4 4368.2.h.h.337.1 2
52.47 even 4 4368.2.h.h.337.2 2
91.34 even 4 3822.2.c.b.883.1 2
91.83 even 4 3822.2.c.b.883.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.c.337.1 2 13.8 odd 4
546.2.c.c.337.2 yes 2 13.5 odd 4
1638.2.c.e.883.1 2 39.5 even 4
1638.2.c.e.883.2 2 39.8 even 4
3822.2.c.b.883.1 2 91.34 even 4
3822.2.c.b.883.2 2 91.83 even 4
4368.2.h.h.337.1 2 52.31 even 4
4368.2.h.h.337.2 2 52.47 even 4
7098.2.a.o.1.1 1 1.1 even 1 trivial
7098.2.a.y.1.1 1 13.12 even 2