# Properties

 Label 336.4.a.n.1.1 Level $336$ Weight $4$ Character 336.1 Self dual yes Analytic conductor $19.825$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.8246417619$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ Defining polynomial: $$x^{2} - x - 44$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$7.15207$$ of defining polynomial Character $$\chi$$ $$=$$ 336.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -6.30413 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -6.30413 q^{5} -7.00000 q^{7} +9.00000 q^{9} -48.9124 q^{11} -2.60827 q^{13} +18.9124 q^{15} +136.737 q^{17} -45.2165 q^{19} +21.0000 q^{21} +38.1289 q^{23} -85.2579 q^{25} -27.0000 q^{27} +52.7835 q^{29} +14.7835 q^{31} +146.737 q^{33} +44.1289 q^{35} +333.908 q^{37} +7.82481 q^{39} +227.263 q^{41} +398.433 q^{43} -56.7372 q^{45} +184.608 q^{47} +49.0000 q^{49} -410.212 q^{51} +359.825 q^{53} +308.350 q^{55} +135.650 q^{57} -99.9075 q^{59} -674.516 q^{61} -63.0000 q^{63} +16.4429 q^{65} +376.959 q^{67} -114.387 q^{69} +1187.60 q^{71} -735.825 q^{73} +255.774 q^{75} +342.387 q^{77} +836.774 q^{79} +81.0000 q^{81} -293.732 q^{83} -862.010 q^{85} -158.350 q^{87} +1298.89 q^{89} +18.2579 q^{91} -44.3504 q^{93} +285.051 q^{95} -201.041 q^{97} -440.212 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 14q^{5} - 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 14q^{5} - 14q^{7} + 18q^{9} - 18q^{11} + 48q^{13} - 42q^{15} + 34q^{17} + 16q^{19} + 42q^{21} - 110q^{23} + 202q^{25} - 54q^{27} + 212q^{29} + 136q^{31} + 54q^{33} - 98q^{35} - 24q^{37} - 144q^{39} + 694q^{41} + 584q^{43} + 126q^{45} + 316q^{47} + 98q^{49} - 102q^{51} + 560q^{53} + 936q^{55} - 48q^{57} + 492q^{59} - 604q^{61} - 126q^{63} + 1044q^{65} + 1020q^{67} + 330q^{69} + 1710q^{71} - 1312q^{73} - 606q^{75} + 126q^{77} + 556q^{79} + 162q^{81} + 264q^{83} - 2948q^{85} - 636q^{87} + 70q^{89} - 336q^{91} - 408q^{93} + 1528q^{95} - 136q^{97} - 162q^{99} + O(q^{100})$$

## 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 0
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −6.30413 −0.563859 −0.281929 0.959435i $$-0.590974\pi$$
−0.281929 + 0.959435i $$0.590974\pi$$
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −48.9124 −1.34069 −0.670347 0.742047i $$-0.733855\pi$$
−0.670347 + 0.742047i $$0.733855\pi$$
$$12$$ 0 0
$$13$$ −2.60827 −0.0556464 −0.0278232 0.999613i $$-0.508858\pi$$
−0.0278232 + 0.999613i $$0.508858\pi$$
$$14$$ 0 0
$$15$$ 18.9124 0.325544
$$16$$ 0 0
$$17$$ 136.737 1.95080 0.975401 0.220436i $$-0.0707482\pi$$
0.975401 + 0.220436i $$0.0707482\pi$$
$$18$$ 0 0
$$19$$ −45.2165 −0.545968 −0.272984 0.962019i $$-0.588011\pi$$
−0.272984 + 0.962019i $$0.588011\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ 0 0
$$23$$ 38.1289 0.345671 0.172836 0.984951i $$-0.444707\pi$$
0.172836 + 0.984951i $$0.444707\pi$$
$$24$$ 0 0
$$25$$ −85.2579 −0.682063
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 52.7835 0.337988 0.168994 0.985617i $$-0.445948\pi$$
0.168994 + 0.985617i $$0.445948\pi$$
$$30$$ 0 0
$$31$$ 14.7835 0.0856512 0.0428256 0.999083i $$-0.486364\pi$$
0.0428256 + 0.999083i $$0.486364\pi$$
$$32$$ 0 0
$$33$$ 146.737 0.774051
$$34$$ 0 0
$$35$$ 44.1289 0.213119
$$36$$ 0 0
$$37$$ 333.908 1.48362 0.741812 0.670608i $$-0.233967\pi$$
0.741812 + 0.670608i $$0.233967\pi$$
$$38$$ 0 0
$$39$$ 7.82481 0.0321275
$$40$$ 0 0
$$41$$ 227.263 0.865670 0.432835 0.901473i $$-0.357513\pi$$
0.432835 + 0.901473i $$0.357513\pi$$
$$42$$ 0 0
$$43$$ 398.433 1.41303 0.706517 0.707696i $$-0.250265\pi$$
0.706517 + 0.707696i $$0.250265\pi$$
$$44$$ 0 0
$$45$$ −56.7372 −0.187953
$$46$$ 0 0
$$47$$ 184.608 0.572934 0.286467 0.958090i $$-0.407519\pi$$
0.286467 + 0.958090i $$0.407519\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ −410.212 −1.12630
$$52$$ 0 0
$$53$$ 359.825 0.932561 0.466281 0.884637i $$-0.345594\pi$$
0.466281 + 0.884637i $$0.345594\pi$$
$$54$$ 0 0
$$55$$ 308.350 0.755963
$$56$$ 0 0
$$57$$ 135.650 0.315215
$$58$$ 0 0
$$59$$ −99.9075 −0.220455 −0.110228 0.993906i $$-0.535158\pi$$
−0.110228 + 0.993906i $$0.535158\pi$$
$$60$$ 0 0
$$61$$ −674.516 −1.41579 −0.707893 0.706320i $$-0.750354\pi$$
−0.707893 + 0.706320i $$0.750354\pi$$
$$62$$ 0 0
$$63$$ −63.0000 −0.125988
$$64$$ 0 0
$$65$$ 16.4429 0.0313767
$$66$$ 0 0
$$67$$ 376.959 0.687356 0.343678 0.939088i $$-0.388327\pi$$
0.343678 + 0.939088i $$0.388327\pi$$
$$68$$ 0 0
$$69$$ −114.387 −0.199573
$$70$$ 0 0
$$71$$ 1187.60 1.98511 0.992553 0.121810i $$-0.0388698\pi$$
0.992553 + 0.121810i $$0.0388698\pi$$
$$72$$ 0 0
$$73$$ −735.825 −1.17975 −0.589875 0.807494i $$-0.700823\pi$$
−0.589875 + 0.807494i $$0.700823\pi$$
$$74$$ 0 0
$$75$$ 255.774 0.393789
$$76$$ 0 0
$$77$$ 342.387 0.506735
$$78$$ 0 0
$$79$$ 836.774 1.19170 0.595851 0.803095i $$-0.296815\pi$$
0.595851 + 0.803095i $$0.296815\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −293.732 −0.388450 −0.194225 0.980957i $$-0.562219\pi$$
−0.194225 + 0.980957i $$0.562219\pi$$
$$84$$ 0 0
$$85$$ −862.010 −1.09998
$$86$$ 0 0
$$87$$ −158.350 −0.195137
$$88$$ 0 0
$$89$$ 1298.89 1.54699 0.773496 0.633801i $$-0.218506\pi$$
0.773496 + 0.633801i $$0.218506\pi$$
$$90$$ 0 0
$$91$$ 18.2579 0.0210324
$$92$$ 0 0
$$93$$ −44.3504 −0.0494508
$$94$$ 0 0
$$95$$ 285.051 0.307849
$$96$$ 0 0
$$97$$ −201.041 −0.210440 −0.105220 0.994449i $$-0.533555\pi$$
−0.105220 + 0.994449i $$0.533555\pi$$
$$98$$ 0 0
$$99$$ −440.212 −0.446898
$$100$$ 0 0
$$101$$ −1053.51 −1.03790 −0.518952 0.854804i $$-0.673678\pi$$
−0.518952 + 0.854804i $$0.673678\pi$$
$$102$$ 0 0
$$103$$ −1025.73 −0.981247 −0.490623 0.871372i $$-0.663231\pi$$
−0.490623 + 0.871372i $$0.663231\pi$$
$$104$$ 0 0
$$105$$ −132.387 −0.123044
$$106$$ 0 0
$$107$$ 103.418 0.0934377 0.0467188 0.998908i $$-0.485124\pi$$
0.0467188 + 0.998908i $$0.485124\pi$$
$$108$$ 0 0
$$109$$ −677.134 −0.595024 −0.297512 0.954718i $$-0.596157\pi$$
−0.297512 + 0.954718i $$0.596157\pi$$
$$110$$ 0 0
$$111$$ −1001.72 −0.856570
$$112$$ 0 0
$$113$$ −452.083 −0.376357 −0.188179 0.982135i $$-0.560258\pi$$
−0.188179 + 0.982135i $$0.560258\pi$$
$$114$$ 0 0
$$115$$ −240.370 −0.194910
$$116$$ 0 0
$$117$$ −23.4744 −0.0185488
$$118$$ 0 0
$$119$$ −957.160 −0.737334
$$120$$ 0 0
$$121$$ 1061.42 0.797463
$$122$$ 0 0
$$123$$ −681.788 −0.499795
$$124$$ 0 0
$$125$$ 1325.49 0.948446
$$126$$ 0 0
$$127$$ 1182.88 0.826482 0.413241 0.910622i $$-0.364397\pi$$
0.413241 + 0.910622i $$0.364397\pi$$
$$128$$ 0 0
$$129$$ −1195.30 −0.815816
$$130$$ 0 0
$$131$$ −1257.75 −0.838857 −0.419429 0.907788i $$-0.637770\pi$$
−0.419429 + 0.907788i $$0.637770\pi$$
$$132$$ 0 0
$$133$$ 316.516 0.206356
$$134$$ 0 0
$$135$$ 170.212 0.108515
$$136$$ 0 0
$$137$$ −56.5256 −0.0352504 −0.0176252 0.999845i $$-0.505611\pi$$
−0.0176252 + 0.999845i $$0.505611\pi$$
$$138$$ 0 0
$$139$$ −2113.57 −1.28972 −0.644858 0.764303i $$-0.723083\pi$$
−0.644858 + 0.764303i $$0.723083\pi$$
$$140$$ 0 0
$$141$$ −553.825 −0.330783
$$142$$ 0 0
$$143$$ 127.577 0.0746049
$$144$$ 0 0
$$145$$ −332.754 −0.190577
$$146$$ 0 0
$$147$$ −147.000 −0.0824786
$$148$$ 0 0
$$149$$ 1794.96 0.986904 0.493452 0.869773i $$-0.335735\pi$$
0.493452 + 0.869773i $$0.335735\pi$$
$$150$$ 0 0
$$151$$ −377.032 −0.203195 −0.101597 0.994826i $$-0.532395\pi$$
−0.101597 + 0.994826i $$0.532395\pi$$
$$152$$ 0 0
$$153$$ 1230.63 0.650268
$$154$$ 0 0
$$155$$ −93.1969 −0.0482952
$$156$$ 0 0
$$157$$ −898.701 −0.456842 −0.228421 0.973562i $$-0.573356\pi$$
−0.228421 + 0.973562i $$0.573356\pi$$
$$158$$ 0 0
$$159$$ −1079.47 −0.538414
$$160$$ 0 0
$$161$$ −266.903 −0.130651
$$162$$ 0 0
$$163$$ −3863.52 −1.85653 −0.928264 0.371922i $$-0.878699\pi$$
−0.928264 + 0.371922i $$0.878699\pi$$
$$164$$ 0 0
$$165$$ −925.051 −0.436455
$$166$$ 0 0
$$167$$ 2861.44 1.32590 0.662948 0.748666i $$-0.269305\pi$$
0.662948 + 0.748666i $$0.269305\pi$$
$$168$$ 0 0
$$169$$ −2190.20 −0.996903
$$170$$ 0 0
$$171$$ −406.949 −0.181989
$$172$$ 0 0
$$173$$ 979.005 0.430245 0.215122 0.976587i $$-0.430985\pi$$
0.215122 + 0.976587i $$0.430985\pi$$
$$174$$ 0 0
$$175$$ 596.805 0.257796
$$176$$ 0 0
$$177$$ 299.723 0.127280
$$178$$ 0 0
$$179$$ 1146.27 0.478640 0.239320 0.970941i $$-0.423075\pi$$
0.239320 + 0.970941i $$0.423075\pi$$
$$180$$ 0 0
$$181$$ 3929.33 1.61362 0.806809 0.590812i $$-0.201192\pi$$
0.806809 + 0.590812i $$0.201192\pi$$
$$182$$ 0 0
$$183$$ 2023.55 0.817404
$$184$$ 0 0
$$185$$ −2105.00 −0.836554
$$186$$ 0 0
$$187$$ −6688.15 −2.61543
$$188$$ 0 0
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ 2937.36 1.11277 0.556386 0.830924i $$-0.312187\pi$$
0.556386 + 0.830924i $$0.312187\pi$$
$$192$$ 0 0
$$193$$ −3533.17 −1.31774 −0.658868 0.752259i $$-0.728964\pi$$
−0.658868 + 0.752259i $$0.728964\pi$$
$$194$$ 0 0
$$195$$ −49.3286 −0.0181154
$$196$$ 0 0
$$197$$ 584.856 0.211519 0.105760 0.994392i $$-0.466273\pi$$
0.105760 + 0.994392i $$0.466273\pi$$
$$198$$ 0 0
$$199$$ −2158.08 −0.768756 −0.384378 0.923176i $$-0.625584\pi$$
−0.384378 + 0.923176i $$0.625584\pi$$
$$200$$ 0 0
$$201$$ −1130.88 −0.396845
$$202$$ 0 0
$$203$$ −369.484 −0.127747
$$204$$ 0 0
$$205$$ −1432.70 −0.488116
$$206$$ 0 0
$$207$$ 343.160 0.115224
$$208$$ 0 0
$$209$$ 2211.65 0.731976
$$210$$ 0 0
$$211$$ 4290.04 1.39971 0.699855 0.714285i $$-0.253248\pi$$
0.699855 + 0.714285i $$0.253248\pi$$
$$212$$ 0 0
$$213$$ −3562.81 −1.14610
$$214$$ 0 0
$$215$$ −2511.78 −0.796752
$$216$$ 0 0
$$217$$ −103.484 −0.0323731
$$218$$ 0 0
$$219$$ 2207.47 0.681129
$$220$$ 0 0
$$221$$ −356.647 −0.108555
$$222$$ 0 0
$$223$$ 2743.78 0.823932 0.411966 0.911199i $$-0.364842\pi$$
0.411966 + 0.911199i $$0.364842\pi$$
$$224$$ 0 0
$$225$$ −767.321 −0.227354
$$226$$ 0 0
$$227$$ 1724.79 0.504311 0.252155 0.967687i $$-0.418861\pi$$
0.252155 + 0.967687i $$0.418861\pi$$
$$228$$ 0 0
$$229$$ 4201.70 1.21247 0.606237 0.795284i $$-0.292678\pi$$
0.606237 + 0.795284i $$0.292678\pi$$
$$230$$ 0 0
$$231$$ −1027.16 −0.292564
$$232$$ 0 0
$$233$$ 1274.94 0.358472 0.179236 0.983806i $$-0.442637\pi$$
0.179236 + 0.983806i $$0.442637\pi$$
$$234$$ 0 0
$$235$$ −1163.80 −0.323054
$$236$$ 0 0
$$237$$ −2510.32 −0.688029
$$238$$ 0 0
$$239$$ 5967.16 1.61499 0.807497 0.589872i $$-0.200822\pi$$
0.807497 + 0.589872i $$0.200822\pi$$
$$240$$ 0 0
$$241$$ 4881.64 1.30479 0.652395 0.757879i $$-0.273764\pi$$
0.652395 + 0.757879i $$0.273764\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −308.903 −0.0805513
$$246$$ 0 0
$$247$$ 117.937 0.0303812
$$248$$ 0 0
$$249$$ 881.197 0.224271
$$250$$ 0 0
$$251$$ −2262.63 −0.568988 −0.284494 0.958678i $$-0.591826\pi$$
−0.284494 + 0.958678i $$0.591826\pi$$
$$252$$ 0 0
$$253$$ −1864.98 −0.463439
$$254$$ 0 0
$$255$$ 2586.03 0.635072
$$256$$ 0 0
$$257$$ −6210.60 −1.50742 −0.753709 0.657209i $$-0.771737\pi$$
−0.753709 + 0.657209i $$0.771737\pi$$
$$258$$ 0 0
$$259$$ −2337.35 −0.560757
$$260$$ 0 0
$$261$$ 475.051 0.112663
$$262$$ 0 0
$$263$$ 2972.69 0.696973 0.348486 0.937314i $$-0.386696\pi$$
0.348486 + 0.937314i $$0.386696\pi$$
$$264$$ 0 0
$$265$$ −2268.38 −0.525833
$$266$$ 0 0
$$267$$ −3896.68 −0.893157
$$268$$ 0 0
$$269$$ −4443.42 −1.00714 −0.503569 0.863955i $$-0.667980\pi$$
−0.503569 + 0.863955i $$0.667980\pi$$
$$270$$ 0 0
$$271$$ 6840.25 1.53327 0.766634 0.642084i $$-0.221930\pi$$
0.766634 + 0.642084i $$0.221930\pi$$
$$272$$ 0 0
$$273$$ −54.7737 −0.0121430
$$274$$ 0 0
$$275$$ 4170.17 0.914439
$$276$$ 0 0
$$277$$ −3228.67 −0.700332 −0.350166 0.936688i $$-0.613875\pi$$
−0.350166 + 0.936688i $$0.613875\pi$$
$$278$$ 0 0
$$279$$ 133.051 0.0285504
$$280$$ 0 0
$$281$$ −6453.83 −1.37012 −0.685059 0.728488i $$-0.740224\pi$$
−0.685059 + 0.728488i $$0.740224\pi$$
$$282$$ 0 0
$$283$$ −3840.72 −0.806739 −0.403369 0.915037i $$-0.632161\pi$$
−0.403369 + 0.915037i $$0.632161\pi$$
$$284$$ 0 0
$$285$$ −855.153 −0.177737
$$286$$ 0 0
$$287$$ −1590.84 −0.327193
$$288$$ 0 0
$$289$$ 13784.1 2.80563
$$290$$ 0 0
$$291$$ 603.124 0.121497
$$292$$ 0 0
$$293$$ 8801.01 1.75481 0.877407 0.479747i $$-0.159272\pi$$
0.877407 + 0.479747i $$0.159272\pi$$
$$294$$ 0 0
$$295$$ 629.830 0.124306
$$296$$ 0 0
$$297$$ 1320.63 0.258017
$$298$$ 0 0
$$299$$ −99.4506 −0.0192354
$$300$$ 0 0
$$301$$ −2789.03 −0.534077
$$302$$ 0 0
$$303$$ 3160.53 0.599234
$$304$$ 0 0
$$305$$ 4252.24 0.798303
$$306$$ 0 0
$$307$$ 3926.72 0.730000 0.365000 0.931008i $$-0.381069\pi$$
0.365000 + 0.931008i $$0.381069\pi$$
$$308$$ 0 0
$$309$$ 3077.20 0.566523
$$310$$ 0 0
$$311$$ 6143.13 1.12008 0.560040 0.828466i $$-0.310786\pi$$
0.560040 + 0.828466i $$0.310786\pi$$
$$312$$ 0 0
$$313$$ 3824.19 0.690594 0.345297 0.938493i $$-0.387778\pi$$
0.345297 + 0.938493i $$0.387778\pi$$
$$314$$ 0 0
$$315$$ 397.160 0.0710396
$$316$$ 0 0
$$317$$ 7949.68 1.40851 0.704257 0.709946i $$-0.251280\pi$$
0.704257 + 0.709946i $$0.251280\pi$$
$$318$$ 0 0
$$319$$ −2581.77 −0.453138
$$320$$ 0 0
$$321$$ −310.255 −0.0539463
$$322$$ 0 0
$$323$$ −6182.78 −1.06508
$$324$$ 0 0
$$325$$ 222.376 0.0379544
$$326$$ 0 0
$$327$$ 2031.40 0.343537
$$328$$ 0 0
$$329$$ −1292.26 −0.216549
$$330$$ 0 0
$$331$$ 11236.4 1.86588 0.932940 0.360032i $$-0.117234\pi$$
0.932940 + 0.360032i $$0.117234\pi$$
$$332$$ 0 0
$$333$$ 3005.17 0.494541
$$334$$ 0 0
$$335$$ −2376.40 −0.387572
$$336$$ 0 0
$$337$$ 8425.41 1.36190 0.680951 0.732329i $$-0.261567\pi$$
0.680951 + 0.732329i $$0.261567\pi$$
$$338$$ 0 0
$$339$$ 1356.25 0.217290
$$340$$ 0 0
$$341$$ −723.095 −0.114832
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 0 0
$$345$$ 721.110 0.112531
$$346$$ 0 0
$$347$$ −7177.15 −1.11034 −0.555172 0.831735i $$-0.687348\pi$$
−0.555172 + 0.831735i $$0.687348\pi$$
$$348$$ 0 0
$$349$$ 1549.41 0.237644 0.118822 0.992916i $$-0.462088\pi$$
0.118822 + 0.992916i $$0.462088\pi$$
$$350$$ 0 0
$$351$$ 70.4233 0.0107092
$$352$$ 0 0
$$353$$ −566.231 −0.0853752 −0.0426876 0.999088i $$-0.513592\pi$$
−0.0426876 + 0.999088i $$0.513592\pi$$
$$354$$ 0 0
$$355$$ −7486.81 −1.11932
$$356$$ 0 0
$$357$$ 2871.48 0.425700
$$358$$ 0 0
$$359$$ −3848.19 −0.565738 −0.282869 0.959159i $$-0.591286\pi$$
−0.282869 + 0.959159i $$0.591286\pi$$
$$360$$ 0 0
$$361$$ −4814.46 −0.701919
$$362$$ 0 0
$$363$$ −3184.27 −0.460415
$$364$$ 0 0
$$365$$ 4638.74 0.665213
$$366$$ 0 0
$$367$$ −12542.2 −1.78392 −0.891960 0.452113i $$-0.850670\pi$$
−0.891960 + 0.452113i $$0.850670\pi$$
$$368$$ 0 0
$$369$$ 2045.37 0.288557
$$370$$ 0 0
$$371$$ −2518.77 −0.352475
$$372$$ 0 0
$$373$$ 6345.87 0.880903 0.440452 0.897776i $$-0.354818\pi$$
0.440452 + 0.897776i $$0.354818\pi$$
$$374$$ 0 0
$$375$$ −3976.48 −0.547586
$$376$$ 0 0
$$377$$ −137.673 −0.0188078
$$378$$ 0 0
$$379$$ 12200.2 1.65351 0.826757 0.562559i $$-0.190183\pi$$
0.826757 + 0.562559i $$0.190183\pi$$
$$380$$ 0 0
$$381$$ −3548.63 −0.477170
$$382$$ 0 0
$$383$$ 2770.91 0.369679 0.184839 0.982769i $$-0.440824\pi$$
0.184839 + 0.982769i $$0.440824\pi$$
$$384$$ 0 0
$$385$$ −2158.45 −0.285727
$$386$$ 0 0
$$387$$ 3585.90 0.471011
$$388$$ 0 0
$$389$$ 1581.89 0.206183 0.103091 0.994672i $$-0.467127\pi$$
0.103091 + 0.994672i $$0.467127\pi$$
$$390$$ 0 0
$$391$$ 5213.65 0.674336
$$392$$ 0 0
$$393$$ 3773.26 0.484314
$$394$$ 0 0
$$395$$ −5275.13 −0.671951
$$396$$ 0 0
$$397$$ −14235.9 −1.79970 −0.899848 0.436203i $$-0.856323\pi$$
−0.899848 + 0.436203i $$0.856323\pi$$
$$398$$ 0 0
$$399$$ −949.547 −0.119140
$$400$$ 0 0
$$401$$ −9556.18 −1.19006 −0.595028 0.803705i $$-0.702859\pi$$
−0.595028 + 0.803705i $$0.702859\pi$$
$$402$$ 0 0
$$403$$ −38.5592 −0.00476619
$$404$$ 0 0
$$405$$ −510.635 −0.0626510
$$406$$ 0 0
$$407$$ −16332.2 −1.98909
$$408$$ 0 0
$$409$$ 2858.17 0.345544 0.172772 0.984962i $$-0.444728\pi$$
0.172772 + 0.984962i $$0.444728\pi$$
$$410$$ 0 0
$$411$$ 169.577 0.0203518
$$412$$ 0 0
$$413$$ 699.353 0.0833242
$$414$$ 0 0
$$415$$ 1851.73 0.219031
$$416$$ 0 0
$$417$$ 6340.70 0.744617
$$418$$ 0 0
$$419$$ 13333.3 1.55460 0.777299 0.629132i $$-0.216589\pi$$
0.777299 + 0.629132i $$0.216589\pi$$
$$420$$ 0 0
$$421$$ −13567.4 −1.57063 −0.785314 0.619098i $$-0.787499\pi$$
−0.785314 + 0.619098i $$0.787499\pi$$
$$422$$ 0 0
$$423$$ 1661.47 0.190978
$$424$$ 0 0
$$425$$ −11657.9 −1.33057
$$426$$ 0 0
$$427$$ 4721.61 0.535116
$$428$$ 0 0
$$429$$ −382.730 −0.0430732
$$430$$ 0 0
$$431$$ −14207.3 −1.58780 −0.793898 0.608051i $$-0.791952\pi$$
−0.793898 + 0.608051i $$0.791952\pi$$
$$432$$ 0 0
$$433$$ −10530.6 −1.16875 −0.584375 0.811484i $$-0.698660\pi$$
−0.584375 + 0.811484i $$0.698660\pi$$
$$434$$ 0 0
$$435$$ 998.262 0.110030
$$436$$ 0 0
$$437$$ −1724.06 −0.188725
$$438$$ 0 0
$$439$$ 4038.23 0.439030 0.219515 0.975609i $$-0.429553\pi$$
0.219515 + 0.975609i $$0.429553\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 0 0
$$443$$ 4574.15 0.490574 0.245287 0.969450i $$-0.421118\pi$$
0.245287 + 0.969450i $$0.421118\pi$$
$$444$$ 0 0
$$445$$ −8188.40 −0.872286
$$446$$ 0 0
$$447$$ −5384.88 −0.569789
$$448$$ 0 0
$$449$$ 14957.1 1.57209 0.786044 0.618171i $$-0.212126\pi$$
0.786044 + 0.618171i $$0.212126\pi$$
$$450$$ 0 0
$$451$$ −11116.0 −1.16060
$$452$$ 0 0
$$453$$ 1131.09 0.117314
$$454$$ 0 0
$$455$$ −115.100 −0.0118593
$$456$$ 0 0
$$457$$ 3027.65 0.309907 0.154954 0.987922i $$-0.450477\pi$$
0.154954 + 0.987922i $$0.450477\pi$$
$$458$$ 0 0
$$459$$ −3691.90 −0.375432
$$460$$ 0 0
$$461$$ −10877.7 −1.09897 −0.549484 0.835504i $$-0.685176\pi$$
−0.549484 + 0.835504i $$0.685176\pi$$
$$462$$ 0 0
$$463$$ 4038.28 0.405345 0.202673 0.979247i $$-0.435037\pi$$
0.202673 + 0.979247i $$0.435037\pi$$
$$464$$ 0 0
$$465$$ 279.591 0.0278833
$$466$$ 0 0
$$467$$ −8411.80 −0.833515 −0.416758 0.909018i $$-0.636834\pi$$
−0.416758 + 0.909018i $$0.636834\pi$$
$$468$$ 0 0
$$469$$ −2638.71 −0.259796
$$470$$ 0 0
$$471$$ 2696.10 0.263758
$$472$$ 0 0
$$473$$ −19488.3 −1.89445
$$474$$ 0 0
$$475$$ 3855.07 0.372384
$$476$$ 0 0
$$477$$ 3238.42 0.310854
$$478$$ 0 0
$$479$$ −7172.70 −0.684194 −0.342097 0.939665i $$-0.611137\pi$$
−0.342097 + 0.939665i $$0.611137\pi$$
$$480$$ 0 0
$$481$$ −870.921 −0.0825584
$$482$$ 0 0
$$483$$ 800.708 0.0754316
$$484$$ 0 0
$$485$$ 1267.39 0.118658
$$486$$ 0 0
$$487$$ 5580.52 0.519256 0.259628 0.965709i $$-0.416400\pi$$
0.259628 + 0.965709i $$0.416400\pi$$
$$488$$ 0 0
$$489$$ 11590.6 1.07187
$$490$$ 0 0
$$491$$ −12489.4 −1.14794 −0.573972 0.818875i $$-0.694598\pi$$
−0.573972 + 0.818875i $$0.694598\pi$$
$$492$$ 0 0
$$493$$ 7217.46 0.659347
$$494$$ 0 0
$$495$$ 2775.15 0.251988
$$496$$ 0 0
$$497$$ −8313.22 −0.750300
$$498$$ 0 0
$$499$$ 15216.6 1.36511 0.682556 0.730834i $$-0.260869\pi$$
0.682556 + 0.730834i $$0.260869\pi$$
$$500$$ 0 0
$$501$$ −8584.31 −0.765506
$$502$$ 0 0
$$503$$ 1814.89 0.160879 0.0804393 0.996760i $$-0.474368\pi$$
0.0804393 + 0.996760i $$0.474368\pi$$
$$504$$ 0 0
$$505$$ 6641.47 0.585231
$$506$$ 0 0
$$507$$ 6570.59 0.575562
$$508$$ 0 0
$$509$$ −4853.68 −0.422663 −0.211332 0.977414i $$-0.567780\pi$$
−0.211332 + 0.977414i $$0.567780\pi$$
$$510$$ 0 0
$$511$$ 5150.77 0.445904
$$512$$ 0 0
$$513$$ 1220.85 0.105072
$$514$$ 0 0
$$515$$ 6466.35 0.553285
$$516$$ 0 0
$$517$$ −9029.63 −0.768129
$$518$$ 0 0
$$519$$ −2937.01 −0.248402
$$520$$ 0 0
$$521$$ −9913.18 −0.833598 −0.416799 0.908999i $$-0.636848\pi$$
−0.416799 + 0.908999i $$0.636848\pi$$
$$522$$ 0 0
$$523$$ 4524.29 0.378267 0.189133 0.981951i $$-0.439432\pi$$
0.189133 + 0.981951i $$0.439432\pi$$
$$524$$ 0 0
$$525$$ −1790.42 −0.148838
$$526$$ 0 0
$$527$$ 2021.45 0.167089
$$528$$ 0 0
$$529$$ −10713.2 −0.880512
$$530$$ 0 0
$$531$$ −899.168 −0.0734850
$$532$$ 0 0
$$533$$ −592.763 −0.0481715
$$534$$ 0 0
$$535$$ −651.963 −0.0526857
$$536$$ 0 0
$$537$$ −3438.82 −0.276343
$$538$$ 0 0
$$539$$ −2396.71 −0.191528
$$540$$ 0 0
$$541$$ 3724.94 0.296022 0.148011 0.988986i $$-0.452713\pi$$
0.148011 + 0.988986i $$0.452713\pi$$
$$542$$ 0 0
$$543$$ −11788.0 −0.931623
$$544$$ 0 0
$$545$$ 4268.74 0.335510
$$546$$ 0 0
$$547$$ −16121.6 −1.26016 −0.630082 0.776528i $$-0.716979\pi$$
−0.630082 + 0.776528i $$0.716979\pi$$
$$548$$ 0 0
$$549$$ −6070.64 −0.471928
$$550$$ 0 0
$$551$$ −2386.69 −0.184530
$$552$$ 0 0
$$553$$ −5857.42 −0.450421
$$554$$ 0 0
$$555$$ 6314.99 0.482985
$$556$$ 0 0
$$557$$ 20451.7 1.55577 0.777887 0.628405i $$-0.216292\pi$$
0.777887 + 0.628405i $$0.216292\pi$$
$$558$$ 0 0
$$559$$ −1039.22 −0.0786303
$$560$$ 0 0
$$561$$ 20064.4 1.51002
$$562$$ 0 0
$$563$$ 10046.7 0.752078 0.376039 0.926604i $$-0.377286\pi$$
0.376039 + 0.926604i $$0.377286\pi$$
$$564$$ 0 0
$$565$$ 2849.99 0.212212
$$566$$ 0 0
$$567$$ −567.000 −0.0419961
$$568$$ 0 0
$$569$$ −15356.4 −1.13141 −0.565705 0.824608i $$-0.691396\pi$$
−0.565705 + 0.824608i $$0.691396\pi$$
$$570$$ 0 0
$$571$$ 19333.6 1.41696 0.708480 0.705731i $$-0.249381\pi$$
0.708480 + 0.705731i $$0.249381\pi$$
$$572$$ 0 0
$$573$$ −8812.07 −0.642460
$$574$$ 0 0
$$575$$ −3250.79 −0.235769
$$576$$ 0 0
$$577$$ 26258.8 1.89458 0.947288 0.320384i $$-0.103812\pi$$
0.947288 + 0.320384i $$0.103812\pi$$
$$578$$ 0 0
$$579$$ 10599.5 0.760795
$$580$$ 0 0
$$581$$ 2056.13 0.146820
$$582$$ 0 0
$$583$$ −17599.9 −1.25028
$$584$$ 0 0
$$585$$ 147.986 0.0104589
$$586$$ 0 0
$$587$$ 4868.98 0.342358 0.171179 0.985240i $$-0.445242\pi$$
0.171179 + 0.985240i $$0.445242\pi$$
$$588$$ 0 0
$$589$$ −668.457 −0.0467628
$$590$$ 0 0
$$591$$ −1754.57 −0.122121
$$592$$ 0 0
$$593$$ −13647.1 −0.945055 −0.472528 0.881316i $$-0.656658\pi$$
−0.472528 + 0.881316i $$0.656658\pi$$
$$594$$ 0 0
$$595$$ 6034.07 0.415752
$$596$$ 0 0
$$597$$ 6474.25 0.443841
$$598$$ 0 0
$$599$$ −7543.11 −0.514529 −0.257265 0.966341i $$-0.582821\pi$$
−0.257265 + 0.966341i $$0.582821\pi$$
$$600$$ 0 0
$$601$$ −19522.4 −1.32501 −0.662507 0.749056i $$-0.730507\pi$$
−0.662507 + 0.749056i $$0.730507\pi$$
$$602$$ 0 0
$$603$$ 3392.63 0.229119
$$604$$ 0 0
$$605$$ −6691.36 −0.449657
$$606$$ 0 0
$$607$$ −13804.5 −0.923079 −0.461539 0.887120i $$-0.652703\pi$$
−0.461539 + 0.887120i $$0.652703\pi$$
$$608$$ 0 0
$$609$$ 1108.45 0.0737550
$$610$$ 0 0
$$611$$ −481.508 −0.0318817
$$612$$ 0 0
$$613$$ 21718.8 1.43102 0.715508 0.698605i $$-0.246195\pi$$
0.715508 + 0.698605i $$0.246195\pi$$
$$614$$ 0 0
$$615$$ 4298.09 0.281814
$$616$$ 0 0
$$617$$ −5183.85 −0.338240 −0.169120 0.985595i $$-0.554093\pi$$
−0.169120 + 0.985595i $$0.554093\pi$$
$$618$$ 0 0
$$619$$ −22003.7 −1.42876 −0.714382 0.699756i $$-0.753292\pi$$
−0.714382 + 0.699756i $$0.753292\pi$$
$$620$$ 0 0
$$621$$ −1029.48 −0.0665244
$$622$$ 0 0
$$623$$ −9092.25 −0.584708
$$624$$ 0 0
$$625$$ 2301.14 0.147273
$$626$$ 0 0
$$627$$ −6634.95 −0.422607
$$628$$ 0 0
$$629$$ 45657.6 2.89426
$$630$$ 0 0
$$631$$ 985.836 0.0621957 0.0310979 0.999516i $$-0.490100\pi$$
0.0310979 + 0.999516i $$0.490100\pi$$
$$632$$ 0 0
$$633$$ −12870.1 −0.808123
$$634$$ 0 0
$$635$$ −7457.01 −0.466020
$$636$$ 0 0
$$637$$ −127.805 −0.00794949
$$638$$ 0 0
$$639$$ 10688.4 0.661702
$$640$$ 0 0
$$641$$ 24282.6 1.49626 0.748131 0.663551i $$-0.230951\pi$$
0.748131 + 0.663551i $$0.230951\pi$$
$$642$$ 0 0
$$643$$ 4743.12 0.290903 0.145451 0.989365i $$-0.453537\pi$$
0.145451 + 0.989365i $$0.453537\pi$$
$$644$$ 0 0
$$645$$ 7535.33 0.460005
$$646$$ 0 0
$$647$$ −29641.1 −1.80110 −0.900549 0.434754i $$-0.856835\pi$$
−0.900549 + 0.434754i $$0.856835\pi$$
$$648$$ 0 0
$$649$$ 4886.72 0.295563
$$650$$ 0 0
$$651$$ 310.453 0.0186906
$$652$$ 0 0
$$653$$ −23046.9 −1.38116 −0.690578 0.723258i $$-0.742644\pi$$
−0.690578 + 0.723258i $$0.742644\pi$$
$$654$$ 0 0
$$655$$ 7929.04 0.472997
$$656$$ 0 0
$$657$$ −6622.42 −0.393250
$$658$$ 0 0
$$659$$ −5795.12 −0.342558 −0.171279 0.985223i $$-0.554790\pi$$
−0.171279 + 0.985223i $$0.554790\pi$$
$$660$$ 0 0
$$661$$ 2592.59 0.152557 0.0762784 0.997087i $$-0.475696\pi$$
0.0762784 + 0.997087i $$0.475696\pi$$
$$662$$ 0 0
$$663$$ 1069.94 0.0626744
$$664$$ 0 0
$$665$$ −1995.36 −0.116356
$$666$$ 0 0
$$667$$ 2012.58 0.116833
$$668$$ 0 0
$$669$$ −8231.33 −0.475697
$$670$$ 0 0
$$671$$ 32992.2 1.89814
$$672$$ 0 0
$$673$$ 28156.0 1.61268 0.806341 0.591451i $$-0.201445\pi$$
0.806341 + 0.591451i $$0.201445\pi$$
$$674$$ 0 0
$$675$$ 2301.96 0.131263
$$676$$ 0 0
$$677$$ 20271.4 1.15080 0.575402 0.817871i $$-0.304846\pi$$
0.575402 + 0.817871i $$0.304846\pi$$
$$678$$ 0 0
$$679$$ 1407.29 0.0795388
$$680$$ 0 0
$$681$$ −5174.38 −0.291164
$$682$$ 0 0
$$683$$ 13267.0 0.743264 0.371632 0.928380i $$-0.378798\pi$$
0.371632 + 0.928380i $$0.378798\pi$$
$$684$$ 0 0
$$685$$ 356.345 0.0198763
$$686$$ 0 0
$$687$$ −12605.1 −0.700022
$$688$$ 0 0
$$689$$ −938.520 −0.0518937
$$690$$ 0 0
$$691$$ 15966.3 0.878995 0.439497 0.898244i $$-0.355157\pi$$
0.439497 + 0.898244i $$0.355157\pi$$
$$692$$ 0 0
$$693$$ 3081.48 0.168912
$$694$$ 0 0
$$695$$ 13324.2 0.727217
$$696$$ 0 0
$$697$$ 31075.3 1.68875
$$698$$ 0 0
$$699$$ −3824.82 −0.206964
$$700$$ 0 0
$$701$$ 7525.29 0.405458 0.202729 0.979235i $$-0.435019\pi$$
0.202729 + 0.979235i $$0.435019\pi$$
$$702$$ 0 0
$$703$$ −15098.1 −0.810010
$$704$$ 0 0
$$705$$ 3491.39 0.186515
$$706$$ 0 0
$$707$$ 7374.58 0.392291
$$708$$ 0 0
$$709$$ 20033.6 1.06118 0.530591 0.847628i $$-0.321970\pi$$
0.530591 + 0.847628i $$0.321970\pi$$
$$710$$ 0 0
$$711$$ 7530.96 0.397234
$$712$$ 0 0
$$713$$ 563.678 0.0296071
$$714$$ 0 0
$$715$$ −804.261 −0.0420666
$$716$$ 0 0
$$717$$ −17901.5 −0.932417
$$718$$ 0 0
$$719$$ 8081.69 0.419188 0.209594 0.977788i $$-0.432786\pi$$
0.209594 + 0.977788i $$0.432786\pi$$
$$720$$ 0 0
$$721$$ 7180.13 0.370876
$$722$$ 0 0
$$723$$ −14644.9 −0.753321
$$724$$ 0 0
$$725$$ −4500.21 −0.230529
$$726$$ 0 0
$$727$$ −34117.8 −1.74052 −0.870262 0.492590i $$-0.836050\pi$$
−0.870262 + 0.492590i $$0.836050\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 54480.6 2.75655
$$732$$ 0 0
$$733$$ 20048.0 1.01022 0.505110 0.863055i $$-0.331452\pi$$
0.505110 + 0.863055i $$0.331452\pi$$
$$734$$ 0 0
$$735$$ 926.708 0.0465063
$$736$$ 0 0
$$737$$ −18438.0 −0.921534
$$738$$ 0 0
$$739$$ 407.607 0.0202897 0.0101448 0.999949i $$-0.496771\pi$$
0.0101448 + 0.999949i $$0.496771\pi$$
$$740$$ 0 0
$$741$$ −353.811 −0.0175406
$$742$$ 0 0
$$743$$ 128.374 0.00633861 0.00316930 0.999995i $$-0.498991\pi$$
0.00316930 + 0.999995i $$0.498991\pi$$
$$744$$ 0 0
$$745$$ −11315.7 −0.556475
$$746$$ 0 0
$$747$$ −2643.59 −0.129483
$$748$$ 0 0
$$749$$ −723.929 −0.0353161
$$750$$ 0 0
$$751$$ −25076.0 −1.21842 −0.609212 0.793007i $$-0.708514\pi$$
−0.609212 + 0.793007i $$0.708514\pi$$
$$752$$ 0 0
$$753$$ 6787.90 0.328506
$$754$$ 0 0
$$755$$ 2376.86 0.114573
$$756$$ 0 0
$$757$$ −21824.1 −1.04783 −0.523916 0.851770i $$-0.675530\pi$$
−0.523916 + 0.851770i $$0.675530\pi$$
$$758$$ 0 0
$$759$$ 5594.93 0.267567
$$760$$ 0 0
$$761$$ −38939.2 −1.85485 −0.927427 0.374004i $$-0.877985\pi$$
−0.927427 + 0.374004i $$0.877985\pi$$
$$762$$ 0 0
$$763$$ 4739.94 0.224898
$$764$$ 0 0
$$765$$ −7758.09 −0.366659
$$766$$ 0 0
$$767$$ 260.586 0.0122675
$$768$$ 0 0
$$769$$ 3079.42 0.144404 0.0722021 0.997390i $$-0.476997\pi$$
0.0722021 + 0.997390i $$0.476997\pi$$
$$770$$ 0 0
$$771$$ 18631.8 0.870308
$$772$$ 0 0
$$773$$ 31770.9 1.47829 0.739146 0.673545i $$-0.235229\pi$$
0.739146 + 0.673545i $$0.235229\pi$$
$$774$$ 0 0
$$775$$ −1260.41 −0.0584195
$$776$$ 0 0
$$777$$ 7012.06 0.323753
$$778$$ 0 0
$$779$$ −10276.0 −0.472628
$$780$$ 0 0
$$781$$ −58088.5 −2.66142
$$782$$ 0 0
$$783$$ −1425.15 −0.0650458
$$784$$ 0 0
$$785$$ 5665.53 0.257594
$$786$$ 0 0
$$787$$ −6736.02 −0.305099 −0.152550 0.988296i $$-0.548748\pi$$
−0.152550 + 0.988296i $$0.548748\pi$$
$$788$$ 0 0
$$789$$ −8918.06 −0.402397
$$790$$ 0 0
$$791$$ 3164.58 0.142250
$$792$$ 0 0
$$793$$ 1759.32 0.0787834
$$794$$ 0 0
$$795$$ 6805.15 0.303590
$$796$$ 0 0
$$797$$ 13445.1 0.597552 0.298776 0.954323i $$-0.403422\pi$$
0.298776 + 0.954323i $$0.403422\pi$$
$$798$$ 0 0
$$799$$ 25242.8 1.11768
$$800$$ 0 0
$$801$$ 11690.0 0.515664
$$802$$ 0 0
$$803$$ 35991.0 1.58169
$$804$$ 0 0
$$805$$ 1682.59 0.0736689
$$806$$ 0 0
$$807$$ 13330.3 0.581472
$$808$$ 0 0
$$809$$ 20337.7 0.883853 0.441927 0.897051i $$-0.354295\pi$$
0.441927 + 0.897051i $$0.354295\pi$$
$$810$$ 0 0
$$811$$ 23069.9 0.998884 0.499442 0.866347i $$-0.333538\pi$$
0.499442 + 0.866347i $$0.333538\pi$$
$$812$$ 0 0
$$813$$ −20520.8 −0.885233
$$814$$ 0 0
$$815$$ 24356.1 1.04682
$$816$$ 0 0
$$817$$ −18015.8 −0.771471
$$818$$ 0 0
$$819$$ 164.321 0.00701079
$$820$$ 0 0
$$821$$ 8230.74 0.349884 0.174942 0.984579i $$-0.444026\pi$$
0.174942 + 0.984579i $$0.444026\pi$$
$$822$$ 0 0
$$823$$ −17577.9 −0.744506 −0.372253 0.928131i $$-0.621415\pi$$
−0.372253 + 0.928131i $$0.621415\pi$$
$$824$$ 0 0
$$825$$ −12510.5 −0.527951
$$826$$ 0 0
$$827$$ 6440.14 0.270793 0.135396 0.990792i $$-0.456769\pi$$
0.135396 + 0.990792i $$0.456769\pi$$
$$828$$ 0 0
$$829$$ 5084.23 0.213007 0.106503 0.994312i $$-0.466034\pi$$
0.106503 + 0.994312i $$0.466034\pi$$
$$830$$ 0 0
$$831$$ 9686.01 0.404337
$$832$$ 0 0
$$833$$ 6700.12 0.278686
$$834$$ 0 0
$$835$$ −18038.9 −0.747618
$$836$$ 0 0
$$837$$ −399.153 −0.0164836
$$838$$ 0 0
$$839$$ 25150.5 1.03491 0.517456 0.855710i $$-0.326879\pi$$
0.517456 + 0.855710i $$0.326879\pi$$
$$840$$ 0 0
$$841$$ −21602.9 −0.885764
$$842$$ 0 0
$$843$$ 19361.5 0.791038
$$844$$ 0 0
$$845$$ 13807.3 0.562113
$$846$$ 0 0
$$847$$ −7429.96 −0.301413
$$848$$ 0 0
$$849$$ 11522.2 0.465771
$$850$$ 0 0
$$851$$ 12731.5 0.512846
$$852$$ 0 0
$$853$$ −6408.37 −0.257232 −0.128616 0.991694i $$-0.541053\pi$$
−0.128616 + 0.991694i $$0.541053\pi$$
$$854$$ 0 0
$$855$$ 2565.46 0.102616
$$856$$ 0 0
$$857$$ 17248.6 0.687515 0.343758 0.939058i $$-0.388300\pi$$
0.343758 + 0.939058i $$0.388300\pi$$
$$858$$ 0 0
$$859$$ −3159.07 −0.125479 −0.0627393 0.998030i $$-0.519984\pi$$
−0.0627393 + 0.998030i $$0.519984\pi$$
$$860$$ 0 0
$$861$$ 4772.52 0.188905
$$862$$ 0 0
$$863$$ −41071.9 −1.62005 −0.810025 0.586395i $$-0.800547\pi$$
−0.810025 + 0.586395i $$0.800547\pi$$
$$864$$ 0 0
$$865$$ −6171.78 −0.242597
$$866$$ 0 0
$$867$$ −41352.2 −1.61983
$$868$$ 0 0
$$869$$ −40928.6 −1.59771
$$870$$ 0 0
$$871$$ −983.210 −0.0382489
$$872$$ 0 0
$$873$$ −1809.37 −0.0701466
$$874$$ 0 0
$$875$$ −9278.46 −0.358479
$$876$$ 0 0
$$877$$ −1034.90 −0.0398472 −0.0199236 0.999802i $$-0.506342\pi$$
−0.0199236 + 0.999802i $$0.506342\pi$$
$$878$$ 0 0
$$879$$ −26403.0 −1.01314
$$880$$ 0 0
$$881$$ −3109.73 −0.118921 −0.0594606 0.998231i $$-0.518938\pi$$
−0.0594606 + 0.998231i $$0.518938\pi$$
$$882$$ 0 0
$$883$$ 19782.8 0.753955 0.376978 0.926222i $$-0.376963\pi$$
0.376978 + 0.926222i $$0.376963\pi$$
$$884$$ 0 0
$$885$$ −1889.49 −0.0717678
$$886$$ 0 0
$$887$$ −9355.56 −0.354148 −0.177074 0.984198i $$-0.556663\pi$$
−0.177074 + 0.984198i $$0.556663\pi$$
$$888$$ 0 0
$$889$$ −8280.13 −0.312381
$$890$$ 0 0
$$891$$ −3961.90 −0.148966
$$892$$ 0 0
$$893$$ −8347.35 −0.312803
$$894$$ 0 0
$$895$$ −7226.27 −0.269886
$$896$$ 0 0
$$897$$ 298.352 0.0111055
$$898$$ 0 0
$$899$$ 780.322 0.0289491
$$900$$ 0 0
$$901$$ 49201.4 1.81924
$$902$$ 0 0
$$903$$ 8367.09 0.308349
$$904$$ 0 0
$$905$$ −24771.0 −0.909853
$$906$$ 0 0
$$907$$ −16211.9 −0.593505 −0.296752 0.954954i $$-0.595904\pi$$
−0.296752 + 0.954954i $$0.595904\pi$$
$$908$$ 0 0
$$909$$ −9481.60 −0.345968
$$910$$ 0 0
$$911$$ −14540.6 −0.528816 −0.264408 0.964411i $$-0.585177\pi$$
−0.264408 + 0.964411i $$0.585177\pi$$
$$912$$ 0 0
$$913$$ 14367.2 0.520792
$$914$$ 0 0
$$915$$ −12756.7 −0.460901
$$916$$ 0 0
$$917$$ 8804.26 0.317058
$$918$$ 0 0
$$919$$ 37239.2 1.33668 0.668339 0.743857i $$-0.267005\pi$$
0.668339 + 0.743857i $$0.267005\pi$$
$$920$$ 0 0
$$921$$ −11780.2 −0.421466
$$922$$ 0 0
$$923$$ −3097.59 −0.110464
$$924$$ 0 0
$$925$$ −28468.2 −1.01192
$$926$$ 0 0
$$927$$ −9231.59 −0.327082
$$928$$ 0 0
$$929$$ 22062.5 0.779167 0.389583 0.920991i $$-0.372619\pi$$
0.389583 + 0.920991i $$0.372619\pi$$
$$930$$ 0 0
$$931$$ −2215.61 −0.0779954
$$932$$ 0 0
$$933$$ −18429.4 −0.646678
$$934$$ 0 0
$$935$$ 42163.0 1.47473
$$936$$ 0 0
$$937$$ −15597.3 −0.543800 −0.271900 0.962326i $$-0.587652\pi$$
−0.271900 + 0.962326i $$0.587652\pi$$
$$938$$ 0 0
$$939$$ −11472.6 −0.398715
$$940$$ 0 0
$$941$$ −15887.1 −0.550377 −0.275188 0.961390i $$-0.588740\pi$$
−0.275188 + 0.961390i $$0.588740\pi$$
$$942$$ 0 0
$$943$$ 8665.29 0.299237
$$944$$ 0 0
$$945$$ −1191.48 −0.0410147
$$946$$ 0 0
$$947$$ −54754.6 −1.87887 −0.939433 0.342733i $$-0.888648\pi$$
−0.939433 + 0.342733i $$0.888648\pi$$
$$948$$ 0 0
$$949$$ 1919.23 0.0656489
$$950$$ 0 0
$$951$$ −23849.0 −0.813205
$$952$$ 0 0
$$953$$ 12091.3 0.410992 0.205496 0.978658i $$-0.434119\pi$$
0.205496 + 0.978658i $$0.434119\pi$$
$$954$$ 0 0
$$955$$ −18517.5 −0.627447
$$956$$ 0 0
$$957$$ 7745.30 0.261620
$$958$$ 0 0
$$959$$ 395.679 0.0133234
$$960$$ 0 0
$$961$$ −29572.4 −0.992664
$$962$$ 0 0
$$963$$ 930.765 0.0311459
$$964$$ 0 0
$$965$$ 22273.6 0.743017
$$966$$ 0 0
$$967$$ 27415.1 0.911695 0.455848 0.890058i $$-0.349336\pi$$
0.455848 + 0.890058i $$0.349336\pi$$
$$968$$ 0 0
$$969$$ 18548.4 0.614921
$$970$$ 0 0
$$971$$ −55397.0 −1.83087 −0.915435 0.402466i $$-0.868153\pi$$
−0.915435 + 0.402466i $$0.868153\pi$$
$$972$$ 0 0
$$973$$ 14795.0 0.487467
$$974$$ 0 0
$$975$$ −667.127 −0.0219130
$$976$$ 0 0
$$977$$ 18294.3 0.599065 0.299532 0.954086i $$-0.403169\pi$$
0.299532 + 0.954086i $$0.403169\pi$$
$$978$$ 0 0
$$979$$ −63532.0 −2.07405
$$980$$ 0 0
$$981$$ −6094.20 −0.198341
$$982$$ 0 0
$$983$$ 23803.2 0.772334 0.386167 0.922429i $$-0.373799\pi$$
0.386167 + 0.922429i $$0.373799\pi$$
$$984$$ 0 0
$$985$$ −3687.01 −0.119267
$$986$$ 0 0
$$987$$ 3876.77 0.125024
$$988$$ 0 0
$$989$$ 15191.8 0.488445
$$990$$ 0 0
$$991$$ 13624.5 0.436726 0.218363 0.975868i $$-0.429928\pi$$
0.218363 + 0.975868i $$0.429928\pi$$
$$992$$ 0 0
$$993$$ −33709.1 −1.07727
$$994$$ 0 0
$$995$$ 13604.8 0.433470
$$996$$ 0 0
$$997$$ −46834.4 −1.48772 −0.743861 0.668334i $$-0.767008\pi$$
−0.743861 + 0.668334i $$0.767008\pi$$
$$998$$ 0 0
$$999$$ −9015.50 −0.285523
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 336.4.a.n.1.1 2
3.2 odd 2 1008.4.a.y.1.2 2
4.3 odd 2 168.4.a.h.1.1 2
7.6 odd 2 2352.4.a.bv.1.2 2
8.3 odd 2 1344.4.a.bd.1.2 2
8.5 even 2 1344.4.a.bl.1.2 2
12.11 even 2 504.4.a.j.1.2 2
28.27 even 2 1176.4.a.p.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.a.h.1.1 2 4.3 odd 2
336.4.a.n.1.1 2 1.1 even 1 trivial
504.4.a.j.1.2 2 12.11 even 2
1008.4.a.y.1.2 2 3.2 odd 2
1176.4.a.p.1.2 2 28.27 even 2
1344.4.a.bd.1.2 2 8.3 odd 2
1344.4.a.bl.1.2 2 8.5 even 2
2352.4.a.bv.1.2 2 7.6 odd 2