Properties

 Label 1176.4.a.p.1.2 Level $1176$ Weight $4$ Character 1176.1 Self dual yes Analytic conductor $69.386$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1176.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$69.3862461668$$ Analytic rank: $$1$$ 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 1.2 Root $$-6.15207$$ of defining polynomial Character $$\chi$$ $$=$$ 1176.1

$q$-expansion

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

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 $$-0.409026\pi$$
0.281929 + 0.959435i $$0.409026\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 48.9124 1.34069 0.670347 0.742047i $$-0.266145\pi$$
0.670347 + 0.742047i $$0.266145\pi$$
$$12$$ 0 0
$$13$$ 2.60827 0.0556464 0.0278232 0.999613i $$-0.491142\pi$$
0.0278232 + 0.999613i $$0.491142\pi$$
$$14$$ 0 0
$$15$$ −18.9124 −0.325544
$$16$$ 0 0
$$17$$ −136.737 −1.95080 −0.975401 0.220436i $$-0.929252\pi$$
−0.975401 + 0.220436i $$0.929252\pi$$
$$18$$ 0 0
$$19$$ −45.2165 −0.545968 −0.272984 0.962019i $$-0.588011\pi$$
−0.272984 + 0.962019i $$0.588011\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −38.1289 −0.345671 −0.172836 0.984951i $$-0.555293\pi$$
−0.172836 + 0.984951i $$0.555293\pi$$
$$24$$ 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
$$35$$ 0 0
$$36$$ 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 $$-0.642487\pi$$
−0.432835 + 0.901473i $$0.642487\pi$$
$$42$$ 0 0
$$43$$ −398.433 −1.41303 −0.706517 0.707696i $$-0.749735\pi$$
−0.706517 + 0.707696i $$0.749735\pi$$
$$44$$ 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
$$49$$ 0 0
$$50$$ 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 $$-0.249646\pi$$
0.707893 + 0.706320i $$0.249646\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 16.4429 0.0313767
$$66$$ 0 0
$$67$$ −376.959 −0.687356 −0.343678 0.939088i $$-0.611673\pi$$
−0.343678 + 0.939088i $$0.611673\pi$$
$$68$$ 0 0
$$69$$ 114.387 0.199573
$$70$$ 0 0
$$71$$ −1187.60 −1.98511 −0.992553 0.121810i $$-0.961130\pi$$
−0.992553 + 0.121810i $$0.961130\pi$$
$$72$$ 0 0
$$73$$ 735.825 1.17975 0.589875 0.807494i $$-0.299177\pi$$
0.589875 + 0.807494i $$0.299177\pi$$
$$74$$ 0 0
$$75$$ 255.774 0.393789
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −836.774 −1.19170 −0.595851 0.803095i $$-0.703185\pi$$
−0.595851 + 0.803095i $$0.703185\pi$$
$$80$$ 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 $$-0.781494\pi$$
−0.773496 + 0.633801i $$0.781494\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 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 $$-0.466445\pi$$
0.105220 + 0.994449i $$0.466445\pi$$
$$98$$ 0 0
$$99$$ 440.212 0.446898
$$100$$ 0 0
$$101$$ 1053.51 1.03790 0.518952 0.854804i $$-0.326322\pi$$
0.518952 + 0.854804i $$0.326322\pi$$
$$102$$ 0 0
$$103$$ −1025.73 −0.981247 −0.490623 0.871372i $$-0.663231\pi$$
−0.490623 + 0.871372i $$0.663231\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −103.418 −0.0934377 −0.0467188 0.998908i $$-0.514876\pi$$
−0.0467188 + 0.998908i $$0.514876\pi$$
$$108$$ 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
$$119$$ 0 0
$$120$$ 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 $$-0.635603\pi$$
−0.413241 + 0.910622i $$0.635603\pi$$
$$128$$ 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
$$133$$ 0 0
$$134$$ 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
$$147$$ 0 0
$$148$$ 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 $$-0.467605\pi$$
0.101597 + 0.994826i $$0.467605\pi$$
$$152$$ 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 $$-0.426644\pi$$
0.228421 + 0.973562i $$0.426644\pi$$
$$158$$ 0 0
$$159$$ −1079.47 −0.538414
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 3863.52 1.85653 0.928264 0.371922i $$-0.121301\pi$$
0.928264 + 0.371922i $$0.121301\pi$$
$$164$$ 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 $$-0.569015\pi$$
−0.215122 + 0.976587i $$0.569015\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 299.723 0.127280
$$178$$ 0 0
$$179$$ −1146.27 −0.478640 −0.239320 0.970941i $$-0.576925\pi$$
−0.239320 + 0.970941i $$0.576925\pi$$
$$180$$ 0 0
$$181$$ −3929.33 −1.61362 −0.806809 0.590812i $$-0.798808\pi$$
−0.806809 + 0.590812i $$0.798808\pi$$
$$182$$ 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
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2937.36 −1.11277 −0.556386 0.830924i $$-0.687813\pi$$
−0.556386 + 0.830924i $$0.687813\pi$$
$$192$$ 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
$$203$$ 0 0
$$204$$ 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 $$-0.746752\pi$$
−0.699855 + 0.714285i $$0.746752\pi$$
$$212$$ 0 0
$$213$$ 3562.81 1.14610
$$214$$ 0 0
$$215$$ −2511.78 −0.796752
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 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 $$-0.707322\pi$$
−0.606237 + 0.795284i $$0.707322\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 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 $$-0.799178\pi$$
−0.807497 + 0.589872i $$0.799178\pi$$
$$240$$ 0 0
$$241$$ −4881.64 −1.30479 −0.652395 0.757879i $$-0.726236\pi$$
−0.652395 + 0.757879i $$0.726236\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 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 $$-0.228263\pi$$
0.753709 + 0.657209i $$0.228263\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 475.051 0.112663
$$262$$ 0 0
$$263$$ −2972.69 −0.696973 −0.348486 0.937314i $$-0.613304\pi$$
−0.348486 + 0.937314i $$0.613304\pi$$
$$264$$ 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 $$-0.332020\pi$$
0.503569 + 0.863955i $$0.332020\pi$$
$$270$$ 0 0
$$271$$ 6840.25 1.53327 0.766634 0.642084i $$-0.221930\pi$$
0.766634 + 0.642084i $$0.221930\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 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
$$287$$ 0 0
$$288$$ 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 $$-0.840728\pi$$
−0.877407 + 0.479747i $$0.840728\pi$$
$$294$$ 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
$$301$$ 0 0
$$302$$ 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 $$-0.612222\pi$$
−0.345297 + 0.938493i $$0.612222\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 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
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −11236.4 −1.86588 −0.932940 0.360032i $$-0.882766\pi$$
−0.932940 + 0.360032i $$0.882766\pi$$
$$332$$ 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
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 721.110 0.112531
$$346$$ 0 0
$$347$$ 7177.15 1.11034 0.555172 0.831735i $$-0.312652\pi$$
0.555172 + 0.831735i $$0.312652\pi$$
$$348$$ 0 0
$$349$$ −1549.41 −0.237644 −0.118822 0.992916i $$-0.537912\pi$$
−0.118822 + 0.992916i $$0.537912\pi$$
$$350$$ 0 0
$$351$$ −70.4233 −0.0107092
$$352$$ 0 0
$$353$$ 566.231 0.0853752 0.0426876 0.999088i $$-0.486408\pi$$
0.0426876 + 0.999088i $$0.486408\pi$$
$$354$$ 0 0
$$355$$ −7486.81 −1.11932
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 3848.19 0.565738 0.282869 0.959159i $$-0.408714\pi$$
0.282869 + 0.959159i $$0.408714\pi$$
$$360$$ 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
$$371$$ 0 0
$$372$$ 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 $$-0.809817\pi$$
−0.826757 + 0.562559i $$0.809817\pi$$
$$380$$ 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
$$385$$ 0 0
$$386$$ 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 $$-0.143677\pi$$
0.899848 + 0.436203i $$0.143677\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 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 $$-0.555272\pi$$
−0.172772 + 0.984962i $$0.555272\pi$$
$$410$$ 0 0
$$411$$ 169.577 0.0203518
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 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
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −382.730 −0.0430732
$$430$$ 0 0
$$431$$ 14207.3 1.58780 0.793898 0.608051i $$-0.208048\pi$$
0.793898 + 0.608051i $$0.208048\pi$$
$$432$$ 0 0
$$433$$ 10530.6 1.16875 0.584375 0.811484i $$-0.301340\pi$$
0.584375 + 0.811484i $$0.301340\pi$$
$$434$$ 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
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −4574.15 −0.490574 −0.245287 0.969450i $$-0.578882\pi$$
−0.245287 + 0.969450i $$0.578882\pi$$
$$444$$ 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
$$455$$ 0 0
$$456$$ 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 $$-0.314824\pi$$
0.549484 + 0.835504i $$0.314824\pi$$
$$462$$ 0 0
$$463$$ −4038.28 −0.405345 −0.202673 0.979247i $$-0.564963\pi$$
−0.202673 + 0.979247i $$0.564963\pi$$
$$464$$ 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
$$469$$ 0 0
$$470$$ 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
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1267.39 0.118658
$$486$$ 0 0
$$487$$ −5580.52 −0.519256 −0.259628 0.965709i $$-0.583600\pi$$
−0.259628 + 0.965709i $$0.583600\pi$$
$$488$$ 0 0
$$489$$ −11590.6 −1.07187
$$490$$ 0 0
$$491$$ 12489.4 1.14794 0.573972 0.818875i $$-0.305402\pi$$
0.573972 + 0.818875i $$0.305402\pi$$
$$492$$ 0 0
$$493$$ −7217.46 −0.659347
$$494$$ 0 0
$$495$$ 2775.15 0.251988
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −15216.6 −1.36511 −0.682556 0.730834i $$-0.739131\pi$$
−0.682556 + 0.730834i $$0.739131\pi$$
$$500$$ 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 $$-0.432220\pi$$
0.211332 + 0.977414i $$0.432220\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 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 $$-0.363152\pi$$
0.416799 + 0.908999i $$0.363152\pi$$
$$522$$ 0 0
$$523$$ 4524.29 0.378267 0.189133 0.981951i $$-0.439432\pi$$
0.189133 + 0.981951i $$0.439432\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 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
$$539$$ 0 0
$$540$$ 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 $$-0.283021\pi$$
0.630082 + 0.776528i $$0.283021\pi$$
$$548$$ 0 0
$$549$$ 6070.64 0.471928
$$550$$ 0 0
$$551$$ −2386.69 −0.184530
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 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
$$567$$ 0 0
$$568$$ 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 $$-0.750619\pi$$
−0.708480 + 0.705731i $$0.750619\pi$$
$$572$$ 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 $$-0.896188\pi$$
−0.947288 + 0.320384i $$0.896188\pi$$
$$578$$ 0 0
$$579$$ 10599.5 0.760795
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 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 $$-0.343342\pi$$
0.472528 + 0.881316i $$0.343342\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 6474.25 0.443841
$$598$$ 0 0
$$599$$ 7543.11 0.514529 0.257265 0.966341i $$-0.417179\pi$$
0.257265 + 0.966341i $$0.417179\pi$$
$$600$$ 0 0
$$601$$ 19522.4 1.32501 0.662507 0.749056i $$-0.269493\pi$$
0.662507 + 0.749056i $$0.269493\pi$$
$$602$$ 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
$$609$$ 0 0
$$610$$ 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
$$623$$ 0 0
$$624$$ 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 $$-0.509900\pi$$
−0.0310979 + 0.999516i $$0.509900\pi$$
$$632$$ 0 0
$$633$$ 12870.1 0.808123
$$634$$ 0 0
$$635$$ −7457.01 −0.466020
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 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
$$651$$ 0 0
$$652$$ 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 $$-0.445210\pi$$
0.171279 + 0.985223i $$0.445210\pi$$
$$660$$ 0 0
$$661$$ −2592.59 −0.152557 −0.0762784 0.997087i $$-0.524304\pi$$
−0.0762784 + 0.997087i $$0.524304\pi$$
$$662$$ 0 0
$$663$$ 1069.94 0.0626744
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 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 $$-0.695154\pi$$
−0.575402 + 0.817871i $$0.695154\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −5174.38 −0.291164
$$682$$ 0 0
$$683$$ −13267.0 −0.743264 −0.371632 0.928380i $$-0.621202\pi$$
−0.371632 + 0.928380i $$0.621202\pi$$
$$684$$ 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
$$693$$ 0 0
$$694$$ 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
$$707$$ 0 0
$$708$$ 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
$$721$$ 0 0
$$722$$ 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 $$-0.668548\pi$$
−0.505110 + 0.863055i $$0.668548\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −18438.0 −0.921534
$$738$$ 0 0
$$739$$ −407.607 −0.0202897 −0.0101448 0.999949i $$-0.503229\pi$$
−0.0101448 + 0.999949i $$0.503229\pi$$
$$740$$ 0 0
$$741$$ 353.811 0.0175406
$$742$$ 0 0
$$743$$ −128.374 −0.00633861 −0.00316930 0.999995i $$-0.501009\pi$$
−0.00316930 + 0.999995i $$0.501009\pi$$
$$744$$ 0 0
$$745$$ 11315.7 0.556475
$$746$$ 0 0
$$747$$ −2643.59 −0.129483
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 25076.0 1.21842 0.609212 0.793007i $$-0.291486\pi$$
0.609212 + 0.793007i $$0.291486\pi$$
$$752$$ 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 $$-0.122015\pi$$
0.927427 + 0.374004i $$0.122015\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 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 $$-0.523003\pi$$
−0.0722021 + 0.997390i $$0.523003\pi$$
$$770$$ 0 0
$$771$$ −18631.8 −0.870308
$$772$$ 0 0
$$773$$ −31770.9 −1.47829 −0.739146 0.673545i $$-0.764771\pi$$
−0.739146 + 0.673545i $$0.764771\pi$$
$$774$$ 0 0
$$775$$ −1260.41 −0.0584195
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 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
$$791$$ 0 0
$$792$$ 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 $$-0.596578\pi$$
−0.298776 + 0.954323i $$0.596578\pi$$
$$798$$ 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
$$805$$ 0 0
$$806$$ 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
$$819$$ 0 0
$$820$$ 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 $$-0.378585\pi$$
0.372253 + 0.928131i $$0.378585\pi$$
$$824$$ 0 0
$$825$$ 12510.5 0.527951
$$826$$ 0 0
$$827$$ −6440.14 −0.270793 −0.135396 0.990792i $$-0.543231\pi$$
−0.135396 + 0.990792i $$0.543231\pi$$
$$828$$ 0 0
$$829$$ −5084.23 −0.213007 −0.106503 0.994312i $$-0.533966\pi$$
−0.106503 + 0.994312i $$0.533966\pi$$
$$830$$ 0 0
$$831$$ 9686.01 0.404337
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 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
$$847$$ 0 0
$$848$$ 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 $$-0.458947\pi$$
0.128616 + 0.991694i $$0.458947\pi$$
$$854$$ 0 0
$$855$$ −2565.46 −0.102616
$$856$$ 0 0
$$857$$ −17248.6 −0.687515 −0.343758 0.939058i $$-0.611700\pi$$
−0.343758 + 0.939058i $$0.611700\pi$$
$$858$$ 0 0
$$859$$ −3159.07 −0.125479 −0.0627393 0.998030i $$-0.519984\pi$$
−0.0627393 + 0.998030i $$0.519984\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 41071.9 1.62005 0.810025 0.586395i $$-0.199453\pi$$
0.810025 + 0.586395i $$0.199453\pi$$
$$864$$ 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
$$875$$ 0 0
$$876$$ 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 $$-0.481062\pi$$
0.0594606 + 0.998231i $$0.481062\pi$$
$$882$$ 0 0
$$883$$ −19782.8 −0.753955 −0.376978 0.926222i $$-0.623037\pi$$
−0.376978 + 0.926222i $$0.623037\pi$$
$$884$$ 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
$$889$$ 0 0
$$890$$ 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
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −24771.0 −0.909853
$$906$$ 0 0
$$907$$ 16211.9 0.593505 0.296752 0.954954i $$-0.404096\pi$$
0.296752 + 0.954954i $$0.404096\pi$$
$$908$$ 0 0
$$909$$ 9481.60 0.345968
$$910$$ 0 0
$$911$$ 14540.6 0.528816 0.264408 0.964411i $$-0.414823\pi$$
0.264408 + 0.964411i $$0.414823\pi$$
$$912$$ 0 0
$$913$$ −14367.2 −0.520792
$$914$$ 0 0
$$915$$ −12756.7 −0.460901
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −37239.2 −1.33668 −0.668339 0.743857i $$-0.732995\pi$$
−0.668339 + 0.743857i $$0.732995\pi$$
$$920$$ 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 $$-0.627381\pi$$
−0.389583 + 0.920991i $$0.627381\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 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 $$-0.412348\pi$$
0.271900 + 0.962326i $$0.412348\pi$$
$$938$$ 0 0
$$939$$ 11472.6 0.398715
$$940$$ 0 0
$$941$$ 15887.1 0.550377 0.275188 0.961390i $$-0.411260\pi$$
0.275188 + 0.961390i $$0.411260\pi$$
$$942$$ 0 0
$$943$$ 8665.29 0.299237
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 54754.6 1.87887 0.939433 0.342733i $$-0.111352\pi$$
0.939433 + 0.342733i $$0.111352\pi$$
$$948$$ 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
$$959$$ 0 0
$$960$$ 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 $$-0.650664\pi$$
−0.455848 + 0.890058i $$0.650664\pi$$
$$968$$ 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
$$973$$ 0 0
$$974$$ 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
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 15191.8 0.488445
$$990$$ 0 0
$$991$$ −13624.5 −0.436726 −0.218363 0.975868i $$-0.570072\pi$$
−0.218363 + 0.975868i $$0.570072\pi$$
$$992$$ 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 $$-0.232992\pi$$
0.743861 + 0.668334i $$0.232992\pi$$
$$998$$ 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 1176.4.a.p.1.2 2
4.3 odd 2 2352.4.a.bv.1.2 2
7.6 odd 2 168.4.a.h.1.1 2
21.20 even 2 504.4.a.j.1.2 2
28.27 even 2 336.4.a.n.1.1 2
56.13 odd 2 1344.4.a.bd.1.2 2
56.27 even 2 1344.4.a.bl.1.2 2
84.83 odd 2 1008.4.a.y.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.a.h.1.1 2 7.6 odd 2
336.4.a.n.1.1 2 28.27 even 2
504.4.a.j.1.2 2 21.20 even 2
1008.4.a.y.1.2 2 84.83 odd 2
1176.4.a.p.1.2 2 1.1 even 1 trivial
1344.4.a.bd.1.2 2 56.13 odd 2
1344.4.a.bl.1.2 2 56.27 even 2
2352.4.a.bv.1.2 2 4.3 odd 2