# Properties

 Label 1856.4.a.s.1.1 Level $1856$ Weight $4$ Character 1856.1 Self dual yes Analytic conductor $109.508$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.507544971$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.19816.1 Defining polynomial: $$x^{3} - x^{2} - 42x - 54$$ x^3 - x^2 - 42*x - 54 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$7.53003$$ of defining polynomial Character $$\chi$$ $$=$$ 1856.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-6.53003 q^{3} -20.9205 q^{5} -8.55839 q^{7} +15.6413 q^{9} +O(q^{10})$$ $$q-6.53003 q^{3} -20.9205 q^{5} -8.55839 q^{7} +15.6413 q^{9} +10.8092 q^{11} -54.7046 q^{13} +136.611 q^{15} -106.127 q^{17} -113.636 q^{19} +55.8865 q^{21} +112.855 q^{23} +312.666 q^{25} +74.1729 q^{27} -29.0000 q^{29} +102.805 q^{31} -70.5845 q^{33} +179.045 q^{35} +105.665 q^{37} +357.223 q^{39} +216.958 q^{41} -102.230 q^{43} -327.222 q^{45} -455.212 q^{47} -269.754 q^{49} +693.011 q^{51} +593.714 q^{53} -226.134 q^{55} +742.048 q^{57} -558.141 q^{59} +473.986 q^{61} -133.864 q^{63} +1144.45 q^{65} +193.132 q^{67} -736.949 q^{69} +2.38155 q^{71} +119.013 q^{73} -2041.71 q^{75} -92.5096 q^{77} +964.306 q^{79} -906.665 q^{81} +1068.19 q^{83} +2220.22 q^{85} +189.371 q^{87} +772.544 q^{89} +468.184 q^{91} -671.318 q^{93} +2377.32 q^{95} +1344.03 q^{97} +169.070 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 20 * q^5 - 24 * q^7 + 5 * q^9 $$3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9} + 10 q^{11} + 4 q^{13} + 130 q^{15} - 66 q^{17} - 164 q^{19} + 88 q^{21} + 204 q^{23} + 79 q^{25} - 142 q^{27} - 87 q^{29} + 86 q^{31} - 130 q^{33} + 24 q^{35} + 42 q^{37} + 394 q^{39} + 562 q^{41} + 18 q^{43} - 422 q^{45} - 654 q^{47} + 539 q^{49} + 556 q^{51} - 712 q^{53} - 142 q^{55} + 828 q^{57} + 184 q^{59} - 322 q^{61} + 784 q^{63} + 1494 q^{65} - 228 q^{67} - 684 q^{69} + 52 q^{71} - 494 q^{73} - 3048 q^{75} - 872 q^{77} + 2110 q^{79} - 1513 q^{81} - 288 q^{83} + 2704 q^{85} - 58 q^{87} + 914 q^{89} - 2984 q^{91} + 62 q^{93} + 1900 q^{95} + 218 q^{97} - 304 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 20 * q^5 - 24 * q^7 + 5 * q^9 + 10 * q^11 + 4 * q^13 + 130 * q^15 - 66 * q^17 - 164 * q^19 + 88 * q^21 + 204 * q^23 + 79 * q^25 - 142 * q^27 - 87 * q^29 + 86 * q^31 - 130 * q^33 + 24 * q^35 + 42 * q^37 + 394 * q^39 + 562 * q^41 + 18 * q^43 - 422 * q^45 - 654 * q^47 + 539 * q^49 + 556 * q^51 - 712 * q^53 - 142 * q^55 + 828 * q^57 + 184 * q^59 - 322 * q^61 + 784 * q^63 + 1494 * q^65 - 228 * q^67 - 684 * q^69 + 52 * q^71 - 494 * q^73 - 3048 * q^75 - 872 * q^77 + 2110 * q^79 - 1513 * q^81 - 288 * q^83 + 2704 * q^85 - 58 * q^87 + 914 * q^89 - 2984 * q^91 + 62 * q^93 + 1900 * q^95 + 218 * q^97 - 304 * q^99

## 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
$$3$$ −6.53003 −1.25670 −0.628352 0.777929i $$-0.716270\pi$$
−0.628352 + 0.777929i $$0.716270\pi$$
$$4$$ 0 0
$$5$$ −20.9205 −1.87118 −0.935591 0.353085i $$-0.885133\pi$$
−0.935591 + 0.353085i $$0.885133\pi$$
$$6$$ 0 0
$$7$$ −8.55839 −0.462110 −0.231055 0.972941i $$-0.574218\pi$$
−0.231055 + 0.972941i $$0.574218\pi$$
$$8$$ 0 0
$$9$$ 15.6413 0.579306
$$10$$ 0 0
$$11$$ 10.8092 0.296282 0.148141 0.988966i $$-0.452671\pi$$
0.148141 + 0.988966i $$0.452671\pi$$
$$12$$ 0 0
$$13$$ −54.7046 −1.16710 −0.583551 0.812076i $$-0.698337\pi$$
−0.583551 + 0.812076i $$0.698337\pi$$
$$14$$ 0 0
$$15$$ 136.611 2.35152
$$16$$ 0 0
$$17$$ −106.127 −1.51409 −0.757045 0.653363i $$-0.773358\pi$$
−0.757045 + 0.653363i $$0.773358\pi$$
$$18$$ 0 0
$$19$$ −113.636 −1.37210 −0.686051 0.727554i $$-0.740657\pi$$
−0.686051 + 0.727554i $$0.740657\pi$$
$$20$$ 0 0
$$21$$ 55.8865 0.580735
$$22$$ 0 0
$$23$$ 112.855 1.02313 0.511564 0.859245i $$-0.329066\pi$$
0.511564 + 0.859245i $$0.329066\pi$$
$$24$$ 0 0
$$25$$ 312.666 2.50132
$$26$$ 0 0
$$27$$ 74.1729 0.528688
$$28$$ 0 0
$$29$$ −29.0000 −0.185695
$$30$$ 0 0
$$31$$ 102.805 0.595622 0.297811 0.954625i $$-0.403743\pi$$
0.297811 + 0.954625i $$0.403743\pi$$
$$32$$ 0 0
$$33$$ −70.5845 −0.372339
$$34$$ 0 0
$$35$$ 179.045 0.864692
$$36$$ 0 0
$$37$$ 105.665 0.469493 0.234746 0.972057i $$-0.424574\pi$$
0.234746 + 0.972057i $$0.424574\pi$$
$$38$$ 0 0
$$39$$ 357.223 1.46670
$$40$$ 0 0
$$41$$ 216.958 0.826417 0.413209 0.910636i $$-0.364408\pi$$
0.413209 + 0.910636i $$0.364408\pi$$
$$42$$ 0 0
$$43$$ −102.230 −0.362555 −0.181278 0.983432i $$-0.558023\pi$$
−0.181278 + 0.983432i $$0.558023\pi$$
$$44$$ 0 0
$$45$$ −327.222 −1.08399
$$46$$ 0 0
$$47$$ −455.212 −1.41275 −0.706377 0.707836i $$-0.749672\pi$$
−0.706377 + 0.707836i $$0.749672\pi$$
$$48$$ 0 0
$$49$$ −269.754 −0.786455
$$50$$ 0 0
$$51$$ 693.011 1.90276
$$52$$ 0 0
$$53$$ 593.714 1.53873 0.769367 0.638807i $$-0.220572\pi$$
0.769367 + 0.638807i $$0.220572\pi$$
$$54$$ 0 0
$$55$$ −226.134 −0.554398
$$56$$ 0 0
$$57$$ 742.048 1.72433
$$58$$ 0 0
$$59$$ −558.141 −1.23159 −0.615794 0.787907i $$-0.711165\pi$$
−0.615794 + 0.787907i $$0.711165\pi$$
$$60$$ 0 0
$$61$$ 473.986 0.994880 0.497440 0.867498i $$-0.334273\pi$$
0.497440 + 0.867498i $$0.334273\pi$$
$$62$$ 0 0
$$63$$ −133.864 −0.267703
$$64$$ 0 0
$$65$$ 1144.45 2.18386
$$66$$ 0 0
$$67$$ 193.132 0.352162 0.176081 0.984376i $$-0.443658\pi$$
0.176081 + 0.984376i $$0.443658\pi$$
$$68$$ 0 0
$$69$$ −736.949 −1.28577
$$70$$ 0 0
$$71$$ 2.38155 0.00398082 0.00199041 0.999998i $$-0.499366\pi$$
0.00199041 + 0.999998i $$0.499366\pi$$
$$72$$ 0 0
$$73$$ 119.013 0.190814 0.0954071 0.995438i $$-0.469585\pi$$
0.0954071 + 0.995438i $$0.469585\pi$$
$$74$$ 0 0
$$75$$ −2041.71 −3.14343
$$76$$ 0 0
$$77$$ −92.5096 −0.136915
$$78$$ 0 0
$$79$$ 964.306 1.37333 0.686664 0.726975i $$-0.259074\pi$$
0.686664 + 0.726975i $$0.259074\pi$$
$$80$$ 0 0
$$81$$ −906.665 −1.24371
$$82$$ 0 0
$$83$$ 1068.19 1.41264 0.706319 0.707893i $$-0.250354\pi$$
0.706319 + 0.707893i $$0.250354\pi$$
$$84$$ 0 0
$$85$$ 2220.22 2.83314
$$86$$ 0 0
$$87$$ 189.371 0.233364
$$88$$ 0 0
$$89$$ 772.544 0.920106 0.460053 0.887891i $$-0.347830\pi$$
0.460053 + 0.887891i $$0.347830\pi$$
$$90$$ 0 0
$$91$$ 468.184 0.539330
$$92$$ 0 0
$$93$$ −671.318 −0.748521
$$94$$ 0 0
$$95$$ 2377.32 2.56745
$$96$$ 0 0
$$97$$ 1344.03 1.40686 0.703431 0.710763i $$-0.251650\pi$$
0.703431 + 0.710763i $$0.251650\pi$$
$$98$$ 0 0
$$99$$ 169.070 0.171638
$$100$$ 0 0
$$101$$ −986.733 −0.972115 −0.486057 0.873927i $$-0.661565\pi$$
−0.486057 + 0.873927i $$0.661565\pi$$
$$102$$ 0 0
$$103$$ −548.272 −0.524493 −0.262247 0.965001i $$-0.584463\pi$$
−0.262247 + 0.965001i $$0.584463\pi$$
$$104$$ 0 0
$$105$$ −1169.17 −1.08666
$$106$$ 0 0
$$107$$ 1387.51 1.25361 0.626803 0.779178i $$-0.284363\pi$$
0.626803 + 0.779178i $$0.284363\pi$$
$$108$$ 0 0
$$109$$ 1293.32 1.13649 0.568246 0.822859i $$-0.307622\pi$$
0.568246 + 0.822859i $$0.307622\pi$$
$$110$$ 0 0
$$111$$ −689.996 −0.590014
$$112$$ 0 0
$$113$$ 302.883 0.252149 0.126075 0.992021i $$-0.459762\pi$$
0.126075 + 0.992021i $$0.459762\pi$$
$$114$$ 0 0
$$115$$ −2360.99 −1.91446
$$116$$ 0 0
$$117$$ −855.650 −0.676110
$$118$$ 0 0
$$119$$ 908.274 0.699676
$$120$$ 0 0
$$121$$ −1214.16 −0.912217
$$122$$ 0 0
$$123$$ −1416.74 −1.03856
$$124$$ 0 0
$$125$$ −3926.05 −2.80925
$$126$$ 0 0
$$127$$ −2021.68 −1.41256 −0.706281 0.707931i $$-0.749629\pi$$
−0.706281 + 0.707931i $$0.749629\pi$$
$$128$$ 0 0
$$129$$ 667.562 0.455625
$$130$$ 0 0
$$131$$ 854.726 0.570059 0.285030 0.958519i $$-0.407997\pi$$
0.285030 + 0.958519i $$0.407997\pi$$
$$132$$ 0 0
$$133$$ 972.543 0.634062
$$134$$ 0 0
$$135$$ −1551.73 −0.989272
$$136$$ 0 0
$$137$$ 365.024 0.227636 0.113818 0.993502i $$-0.463692\pi$$
0.113818 + 0.993502i $$0.463692\pi$$
$$138$$ 0 0
$$139$$ −1010.10 −0.616370 −0.308185 0.951326i $$-0.599722\pi$$
−0.308185 + 0.951326i $$0.599722\pi$$
$$140$$ 0 0
$$141$$ 2972.55 1.77541
$$142$$ 0 0
$$143$$ −591.315 −0.345792
$$144$$ 0 0
$$145$$ 606.693 0.347470
$$146$$ 0 0
$$147$$ 1761.50 0.988341
$$148$$ 0 0
$$149$$ −819.765 −0.450723 −0.225362 0.974275i $$-0.572356\pi$$
−0.225362 + 0.974275i $$0.572356\pi$$
$$150$$ 0 0
$$151$$ −1000.25 −0.539068 −0.269534 0.962991i $$-0.586870\pi$$
−0.269534 + 0.962991i $$0.586870\pi$$
$$152$$ 0 0
$$153$$ −1659.96 −0.877121
$$154$$ 0 0
$$155$$ −2150.72 −1.11452
$$156$$ 0 0
$$157$$ 1702.38 0.865382 0.432691 0.901542i $$-0.357564\pi$$
0.432691 + 0.901542i $$0.357564\pi$$
$$158$$ 0 0
$$159$$ −3876.97 −1.93373
$$160$$ 0 0
$$161$$ −965.860 −0.472798
$$162$$ 0 0
$$163$$ −3451.76 −1.65867 −0.829334 0.558753i $$-0.811280\pi$$
−0.829334 + 0.558753i $$0.811280\pi$$
$$164$$ 0 0
$$165$$ 1476.66 0.696714
$$166$$ 0 0
$$167$$ −1409.23 −0.652990 −0.326495 0.945199i $$-0.605868\pi$$
−0.326495 + 0.945199i $$0.605868\pi$$
$$168$$ 0 0
$$169$$ 795.598 0.362129
$$170$$ 0 0
$$171$$ −1777.41 −0.794867
$$172$$ 0 0
$$173$$ −1358.39 −0.596973 −0.298487 0.954414i $$-0.596482\pi$$
−0.298487 + 0.954414i $$0.596482\pi$$
$$174$$ 0 0
$$175$$ −2675.91 −1.15589
$$176$$ 0 0
$$177$$ 3644.67 1.54774
$$178$$ 0 0
$$179$$ −1702.56 −0.710924 −0.355462 0.934691i $$-0.615677\pi$$
−0.355462 + 0.934691i $$0.615677\pi$$
$$180$$ 0 0
$$181$$ 7.33698 0.00301300 0.00150650 0.999999i $$-0.499520\pi$$
0.00150650 + 0.999999i $$0.499520\pi$$
$$182$$ 0 0
$$183$$ −3095.14 −1.25027
$$184$$ 0 0
$$185$$ −2210.56 −0.878507
$$186$$ 0 0
$$187$$ −1147.15 −0.448598
$$188$$ 0 0
$$189$$ −634.801 −0.244312
$$190$$ 0 0
$$191$$ −1324.78 −0.501873 −0.250936 0.968004i $$-0.580738\pi$$
−0.250936 + 0.968004i $$0.580738\pi$$
$$192$$ 0 0
$$193$$ 1834.47 0.684187 0.342094 0.939666i $$-0.388864\pi$$
0.342094 + 0.939666i $$0.388864\pi$$
$$194$$ 0 0
$$195$$ −7473.26 −2.74447
$$196$$ 0 0
$$197$$ 4949.99 1.79021 0.895107 0.445851i $$-0.147099\pi$$
0.895107 + 0.445851i $$0.147099\pi$$
$$198$$ 0 0
$$199$$ −3554.04 −1.26603 −0.633014 0.774140i $$-0.718182\pi$$
−0.633014 + 0.774140i $$0.718182\pi$$
$$200$$ 0 0
$$201$$ −1261.16 −0.442563
$$202$$ 0 0
$$203$$ 248.193 0.0858116
$$204$$ 0 0
$$205$$ −4538.85 −1.54638
$$206$$ 0 0
$$207$$ 1765.20 0.592705
$$208$$ 0 0
$$209$$ −1228.32 −0.406529
$$210$$ 0 0
$$211$$ 2475.23 0.807590 0.403795 0.914849i $$-0.367691\pi$$
0.403795 + 0.914849i $$0.367691\pi$$
$$212$$ 0 0
$$213$$ −15.5516 −0.00500271
$$214$$ 0 0
$$215$$ 2138.69 0.678407
$$216$$ 0 0
$$217$$ −879.844 −0.275243
$$218$$ 0 0
$$219$$ −777.159 −0.239797
$$220$$ 0 0
$$221$$ 5805.63 1.76710
$$222$$ 0 0
$$223$$ −2381.72 −0.715211 −0.357605 0.933873i $$-0.616407\pi$$
−0.357605 + 0.933873i $$0.616407\pi$$
$$224$$ 0 0
$$225$$ 4890.48 1.44903
$$226$$ 0 0
$$227$$ 5452.74 1.59432 0.797160 0.603768i $$-0.206334\pi$$
0.797160 + 0.603768i $$0.206334\pi$$
$$228$$ 0 0
$$229$$ −596.232 −0.172053 −0.0860264 0.996293i $$-0.527417\pi$$
−0.0860264 + 0.996293i $$0.527417\pi$$
$$230$$ 0 0
$$231$$ 604.090 0.172062
$$232$$ 0 0
$$233$$ −5623.04 −1.58102 −0.790509 0.612450i $$-0.790184\pi$$
−0.790509 + 0.612450i $$0.790184\pi$$
$$234$$ 0 0
$$235$$ 9523.24 2.64352
$$236$$ 0 0
$$237$$ −6296.95 −1.72587
$$238$$ 0 0
$$239$$ −1564.27 −0.423366 −0.211683 0.977338i $$-0.567894\pi$$
−0.211683 + 0.977338i $$0.567894\pi$$
$$240$$ 0 0
$$241$$ −730.326 −0.195205 −0.0976026 0.995225i $$-0.531117\pi$$
−0.0976026 + 0.995225i $$0.531117\pi$$
$$242$$ 0 0
$$243$$ 3917.88 1.03429
$$244$$ 0 0
$$245$$ 5643.38 1.47160
$$246$$ 0 0
$$247$$ 6216.43 1.60138
$$248$$ 0 0
$$249$$ −6975.31 −1.77527
$$250$$ 0 0
$$251$$ −4244.39 −1.06735 −0.533673 0.845691i $$-0.679189\pi$$
−0.533673 + 0.845691i $$0.679189\pi$$
$$252$$ 0 0
$$253$$ 1219.88 0.303135
$$254$$ 0 0
$$255$$ −14498.1 −3.56042
$$256$$ 0 0
$$257$$ −235.374 −0.0571292 −0.0285646 0.999592i $$-0.509094\pi$$
−0.0285646 + 0.999592i $$0.509094\pi$$
$$258$$ 0 0
$$259$$ −904.323 −0.216957
$$260$$ 0 0
$$261$$ −453.597 −0.107574
$$262$$ 0 0
$$263$$ −3453.63 −0.809734 −0.404867 0.914376i $$-0.632682\pi$$
−0.404867 + 0.914376i $$0.632682\pi$$
$$264$$ 0 0
$$265$$ −12420.8 −2.87925
$$266$$ 0 0
$$267$$ −5044.73 −1.15630
$$268$$ 0 0
$$269$$ 1921.31 0.435481 0.217741 0.976007i $$-0.430131\pi$$
0.217741 + 0.976007i $$0.430131\pi$$
$$270$$ 0 0
$$271$$ 2480.09 0.555921 0.277961 0.960592i $$-0.410342\pi$$
0.277961 + 0.960592i $$0.410342\pi$$
$$272$$ 0 0
$$273$$ −3057.25 −0.677778
$$274$$ 0 0
$$275$$ 3379.67 0.741098
$$276$$ 0 0
$$277$$ 4766.90 1.03399 0.516995 0.855989i $$-0.327051\pi$$
0.516995 + 0.855989i $$0.327051\pi$$
$$278$$ 0 0
$$279$$ 1608.00 0.345047
$$280$$ 0 0
$$281$$ 194.329 0.0412552 0.0206276 0.999787i $$-0.493434\pi$$
0.0206276 + 0.999787i $$0.493434\pi$$
$$282$$ 0 0
$$283$$ −2817.42 −0.591797 −0.295898 0.955219i $$-0.595619\pi$$
−0.295898 + 0.955219i $$0.595619\pi$$
$$284$$ 0 0
$$285$$ −15524.0 −3.22653
$$286$$ 0 0
$$287$$ −1856.81 −0.381895
$$288$$ 0 0
$$289$$ 6349.89 1.29247
$$290$$ 0 0
$$291$$ −8776.56 −1.76801
$$292$$ 0 0
$$293$$ −4059.12 −0.809339 −0.404670 0.914463i $$-0.632614\pi$$
−0.404670 + 0.914463i $$0.632614\pi$$
$$294$$ 0 0
$$295$$ 11676.6 2.30453
$$296$$ 0 0
$$297$$ 801.751 0.156641
$$298$$ 0 0
$$299$$ −6173.71 −1.19410
$$300$$ 0 0
$$301$$ 874.921 0.167540
$$302$$ 0 0
$$303$$ 6443.39 1.22166
$$304$$ 0 0
$$305$$ −9916.01 −1.86160
$$306$$ 0 0
$$307$$ −131.572 −0.0244600 −0.0122300 0.999925i $$-0.503893\pi$$
−0.0122300 + 0.999925i $$0.503893\pi$$
$$308$$ 0 0
$$309$$ 3580.23 0.659133
$$310$$ 0 0
$$311$$ −1517.84 −0.276748 −0.138374 0.990380i $$-0.544188\pi$$
−0.138374 + 0.990380i $$0.544188\pi$$
$$312$$ 0 0
$$313$$ 4244.17 0.766436 0.383218 0.923658i $$-0.374816\pi$$
0.383218 + 0.923658i $$0.374816\pi$$
$$314$$ 0 0
$$315$$ 2800.50 0.500921
$$316$$ 0 0
$$317$$ −5596.56 −0.991590 −0.495795 0.868439i $$-0.665123\pi$$
−0.495795 + 0.868439i $$0.665123\pi$$
$$318$$ 0 0
$$319$$ −313.467 −0.0550182
$$320$$ 0 0
$$321$$ −9060.50 −1.57541
$$322$$ 0 0
$$323$$ 12059.8 2.07748
$$324$$ 0 0
$$325$$ −17104.3 −2.91930
$$326$$ 0 0
$$327$$ −8445.42 −1.42824
$$328$$ 0 0
$$329$$ 3895.88 0.652848
$$330$$ 0 0
$$331$$ −8383.85 −1.39220 −0.696100 0.717945i $$-0.745083\pi$$
−0.696100 + 0.717945i $$0.745083\pi$$
$$332$$ 0 0
$$333$$ 1652.73 0.271980
$$334$$ 0 0
$$335$$ −4040.41 −0.658959
$$336$$ 0 0
$$337$$ 9887.12 1.59818 0.799089 0.601213i $$-0.205316\pi$$
0.799089 + 0.601213i $$0.205316\pi$$
$$338$$ 0 0
$$339$$ −1977.84 −0.316877
$$340$$ 0 0
$$341$$ 1111.24 0.176472
$$342$$ 0 0
$$343$$ 5244.19 0.825538
$$344$$ 0 0
$$345$$ 15417.3 2.40591
$$346$$ 0 0
$$347$$ 10678.8 1.65206 0.826032 0.563623i $$-0.190593\pi$$
0.826032 + 0.563623i $$0.190593\pi$$
$$348$$ 0 0
$$349$$ 1457.88 0.223605 0.111803 0.993730i $$-0.464338\pi$$
0.111803 + 0.993730i $$0.464338\pi$$
$$350$$ 0 0
$$351$$ −4057.60 −0.617033
$$352$$ 0 0
$$353$$ 6737.20 1.01582 0.507911 0.861410i $$-0.330418\pi$$
0.507911 + 0.861410i $$0.330418\pi$$
$$354$$ 0 0
$$355$$ −49.8232 −0.00744884
$$356$$ 0 0
$$357$$ −5931.06 −0.879285
$$358$$ 0 0
$$359$$ 3539.85 0.520407 0.260204 0.965554i $$-0.416210\pi$$
0.260204 + 0.965554i $$0.416210\pi$$
$$360$$ 0 0
$$361$$ 6054.19 0.882663
$$362$$ 0 0
$$363$$ 7928.50 1.14639
$$364$$ 0 0
$$365$$ −2489.81 −0.357048
$$366$$ 0 0
$$367$$ 4917.53 0.699436 0.349718 0.936855i $$-0.386277\pi$$
0.349718 + 0.936855i $$0.386277\pi$$
$$368$$ 0 0
$$369$$ 3393.49 0.478748
$$370$$ 0 0
$$371$$ −5081.24 −0.711064
$$372$$ 0 0
$$373$$ −2032.31 −0.282115 −0.141057 0.990001i $$-0.545050\pi$$
−0.141057 + 0.990001i $$0.545050\pi$$
$$374$$ 0 0
$$375$$ 25637.2 3.53040
$$376$$ 0 0
$$377$$ 1586.43 0.216726
$$378$$ 0 0
$$379$$ −7051.47 −0.955699 −0.477849 0.878442i $$-0.658584\pi$$
−0.477849 + 0.878442i $$0.658584\pi$$
$$380$$ 0 0
$$381$$ 13201.7 1.77517
$$382$$ 0 0
$$383$$ 9334.72 1.24538 0.622691 0.782467i $$-0.286039\pi$$
0.622691 + 0.782467i $$0.286039\pi$$
$$384$$ 0 0
$$385$$ 1935.34 0.256193
$$386$$ 0 0
$$387$$ −1599.00 −0.210030
$$388$$ 0 0
$$389$$ 1901.32 0.247816 0.123908 0.992294i $$-0.460457\pi$$
0.123908 + 0.992294i $$0.460457\pi$$
$$390$$ 0 0
$$391$$ −11977.0 −1.54911
$$392$$ 0 0
$$393$$ −5581.38 −0.716396
$$394$$ 0 0
$$395$$ −20173.7 −2.56975
$$396$$ 0 0
$$397$$ −1995.81 −0.252309 −0.126155 0.992011i $$-0.540264\pi$$
−0.126155 + 0.992011i $$0.540264\pi$$
$$398$$ 0 0
$$399$$ −6350.73 −0.796828
$$400$$ 0 0
$$401$$ 12920.6 1.60904 0.804520 0.593925i $$-0.202422\pi$$
0.804520 + 0.593925i $$0.202422\pi$$
$$402$$ 0 0
$$403$$ −5623.90 −0.695152
$$404$$ 0 0
$$405$$ 18967.8 2.32721
$$406$$ 0 0
$$407$$ 1142.16 0.139102
$$408$$ 0 0
$$409$$ −9713.54 −1.17434 −0.587168 0.809465i $$-0.699757\pi$$
−0.587168 + 0.809465i $$0.699757\pi$$
$$410$$ 0 0
$$411$$ −2383.62 −0.286071
$$412$$ 0 0
$$413$$ 4776.79 0.569129
$$414$$ 0 0
$$415$$ −22347.0 −2.64330
$$416$$ 0 0
$$417$$ 6595.97 0.774595
$$418$$ 0 0
$$419$$ 15925.4 1.85682 0.928411 0.371555i $$-0.121175\pi$$
0.928411 + 0.371555i $$0.121175\pi$$
$$420$$ 0 0
$$421$$ −10849.9 −1.25604 −0.628019 0.778198i $$-0.716134\pi$$
−0.628019 + 0.778198i $$0.716134\pi$$
$$422$$ 0 0
$$423$$ −7120.09 −0.818417
$$424$$ 0 0
$$425$$ −33182.2 −3.78723
$$426$$ 0 0
$$427$$ −4056.56 −0.459744
$$428$$ 0 0
$$429$$ 3861.30 0.434558
$$430$$ 0 0
$$431$$ 532.335 0.0594934 0.0297467 0.999557i $$-0.490530\pi$$
0.0297467 + 0.999557i $$0.490530\pi$$
$$432$$ 0 0
$$433$$ −6995.94 −0.776451 −0.388225 0.921564i $$-0.626912\pi$$
−0.388225 + 0.921564i $$0.626912\pi$$
$$434$$ 0 0
$$435$$ −3961.72 −0.436667
$$436$$ 0 0
$$437$$ −12824.5 −1.40384
$$438$$ 0 0
$$439$$ −3272.22 −0.355750 −0.177875 0.984053i $$-0.556922\pi$$
−0.177875 + 0.984053i $$0.556922\pi$$
$$440$$ 0 0
$$441$$ −4219.29 −0.455598
$$442$$ 0 0
$$443$$ −3818.14 −0.409493 −0.204746 0.978815i $$-0.565637\pi$$
−0.204746 + 0.978815i $$0.565637\pi$$
$$444$$ 0 0
$$445$$ −16162.0 −1.72169
$$446$$ 0 0
$$447$$ 5353.09 0.566426
$$448$$ 0 0
$$449$$ −4323.19 −0.454396 −0.227198 0.973849i $$-0.572957\pi$$
−0.227198 + 0.973849i $$0.572957\pi$$
$$450$$ 0 0
$$451$$ 2345.14 0.244853
$$452$$ 0 0
$$453$$ 6531.66 0.677449
$$454$$ 0 0
$$455$$ −9794.62 −1.00918
$$456$$ 0 0
$$457$$ −8367.43 −0.856481 −0.428240 0.903665i $$-0.640866\pi$$
−0.428240 + 0.903665i $$0.640866\pi$$
$$458$$ 0 0
$$459$$ −7871.73 −0.800481
$$460$$ 0 0
$$461$$ 17249.0 1.74266 0.871328 0.490701i $$-0.163259\pi$$
0.871328 + 0.490701i $$0.163259\pi$$
$$462$$ 0 0
$$463$$ 16774.3 1.68373 0.841864 0.539690i $$-0.181458\pi$$
0.841864 + 0.539690i $$0.181458\pi$$
$$464$$ 0 0
$$465$$ 14044.3 1.40062
$$466$$ 0 0
$$467$$ 7701.05 0.763088 0.381544 0.924351i $$-0.375392\pi$$
0.381544 + 0.924351i $$0.375392\pi$$
$$468$$ 0 0
$$469$$ −1652.90 −0.162737
$$470$$ 0 0
$$471$$ −11116.6 −1.08753
$$472$$ 0 0
$$473$$ −1105.02 −0.107419
$$474$$ 0 0
$$475$$ −35530.1 −3.43207
$$476$$ 0 0
$$477$$ 9286.44 0.891398
$$478$$ 0 0
$$479$$ −5988.77 −0.571261 −0.285630 0.958340i $$-0.592203\pi$$
−0.285630 + 0.958340i $$0.592203\pi$$
$$480$$ 0 0
$$481$$ −5780.37 −0.547946
$$482$$ 0 0
$$483$$ 6307.09 0.594167
$$484$$ 0 0
$$485$$ −28117.7 −2.63250
$$486$$ 0 0
$$487$$ −5790.34 −0.538779 −0.269390 0.963031i $$-0.586822\pi$$
−0.269390 + 0.963031i $$0.586822\pi$$
$$488$$ 0 0
$$489$$ 22540.1 2.08446
$$490$$ 0 0
$$491$$ 19228.8 1.76738 0.883692 0.468069i $$-0.155050\pi$$
0.883692 + 0.468069i $$0.155050\pi$$
$$492$$ 0 0
$$493$$ 3077.68 0.281159
$$494$$ 0 0
$$495$$ −3537.02 −0.321166
$$496$$ 0 0
$$497$$ −20.3823 −0.00183958
$$498$$ 0 0
$$499$$ −7081.65 −0.635307 −0.317653 0.948207i $$-0.602895\pi$$
−0.317653 + 0.948207i $$0.602895\pi$$
$$500$$ 0 0
$$501$$ 9202.29 0.820615
$$502$$ 0 0
$$503$$ 5312.64 0.470932 0.235466 0.971883i $$-0.424338\pi$$
0.235466 + 0.971883i $$0.424338\pi$$
$$504$$ 0 0
$$505$$ 20642.9 1.81900
$$506$$ 0 0
$$507$$ −5195.28 −0.455089
$$508$$ 0 0
$$509$$ 13862.7 1.20717 0.603587 0.797297i $$-0.293738\pi$$
0.603587 + 0.797297i $$0.293738\pi$$
$$510$$ 0 0
$$511$$ −1018.56 −0.0881771
$$512$$ 0 0
$$513$$ −8428.72 −0.725414
$$514$$ 0 0
$$515$$ 11470.1 0.981423
$$516$$ 0 0
$$517$$ −4920.49 −0.418574
$$518$$ 0 0
$$519$$ 8870.32 0.750219
$$520$$ 0 0
$$521$$ 15447.5 1.29897 0.649487 0.760373i $$-0.274984\pi$$
0.649487 + 0.760373i $$0.274984\pi$$
$$522$$ 0 0
$$523$$ 4349.54 0.363656 0.181828 0.983330i $$-0.441799\pi$$
0.181828 + 0.983330i $$0.441799\pi$$
$$524$$ 0 0
$$525$$ 17473.8 1.45261
$$526$$ 0 0
$$527$$ −10910.3 −0.901825
$$528$$ 0 0
$$529$$ 569.329 0.0467928
$$530$$ 0 0
$$531$$ −8730.03 −0.713467
$$532$$ 0 0
$$533$$ −11868.6 −0.964514
$$534$$ 0 0
$$535$$ −29027.4 −2.34573
$$536$$ 0 0
$$537$$ 11117.8 0.893422
$$538$$ 0 0
$$539$$ −2915.83 −0.233012
$$540$$ 0 0
$$541$$ −3206.84 −0.254848 −0.127424 0.991848i $$-0.540671\pi$$
−0.127424 + 0.991848i $$0.540671\pi$$
$$542$$ 0 0
$$543$$ −47.9107 −0.00378645
$$544$$ 0 0
$$545$$ −27056.9 −2.12658
$$546$$ 0 0
$$547$$ −3289.81 −0.257152 −0.128576 0.991700i $$-0.541041\pi$$
−0.128576 + 0.991700i $$0.541041\pi$$
$$548$$ 0 0
$$549$$ 7413.74 0.576340
$$550$$ 0 0
$$551$$ 3295.45 0.254793
$$552$$ 0 0
$$553$$ −8252.91 −0.634628
$$554$$ 0 0
$$555$$ 14435.0 1.10402
$$556$$ 0 0
$$557$$ 313.140 0.0238207 0.0119104 0.999929i $$-0.496209\pi$$
0.0119104 + 0.999929i $$0.496209\pi$$
$$558$$ 0 0
$$559$$ 5592.43 0.423139
$$560$$ 0 0
$$561$$ 7490.91 0.563755
$$562$$ 0 0
$$563$$ −3425.84 −0.256451 −0.128225 0.991745i $$-0.540928\pi$$
−0.128225 + 0.991745i $$0.540928\pi$$
$$564$$ 0 0
$$565$$ −6336.46 −0.471818
$$566$$ 0 0
$$567$$ 7759.60 0.574731
$$568$$ 0 0
$$569$$ 17763.8 1.30878 0.654390 0.756158i $$-0.272926\pi$$
0.654390 + 0.756158i $$0.272926\pi$$
$$570$$ 0 0
$$571$$ 5741.80 0.420818 0.210409 0.977613i $$-0.432520\pi$$
0.210409 + 0.977613i $$0.432520\pi$$
$$572$$ 0 0
$$573$$ 8650.85 0.630706
$$574$$ 0 0
$$575$$ 35286.0 2.55918
$$576$$ 0 0
$$577$$ 14477.5 1.04455 0.522275 0.852777i $$-0.325083\pi$$
0.522275 + 0.852777i $$0.325083\pi$$
$$578$$ 0 0
$$579$$ −11979.1 −0.859821
$$580$$ 0 0
$$581$$ −9141.98 −0.652794
$$582$$ 0 0
$$583$$ 6417.59 0.455899
$$584$$ 0 0
$$585$$ 17900.6 1.26512
$$586$$ 0 0
$$587$$ −18082.8 −1.27148 −0.635738 0.771905i $$-0.719304\pi$$
−0.635738 + 0.771905i $$0.719304\pi$$
$$588$$ 0 0
$$589$$ −11682.3 −0.817254
$$590$$ 0 0
$$591$$ −32323.6 −2.24977
$$592$$ 0 0
$$593$$ 1731.57 0.119911 0.0599553 0.998201i $$-0.480904\pi$$
0.0599553 + 0.998201i $$0.480904\pi$$
$$594$$ 0 0
$$595$$ −19001.5 −1.30922
$$596$$ 0 0
$$597$$ 23208.0 1.59102
$$598$$ 0 0
$$599$$ 11480.5 0.783105 0.391552 0.920156i $$-0.371938\pi$$
0.391552 + 0.920156i $$0.371938\pi$$
$$600$$ 0 0
$$601$$ 2924.56 0.198495 0.0992474 0.995063i $$-0.468356\pi$$
0.0992474 + 0.995063i $$0.468356\pi$$
$$602$$ 0 0
$$603$$ 3020.83 0.204009
$$604$$ 0 0
$$605$$ 25400.8 1.70692
$$606$$ 0 0
$$607$$ −3586.09 −0.239794 −0.119897 0.992786i $$-0.538256\pi$$
−0.119897 + 0.992786i $$0.538256\pi$$
$$608$$ 0 0
$$609$$ −1620.71 −0.107840
$$610$$ 0 0
$$611$$ 24902.2 1.64883
$$612$$ 0 0
$$613$$ 11679.8 0.769561 0.384781 0.923008i $$-0.374277\pi$$
0.384781 + 0.923008i $$0.374277\pi$$
$$614$$ 0 0
$$615$$ 29638.8 1.94334
$$616$$ 0 0
$$617$$ 11063.0 0.721847 0.360924 0.932595i $$-0.382461\pi$$
0.360924 + 0.932595i $$0.382461\pi$$
$$618$$ 0 0
$$619$$ −2463.60 −0.159969 −0.0799843 0.996796i $$-0.525487\pi$$
−0.0799843 + 0.996796i $$0.525487\pi$$
$$620$$ 0 0
$$621$$ 8370.81 0.540916
$$622$$ 0 0
$$623$$ −6611.73 −0.425190
$$624$$ 0 0
$$625$$ 43051.5 2.75530
$$626$$ 0 0
$$627$$ 8020.96 0.510887
$$628$$ 0 0
$$629$$ −11213.9 −0.710854
$$630$$ 0 0
$$631$$ −1032.43 −0.0651352 −0.0325676 0.999470i $$-0.510368\pi$$
−0.0325676 + 0.999470i $$0.510368\pi$$
$$632$$ 0 0
$$633$$ −16163.3 −1.01490
$$634$$ 0 0
$$635$$ 42294.6 2.64316
$$636$$ 0 0
$$637$$ 14756.8 0.917873
$$638$$ 0 0
$$639$$ 37.2505 0.00230611
$$640$$ 0 0
$$641$$ 12841.6 0.791282 0.395641 0.918405i $$-0.370522\pi$$
0.395641 + 0.918405i $$0.370522\pi$$
$$642$$ 0 0
$$643$$ 8449.14 0.518198 0.259099 0.965851i $$-0.416574\pi$$
0.259099 + 0.965851i $$0.416574\pi$$
$$644$$ 0 0
$$645$$ −13965.7 −0.852557
$$646$$ 0 0
$$647$$ 27036.8 1.64285 0.821426 0.570315i $$-0.193179\pi$$
0.821426 + 0.570315i $$0.193179\pi$$
$$648$$ 0 0
$$649$$ −6033.07 −0.364898
$$650$$ 0 0
$$651$$ 5745.40 0.345899
$$652$$ 0 0
$$653$$ 27105.2 1.62436 0.812181 0.583405i $$-0.198280\pi$$
0.812181 + 0.583405i $$0.198280\pi$$
$$654$$ 0 0
$$655$$ −17881.3 −1.06668
$$656$$ 0 0
$$657$$ 1861.52 0.110540
$$658$$ 0 0
$$659$$ −22622.4 −1.33724 −0.668621 0.743603i $$-0.733115\pi$$
−0.668621 + 0.743603i $$0.733115\pi$$
$$660$$ 0 0
$$661$$ 21000.2 1.23572 0.617862 0.786287i $$-0.287999\pi$$
0.617862 + 0.786287i $$0.287999\pi$$
$$662$$ 0 0
$$663$$ −37910.9 −2.22072
$$664$$ 0 0
$$665$$ −20346.0 −1.18645
$$666$$ 0 0
$$667$$ −3272.80 −0.189990
$$668$$ 0 0
$$669$$ 15552.7 0.898809
$$670$$ 0 0
$$671$$ 5123.42 0.294765
$$672$$ 0 0
$$673$$ −13691.2 −0.784186 −0.392093 0.919926i $$-0.628249\pi$$
−0.392093 + 0.919926i $$0.628249\pi$$
$$674$$ 0 0
$$675$$ 23191.3 1.32242
$$676$$ 0 0
$$677$$ −9694.60 −0.550360 −0.275180 0.961393i $$-0.588737\pi$$
−0.275180 + 0.961393i $$0.588737\pi$$
$$678$$ 0 0
$$679$$ −11502.7 −0.650125
$$680$$ 0 0
$$681$$ −35606.5 −2.00359
$$682$$ 0 0
$$683$$ 6012.25 0.336826 0.168413 0.985717i $$-0.446136\pi$$
0.168413 + 0.985717i $$0.446136\pi$$
$$684$$ 0 0
$$685$$ −7636.47 −0.425948
$$686$$ 0 0
$$687$$ 3893.41 0.216220
$$688$$ 0 0
$$689$$ −32478.9 −1.79586
$$690$$ 0 0
$$691$$ −10304.9 −0.567317 −0.283659 0.958925i $$-0.591548\pi$$
−0.283659 + 0.958925i $$0.591548\pi$$
$$692$$ 0 0
$$693$$ −1446.97 −0.0793156
$$694$$ 0 0
$$695$$ 21131.7 1.15334
$$696$$ 0 0
$$697$$ −23025.0 −1.25127
$$698$$ 0 0
$$699$$ 36718.6 1.98687
$$700$$ 0 0
$$701$$ −11785.8 −0.635011 −0.317506 0.948256i $$-0.602845\pi$$
−0.317506 + 0.948256i $$0.602845\pi$$
$$702$$ 0 0
$$703$$ −12007.4 −0.644192
$$704$$ 0 0
$$705$$ −62187.0 −3.32212
$$706$$ 0 0
$$707$$ 8444.85 0.449224
$$708$$ 0 0
$$709$$ −29226.1 −1.54811 −0.774054 0.633120i $$-0.781774\pi$$
−0.774054 + 0.633120i $$0.781774\pi$$
$$710$$ 0 0
$$711$$ 15083.0 0.795577
$$712$$ 0 0
$$713$$ 11602.1 0.609398
$$714$$ 0 0
$$715$$ 12370.6 0.647040
$$716$$ 0 0
$$717$$ 10214.7 0.532045
$$718$$ 0 0
$$719$$ −14706.5 −0.762809 −0.381404 0.924408i $$-0.624559\pi$$
−0.381404 + 0.924408i $$0.624559\pi$$
$$720$$ 0 0
$$721$$ 4692.32 0.242374
$$722$$ 0 0
$$723$$ 4769.05 0.245315
$$724$$ 0 0
$$725$$ −9067.30 −0.464484
$$726$$ 0 0
$$727$$ 35975.1 1.83527 0.917637 0.397419i $$-0.130094\pi$$
0.917637 + 0.397419i $$0.130094\pi$$
$$728$$ 0 0
$$729$$ −1103.91 −0.0560843
$$730$$ 0 0
$$731$$ 10849.3 0.548941
$$732$$ 0 0
$$733$$ −10872.8 −0.547879 −0.273939 0.961747i $$-0.588327\pi$$
−0.273939 + 0.961747i $$0.588327\pi$$
$$734$$ 0 0
$$735$$ −36851.4 −1.84937
$$736$$ 0 0
$$737$$ 2087.61 0.104339
$$738$$ 0 0
$$739$$ −1078.25 −0.0536727 −0.0268363 0.999640i $$-0.508543\pi$$
−0.0268363 + 0.999640i $$0.508543\pi$$
$$740$$ 0 0
$$741$$ −40593.4 −2.01247
$$742$$ 0 0
$$743$$ 23176.4 1.14436 0.572180 0.820128i $$-0.306098\pi$$
0.572180 + 0.820128i $$0.306098\pi$$
$$744$$ 0 0
$$745$$ 17149.9 0.843385
$$746$$ 0 0
$$747$$ 16707.8 0.818350
$$748$$ 0 0
$$749$$ −11874.9 −0.579304
$$750$$ 0 0
$$751$$ −8738.85 −0.424614 −0.212307 0.977203i $$-0.568098\pi$$
−0.212307 + 0.977203i $$0.568098\pi$$
$$752$$ 0 0
$$753$$ 27716.0 1.34134
$$754$$ 0 0
$$755$$ 20925.7 1.00869
$$756$$ 0 0
$$757$$ −4121.15 −0.197868 −0.0989338 0.995094i $$-0.531543\pi$$
−0.0989338 + 0.995094i $$0.531543\pi$$
$$758$$ 0 0
$$759$$ −7965.84 −0.380951
$$760$$ 0 0
$$761$$ 8706.54 0.414733 0.207367 0.978263i $$-0.433511\pi$$
0.207367 + 0.978263i $$0.433511\pi$$
$$762$$ 0 0
$$763$$ −11068.7 −0.525184
$$764$$ 0 0
$$765$$ 34727.0 1.64125
$$766$$ 0 0
$$767$$ 30532.9 1.43739
$$768$$ 0 0
$$769$$ −32258.8 −1.51272 −0.756361 0.654154i $$-0.773025\pi$$
−0.756361 + 0.654154i $$0.773025\pi$$
$$770$$ 0 0
$$771$$ 1537.00 0.0717945
$$772$$ 0 0
$$773$$ −24794.1 −1.15366 −0.576832 0.816863i $$-0.695711\pi$$
−0.576832 + 0.816863i $$0.695711\pi$$
$$774$$ 0 0
$$775$$ 32143.5 1.48984
$$776$$ 0 0
$$777$$ 5905.25 0.272651
$$778$$ 0 0
$$779$$ −24654.2 −1.13393
$$780$$ 0 0
$$781$$ 25.7427 0.00117945
$$782$$ 0 0
$$783$$ −2151.01 −0.0981749
$$784$$ 0 0
$$785$$ −35614.6 −1.61929
$$786$$ 0 0
$$787$$ −11114.7 −0.503427 −0.251714 0.967802i $$-0.580994\pi$$
−0.251714 + 0.967802i $$0.580994\pi$$
$$788$$ 0 0
$$789$$ 22552.3 1.01760
$$790$$ 0 0
$$791$$ −2592.20 −0.116521
$$792$$ 0 0
$$793$$ −25929.2 −1.16113
$$794$$ 0 0
$$795$$ 81108.0 3.61837
$$796$$ 0 0
$$797$$ 12329.8 0.547986 0.273993 0.961732i $$-0.411656\pi$$
0.273993 + 0.961732i $$0.411656\pi$$
$$798$$ 0 0
$$799$$ 48310.1 2.13904
$$800$$ 0 0
$$801$$ 12083.6 0.533023
$$802$$ 0 0
$$803$$ 1286.44 0.0565348
$$804$$ 0 0
$$805$$ 20206.2 0.884691
$$806$$ 0 0
$$807$$ −12546.2 −0.547271
$$808$$ 0 0
$$809$$ −36175.5 −1.57214 −0.786070 0.618138i $$-0.787887\pi$$
−0.786070 + 0.618138i $$0.787887\pi$$
$$810$$ 0 0
$$811$$ −26581.0 −1.15091 −0.575453 0.817835i $$-0.695174\pi$$
−0.575453 + 0.817835i $$0.695174\pi$$
$$812$$ 0 0
$$813$$ −16195.0 −0.698629
$$814$$ 0 0
$$815$$ 72212.5 3.10367
$$816$$ 0 0
$$817$$ 11617.0 0.497462
$$818$$ 0 0
$$819$$ 7322.98 0.312437
$$820$$ 0 0
$$821$$ −10811.1 −0.459574 −0.229787 0.973241i $$-0.573803\pi$$
−0.229787 + 0.973241i $$0.573803\pi$$
$$822$$ 0 0
$$823$$ −28625.1 −1.21240 −0.606202 0.795311i $$-0.707308\pi$$
−0.606202 + 0.795311i $$0.707308\pi$$
$$824$$ 0 0
$$825$$ −22069.3 −0.931341
$$826$$ 0 0
$$827$$ −9970.68 −0.419244 −0.209622 0.977783i $$-0.567223\pi$$
−0.209622 + 0.977783i $$0.567223\pi$$
$$828$$ 0 0
$$829$$ −20201.0 −0.846334 −0.423167 0.906052i $$-0.639082\pi$$
−0.423167 + 0.906052i $$0.639082\pi$$
$$830$$ 0 0
$$831$$ −31128.0 −1.29942
$$832$$ 0 0
$$833$$ 28628.1 1.19076
$$834$$ 0 0
$$835$$ 29481.7 1.22186
$$836$$ 0 0
$$837$$ 7625.33 0.314898
$$838$$ 0 0
$$839$$ −28023.4 −1.15313 −0.576564 0.817052i $$-0.695607\pi$$
−0.576564 + 0.817052i $$0.695607\pi$$
$$840$$ 0 0
$$841$$ 841.000 0.0344828
$$842$$ 0 0
$$843$$ −1268.98 −0.0518456
$$844$$ 0 0
$$845$$ −16644.3 −0.677610
$$846$$ 0 0
$$847$$ 10391.3 0.421544
$$848$$ 0 0
$$849$$ 18397.9 0.743714
$$850$$ 0 0
$$851$$ 11924.9 0.480352
$$852$$ 0 0
$$853$$ 26333.8 1.05704 0.528518 0.848922i $$-0.322748\pi$$
0.528518 + 0.848922i $$0.322748\pi$$
$$854$$ 0 0
$$855$$ 37184.3 1.48734
$$856$$ 0 0
$$857$$ 36388.6 1.45042 0.725211 0.688527i $$-0.241742\pi$$
0.725211 + 0.688527i $$0.241742\pi$$
$$858$$ 0 0
$$859$$ 14974.0 0.594770 0.297385 0.954758i $$-0.403885\pi$$
0.297385 + 0.954758i $$0.403885\pi$$
$$860$$ 0 0
$$861$$ 12125.0 0.479930
$$862$$ 0 0
$$863$$ 40168.2 1.58440 0.792202 0.610259i $$-0.208935\pi$$
0.792202 + 0.610259i $$0.208935\pi$$
$$864$$ 0 0
$$865$$ 28418.1 1.11705
$$866$$ 0 0
$$867$$ −41465.0 −1.62425
$$868$$ 0 0
$$869$$ 10423.4 0.406893
$$870$$ 0 0
$$871$$ −10565.2 −0.411009
$$872$$ 0 0
$$873$$ 21022.3 0.815004
$$874$$ 0 0
$$875$$ 33600.7 1.29818
$$876$$ 0 0
$$877$$ −32161.7 −1.23834 −0.619169 0.785258i $$-0.712531\pi$$
−0.619169 + 0.785258i $$0.712531\pi$$
$$878$$ 0 0
$$879$$ 26506.2 1.01710
$$880$$ 0 0
$$881$$ −8870.49 −0.339222 −0.169611 0.985511i $$-0.554251\pi$$
−0.169611 + 0.985511i $$0.554251\pi$$
$$882$$ 0 0
$$883$$ −24647.3 −0.939351 −0.469675 0.882839i $$-0.655629\pi$$
−0.469675 + 0.882839i $$0.655629\pi$$
$$884$$ 0 0
$$885$$ −76248.3 −2.89611
$$886$$ 0 0
$$887$$ −32371.7 −1.22541 −0.612703 0.790313i $$-0.709918\pi$$
−0.612703 + 0.790313i $$0.709918\pi$$
$$888$$ 0 0
$$889$$ 17302.4 0.652759
$$890$$ 0 0
$$891$$ −9800.35 −0.368489
$$892$$ 0 0
$$893$$ 51728.5 1.93844
$$894$$ 0 0
$$895$$ 35618.4 1.33027
$$896$$ 0 0
$$897$$ 40314.5 1.50063
$$898$$ 0 0
$$899$$ −2981.34 −0.110604
$$900$$ 0 0
$$901$$ −63008.9 −2.32978
$$902$$ 0 0
$$903$$ −5713.26 −0.210549
$$904$$ 0 0
$$905$$ −153.493 −0.00563787
$$906$$ 0 0
$$907$$ −43728.0 −1.60084 −0.800422 0.599437i $$-0.795391\pi$$
−0.800422 + 0.599437i $$0.795391\pi$$
$$908$$ 0 0
$$909$$ −15433.7 −0.563152
$$910$$ 0 0
$$911$$ −16033.6 −0.583115 −0.291557 0.956553i $$-0.594173\pi$$
−0.291557 + 0.956553i $$0.594173\pi$$
$$912$$ 0 0
$$913$$ 11546.3 0.418540
$$914$$ 0 0
$$915$$ 64751.8 2.33948
$$916$$ 0 0
$$917$$ −7315.08 −0.263430
$$918$$ 0 0
$$919$$ −29331.1 −1.05282 −0.526411 0.850230i $$-0.676463\pi$$
−0.526411 + 0.850230i $$0.676463\pi$$
$$920$$ 0 0
$$921$$ 859.170 0.0307390
$$922$$ 0 0
$$923$$ −130.282 −0.00464603
$$924$$ 0 0
$$925$$ 33037.8 1.17435
$$926$$ 0 0
$$927$$ −8575.66 −0.303842
$$928$$ 0 0
$$929$$ −10024.6 −0.354034 −0.177017 0.984208i $$-0.556645\pi$$
−0.177017 + 0.984208i $$0.556645\pi$$
$$930$$ 0 0
$$931$$ 30653.8 1.07910
$$932$$ 0 0
$$933$$ 9911.51 0.347790
$$934$$ 0 0
$$935$$ 23998.9 0.839408
$$936$$ 0 0
$$937$$ 21241.3 0.740578 0.370289 0.928917i $$-0.379259\pi$$
0.370289 + 0.928917i $$0.379259\pi$$
$$938$$ 0 0
$$939$$ −27714.5 −0.963184
$$940$$ 0 0
$$941$$ −11953.8 −0.414115 −0.207057 0.978329i $$-0.566389\pi$$
−0.207057 + 0.978329i $$0.566389\pi$$
$$942$$ 0 0
$$943$$ 24484.8 0.845531
$$944$$ 0 0
$$945$$ 13280.3 0.457152
$$946$$ 0 0
$$947$$ −31650.6 −1.08607 −0.543034 0.839711i $$-0.682725\pi$$
−0.543034 + 0.839711i $$0.682725\pi$$
$$948$$ 0 0
$$949$$ −6510.57 −0.222700
$$950$$ 0 0
$$951$$ 36545.7 1.24614
$$952$$ 0 0
$$953$$ −17373.5 −0.590539 −0.295269 0.955414i $$-0.595409\pi$$
−0.295269 + 0.955414i $$0.595409\pi$$
$$954$$ 0 0
$$955$$ 27715.0 0.939095
$$956$$ 0 0
$$957$$ 2046.95 0.0691416
$$958$$ 0 0
$$959$$ −3124.02 −0.105193
$$960$$ 0 0
$$961$$ −19222.2 −0.645234
$$962$$ 0 0
$$963$$ 21702.4 0.726222
$$964$$ 0 0
$$965$$ −38378.0 −1.28024
$$966$$ 0 0
$$967$$ 11017.6 0.366394 0.183197 0.983076i $$-0.441355\pi$$
0.183197 + 0.983076i $$0.441355\pi$$
$$968$$ 0 0
$$969$$ −78751.1 −2.61078
$$970$$ 0 0
$$971$$ 17365.0 0.573912 0.286956 0.957944i $$-0.407357\pi$$
0.286956 + 0.957944i $$0.407357\pi$$
$$972$$ 0 0
$$973$$ 8644.82 0.284831
$$974$$ 0 0
$$975$$ 111691. 3.66870
$$976$$ 0 0
$$977$$ −29193.4 −0.955967 −0.477983 0.878369i $$-0.658632\pi$$
−0.477983 + 0.878369i $$0.658632\pi$$
$$978$$ 0 0
$$979$$ 8350.60 0.272611
$$980$$ 0 0
$$981$$ 20229.2 0.658377
$$982$$ 0 0
$$983$$ −50359.8 −1.63401 −0.817003 0.576634i $$-0.804366\pi$$
−0.817003 + 0.576634i $$0.804366\pi$$
$$984$$ 0 0
$$985$$ −103556. −3.34982
$$986$$ 0 0
$$987$$ −25440.2 −0.820436
$$988$$ 0 0
$$989$$ −11537.2 −0.370941
$$990$$ 0 0
$$991$$ 20823.5 0.667488 0.333744 0.942664i $$-0.391688\pi$$
0.333744 + 0.942664i $$0.391688\pi$$
$$992$$ 0 0
$$993$$ 54746.8 1.74958
$$994$$ 0 0
$$995$$ 74352.2 2.36897
$$996$$ 0 0
$$997$$ 27651.2 0.878356 0.439178 0.898400i $$-0.355270\pi$$
0.439178 + 0.898400i $$0.355270\pi$$
$$998$$ 0 0
$$999$$ 7837.48 0.248215
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 1856.4.a.s.1.1 3
4.3 odd 2 1856.4.a.r.1.3 3
8.3 odd 2 58.4.a.d.1.1 3
8.5 even 2 464.4.a.i.1.3 3
24.11 even 2 522.4.a.k.1.1 3
40.19 odd 2 1450.4.a.h.1.3 3
232.115 odd 2 1682.4.a.d.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
58.4.a.d.1.1 3 8.3 odd 2
464.4.a.i.1.3 3 8.5 even 2
522.4.a.k.1.1 3 24.11 even 2
1450.4.a.h.1.3 3 40.19 odd 2
1682.4.a.d.1.3 3 232.115 odd 2
1856.4.a.r.1.3 3 4.3 odd 2
1856.4.a.s.1.1 3 1.1 even 1 trivial