Properties

 Label 441.4.a.v.1.1 Level $441$ Weight $4$ Character 441.1 Self dual yes Analytic conductor $26.020$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$26.0198423125$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.6257832.1 Defining polynomial: $$x^{4} - 19 x^{2} + 42$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-4.05539$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-4.05539 q^{2} +8.44622 q^{4} +9.92039 q^{5} -1.80961 q^{8} +O(q^{10})$$ $$q-4.05539 q^{2} +8.44622 q^{4} +9.92039 q^{5} -1.80961 q^{8} -40.2311 q^{10} -13.5396 q^{11} -18.5538 q^{13} -60.2311 q^{16} +93.7102 q^{17} -131.908 q^{19} +83.7899 q^{20} +54.9084 q^{22} -198.278 q^{23} -26.5858 q^{25} +75.2429 q^{26} +188.358 q^{29} +83.9244 q^{31} +258.738 q^{32} -380.032 q^{34} +80.1555 q^{37} +534.941 q^{38} -17.9520 q^{40} +385.828 q^{41} -397.048 q^{43} -114.359 q^{44} +804.096 q^{46} -272.277 q^{47} +107.816 q^{50} -156.709 q^{52} -36.9996 q^{53} -134.318 q^{55} -763.865 q^{58} -395.749 q^{59} -13.4738 q^{61} -340.347 q^{62} -567.435 q^{64} -184.061 q^{65} +340.522 q^{67} +791.498 q^{68} -211.140 q^{71} -486.124 q^{73} -325.062 q^{74} -1114.13 q^{76} +293.741 q^{79} -597.516 q^{80} -1564.69 q^{82} -889.635 q^{83} +929.643 q^{85} +1610.19 q^{86} +24.5014 q^{88} -1144.36 q^{89} -1674.70 q^{92} +1104.19 q^{94} -1308.58 q^{95} -1384.61 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{4} + O(q^{10})$$ $$4q + 6q^{4} - 22q^{10} - 102q^{13} - 102q^{16} - 222q^{19} - 86q^{22} + 366q^{25} - 220q^{31} - 1020q^{34} - 374q^{37} - 822q^{40} - 838q^{43} + 1716q^{46} + 40q^{52} - 2510q^{55} - 1694q^{58} - 1332q^{61} - 686q^{64} + 1890q^{67} - 1750q^{73} - 2456q^{76} + 8q^{79} - 2480q^{82} - 1116q^{85} + 2682q^{88} + 1416q^{94} - 3010q^{97} + O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.05539 −1.43380 −0.716899 0.697177i $$-0.754439\pi$$
−0.716899 + 0.697177i $$0.754439\pi$$
$$3$$ 0 0
$$4$$ 8.44622 1.05578
$$5$$ 9.92039 0.887307 0.443654 0.896198i $$-0.353682\pi$$
0.443654 + 0.896198i $$0.353682\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.80961 −0.0799740
$$9$$ 0 0
$$10$$ −40.2311 −1.27222
$$11$$ −13.5396 −0.371122 −0.185561 0.982633i $$-0.559410\pi$$
−0.185561 + 0.982633i $$0.559410\pi$$
$$12$$ 0 0
$$13$$ −18.5538 −0.395838 −0.197919 0.980218i $$-0.563418\pi$$
−0.197919 + 0.980218i $$0.563418\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −60.2311 −0.941111
$$17$$ 93.7102 1.33695 0.668473 0.743737i $$-0.266948\pi$$
0.668473 + 0.743737i $$0.266948\pi$$
$$18$$ 0 0
$$19$$ −131.908 −1.59273 −0.796365 0.604816i $$-0.793247\pi$$
−0.796365 + 0.604816i $$0.793247\pi$$
$$20$$ 83.7899 0.936799
$$21$$ 0 0
$$22$$ 54.9084 0.532115
$$23$$ −198.278 −1.79756 −0.898779 0.438401i $$-0.855545\pi$$
−0.898779 + 0.438401i $$0.855545\pi$$
$$24$$ 0 0
$$25$$ −26.5858 −0.212686
$$26$$ 75.2429 0.567552
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 188.358 1.20611 0.603054 0.797700i $$-0.293950\pi$$
0.603054 + 0.797700i $$0.293950\pi$$
$$30$$ 0 0
$$31$$ 83.9244 0.486235 0.243117 0.969997i $$-0.421830\pi$$
0.243117 + 0.969997i $$0.421830\pi$$
$$32$$ 258.738 1.42934
$$33$$ 0 0
$$34$$ −380.032 −1.91691
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 80.1555 0.356148 0.178074 0.984017i $$-0.443013\pi$$
0.178074 + 0.984017i $$0.443013\pi$$
$$38$$ 534.941 2.28365
$$39$$ 0 0
$$40$$ −17.9520 −0.0709615
$$41$$ 385.828 1.46967 0.734833 0.678249i $$-0.237261\pi$$
0.734833 + 0.678249i $$0.237261\pi$$
$$42$$ 0 0
$$43$$ −397.048 −1.40812 −0.704061 0.710139i $$-0.748632\pi$$
−0.704061 + 0.710139i $$0.748632\pi$$
$$44$$ −114.359 −0.391823
$$45$$ 0 0
$$46$$ 804.096 2.57734
$$47$$ −272.277 −0.845016 −0.422508 0.906359i $$-0.638850\pi$$
−0.422508 + 0.906359i $$0.638850\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 107.816 0.304949
$$51$$ 0 0
$$52$$ −156.709 −0.417917
$$53$$ −36.9996 −0.0958922 −0.0479461 0.998850i $$-0.515268\pi$$
−0.0479461 + 0.998850i $$0.515268\pi$$
$$54$$ 0 0
$$55$$ −134.318 −0.329299
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −763.865 −1.72932
$$59$$ −395.749 −0.873256 −0.436628 0.899642i $$-0.643827\pi$$
−0.436628 + 0.899642i $$0.643827\pi$$
$$60$$ 0 0
$$61$$ −13.4738 −0.0282810 −0.0141405 0.999900i $$-0.504501\pi$$
−0.0141405 + 0.999900i $$0.504501\pi$$
$$62$$ −340.347 −0.697162
$$63$$ 0 0
$$64$$ −567.435 −1.10827
$$65$$ −184.061 −0.351230
$$66$$ 0 0
$$67$$ 340.522 0.620916 0.310458 0.950587i $$-0.399518\pi$$
0.310458 + 0.950587i $$0.399518\pi$$
$$68$$ 791.498 1.41152
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −211.140 −0.352925 −0.176463 0.984307i $$-0.556466\pi$$
−0.176463 + 0.984307i $$0.556466\pi$$
$$72$$ 0 0
$$73$$ −486.124 −0.779404 −0.389702 0.920941i $$-0.627422\pi$$
−0.389702 + 0.920941i $$0.627422\pi$$
$$74$$ −325.062 −0.510645
$$75$$ 0 0
$$76$$ −1114.13 −1.68157
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 293.741 0.418335 0.209168 0.977880i $$-0.432925\pi$$
0.209168 + 0.977880i $$0.432925\pi$$
$$80$$ −597.516 −0.835055
$$81$$ 0 0
$$82$$ −1564.69 −2.10720
$$83$$ −889.635 −1.17651 −0.588253 0.808677i $$-0.700184\pi$$
−0.588253 + 0.808677i $$0.700184\pi$$
$$84$$ 0 0
$$85$$ 929.643 1.18628
$$86$$ 1610.19 2.01896
$$87$$ 0 0
$$88$$ 24.5014 0.0296801
$$89$$ −1144.36 −1.36295 −0.681474 0.731843i $$-0.738661\pi$$
−0.681474 + 0.731843i $$0.738661\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1674.70 −1.89782
$$93$$ 0 0
$$94$$ 1104.19 1.21158
$$95$$ −1308.58 −1.41324
$$96$$ 0 0
$$97$$ −1384.61 −1.44933 −0.724667 0.689099i $$-0.758007\pi$$
−0.724667 + 0.689099i $$0.758007\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −224.549 −0.224549
$$101$$ −1785.99 −1.75953 −0.879765 0.475409i $$-0.842300\pi$$
−0.879765 + 0.475409i $$0.842300\pi$$
$$102$$ 0 0
$$103$$ −489.661 −0.468425 −0.234212 0.972185i $$-0.575251\pi$$
−0.234212 + 0.972185i $$0.575251\pi$$
$$104$$ 33.5750 0.0316568
$$105$$ 0 0
$$106$$ 150.048 0.137490
$$107$$ 282.068 0.254846 0.127423 0.991848i $$-0.459329\pi$$
0.127423 + 0.991848i $$0.459329\pi$$
$$108$$ 0 0
$$109$$ −287.231 −0.252401 −0.126201 0.992005i $$-0.540278\pi$$
−0.126201 + 0.992005i $$0.540278\pi$$
$$110$$ 544.713 0.472149
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1895.21 1.57776 0.788879 0.614548i $$-0.210662\pi$$
0.788879 + 0.614548i $$0.210662\pi$$
$$114$$ 0 0
$$115$$ −1967.00 −1.59499
$$116$$ 1590.91 1.27338
$$117$$ 0 0
$$118$$ 1604.92 1.25207
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1147.68 −0.862268
$$122$$ 54.6415 0.0405493
$$123$$ 0 0
$$124$$ 708.844 0.513356
$$125$$ −1503.79 −1.07603
$$126$$ 0 0
$$127$$ −1222.92 −0.854463 −0.427231 0.904142i $$-0.640511\pi$$
−0.427231 + 0.904142i $$0.640511\pi$$
$$128$$ 231.269 0.159699
$$129$$ 0 0
$$130$$ 746.439 0.503593
$$131$$ −1190.61 −0.794074 −0.397037 0.917803i $$-0.629962\pi$$
−0.397037 + 0.917803i $$0.629962\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −1380.95 −0.890268
$$135$$ 0 0
$$136$$ −169.579 −0.106921
$$137$$ 106.961 0.0667031 0.0333516 0.999444i $$-0.489382\pi$$
0.0333516 + 0.999444i $$0.489382\pi$$
$$138$$ 0 0
$$139$$ 1096.78 0.669262 0.334631 0.942349i $$-0.391388\pi$$
0.334631 + 0.942349i $$0.391388\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 856.256 0.506024
$$143$$ 251.211 0.146904
$$144$$ 0 0
$$145$$ 1868.58 1.07019
$$146$$ 1971.42 1.11751
$$147$$ 0 0
$$148$$ 677.012 0.376014
$$149$$ 716.662 0.394035 0.197018 0.980400i $$-0.436874\pi$$
0.197018 + 0.980400i $$0.436874\pi$$
$$150$$ 0 0
$$151$$ 154.368 0.0831939 0.0415970 0.999134i $$-0.486755\pi$$
0.0415970 + 0.999134i $$0.486755\pi$$
$$152$$ 238.702 0.127377
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 832.564 0.431439
$$156$$ 0 0
$$157$$ 2093.75 1.06433 0.532163 0.846642i $$-0.321379\pi$$
0.532163 + 0.846642i $$0.321379\pi$$
$$158$$ −1191.24 −0.599808
$$159$$ 0 0
$$160$$ 2566.78 1.26826
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 3007.86 1.44536 0.722680 0.691183i $$-0.242910\pi$$
0.722680 + 0.691183i $$0.242910\pi$$
$$164$$ 3258.79 1.55164
$$165$$ 0 0
$$166$$ 3607.82 1.68687
$$167$$ 2230.43 1.03351 0.516754 0.856134i $$-0.327140\pi$$
0.516754 + 0.856134i $$0.327140\pi$$
$$168$$ 0 0
$$169$$ −1852.76 −0.843312
$$170$$ −3770.07 −1.70089
$$171$$ 0 0
$$172$$ −3353.56 −1.48666
$$173$$ 563.170 0.247497 0.123749 0.992314i $$-0.460508\pi$$
0.123749 + 0.992314i $$0.460508\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 815.506 0.349267
$$177$$ 0 0
$$178$$ 4640.85 1.95419
$$179$$ 1839.50 0.768104 0.384052 0.923312i $$-0.374528\pi$$
0.384052 + 0.923312i $$0.374528\pi$$
$$180$$ 0 0
$$181$$ −2324.71 −0.954664 −0.477332 0.878723i $$-0.658396\pi$$
−0.477332 + 0.878723i $$0.658396\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 358.805 0.143758
$$185$$ 795.175 0.316013
$$186$$ 0 0
$$187$$ −1268.80 −0.496170
$$188$$ −2299.71 −0.892149
$$189$$ 0 0
$$190$$ 5306.82 2.02630
$$191$$ −3127.27 −1.18472 −0.592360 0.805673i $$-0.701804\pi$$
−0.592360 + 0.805673i $$0.701804\pi$$
$$192$$ 0 0
$$193$$ −3709.29 −1.38342 −0.691711 0.722175i $$-0.743143\pi$$
−0.691711 + 0.722175i $$0.743143\pi$$
$$194$$ 5615.12 2.07805
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 851.150 0.307827 0.153913 0.988084i $$-0.450812\pi$$
0.153913 + 0.988084i $$0.450812\pi$$
$$198$$ 0 0
$$199$$ −3397.78 −1.21036 −0.605182 0.796087i $$-0.706900\pi$$
−0.605182 + 0.796087i $$0.706900\pi$$
$$200$$ 48.1098 0.0170094
$$201$$ 0 0
$$202$$ 7242.89 2.52281
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 3827.57 1.30404
$$206$$ 1985.77 0.671627
$$207$$ 0 0
$$208$$ 1117.51 0.372527
$$209$$ 1785.99 0.591098
$$210$$ 0 0
$$211$$ 216.732 0.0707132 0.0353566 0.999375i $$-0.488743\pi$$
0.0353566 + 0.999375i $$0.488743\pi$$
$$212$$ −312.507 −0.101241
$$213$$ 0 0
$$214$$ −1143.90 −0.365398
$$215$$ −3938.87 −1.24944
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 1164.84 0.361893
$$219$$ 0 0
$$220$$ −1134.48 −0.347667
$$221$$ −1738.68 −0.529214
$$222$$ 0 0
$$223$$ 2254.86 0.677115 0.338558 0.940946i $$-0.390061\pi$$
0.338558 + 0.940946i $$0.390061\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −7685.84 −2.26219
$$227$$ 3390.80 0.991433 0.495716 0.868484i $$-0.334906\pi$$
0.495716 + 0.868484i $$0.334906\pi$$
$$228$$ 0 0
$$229$$ −2587.85 −0.746768 −0.373384 0.927677i $$-0.621803\pi$$
−0.373384 + 0.927677i $$0.621803\pi$$
$$230$$ 7976.95 2.28689
$$231$$ 0 0
$$232$$ −340.853 −0.0964574
$$233$$ −954.420 −0.268353 −0.134176 0.990957i $$-0.542839\pi$$
−0.134176 + 0.990957i $$0.542839\pi$$
$$234$$ 0 0
$$235$$ −2701.10 −0.749788
$$236$$ −3342.58 −0.921964
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 199.504 0.0539951 0.0269976 0.999635i $$-0.491405\pi$$
0.0269976 + 0.999635i $$0.491405\pi$$
$$240$$ 0 0
$$241$$ −4794.43 −1.28148 −0.640739 0.767759i $$-0.721372\pi$$
−0.640739 + 0.767759i $$0.721372\pi$$
$$242$$ 4654.29 1.23632
$$243$$ 0 0
$$244$$ −113.803 −0.0298585
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2447.40 0.630463
$$248$$ −151.870 −0.0388862
$$249$$ 0 0
$$250$$ 6098.46 1.54280
$$251$$ −6249.73 −1.57163 −0.785816 0.618460i $$-0.787757\pi$$
−0.785816 + 0.618460i $$0.787757\pi$$
$$252$$ 0 0
$$253$$ 2684.61 0.667114
$$254$$ 4959.43 1.22513
$$255$$ 0 0
$$256$$ 3601.59 0.879294
$$257$$ 3837.85 0.931512 0.465756 0.884913i $$-0.345782\pi$$
0.465756 + 0.884913i $$0.345782\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −1554.62 −0.370821
$$261$$ 0 0
$$262$$ 4828.38 1.13854
$$263$$ 207.203 0.0485806 0.0242903 0.999705i $$-0.492267\pi$$
0.0242903 + 0.999705i $$0.492267\pi$$
$$264$$ 0 0
$$265$$ −367.051 −0.0850858
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 2876.12 0.655549
$$269$$ −4804.08 −1.08888 −0.544442 0.838798i $$-0.683259\pi$$
−0.544442 + 0.838798i $$0.683259\pi$$
$$270$$ 0 0
$$271$$ −3215.75 −0.720823 −0.360411 0.932793i $$-0.617364\pi$$
−0.360411 + 0.932793i $$0.617364\pi$$
$$272$$ −5644.27 −1.25821
$$273$$ 0 0
$$274$$ −433.771 −0.0956388
$$275$$ 359.961 0.0789326
$$276$$ 0 0
$$277$$ 2054.39 0.445619 0.222810 0.974862i $$-0.428477\pi$$
0.222810 + 0.974862i $$0.428477\pi$$
$$278$$ −4447.87 −0.959587
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1768.61 0.375468 0.187734 0.982220i $$-0.439886\pi$$
0.187734 + 0.982220i $$0.439886\pi$$
$$282$$ 0 0
$$283$$ −2340.53 −0.491625 −0.245813 0.969317i $$-0.579055\pi$$
−0.245813 + 0.969317i $$0.579055\pi$$
$$284$$ −1783.34 −0.372611
$$285$$ 0 0
$$286$$ −1018.76 −0.210631
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 3868.61 0.787423
$$290$$ −7577.84 −1.53443
$$291$$ 0 0
$$292$$ −4105.91 −0.822877
$$293$$ 3633.47 0.724470 0.362235 0.932087i $$-0.382014\pi$$
0.362235 + 0.932087i $$0.382014\pi$$
$$294$$ 0 0
$$295$$ −3925.98 −0.774846
$$296$$ −145.050 −0.0284826
$$297$$ 0 0
$$298$$ −2906.35 −0.564967
$$299$$ 3678.81 0.711542
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −626.023 −0.119283
$$303$$ 0 0
$$304$$ 7944.99 1.49894
$$305$$ −133.665 −0.0250939
$$306$$ 0 0
$$307$$ −5954.32 −1.10694 −0.553471 0.832868i $$-0.686697\pi$$
−0.553471 + 0.832868i $$0.686697\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −3376.37 −0.618597
$$311$$ −1180.09 −0.215167 −0.107584 0.994196i $$-0.534311\pi$$
−0.107584 + 0.994196i $$0.534311\pi$$
$$312$$ 0 0
$$313$$ 9746.25 1.76003 0.880017 0.474943i $$-0.157531\pi$$
0.880017 + 0.474943i $$0.157531\pi$$
$$314$$ −8490.97 −1.52603
$$315$$ 0 0
$$316$$ 2481.00 0.441669
$$317$$ 8591.91 1.52230 0.761151 0.648574i $$-0.224634\pi$$
0.761151 + 0.648574i $$0.224634\pi$$
$$318$$ 0 0
$$319$$ −2550.29 −0.447614
$$320$$ −5629.18 −0.983377
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −12361.2 −2.12939
$$324$$ 0 0
$$325$$ 493.267 0.0841892
$$326$$ −12198.0 −2.07235
$$327$$ 0 0
$$328$$ −698.197 −0.117535
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −5251.20 −0.872000 −0.436000 0.899947i $$-0.643605\pi$$
−0.436000 + 0.899947i $$0.643605\pi$$
$$332$$ −7514.05 −1.24213
$$333$$ 0 0
$$334$$ −9045.28 −1.48184
$$335$$ 3378.11 0.550943
$$336$$ 0 0
$$337$$ −8496.45 −1.37339 −0.686693 0.726947i $$-0.740938\pi$$
−0.686693 + 0.726947i $$0.740938\pi$$
$$338$$ 7513.66 1.20914
$$339$$ 0 0
$$340$$ 7851.97 1.25245
$$341$$ −1136.30 −0.180453
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 718.500 0.112613
$$345$$ 0 0
$$346$$ −2283.88 −0.354861
$$347$$ −5830.15 −0.901956 −0.450978 0.892535i $$-0.648925\pi$$
−0.450978 + 0.892535i $$0.648925\pi$$
$$348$$ 0 0
$$349$$ −1811.13 −0.277786 −0.138893 0.990307i $$-0.544354\pi$$
−0.138893 + 0.990307i $$0.544354\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −3503.21 −0.530459
$$353$$ −3327.76 −0.501753 −0.250876 0.968019i $$-0.580719\pi$$
−0.250876 + 0.968019i $$0.580719\pi$$
$$354$$ 0 0
$$355$$ −2094.59 −0.313153
$$356$$ −9665.55 −1.43897
$$357$$ 0 0
$$358$$ −7459.89 −1.10131
$$359$$ −870.861 −0.128029 −0.0640143 0.997949i $$-0.520390\pi$$
−0.0640143 + 0.997949i $$0.520390\pi$$
$$360$$ 0 0
$$361$$ 10540.8 1.53679
$$362$$ 9427.61 1.36880
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4822.54 −0.691570
$$366$$ 0 0
$$367$$ −1174.72 −0.167084 −0.0835418 0.996504i $$-0.526623\pi$$
−0.0835418 + 0.996504i $$0.526623\pi$$
$$368$$ 11942.5 1.69170
$$369$$ 0 0
$$370$$ −3224.75 −0.453099
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3628.33 0.503667 0.251834 0.967771i $$-0.418966\pi$$
0.251834 + 0.967771i $$0.418966\pi$$
$$374$$ 5145.48 0.711408
$$375$$ 0 0
$$376$$ 492.715 0.0675793
$$377$$ −3494.75 −0.477424
$$378$$ 0 0
$$379$$ 7321.99 0.992362 0.496181 0.868219i $$-0.334735\pi$$
0.496181 + 0.868219i $$0.334735\pi$$
$$380$$ −11052.6 −1.49207
$$381$$ 0 0
$$382$$ 12682.3 1.69865
$$383$$ 7354.89 0.981247 0.490623 0.871372i $$-0.336769\pi$$
0.490623 + 0.871372i $$0.336769\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 15042.6 1.98355
$$387$$ 0 0
$$388$$ −11694.7 −1.53018
$$389$$ 9069.62 1.18213 0.591064 0.806624i $$-0.298708\pi$$
0.591064 + 0.806624i $$0.298708\pi$$
$$390$$ 0 0
$$391$$ −18580.7 −2.40324
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −3451.75 −0.441362
$$395$$ 2914.03 0.371192
$$396$$ 0 0
$$397$$ −7376.83 −0.932575 −0.466288 0.884633i $$-0.654409\pi$$
−0.466288 + 0.884633i $$0.654409\pi$$
$$398$$ 13779.4 1.73542
$$399$$ 0 0
$$400$$ 1601.29 0.200161
$$401$$ −2853.51 −0.355356 −0.177678 0.984089i $$-0.556858\pi$$
−0.177678 + 0.984089i $$0.556858\pi$$
$$402$$ 0 0
$$403$$ −1557.12 −0.192470
$$404$$ −15084.9 −1.85767
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1085.27 −0.132175
$$408$$ 0 0
$$409$$ −11260.1 −1.36130 −0.680652 0.732607i $$-0.738304\pi$$
−0.680652 + 0.732607i $$0.738304\pi$$
$$410$$ −15522.3 −1.86974
$$411$$ 0 0
$$412$$ −4135.79 −0.494553
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8825.53 −1.04392
$$416$$ −4800.56 −0.565786
$$417$$ 0 0
$$418$$ −7242.89 −0.847515
$$419$$ 9221.47 1.07517 0.537587 0.843208i $$-0.319336\pi$$
0.537587 + 0.843208i $$0.319336\pi$$
$$420$$ 0 0
$$421$$ −8520.28 −0.986349 −0.493175 0.869930i $$-0.664164\pi$$
−0.493175 + 0.869930i $$0.664164\pi$$
$$422$$ −878.936 −0.101388
$$423$$ 0 0
$$424$$ 66.9547 0.00766889
$$425$$ −2491.36 −0.284350
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 2382.41 0.269061
$$429$$ 0 0
$$430$$ 15973.7 1.79144
$$431$$ 9162.10 1.02395 0.511976 0.859000i $$-0.328914\pi$$
0.511976 + 0.859000i $$0.328914\pi$$
$$432$$ 0 0
$$433$$ 10976.2 1.21820 0.609100 0.793093i $$-0.291531\pi$$
0.609100 + 0.793093i $$0.291531\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2426.02 −0.266480
$$437$$ 26154.6 2.86303
$$438$$ 0 0
$$439$$ −3983.18 −0.433045 −0.216523 0.976278i $$-0.569472\pi$$
−0.216523 + 0.976278i $$0.569472\pi$$
$$440$$ 243.063 0.0263354
$$441$$ 0 0
$$442$$ 7051.03 0.758786
$$443$$ 4524.45 0.485244 0.242622 0.970121i $$-0.421993\pi$$
0.242622 + 0.970121i $$0.421993\pi$$
$$444$$ 0 0
$$445$$ −11352.5 −1.20935
$$446$$ −9144.35 −0.970847
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2076.49 −0.218253 −0.109127 0.994028i $$-0.534805\pi$$
−0.109127 + 0.994028i $$0.534805\pi$$
$$450$$ 0 0
$$451$$ −5223.96 −0.545425
$$452$$ 16007.4 1.66576
$$453$$ 0 0
$$454$$ −13751.0 −1.42151
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1847.59 −0.189117 −0.0945587 0.995519i $$-0.530144\pi$$
−0.0945587 + 0.995519i $$0.530144\pi$$
$$458$$ 10494.7 1.07071
$$459$$ 0 0
$$460$$ −16613.7 −1.68395
$$461$$ −876.945 −0.0885974 −0.0442987 0.999018i $$-0.514105\pi$$
−0.0442987 + 0.999018i $$0.514105\pi$$
$$462$$ 0 0
$$463$$ 16245.2 1.63062 0.815310 0.579025i $$-0.196566\pi$$
0.815310 + 0.579025i $$0.196566\pi$$
$$464$$ −11345.0 −1.13508
$$465$$ 0 0
$$466$$ 3870.55 0.384763
$$467$$ 18961.8 1.87890 0.939449 0.342689i $$-0.111338\pi$$
0.939449 + 0.342689i $$0.111338\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 10954.0 1.07505
$$471$$ 0 0
$$472$$ 716.149 0.0698378
$$473$$ 5375.87 0.522585
$$474$$ 0 0
$$475$$ 3506.89 0.338752
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −809.067 −0.0774181
$$479$$ 8377.76 0.799143 0.399572 0.916702i $$-0.369159\pi$$
0.399572 + 0.916702i $$0.369159\pi$$
$$480$$ 0 0
$$481$$ −1487.19 −0.140977
$$482$$ 19443.3 1.83738
$$483$$ 0 0
$$484$$ −9693.55 −0.910364
$$485$$ −13735.8 −1.28600
$$486$$ 0 0
$$487$$ −4558.85 −0.424191 −0.212096 0.977249i $$-0.568029\pi$$
−0.212096 + 0.977249i $$0.568029\pi$$
$$488$$ 24.3822 0.00226175
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 15809.9 1.45314 0.726570 0.687092i $$-0.241113\pi$$
0.726570 + 0.687092i $$0.241113\pi$$
$$492$$ 0 0
$$493$$ 17651.1 1.61250
$$494$$ −9925.17 −0.903957
$$495$$ 0 0
$$496$$ −5054.86 −0.457601
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 13386.1 1.20089 0.600444 0.799667i $$-0.294990\pi$$
0.600444 + 0.799667i $$0.294990\pi$$
$$500$$ −12701.3 −1.13604
$$501$$ 0 0
$$502$$ 25345.1 2.25340
$$503$$ −5720.55 −0.507091 −0.253545 0.967323i $$-0.581597\pi$$
−0.253545 + 0.967323i $$0.581597\pi$$
$$504$$ 0 0
$$505$$ −17717.7 −1.56124
$$506$$ −10887.1 −0.956507
$$507$$ 0 0
$$508$$ −10329.1 −0.902123
$$509$$ 15293.2 1.33175 0.665873 0.746065i $$-0.268059\pi$$
0.665873 + 0.746065i $$0.268059\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −16456.0 −1.42043
$$513$$ 0 0
$$514$$ −15564.0 −1.33560
$$515$$ −4857.63 −0.415637
$$516$$ 0 0
$$517$$ 3686.53 0.313604
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 333.078 0.0280893
$$521$$ −4368.69 −0.367363 −0.183681 0.982986i $$-0.558801\pi$$
−0.183681 + 0.982986i $$0.558801\pi$$
$$522$$ 0 0
$$523$$ 2422.17 0.202512 0.101256 0.994860i $$-0.467714\pi$$
0.101256 + 0.994860i $$0.467714\pi$$
$$524$$ −10056.1 −0.838366
$$525$$ 0 0
$$526$$ −840.291 −0.0696548
$$527$$ 7864.58 0.650069
$$528$$ 0 0
$$529$$ 27147.2 2.23122
$$530$$ 1488.54 0.121996
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −7158.57 −0.581749
$$534$$ 0 0
$$535$$ 2798.23 0.226127
$$536$$ −616.210 −0.0496571
$$537$$ 0 0
$$538$$ 19482.4 1.56124
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 13162.5 1.04603 0.523014 0.852324i $$-0.324807\pi$$
0.523014 + 0.852324i $$0.324807\pi$$
$$542$$ 13041.1 1.03351
$$543$$ 0 0
$$544$$ 24246.4 1.91095
$$545$$ −2849.45 −0.223958
$$546$$ 0 0
$$547$$ 12112.4 0.946778 0.473389 0.880853i $$-0.343031\pi$$
0.473389 + 0.880853i $$0.343031\pi$$
$$548$$ 903.420 0.0704237
$$549$$ 0 0
$$550$$ −1459.78 −0.113173
$$551$$ −24846.0 −1.92101
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −8331.38 −0.638928
$$555$$ 0 0
$$556$$ 9263.63 0.706592
$$557$$ −8359.65 −0.635924 −0.317962 0.948103i $$-0.602998\pi$$
−0.317962 + 0.948103i $$0.602998\pi$$
$$558$$ 0 0
$$559$$ 7366.74 0.557388
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −7172.42 −0.538346
$$563$$ −13638.4 −1.02094 −0.510471 0.859895i $$-0.670529\pi$$
−0.510471 + 0.859895i $$0.670529\pi$$
$$564$$ 0 0
$$565$$ 18801.3 1.39996
$$566$$ 9491.76 0.704891
$$567$$ 0 0
$$568$$ 382.080 0.0282249
$$569$$ 15491.3 1.14135 0.570677 0.821175i $$-0.306681\pi$$
0.570677 + 0.821175i $$0.306681\pi$$
$$570$$ 0 0
$$571$$ −4648.15 −0.340664 −0.170332 0.985387i $$-0.554484\pi$$
−0.170332 + 0.985387i $$0.554484\pi$$
$$572$$ 2121.78 0.155098
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 5271.38 0.382316
$$576$$ 0 0
$$577$$ 5479.06 0.395314 0.197657 0.980271i $$-0.436667\pi$$
0.197657 + 0.980271i $$0.436667\pi$$
$$578$$ −15688.7 −1.12901
$$579$$ 0 0
$$580$$ 15782.5 1.12988
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 500.960 0.0355877
$$584$$ 879.692 0.0623321
$$585$$ 0 0
$$586$$ −14735.2 −1.03874
$$587$$ −4408.22 −0.309960 −0.154980 0.987918i $$-0.549531\pi$$
−0.154980 + 0.987918i $$0.549531\pi$$
$$588$$ 0 0
$$589$$ −11070.3 −0.774441
$$590$$ 15921.4 1.11097
$$591$$ 0 0
$$592$$ −4827.86 −0.335175
$$593$$ −2815.26 −0.194956 −0.0974779 0.995238i $$-0.531078\pi$$
−0.0974779 + 0.995238i $$0.531078\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6053.09 0.416014
$$597$$ 0 0
$$598$$ −14919.0 −1.02021
$$599$$ 19719.1 1.34507 0.672537 0.740064i $$-0.265205\pi$$
0.672537 + 0.740064i $$0.265205\pi$$
$$600$$ 0 0
$$601$$ 13982.8 0.949033 0.474517 0.880247i $$-0.342623\pi$$
0.474517 + 0.880247i $$0.342623\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 1303.83 0.0878343
$$605$$ −11385.4 −0.765097
$$606$$ 0 0
$$607$$ −13388.3 −0.895245 −0.447622 0.894223i $$-0.647729\pi$$
−0.447622 + 0.894223i $$0.647729\pi$$
$$608$$ −34129.7 −2.27655
$$609$$ 0 0
$$610$$ 542.065 0.0359797
$$611$$ 5051.77 0.334489
$$612$$ 0 0
$$613$$ −27791.9 −1.83116 −0.915582 0.402131i $$-0.868269\pi$$
−0.915582 + 0.402131i $$0.868269\pi$$
$$614$$ 24147.1 1.58713
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −19107.2 −1.24672 −0.623361 0.781935i $$-0.714233\pi$$
−0.623361 + 0.781935i $$0.714233\pi$$
$$618$$ 0 0
$$619$$ −1092.94 −0.0709675 −0.0354837 0.999370i $$-0.511297\pi$$
−0.0354837 + 0.999370i $$0.511297\pi$$
$$620$$ 7032.02 0.455504
$$621$$ 0 0
$$622$$ 4785.75 0.308507
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11595.0 −0.742078
$$626$$ −39524.9 −2.52353
$$627$$ 0 0
$$628$$ 17684.3 1.12369
$$629$$ 7511.40 0.476151
$$630$$ 0 0
$$631$$ 19235.2 1.21353 0.606767 0.794879i $$-0.292466\pi$$
0.606767 + 0.794879i $$0.292466\pi$$
$$632$$ −531.556 −0.0334560
$$633$$ 0 0
$$634$$ −34843.6 −2.18267
$$635$$ −12131.9 −0.758171
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 10342.4 0.641788
$$639$$ 0 0
$$640$$ 2294.28 0.141702
$$641$$ −19950.7 −1.22933 −0.614667 0.788787i $$-0.710710\pi$$
−0.614667 + 0.788787i $$0.710710\pi$$
$$642$$ 0 0
$$643$$ −688.125 −0.0422037 −0.0211019 0.999777i $$-0.506717\pi$$
−0.0211019 + 0.999777i $$0.506717\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 50129.4 3.05312
$$647$$ −10966.2 −0.666345 −0.333173 0.942866i $$-0.608119\pi$$
−0.333173 + 0.942866i $$0.608119\pi$$
$$648$$ 0 0
$$649$$ 5358.28 0.324085
$$650$$ −2000.39 −0.120710
$$651$$ 0 0
$$652$$ 25405.0 1.52598
$$653$$ −12925.1 −0.774575 −0.387287 0.921959i $$-0.626588\pi$$
−0.387287 + 0.921959i $$0.626588\pi$$
$$654$$ 0 0
$$655$$ −11811.3 −0.704588
$$656$$ −23238.9 −1.38312
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −11779.0 −0.696273 −0.348137 0.937444i $$-0.613185\pi$$
−0.348137 + 0.937444i $$0.613185\pi$$
$$660$$ 0 0
$$661$$ 25040.1 1.47344 0.736721 0.676196i $$-0.236373\pi$$
0.736721 + 0.676196i $$0.236373\pi$$
$$662$$ 21295.7 1.25027
$$663$$ 0 0
$$664$$ 1609.89 0.0940900
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −37347.2 −2.16805
$$668$$ 18838.7 1.09116
$$669$$ 0 0
$$670$$ −13699.6 −0.789941
$$671$$ 182.430 0.0104957
$$672$$ 0 0
$$673$$ 4104.64 0.235100 0.117550 0.993067i $$-0.462496\pi$$
0.117550 + 0.993067i $$0.462496\pi$$
$$674$$ 34456.5 1.96916
$$675$$ 0 0
$$676$$ −15648.8 −0.890350
$$677$$ −12153.4 −0.689945 −0.344973 0.938613i $$-0.612112\pi$$
−0.344973 + 0.938613i $$0.612112\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −1682.29 −0.0948717
$$681$$ 0 0
$$682$$ 4608.16 0.258733
$$683$$ −20414.0 −1.14366 −0.571830 0.820372i $$-0.693766\pi$$
−0.571830 + 0.820372i $$0.693766\pi$$
$$684$$ 0 0
$$685$$ 1061.10 0.0591862
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 23914.6 1.32520
$$689$$ 686.483 0.0379578
$$690$$ 0 0
$$691$$ −16093.5 −0.886001 −0.443000 0.896521i $$-0.646086\pi$$
−0.443000 + 0.896521i $$0.646086\pi$$
$$692$$ 4756.66 0.261302
$$693$$ 0 0
$$694$$ 23643.6 1.29322
$$695$$ 10880.5 0.593841
$$696$$ 0 0
$$697$$ 36156.1 1.96486
$$698$$ 7344.83 0.398290
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 20803.0 1.12085 0.560426 0.828204i $$-0.310637\pi$$
0.560426 + 0.828204i $$0.310637\pi$$
$$702$$ 0 0
$$703$$ −10573.2 −0.567248
$$704$$ 7682.84 0.411304
$$705$$ 0 0
$$706$$ 13495.4 0.719412
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 141.492 0.00749484 0.00374742 0.999993i $$-0.498807\pi$$
0.00374742 + 0.999993i $$0.498807\pi$$
$$710$$ 8494.40 0.448999
$$711$$ 0 0
$$712$$ 2070.85 0.109000
$$713$$ −16640.4 −0.874035
$$714$$ 0 0
$$715$$ 2492.11 0.130349
$$716$$ 15536.8 0.810947
$$717$$ 0 0
$$718$$ 3531.68 0.183567
$$719$$ 6664.46 0.345678 0.172839 0.984950i $$-0.444706\pi$$
0.172839 + 0.984950i $$0.444706\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −42747.2 −2.20345
$$723$$ 0 0
$$724$$ −19635.0 −1.00791
$$725$$ −5007.64 −0.256523
$$726$$ 0 0
$$727$$ 4837.23 0.246772 0.123386 0.992359i $$-0.460625\pi$$
0.123386 + 0.992359i $$0.460625\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 19557.3 0.991572
$$731$$ −37207.5 −1.88258
$$732$$ 0 0
$$733$$ 31602.5 1.59245 0.796225 0.605001i $$-0.206827\pi$$
0.796225 + 0.605001i $$0.206827\pi$$
$$734$$ 4763.93 0.239564
$$735$$ 0 0
$$736$$ −51302.0 −2.56932
$$737$$ −4610.53 −0.230436
$$738$$ 0 0
$$739$$ −8227.58 −0.409549 −0.204774 0.978809i $$-0.565646\pi$$
−0.204774 + 0.978809i $$0.565646\pi$$
$$740$$ 6716.22 0.333639
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 37020.3 1.82792 0.913959 0.405805i $$-0.133009\pi$$
0.913959 + 0.405805i $$0.133009\pi$$
$$744$$ 0 0
$$745$$ 7109.57 0.349630
$$746$$ −14714.3 −0.722158
$$747$$ 0 0
$$748$$ −10716.6 −0.523846
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −25149.9 −1.22202 −0.611008 0.791625i $$-0.709236\pi$$
−0.611008 + 0.791625i $$0.709236\pi$$
$$752$$ 16399.6 0.795254
$$753$$ 0 0
$$754$$ 14172.6 0.684529
$$755$$ 1531.39 0.0738186
$$756$$ 0 0
$$757$$ 20460.8 0.982377 0.491189 0.871053i $$-0.336563\pi$$
0.491189 + 0.871053i $$0.336563\pi$$
$$758$$ −29693.6 −1.42285
$$759$$ 0 0
$$760$$ 2368.02 0.113023
$$761$$ −32659.1 −1.55571 −0.777853 0.628447i $$-0.783691\pi$$
−0.777853 + 0.628447i $$0.783691\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −26413.7 −1.25080
$$765$$ 0 0
$$766$$ −29827.0 −1.40691
$$767$$ 7342.64 0.345668
$$768$$ 0 0
$$769$$ 11005.3 0.516075 0.258037 0.966135i $$-0.416924\pi$$
0.258037 + 0.966135i $$0.416924\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −31329.5 −1.46059
$$773$$ 3038.18 0.141366 0.0706829 0.997499i $$-0.477482\pi$$
0.0706829 + 0.997499i $$0.477482\pi$$
$$774$$ 0 0
$$775$$ −2231.20 −0.103415
$$776$$ 2505.59 0.115909
$$777$$ 0 0
$$778$$ −36780.9 −1.69493
$$779$$ −50894.0 −2.34078
$$780$$ 0 0
$$781$$ 2858.75 0.130979
$$782$$ 75352.0 3.44576
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 20770.8 0.944384
$$786$$ 0 0
$$787$$ 12306.8 0.557419 0.278710 0.960375i $$-0.410093\pi$$
0.278710 + 0.960375i $$0.410093\pi$$
$$788$$ 7189.00 0.324997
$$789$$ 0 0
$$790$$ −11817.5 −0.532214
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 249.990 0.0111947
$$794$$ 29915.9 1.33712
$$795$$ 0 0
$$796$$ −28698.4 −1.27788
$$797$$ −3007.06 −0.133646 −0.0668228 0.997765i $$-0.521286\pi$$
−0.0668228 + 0.997765i $$0.521286\pi$$
$$798$$ 0 0
$$799$$ −25515.2 −1.12974
$$800$$ −6878.74 −0.304000
$$801$$ 0 0
$$802$$ 11572.1 0.509508
$$803$$ 6581.92 0.289254
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 6314.72 0.275963
$$807$$ 0 0
$$808$$ 3231.94 0.140717
$$809$$ −10585.0 −0.460009 −0.230005 0.973190i $$-0.573874\pi$$
−0.230005 + 0.973190i $$0.573874\pi$$
$$810$$ 0 0
$$811$$ 18217.8 0.788796 0.394398 0.918940i $$-0.370953\pi$$
0.394398 + 0.918940i $$0.370953\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 4401.22 0.189512
$$815$$ 29839.1 1.28248
$$816$$ 0 0
$$817$$ 52374.0 2.24276
$$818$$ 45663.9 1.95184
$$819$$ 0 0
$$820$$ 32328.5 1.37678
$$821$$ −25596.5 −1.08809 −0.544047 0.839055i $$-0.683109\pi$$
−0.544047 + 0.839055i $$0.683109\pi$$
$$822$$ 0 0
$$823$$ 43778.9 1.85424 0.927118 0.374769i $$-0.122278\pi$$
0.927118 + 0.374769i $$0.122278\pi$$
$$824$$ 886.094 0.0374618
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2735.78 0.115033 0.0575166 0.998345i $$-0.481682\pi$$
0.0575166 + 0.998345i $$0.481682\pi$$
$$828$$ 0 0
$$829$$ 31144.2 1.30480 0.652402 0.757873i $$-0.273761\pi$$
0.652402 + 0.757873i $$0.273761\pi$$
$$830$$ 35791.0 1.49677
$$831$$ 0 0
$$832$$ 10528.1 0.438696
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 22126.8 0.917039
$$836$$ 15084.9 0.624068
$$837$$ 0 0
$$838$$ −37396.7 −1.54158
$$839$$ 14977.3 0.616300 0.308150 0.951338i $$-0.400290\pi$$
0.308150 + 0.951338i $$0.400290\pi$$
$$840$$ 0 0
$$841$$ 11089.6 0.454698
$$842$$ 34553.1 1.41423
$$843$$ 0 0
$$844$$ 1830.57 0.0746574
$$845$$ −18380.1 −0.748277
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2228.53 0.0902452
$$849$$ 0 0
$$850$$ 10103.4 0.407700
$$851$$ −15893.1 −0.640198
$$852$$ 0 0
$$853$$ −42861.7 −1.72047 −0.860233 0.509901i $$-0.829682\pi$$
−0.860233 + 0.509901i $$0.829682\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −510.432 −0.0203811
$$857$$ 9390.00 0.374278 0.187139 0.982333i $$-0.440079\pi$$
0.187139 + 0.982333i $$0.440079\pi$$
$$858$$ 0 0
$$859$$ −33561.2 −1.33305 −0.666526 0.745482i $$-0.732219\pi$$
−0.666526 + 0.745482i $$0.732219\pi$$
$$860$$ −33268.6 −1.31913
$$861$$ 0 0
$$862$$ −37155.9 −1.46814
$$863$$ 25691.6 1.01339 0.506693 0.862127i $$-0.330868\pi$$
0.506693 + 0.862127i $$0.330868\pi$$
$$864$$ 0 0
$$865$$ 5586.87 0.219606
$$866$$ −44512.7 −1.74665
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3977.14 −0.155254
$$870$$ 0 0
$$871$$ −6317.97 −0.245782
$$872$$ 519.775 0.0201856
$$873$$ 0 0
$$874$$ −106067. −4.10500
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −5351.52 −0.206052 −0.103026 0.994679i $$-0.532853\pi$$
−0.103026 + 0.994679i $$0.532853\pi$$
$$878$$ 16153.4 0.620900
$$879$$ 0 0
$$880$$ 8090.14 0.309907
$$881$$ 34212.7 1.30835 0.654174 0.756344i $$-0.273016\pi$$
0.654174 + 0.756344i $$0.273016\pi$$
$$882$$ 0 0
$$883$$ 17149.2 0.653587 0.326794 0.945096i $$-0.394032\pi$$
0.326794 + 0.945096i $$0.394032\pi$$
$$884$$ −14685.3 −0.558732
$$885$$ 0 0
$$886$$ −18348.4 −0.695742
$$887$$ −4020.87 −0.152207 −0.0761035 0.997100i $$-0.524248\pi$$
−0.0761035 + 0.997100i $$0.524248\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 46039.0 1.73397
$$891$$ 0 0
$$892$$ 19045.1 0.714883
$$893$$ 35915.7 1.34588
$$894$$ 0 0
$$895$$ 18248.5 0.681544
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 8420.99 0.312931
$$899$$ 15807.8 0.586452
$$900$$ 0 0
$$901$$ −3467.24 −0.128203
$$902$$ 21185.2 0.782030
$$903$$ 0 0
$$904$$ −3429.59 −0.126180
$$905$$ −23062.0 −0.847080
$$906$$ 0 0
$$907$$ −22967.2 −0.840810 −0.420405 0.907337i $$-0.638112\pi$$
−0.420405 + 0.907337i $$0.638112\pi$$
$$908$$ 28639.4 1.04673
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 9860.77 0.358619 0.179309 0.983793i $$-0.442614\pi$$
0.179309 + 0.983793i $$0.442614\pi$$
$$912$$ 0 0
$$913$$ 12045.3 0.436628
$$914$$ 7492.71 0.271156
$$915$$ 0 0
$$916$$ −21857.5 −0.788421
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −5271.55 −0.189219 −0.0946096 0.995514i $$-0.530160\pi$$
−0.0946096 + 0.995514i $$0.530160\pi$$
$$920$$ 3559.49 0.127558
$$921$$ 0 0
$$922$$ 3556.36 0.127031
$$923$$ 3917.44 0.139701
$$924$$ 0 0
$$925$$ −2131.00 −0.0757478
$$926$$ −65880.6 −2.33798
$$927$$ 0 0
$$928$$ 48735.3 1.72394
$$929$$ 14902.6 0.526307 0.263153 0.964754i $$-0.415238\pi$$
0.263153 + 0.964754i $$0.415238\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −8061.24 −0.283321
$$933$$ 0 0
$$934$$ −76897.4 −2.69396
$$935$$ −12587.0 −0.440255
$$936$$ 0 0
$$937$$ 21934.8 0.764757 0.382378 0.924006i $$-0.375105\pi$$
0.382378 + 0.924006i $$0.375105\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −22814.1 −0.791610
$$941$$ −14521.5 −0.503069 −0.251535 0.967848i $$-0.580935\pi$$
−0.251535 + 0.967848i $$0.580935\pi$$
$$942$$ 0 0
$$943$$ −76501.3 −2.64181
$$944$$ 23836.4 0.821831
$$945$$ 0 0
$$946$$ −21801.3 −0.749282
$$947$$ 1943.73 0.0666979 0.0333489 0.999444i $$-0.489383\pi$$
0.0333489 + 0.999444i $$0.489383\pi$$
$$948$$ 0 0
$$949$$ 9019.43 0.308517
$$950$$ −14221.8 −0.485702
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −16904.1 −0.574583 −0.287292 0.957843i $$-0.592755\pi$$
−0.287292 + 0.957843i $$0.592755\pi$$
$$954$$ 0 0
$$955$$ −31023.8 −1.05121
$$956$$ 1685.05 0.0570068
$$957$$ 0 0
$$958$$ −33975.1 −1.14581
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −22747.7 −0.763576
$$962$$ 6031.13 0.202133
$$963$$ 0 0
$$964$$ −40494.8 −1.35296
$$965$$ −36797.6 −1.22752
$$966$$ 0 0
$$967$$ 26699.3 0.887891 0.443946 0.896054i $$-0.353578\pi$$
0.443946 + 0.896054i $$0.353578\pi$$
$$968$$ 2076.85 0.0689591
$$969$$ 0 0
$$970$$ 55704.2 1.84387
$$971$$ −11089.5 −0.366507 −0.183253 0.983066i $$-0.558663\pi$$
−0.183253 + 0.983066i $$0.558663\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 18487.9 0.608205
$$975$$ 0 0
$$976$$ 811.541 0.0266156
$$977$$ 51891.9 1.69925 0.849625 0.527387i $$-0.176828\pi$$
0.849625 + 0.527387i $$0.176828\pi$$
$$978$$ 0 0
$$979$$ 15494.2 0.505820
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −64115.5 −2.08351
$$983$$ −24560.7 −0.796911 −0.398456 0.917188i $$-0.630454\pi$$
−0.398456 + 0.917188i $$0.630454\pi$$
$$984$$ 0 0
$$985$$ 8443.74 0.273137
$$986$$ −71582.0 −2.31200
$$987$$ 0 0
$$988$$ 20671.3 0.665629
$$989$$ 78725.9 2.53118
$$990$$ 0 0
$$991$$ −18507.5 −0.593250 −0.296625 0.954994i $$-0.595861\pi$$
−0.296625 + 0.954994i $$0.595861\pi$$
$$992$$ 21714.4 0.694993
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −33707.4 −1.07397
$$996$$ 0 0
$$997$$ 42966.2 1.36485 0.682424 0.730957i $$-0.260926\pi$$
0.682424 + 0.730957i $$0.260926\pi$$
$$998$$ −54285.8 −1.72183
$$999$$ 0 0
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 441.4.a.v.1.1 4
3.2 odd 2 inner 441.4.a.v.1.4 4
7.2 even 3 441.4.e.x.361.4 8
7.3 odd 6 63.4.e.d.37.4 yes 8
7.4 even 3 441.4.e.x.226.4 8
7.5 odd 6 63.4.e.d.46.4 yes 8
7.6 odd 2 441.4.a.w.1.1 4
21.2 odd 6 441.4.e.x.361.1 8
21.5 even 6 63.4.e.d.46.1 yes 8
21.11 odd 6 441.4.e.x.226.1 8
21.17 even 6 63.4.e.d.37.1 8
21.20 even 2 441.4.a.w.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.e.d.37.1 8 21.17 even 6
63.4.e.d.37.4 yes 8 7.3 odd 6
63.4.e.d.46.1 yes 8 21.5 even 6
63.4.e.d.46.4 yes 8 7.5 odd 6
441.4.a.v.1.1 4 1.1 even 1 trivial
441.4.a.v.1.4 4 3.2 odd 2 inner
441.4.a.w.1.1 4 7.6 odd 2
441.4.a.w.1.4 4 21.20 even 2
441.4.e.x.226.1 8 21.11 odd 6
441.4.e.x.226.4 8 7.4 even 3
441.4.e.x.361.1 8 21.2 odd 6
441.4.e.x.361.4 8 7.2 even 3