# Properties

 Label 441.4.a.u.1.4 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: $$\Q(\sqrt{2}, \sqrt{65})$$ Defining polynomial: $$x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-2.11692$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.53113 q^{2} +4.46887 q^{4} +2.07730 q^{5} -12.4689 q^{8} +O(q^{10})$$ $$q+3.53113 q^{2} +4.46887 q^{4} +2.07730 q^{5} -12.4689 q^{8} +7.33521 q^{10} -49.1868 q^{11} -44.8559 q^{13} -79.7802 q^{16} -26.5179 q^{17} +77.7350 q^{19} +9.28317 q^{20} -173.685 q^{22} -55.7510 q^{23} -120.685 q^{25} -158.392 q^{26} -121.436 q^{29} +305.553 q^{31} -181.963 q^{32} -93.6380 q^{34} +77.1868 q^{37} +274.492 q^{38} -25.9016 q^{40} -248.720 q^{41} -147.179 q^{43} -219.809 q^{44} -196.864 q^{46} -269.851 q^{47} -426.154 q^{50} -200.455 q^{52} +141.121 q^{53} -102.176 q^{55} -428.805 q^{58} +424.834 q^{59} -587.996 q^{61} +1078.95 q^{62} -4.29373 q^{64} -93.1790 q^{65} -179.634 q^{67} -118.505 q^{68} -674.872 q^{71} +237.489 q^{73} +272.556 q^{74} +347.388 q^{76} +495.852 q^{79} -165.727 q^{80} -878.262 q^{82} +24.4406 q^{83} -55.0855 q^{85} -519.708 q^{86} +613.304 q^{88} -1072.29 q^{89} -249.144 q^{92} -952.877 q^{94} +161.479 q^{95} +1667.43 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 34q^{4} - 66q^{8} + O(q^{10})$$ $$4q - 2q^{2} + 34q^{4} - 66q^{8} - 100q^{11} - 174q^{16} - 340q^{22} - 352q^{23} - 128q^{25} - 260q^{29} + 30q^{32} + 212q^{37} + 540q^{43} - 460q^{44} + 696q^{46} - 1366q^{50} - 16q^{53} - 780q^{58} - 1678q^{64} + 756q^{65} - 1944q^{67} - 2248q^{71} + 284q^{74} - 1048q^{79} - 3284q^{85} - 4820q^{86} + 1260q^{88} - 3512q^{92} - 2192q^{95} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.53113 1.24844 0.624221 0.781248i $$-0.285416\pi$$
0.624221 + 0.781248i $$0.285416\pi$$
$$3$$ 0 0
$$4$$ 4.46887 0.558609
$$5$$ 2.07730 0.185799 0.0928996 0.995675i $$-0.470386\pi$$
0.0928996 + 0.995675i $$0.470386\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −12.4689 −0.551051
$$9$$ 0 0
$$10$$ 7.33521 0.231960
$$11$$ −49.1868 −1.34822 −0.674108 0.738633i $$-0.735472\pi$$
−0.674108 + 0.738633i $$0.735472\pi$$
$$12$$ 0 0
$$13$$ −44.8559 −0.956983 −0.478492 0.878092i $$-0.658816\pi$$
−0.478492 + 0.878092i $$0.658816\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −79.7802 −1.24656
$$17$$ −26.5179 −0.378325 −0.189163 0.981946i $$-0.560577\pi$$
−0.189163 + 0.981946i $$0.560577\pi$$
$$18$$ 0 0
$$19$$ 77.7350 0.938612 0.469306 0.883036i $$-0.344504\pi$$
0.469306 + 0.883036i $$0.344504\pi$$
$$20$$ 9.28317 0.103789
$$21$$ 0 0
$$22$$ −173.685 −1.68317
$$23$$ −55.7510 −0.505430 −0.252715 0.967541i $$-0.581323\pi$$
−0.252715 + 0.967541i $$0.581323\pi$$
$$24$$ 0 0
$$25$$ −120.685 −0.965479
$$26$$ −158.392 −1.19474
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −121.436 −0.777588 −0.388794 0.921325i $$-0.627108\pi$$
−0.388794 + 0.921325i $$0.627108\pi$$
$$30$$ 0 0
$$31$$ 305.553 1.77029 0.885143 0.465319i $$-0.154060\pi$$
0.885143 + 0.465319i $$0.154060\pi$$
$$32$$ −181.963 −1.00521
$$33$$ 0 0
$$34$$ −93.6380 −0.472317
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 77.1868 0.342957 0.171479 0.985188i $$-0.445146\pi$$
0.171479 + 0.985188i $$0.445146\pi$$
$$38$$ 274.492 1.17180
$$39$$ 0 0
$$40$$ −25.9016 −0.102385
$$41$$ −248.720 −0.947403 −0.473702 0.880685i $$-0.657083\pi$$
−0.473702 + 0.880685i $$0.657083\pi$$
$$42$$ 0 0
$$43$$ −147.179 −0.521967 −0.260984 0.965343i $$-0.584047\pi$$
−0.260984 + 0.965343i $$0.584047\pi$$
$$44$$ −219.809 −0.753125
$$45$$ 0 0
$$46$$ −196.864 −0.631000
$$47$$ −269.851 −0.837484 −0.418742 0.908105i $$-0.637529\pi$$
−0.418742 + 0.908105i $$0.637529\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −426.154 −1.20534
$$51$$ 0 0
$$52$$ −200.455 −0.534579
$$53$$ 141.121 0.365744 0.182872 0.983137i $$-0.441461\pi$$
0.182872 + 0.983137i $$0.441461\pi$$
$$54$$ 0 0
$$55$$ −102.176 −0.250497
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −428.805 −0.970774
$$59$$ 424.834 0.937434 0.468717 0.883348i $$-0.344716\pi$$
0.468717 + 0.883348i $$0.344716\pi$$
$$60$$ 0 0
$$61$$ −587.996 −1.23418 −0.617092 0.786891i $$-0.711689\pi$$
−0.617092 + 0.786891i $$0.711689\pi$$
$$62$$ 1078.95 2.21010
$$63$$ 0 0
$$64$$ −4.29373 −0.00838618
$$65$$ −93.1790 −0.177807
$$66$$ 0 0
$$67$$ −179.634 −0.327549 −0.163775 0.986498i $$-0.552367\pi$$
−0.163775 + 0.986498i $$0.552367\pi$$
$$68$$ −118.505 −0.211336
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −674.872 −1.12806 −0.564032 0.825753i $$-0.690750\pi$$
−0.564032 + 0.825753i $$0.690750\pi$$
$$72$$ 0 0
$$73$$ 237.489 0.380767 0.190383 0.981710i $$-0.439027\pi$$
0.190383 + 0.981710i $$0.439027\pi$$
$$74$$ 272.556 0.428163
$$75$$ 0 0
$$76$$ 347.388 0.524317
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 495.852 0.706174 0.353087 0.935591i $$-0.385132\pi$$
0.353087 + 0.935591i $$0.385132\pi$$
$$80$$ −165.727 −0.231611
$$81$$ 0 0
$$82$$ −878.262 −1.18278
$$83$$ 24.4406 0.0323217 0.0161609 0.999869i $$-0.494856\pi$$
0.0161609 + 0.999869i $$0.494856\pi$$
$$84$$ 0 0
$$85$$ −55.0855 −0.0702925
$$86$$ −519.708 −0.651646
$$87$$ 0 0
$$88$$ 613.304 0.742936
$$89$$ −1072.29 −1.27710 −0.638552 0.769579i $$-0.720466\pi$$
−0.638552 + 0.769579i $$0.720466\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −249.144 −0.282337
$$93$$ 0 0
$$94$$ −952.877 −1.04555
$$95$$ 161.479 0.174393
$$96$$ 0 0
$$97$$ 1667.43 1.74538 0.872690 0.488275i $$-0.162374\pi$$
0.872690 + 0.488275i $$0.162374\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −539.325 −0.539325
$$101$$ 77.1187 0.0759762 0.0379881 0.999278i $$-0.487905\pi$$
0.0379881 + 0.999278i $$0.487905\pi$$
$$102$$ 0 0
$$103$$ 164.693 0.157550 0.0787749 0.996892i $$-0.474899\pi$$
0.0787749 + 0.996892i $$0.474899\pi$$
$$104$$ 559.302 0.527347
$$105$$ 0 0
$$106$$ 498.315 0.456610
$$107$$ −1022.62 −0.923931 −0.461966 0.886898i $$-0.652856\pi$$
−0.461966 + 0.886898i $$0.652856\pi$$
$$108$$ 0 0
$$109$$ 1362.52 1.19730 0.598649 0.801011i $$-0.295704\pi$$
0.598649 + 0.801011i $$0.295704\pi$$
$$110$$ −360.795 −0.312731
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1538.41 1.28072 0.640360 0.768075i $$-0.278785\pi$$
0.640360 + 0.768075i $$0.278785\pi$$
$$114$$ 0 0
$$115$$ −115.811 −0.0939084
$$116$$ −542.681 −0.434368
$$117$$ 0 0
$$118$$ 1500.14 1.17033
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1088.34 0.817685
$$122$$ −2076.29 −1.54081
$$123$$ 0 0
$$124$$ 1365.48 0.988898
$$125$$ −510.360 −0.365184
$$126$$ 0 0
$$127$$ −170.358 −0.119030 −0.0595151 0.998227i $$-0.518955\pi$$
−0.0595151 + 0.998227i $$0.518955\pi$$
$$128$$ 1440.54 0.994744
$$129$$ 0 0
$$130$$ −329.027 −0.221981
$$131$$ 751.935 0.501503 0.250751 0.968051i $$-0.419322\pi$$
0.250751 + 0.968051i $$0.419322\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −634.312 −0.408927
$$135$$ 0 0
$$136$$ 330.648 0.208477
$$137$$ −518.623 −0.323423 −0.161711 0.986838i $$-0.551701\pi$$
−0.161711 + 0.986838i $$0.551701\pi$$
$$138$$ 0 0
$$139$$ −2975.72 −1.81581 −0.907905 0.419177i $$-0.862319\pi$$
−0.907905 + 0.419177i $$0.862319\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2383.06 −1.40832
$$143$$ 2206.32 1.29022
$$144$$ 0 0
$$145$$ −252.258 −0.144475
$$146$$ 838.604 0.475365
$$147$$ 0 0
$$148$$ 344.938 0.191579
$$149$$ 2717.94 1.49438 0.747188 0.664612i $$-0.231403\pi$$
0.747188 + 0.664612i $$0.231403\pi$$
$$150$$ 0 0
$$151$$ 707.650 0.381376 0.190688 0.981651i $$-0.438928\pi$$
0.190688 + 0.981651i $$0.438928\pi$$
$$152$$ −969.267 −0.517224
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 634.724 0.328918
$$156$$ 0 0
$$157$$ 3117.91 1.58495 0.792473 0.609906i $$-0.208793\pi$$
0.792473 + 0.609906i $$0.208793\pi$$
$$158$$ 1750.92 0.881618
$$159$$ 0 0
$$160$$ −377.991 −0.186768
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1808.77 0.869167 0.434583 0.900632i $$-0.356896\pi$$
0.434583 + 0.900632i $$0.356896\pi$$
$$164$$ −1111.50 −0.529228
$$165$$ 0 0
$$166$$ 86.3028 0.0403518
$$167$$ −3147.38 −1.45839 −0.729197 0.684303i $$-0.760106\pi$$
−0.729197 + 0.684303i $$0.760106\pi$$
$$168$$ 0 0
$$169$$ −184.949 −0.0841827
$$170$$ −194.514 −0.0877562
$$171$$ 0 0
$$172$$ −657.724 −0.291576
$$173$$ 3284.36 1.44338 0.721691 0.692215i $$-0.243365\pi$$
0.721691 + 0.692215i $$0.243365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3924.13 1.68064
$$177$$ 0 0
$$178$$ −3786.39 −1.59439
$$179$$ −2798.83 −1.16868 −0.584341 0.811508i $$-0.698647\pi$$
−0.584341 + 0.811508i $$0.698647\pi$$
$$180$$ 0 0
$$181$$ −3723.04 −1.52890 −0.764451 0.644682i $$-0.776990\pi$$
−0.764451 + 0.644682i $$0.776990\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 695.152 0.278518
$$185$$ 160.340 0.0637212
$$186$$ 0 0
$$187$$ 1304.33 0.510064
$$188$$ −1205.93 −0.467826
$$189$$ 0 0
$$190$$ 570.202 0.217720
$$191$$ −959.650 −0.363549 −0.181774 0.983340i $$-0.558184\pi$$
−0.181774 + 0.983340i $$0.558184\pi$$
$$192$$ 0 0
$$193$$ −3790.25 −1.41362 −0.706808 0.707406i $$-0.749865\pi$$
−0.706808 + 0.707406i $$0.749865\pi$$
$$194$$ 5887.91 2.17901
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −5117.99 −1.85097 −0.925487 0.378779i $$-0.876344\pi$$
−0.925487 + 0.378779i $$0.876344\pi$$
$$198$$ 0 0
$$199$$ 864.855 0.308080 0.154040 0.988065i $$-0.450772\pi$$
0.154040 + 0.988065i $$0.450772\pi$$
$$200$$ 1504.80 0.532028
$$201$$ 0 0
$$202$$ 272.316 0.0948519
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −516.665 −0.176027
$$206$$ 581.550 0.196692
$$207$$ 0 0
$$208$$ 3578.61 1.19294
$$209$$ −3823.53 −1.26545
$$210$$ 0 0
$$211$$ −1344.61 −0.438707 −0.219353 0.975645i $$-0.570395\pi$$
−0.219353 + 0.975645i $$0.570395\pi$$
$$212$$ 630.650 0.204308
$$213$$ 0 0
$$214$$ −3611.01 −1.15348
$$215$$ −305.735 −0.0969811
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4811.23 1.49476
$$219$$ 0 0
$$220$$ −456.609 −0.139930
$$221$$ 1189.48 0.362051
$$222$$ 0 0
$$223$$ −864.916 −0.259727 −0.129863 0.991532i $$-0.541454\pi$$
−0.129863 + 0.991532i $$0.541454\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 5432.32 1.59890
$$227$$ −1715.34 −0.501548 −0.250774 0.968046i $$-0.580685\pi$$
−0.250774 + 0.968046i $$0.580685\pi$$
$$228$$ 0 0
$$229$$ 1045.46 0.301685 0.150842 0.988558i $$-0.451801\pi$$
0.150842 + 0.988558i $$0.451801\pi$$
$$230$$ −408.945 −0.117239
$$231$$ 0 0
$$232$$ 1514.17 0.428491
$$233$$ −1448.67 −0.407320 −0.203660 0.979042i $$-0.565284\pi$$
−0.203660 + 0.979042i $$0.565284\pi$$
$$234$$ 0 0
$$235$$ −560.560 −0.155604
$$236$$ 1898.53 0.523659
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3153.12 0.853383 0.426691 0.904397i $$-0.359679\pi$$
0.426691 + 0.904397i $$0.359679\pi$$
$$240$$ 0 0
$$241$$ 381.012 0.101839 0.0509194 0.998703i $$-0.483785\pi$$
0.0509194 + 0.998703i $$0.483785\pi$$
$$242$$ 3843.06 1.02083
$$243$$ 0 0
$$244$$ −2627.68 −0.689426
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3486.87 −0.898236
$$248$$ −3809.90 −0.975519
$$249$$ 0 0
$$250$$ −1802.15 −0.455912
$$251$$ 3776.23 0.949617 0.474808 0.880089i $$-0.342517\pi$$
0.474808 + 0.880089i $$0.342517\pi$$
$$252$$ 0 0
$$253$$ 2742.21 0.681428
$$254$$ −601.556 −0.148602
$$255$$ 0 0
$$256$$ 5121.09 1.25027
$$257$$ 4258.42 1.03359 0.516795 0.856109i $$-0.327125\pi$$
0.516795 + 0.856109i $$0.327125\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −416.405 −0.0993244
$$261$$ 0 0
$$262$$ 2655.18 0.626098
$$263$$ −4198.83 −0.984451 −0.492226 0.870468i $$-0.663816\pi$$
−0.492226 + 0.870468i $$0.663816\pi$$
$$264$$ 0 0
$$265$$ 293.150 0.0679548
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −802.762 −0.182972
$$269$$ 3740.59 0.847835 0.423917 0.905701i $$-0.360655\pi$$
0.423917 + 0.905701i $$0.360655\pi$$
$$270$$ 0 0
$$271$$ −4356.30 −0.976480 −0.488240 0.872709i $$-0.662361\pi$$
−0.488240 + 0.872709i $$0.662361\pi$$
$$272$$ 2115.60 0.471607
$$273$$ 0 0
$$274$$ −1831.32 −0.403775
$$275$$ 5936.10 1.30167
$$276$$ 0 0
$$277$$ −1344.30 −0.291593 −0.145797 0.989315i $$-0.546575\pi$$
−0.145797 + 0.989315i $$0.546575\pi$$
$$278$$ −10507.7 −2.26693
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4205.54 −0.892817 −0.446408 0.894829i $$-0.647297\pi$$
−0.446408 + 0.894829i $$0.647297\pi$$
$$282$$ 0 0
$$283$$ −4752.03 −0.998159 −0.499079 0.866556i $$-0.666328\pi$$
−0.499079 + 0.866556i $$0.666328\pi$$
$$284$$ −3015.91 −0.630146
$$285$$ 0 0
$$286$$ 7790.79 1.61077
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4209.80 −0.856870
$$290$$ −890.757 −0.180369
$$291$$ 0 0
$$292$$ 1061.31 0.212700
$$293$$ −4961.17 −0.989196 −0.494598 0.869122i $$-0.664685\pi$$
−0.494598 + 0.869122i $$0.664685\pi$$
$$294$$ 0 0
$$295$$ 882.506 0.174174
$$296$$ −962.432 −0.188987
$$297$$ 0 0
$$298$$ 9597.39 1.86564
$$299$$ 2500.76 0.483688
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 2498.80 0.476126
$$303$$ 0 0
$$304$$ −6201.71 −1.17004
$$305$$ −1221.44 −0.229310
$$306$$ 0 0
$$307$$ 4234.00 0.787124 0.393562 0.919298i $$-0.371243\pi$$
0.393562 + 0.919298i $$0.371243\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 2241.29 0.410635
$$311$$ −684.700 −0.124842 −0.0624209 0.998050i $$-0.519882\pi$$
−0.0624209 + 0.998050i $$0.519882\pi$$
$$312$$ 0 0
$$313$$ 5944.07 1.07341 0.536707 0.843768i $$-0.319668\pi$$
0.536707 + 0.843768i $$0.319668\pi$$
$$314$$ 11009.8 1.97872
$$315$$ 0 0
$$316$$ 2215.90 0.394475
$$317$$ 2823.89 0.500333 0.250166 0.968203i $$-0.419515\pi$$
0.250166 + 0.968203i $$0.419515\pi$$
$$318$$ 0 0
$$319$$ 5973.04 1.04836
$$320$$ −8.91935 −0.00155815
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2061.37 −0.355101
$$324$$ 0 0
$$325$$ 5413.43 0.923947
$$326$$ 6387.02 1.08510
$$327$$ 0 0
$$328$$ 3101.26 0.522068
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2812.97 −0.467114 −0.233557 0.972343i $$-0.575037\pi$$
−0.233557 + 0.972343i $$0.575037\pi$$
$$332$$ 109.222 0.0180552
$$333$$ 0 0
$$334$$ −11113.8 −1.82072
$$335$$ −373.154 −0.0608584
$$336$$ 0 0
$$337$$ 4260.10 0.688612 0.344306 0.938857i $$-0.388114\pi$$
0.344306 + 0.938857i $$0.388114\pi$$
$$338$$ −653.080 −0.105097
$$339$$ 0 0
$$340$$ −246.170 −0.0392660
$$341$$ −15029.2 −2.38673
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 1835.16 0.287631
$$345$$ 0 0
$$346$$ 11597.5 1.80198
$$347$$ −36.0584 −0.00557843 −0.00278922 0.999996i $$-0.500888\pi$$
−0.00278922 + 0.999996i $$0.500888\pi$$
$$348$$ 0 0
$$349$$ 242.692 0.0372236 0.0186118 0.999827i $$-0.494075\pi$$
0.0186118 + 0.999827i $$0.494075\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 8950.18 1.35524
$$353$$ −109.990 −0.0165840 −0.00829201 0.999966i $$-0.502639\pi$$
−0.00829201 + 0.999966i $$0.502639\pi$$
$$354$$ 0 0
$$355$$ −1401.91 −0.209593
$$356$$ −4791.92 −0.713402
$$357$$ 0 0
$$358$$ −9883.01 −1.45903
$$359$$ −12404.5 −1.82363 −0.911814 0.410604i $$-0.865318\pi$$
−0.911814 + 0.410604i $$0.865318\pi$$
$$360$$ 0 0
$$361$$ −816.273 −0.119008
$$362$$ −13146.5 −1.90875
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 493.335 0.0707461
$$366$$ 0 0
$$367$$ −13859.6 −1.97130 −0.985649 0.168807i $$-0.946009\pi$$
−0.985649 + 0.168807i $$0.946009\pi$$
$$368$$ 4447.82 0.630051
$$369$$ 0 0
$$370$$ 566.181 0.0795523
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 4898.06 0.679925 0.339963 0.940439i $$-0.389586\pi$$
0.339963 + 0.940439i $$0.389586\pi$$
$$374$$ 4605.75 0.636786
$$375$$ 0 0
$$376$$ 3364.73 0.461497
$$377$$ 5447.11 0.744139
$$378$$ 0 0
$$379$$ −9806.25 −1.32906 −0.664530 0.747262i $$-0.731368\pi$$
−0.664530 + 0.747262i $$0.731368\pi$$
$$380$$ 721.627 0.0974176
$$381$$ 0 0
$$382$$ −3388.65 −0.453870
$$383$$ 10729.7 1.43149 0.715746 0.698361i $$-0.246087\pi$$
0.715746 + 0.698361i $$0.246087\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −13383.8 −1.76482
$$387$$ 0 0
$$388$$ 7451.53 0.974984
$$389$$ −5264.05 −0.686113 −0.343057 0.939315i $$-0.611462\pi$$
−0.343057 + 0.939315i $$0.611462\pi$$
$$390$$ 0 0
$$391$$ 1478.40 0.191217
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −18072.3 −2.31083
$$395$$ 1030.03 0.131206
$$396$$ 0 0
$$397$$ −1214.90 −0.153587 −0.0767935 0.997047i $$-0.524468\pi$$
−0.0767935 + 0.997047i $$0.524468\pi$$
$$398$$ 3053.91 0.384620
$$399$$ 0 0
$$400$$ 9628.26 1.20353
$$401$$ −2295.45 −0.285859 −0.142929 0.989733i $$-0.545652\pi$$
−0.142929 + 0.989733i $$0.545652\pi$$
$$402$$ 0 0
$$403$$ −13705.8 −1.69413
$$404$$ 344.633 0.0424410
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −3796.57 −0.462381
$$408$$ 0 0
$$409$$ −4646.54 −0.561753 −0.280876 0.959744i $$-0.590625\pi$$
−0.280876 + 0.959744i $$0.590625\pi$$
$$410$$ −1824.41 −0.219759
$$411$$ 0 0
$$412$$ 735.990 0.0880087
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 50.7703 0.00600535
$$416$$ 8162.11 0.961973
$$417$$ 0 0
$$418$$ −13501.4 −1.57984
$$419$$ 7541.24 0.879269 0.439634 0.898177i $$-0.355108\pi$$
0.439634 + 0.898177i $$0.355108\pi$$
$$420$$ 0 0
$$421$$ −6243.63 −0.722794 −0.361397 0.932412i $$-0.617700\pi$$
−0.361397 + 0.932412i $$0.617700\pi$$
$$422$$ −4748.01 −0.547700
$$423$$ 0 0
$$424$$ −1759.62 −0.201544
$$425$$ 3200.31 0.365265
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −4569.97 −0.516116
$$429$$ 0 0
$$430$$ −1079.59 −0.121075
$$431$$ −11465.8 −1.28141 −0.640706 0.767786i $$-0.721358\pi$$
−0.640706 + 0.767786i $$0.721358\pi$$
$$432$$ 0 0
$$433$$ −5156.40 −0.572289 −0.286144 0.958187i $$-0.592374\pi$$
−0.286144 + 0.958187i $$0.592374\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6088.92 0.668822
$$437$$ −4333.80 −0.474402
$$438$$ 0 0
$$439$$ 5064.25 0.550577 0.275289 0.961362i $$-0.411227\pi$$
0.275289 + 0.961362i $$0.411227\pi$$
$$440$$ 1274.01 0.138037
$$441$$ 0 0
$$442$$ 4200.22 0.452000
$$443$$ −12703.6 −1.36246 −0.681228 0.732071i $$-0.738554\pi$$
−0.681228 + 0.732071i $$0.738554\pi$$
$$444$$ 0 0
$$445$$ −2227.46 −0.237285
$$446$$ −3054.13 −0.324254
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −13942.2 −1.46542 −0.732709 0.680542i $$-0.761744\pi$$
−0.732709 + 0.680542i $$0.761744\pi$$
$$450$$ 0 0
$$451$$ 12233.7 1.27730
$$452$$ 6874.95 0.715421
$$453$$ 0 0
$$454$$ −6057.10 −0.626154
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −15214.0 −1.55729 −0.778646 0.627464i $$-0.784093\pi$$
−0.778646 + 0.627464i $$0.784093\pi$$
$$458$$ 3691.65 0.376636
$$459$$ 0 0
$$460$$ −517.546 −0.0524581
$$461$$ −11430.2 −1.15479 −0.577394 0.816465i $$-0.695930\pi$$
−0.577394 + 0.816465i $$0.695930\pi$$
$$462$$ 0 0
$$463$$ −9347.88 −0.938300 −0.469150 0.883119i $$-0.655440\pi$$
−0.469150 + 0.883119i $$0.655440\pi$$
$$464$$ 9688.17 0.969314
$$465$$ 0 0
$$466$$ −5115.44 −0.508515
$$467$$ 3630.84 0.359776 0.179888 0.983687i $$-0.442427\pi$$
0.179888 + 0.983687i $$0.442427\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −1979.41 −0.194262
$$471$$ 0 0
$$472$$ −5297.20 −0.516575
$$473$$ 7239.26 0.703724
$$474$$ 0 0
$$475$$ −9381.43 −0.906210
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 11134.1 1.06540
$$479$$ 6521.25 0.622053 0.311027 0.950401i $$-0.399327\pi$$
0.311027 + 0.950401i $$0.399327\pi$$
$$480$$ 0 0
$$481$$ −3462.28 −0.328205
$$482$$ 1345.40 0.127140
$$483$$ 0 0
$$484$$ 4863.65 0.456766
$$485$$ 3463.75 0.324290
$$486$$ 0 0
$$487$$ −3666.29 −0.341140 −0.170570 0.985346i $$-0.554561\pi$$
−0.170570 + 0.985346i $$0.554561\pi$$
$$488$$ 7331.65 0.680099
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12470.7 1.14623 0.573113 0.819476i $$-0.305736\pi$$
0.573113 + 0.819476i $$0.305736\pi$$
$$492$$ 0 0
$$493$$ 3220.22 0.294181
$$494$$ −12312.6 −1.12140
$$495$$ 0 0
$$496$$ −24377.0 −2.20678
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −2303.93 −0.206690 −0.103345 0.994646i $$-0.532955\pi$$
−0.103345 + 0.994646i $$0.532955\pi$$
$$500$$ −2280.74 −0.203995
$$501$$ 0 0
$$502$$ 13334.4 1.18554
$$503$$ −10520.4 −0.932570 −0.466285 0.884635i $$-0.654408\pi$$
−0.466285 + 0.884635i $$0.654408\pi$$
$$504$$ 0 0
$$505$$ 160.198 0.0141163
$$506$$ 9683.10 0.850724
$$507$$ 0 0
$$508$$ −761.308 −0.0664913
$$509$$ −9662.22 −0.841395 −0.420698 0.907201i $$-0.638215\pi$$
−0.420698 + 0.907201i $$0.638215\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 6558.89 0.566142
$$513$$ 0 0
$$514$$ 15037.0 1.29038
$$515$$ 342.115 0.0292726
$$516$$ 0 0
$$517$$ 13273.1 1.12911
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 1161.84 0.0979806
$$521$$ 8607.81 0.723829 0.361914 0.932211i $$-0.382123\pi$$
0.361914 + 0.932211i $$0.382123\pi$$
$$522$$ 0 0
$$523$$ 10482.7 0.876439 0.438219 0.898868i $$-0.355609\pi$$
0.438219 + 0.898868i $$0.355609\pi$$
$$524$$ 3360.30 0.280144
$$525$$ 0 0
$$526$$ −14826.6 −1.22903
$$527$$ −8102.61 −0.669744
$$528$$ 0 0
$$529$$ −9058.83 −0.744541
$$530$$ 1035.15 0.0848377
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 11156.6 0.906649
$$534$$ 0 0
$$535$$ −2124.29 −0.171666
$$536$$ 2239.84 0.180497
$$537$$ 0 0
$$538$$ 13208.5 1.05847
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 20722.6 1.64683 0.823416 0.567438i $$-0.192066\pi$$
0.823416 + 0.567438i $$0.192066\pi$$
$$542$$ −15382.6 −1.21908
$$543$$ 0 0
$$544$$ 4825.27 0.380298
$$545$$ 2830.35 0.222457
$$546$$ 0 0
$$547$$ −4175.09 −0.326351 −0.163176 0.986597i $$-0.552174\pi$$
−0.163176 + 0.986597i $$0.552174\pi$$
$$548$$ −2317.66 −0.180667
$$549$$ 0 0
$$550$$ 20961.1 1.62506
$$551$$ −9439.81 −0.729854
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −4746.91 −0.364038
$$555$$ 0 0
$$556$$ −13298.1 −1.01433
$$557$$ 10161.7 0.773011 0.386505 0.922287i $$-0.373682\pi$$
0.386505 + 0.922287i $$0.373682\pi$$
$$558$$ 0 0
$$559$$ 6601.85 0.499514
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −14850.3 −1.11463
$$563$$ −17104.4 −1.28040 −0.640201 0.768208i $$-0.721149\pi$$
−0.640201 + 0.768208i $$0.721149\pi$$
$$564$$ 0 0
$$565$$ 3195.73 0.237957
$$566$$ −16780.0 −1.24614
$$567$$ 0 0
$$568$$ 8414.89 0.621621
$$569$$ 18257.6 1.34516 0.672581 0.740023i $$-0.265185\pi$$
0.672581 + 0.740023i $$0.265185\pi$$
$$570$$ 0 0
$$571$$ 13630.5 0.998982 0.499491 0.866319i $$-0.333520\pi$$
0.499491 + 0.866319i $$0.333520\pi$$
$$572$$ 9859.74 0.720728
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6728.30 0.487981
$$576$$ 0 0
$$577$$ −4442.08 −0.320496 −0.160248 0.987077i $$-0.551229\pi$$
−0.160248 + 0.987077i $$0.551229\pi$$
$$578$$ −14865.4 −1.06975
$$579$$ 0 0
$$580$$ −1127.31 −0.0807052
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −6941.27 −0.493101
$$584$$ −2961.22 −0.209822
$$585$$ 0 0
$$586$$ −17518.5 −1.23495
$$587$$ 3103.38 0.218211 0.109106 0.994030i $$-0.465201\pi$$
0.109106 + 0.994030i $$0.465201\pi$$
$$588$$ 0 0
$$589$$ 23752.1 1.66161
$$590$$ 3116.24 0.217447
$$591$$ 0 0
$$592$$ −6157.97 −0.427519
$$593$$ 5937.71 0.411185 0.205592 0.978638i $$-0.434088\pi$$
0.205592 + 0.978638i $$0.434088\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 12146.1 0.834772
$$597$$ 0 0
$$598$$ 8830.50 0.603856
$$599$$ 2600.33 0.177373 0.0886866 0.996060i $$-0.471733\pi$$
0.0886866 + 0.996060i $$0.471733\pi$$
$$600$$ 0 0
$$601$$ 13881.4 0.942156 0.471078 0.882092i $$-0.343865\pi$$
0.471078 + 0.882092i $$0.343865\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 3162.40 0.213040
$$605$$ 2260.80 0.151925
$$606$$ 0 0
$$607$$ 12284.6 0.821442 0.410721 0.911761i $$-0.365277\pi$$
0.410721 + 0.911761i $$0.365277\pi$$
$$608$$ −14144.9 −0.943505
$$609$$ 0 0
$$610$$ −4313.07 −0.286281
$$611$$ 12104.4 0.801459
$$612$$ 0 0
$$613$$ 22062.0 1.45363 0.726815 0.686833i $$-0.241000\pi$$
0.726815 + 0.686833i $$0.241000\pi$$
$$614$$ 14950.8 0.982679
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12182.2 0.794871 0.397436 0.917630i $$-0.369900\pi$$
0.397436 + 0.917630i $$0.369900\pi$$
$$618$$ 0 0
$$619$$ −23248.6 −1.50960 −0.754799 0.655956i $$-0.772266\pi$$
−0.754799 + 0.655956i $$0.772266\pi$$
$$620$$ 2836.50 0.183736
$$621$$ 0 0
$$622$$ −2417.76 −0.155858
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 14025.4 0.897628
$$626$$ 20989.3 1.34010
$$627$$ 0 0
$$628$$ 13933.6 0.885365
$$629$$ −2046.83 −0.129749
$$630$$ 0 0
$$631$$ 19184.4 1.21033 0.605165 0.796100i $$-0.293107\pi$$
0.605165 + 0.796100i $$0.293107\pi$$
$$632$$ −6182.72 −0.389138
$$633$$ 0 0
$$634$$ 9971.52 0.624637
$$635$$ −353.884 −0.0221157
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 21091.6 1.30881
$$639$$ 0 0
$$640$$ 2992.44 0.184823
$$641$$ 19433.4 1.19746 0.598730 0.800951i $$-0.295672\pi$$
0.598730 + 0.800951i $$0.295672\pi$$
$$642$$ 0 0
$$643$$ −5777.47 −0.354341 −0.177170 0.984180i $$-0.556694\pi$$
−0.177170 + 0.984180i $$0.556694\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −7278.95 −0.443323
$$647$$ −29231.5 −1.77621 −0.888106 0.459640i $$-0.847979\pi$$
−0.888106 + 0.459640i $$0.847979\pi$$
$$648$$ 0 0
$$649$$ −20896.2 −1.26386
$$650$$ 19115.5 1.15349
$$651$$ 0 0
$$652$$ 8083.18 0.485524
$$653$$ 7093.27 0.425086 0.212543 0.977152i $$-0.431826\pi$$
0.212543 + 0.977152i $$0.431826\pi$$
$$654$$ 0 0
$$655$$ 1561.99 0.0931788
$$656$$ 19842.9 1.18100
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −19014.2 −1.12396 −0.561980 0.827151i $$-0.689960\pi$$
−0.561980 + 0.827151i $$0.689960\pi$$
$$660$$ 0 0
$$661$$ 21058.4 1.23915 0.619573 0.784939i $$-0.287306\pi$$
0.619573 + 0.784939i $$0.287306\pi$$
$$662$$ −9932.96 −0.583165
$$663$$ 0 0
$$664$$ −304.746 −0.0178109
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6770.16 0.393016
$$668$$ −14065.3 −0.814672
$$669$$ 0 0
$$670$$ −1317.65 −0.0759782
$$671$$ 28921.6 1.66395
$$672$$ 0 0
$$673$$ 9634.87 0.551853 0.275926 0.961179i $$-0.411015\pi$$
0.275926 + 0.961179i $$0.411015\pi$$
$$674$$ 15043.0 0.859693
$$675$$ 0 0
$$676$$ −826.515 −0.0470252
$$677$$ 8371.31 0.475237 0.237619 0.971359i $$-0.423633\pi$$
0.237619 + 0.971359i $$0.423633\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 686.854 0.0387348
$$681$$ 0 0
$$682$$ −53069.9 −2.97969
$$683$$ 12068.8 0.676137 0.338069 0.941121i $$-0.390226\pi$$
0.338069 + 0.941121i $$0.390226\pi$$
$$684$$ 0 0
$$685$$ −1077.33 −0.0600917
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 11742.0 0.650666
$$689$$ −6330.09 −0.350011
$$690$$ 0 0
$$691$$ −2981.29 −0.164130 −0.0820648 0.996627i $$-0.526151\pi$$
−0.0820648 + 0.996627i $$0.526151\pi$$
$$692$$ 14677.4 0.806286
$$693$$ 0 0
$$694$$ −127.327 −0.00696435
$$695$$ −6181.46 −0.337376
$$696$$ 0 0
$$697$$ 6595.53 0.358427
$$698$$ 856.978 0.0464715
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 28978.0 1.56132 0.780660 0.624956i $$-0.214883\pi$$
0.780660 + 0.624956i $$0.214883\pi$$
$$702$$ 0 0
$$703$$ 6000.11 0.321904
$$704$$ 211.195 0.0113064
$$705$$ 0 0
$$706$$ −388.388 −0.0207042
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −16372.4 −0.867249 −0.433625 0.901094i $$-0.642766\pi$$
−0.433625 + 0.901094i $$0.642766\pi$$
$$710$$ −4950.32 −0.261665
$$711$$ 0 0
$$712$$ 13370.2 0.703750
$$713$$ −17034.9 −0.894755
$$714$$ 0 0
$$715$$ 4583.18 0.239722
$$716$$ −12507.6 −0.652836
$$717$$ 0 0
$$718$$ −43801.7 −2.27669
$$719$$ 23010.5 1.19353 0.596765 0.802416i $$-0.296453\pi$$
0.596765 + 0.802416i $$0.296453\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −2882.36 −0.148574
$$723$$ 0 0
$$724$$ −16637.8 −0.854058
$$725$$ 14655.5 0.750745
$$726$$ 0 0
$$727$$ 24636.8 1.25685 0.628423 0.777872i $$-0.283701\pi$$
0.628423 + 0.777872i $$0.283701\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 1742.03 0.0883225
$$731$$ 3902.87 0.197473
$$732$$ 0 0
$$733$$ −6904.76 −0.347931 −0.173965 0.984752i $$-0.555658\pi$$
−0.173965 + 0.984752i $$0.555658\pi$$
$$734$$ −48940.1 −2.46105
$$735$$ 0 0
$$736$$ 10144.6 0.508065
$$737$$ 8835.63 0.441607
$$738$$ 0 0
$$739$$ −9234.89 −0.459690 −0.229845 0.973227i $$-0.573822\pi$$
−0.229845 + 0.973227i $$0.573822\pi$$
$$740$$ 716.538 0.0355952
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −20216.9 −0.998232 −0.499116 0.866535i $$-0.666342\pi$$
−0.499116 + 0.866535i $$0.666342\pi$$
$$744$$ 0 0
$$745$$ 5645.97 0.277654
$$746$$ 17295.7 0.848847
$$747$$ 0 0
$$748$$ 5828.88 0.284926
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 24054.9 1.16881 0.584405 0.811462i $$-0.301328\pi$$
0.584405 + 0.811462i $$0.301328\pi$$
$$752$$ 21528.7 1.04398
$$753$$ 0 0
$$754$$ 19234.5 0.929015
$$755$$ 1470.00 0.0708593
$$756$$ 0 0
$$757$$ −30328.2 −1.45614 −0.728069 0.685504i $$-0.759582\pi$$
−0.728069 + 0.685504i $$0.759582\pi$$
$$758$$ −34627.1 −1.65925
$$759$$ 0 0
$$760$$ −2013.46 −0.0960997
$$761$$ −33834.1 −1.61168 −0.805839 0.592135i $$-0.798285\pi$$
−0.805839 + 0.592135i $$0.798285\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −4288.55 −0.203082
$$765$$ 0 0
$$766$$ 37887.9 1.78713
$$767$$ −19056.3 −0.897109
$$768$$ 0 0
$$769$$ −31738.1 −1.48830 −0.744151 0.668011i $$-0.767146\pi$$
−0.744151 + 0.668011i $$0.767146\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −16938.1 −0.789658
$$773$$ 27494.2 1.27930 0.639650 0.768667i $$-0.279080\pi$$
0.639650 + 0.768667i $$0.279080\pi$$
$$774$$ 0 0
$$775$$ −36875.6 −1.70917
$$776$$ −20791.0 −0.961794
$$777$$ 0 0
$$778$$ −18588.0 −0.856573
$$779$$ −19334.2 −0.889244
$$780$$ 0 0
$$781$$ 33194.8 1.52087
$$782$$ 5220.41 0.238723
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 6476.84 0.294482
$$786$$ 0 0
$$787$$ 468.356 0.0212136 0.0106068 0.999944i $$-0.496624\pi$$
0.0106068 + 0.999944i $$0.496624\pi$$
$$788$$ −22871.6 −1.03397
$$789$$ 0 0
$$790$$ 3637.18 0.163804
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 26375.1 1.18109
$$794$$ −4289.97 −0.191745
$$795$$ 0 0
$$796$$ 3864.93 0.172096
$$797$$ 37723.8 1.67659 0.838297 0.545214i $$-0.183551\pi$$
0.838297 + 0.545214i $$0.183551\pi$$
$$798$$ 0 0
$$799$$ 7155.87 0.316841
$$800$$ 21960.2 0.970512
$$801$$ 0 0
$$802$$ −8105.53 −0.356878
$$803$$ −11681.3 −0.513356
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −48397.1 −2.11503
$$807$$ 0 0
$$808$$ −961.583 −0.0418668
$$809$$ 7797.13 0.338854 0.169427 0.985543i $$-0.445808\pi$$
0.169427 + 0.985543i $$0.445808\pi$$
$$810$$ 0 0
$$811$$ −16925.9 −0.732860 −0.366430 0.930446i $$-0.619420\pi$$
−0.366430 + 0.930446i $$0.619420\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −13406.2 −0.577256
$$815$$ 3757.36 0.161490
$$816$$ 0 0
$$817$$ −11441.0 −0.489925
$$818$$ −16407.5 −0.701316
$$819$$ 0 0
$$820$$ −2308.91 −0.0983301
$$821$$ 30009.3 1.27568 0.637840 0.770169i $$-0.279828\pi$$
0.637840 + 0.770169i $$0.279828\pi$$
$$822$$ 0 0
$$823$$ −23385.6 −0.990486 −0.495243 0.868754i $$-0.664921\pi$$
−0.495243 + 0.868754i $$0.664921\pi$$
$$824$$ −2053.53 −0.0868181
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 37325.9 1.56947 0.784734 0.619833i $$-0.212800\pi$$
0.784734 + 0.619833i $$0.212800\pi$$
$$828$$ 0 0
$$829$$ 24671.3 1.03362 0.516809 0.856100i $$-0.327120\pi$$
0.516809 + 0.856100i $$0.327120\pi$$
$$830$$ 179.277 0.00749733
$$831$$ 0 0
$$832$$ 192.599 0.00802544
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −6538.05 −0.270969
$$836$$ −17086.9 −0.706892
$$837$$ 0 0
$$838$$ 26629.1 1.09772
$$839$$ 14147.4 0.582147 0.291074 0.956701i $$-0.405987\pi$$
0.291074 + 0.956701i $$0.405987\pi$$
$$840$$ 0 0
$$841$$ −9642.35 −0.395356
$$842$$ −22047.1 −0.902366
$$843$$ 0 0
$$844$$ −6008.91 −0.245065
$$845$$ −384.195 −0.0156411
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −11258.6 −0.455923
$$849$$ 0 0
$$850$$ 11300.7 0.456012
$$851$$ −4303.24 −0.173341
$$852$$ 0 0
$$853$$ −27963.6 −1.12246 −0.561229 0.827661i $$-0.689671\pi$$
−0.561229 + 0.827661i $$0.689671\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12750.9 0.509134
$$857$$ 33855.8 1.34947 0.674734 0.738061i $$-0.264259\pi$$
0.674734 + 0.738061i $$0.264259\pi$$
$$858$$ 0 0
$$859$$ 24282.7 0.964511 0.482255 0.876031i $$-0.339818\pi$$
0.482255 + 0.876031i $$0.339818\pi$$
$$860$$ −1366.29 −0.0541745
$$861$$ 0 0
$$862$$ −40487.2 −1.59977
$$863$$ 19667.0 0.775750 0.387875 0.921712i $$-0.373209\pi$$
0.387875 + 0.921712i $$0.373209\pi$$
$$864$$ 0 0
$$865$$ 6822.59 0.268179
$$866$$ −18207.9 −0.714469
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −24389.4 −0.952075
$$870$$ 0 0
$$871$$ 8057.65 0.313459
$$872$$ −16989.1 −0.659773
$$873$$ 0 0
$$874$$ −15303.2 −0.592264
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −36061.0 −1.38848 −0.694238 0.719745i $$-0.744259\pi$$
−0.694238 + 0.719745i $$0.744259\pi$$
$$878$$ 17882.5 0.687364
$$879$$ 0 0
$$880$$ 8151.58 0.312261
$$881$$ −15889.7 −0.607646 −0.303823 0.952728i $$-0.598263\pi$$
−0.303823 + 0.952728i $$0.598263\pi$$
$$882$$ 0 0
$$883$$ 14861.3 0.566390 0.283195 0.959062i $$-0.408606\pi$$
0.283195 + 0.959062i $$0.408606\pi$$
$$884$$ 5315.65 0.202245
$$885$$ 0 0
$$886$$ −44858.2 −1.70095
$$887$$ −38189.9 −1.44565 −0.722824 0.691032i $$-0.757156\pi$$
−0.722824 + 0.691032i $$0.757156\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −7865.45 −0.296237
$$891$$ 0 0
$$892$$ −3865.20 −0.145086
$$893$$ −20976.8 −0.786073
$$894$$ 0 0
$$895$$ −5813.99 −0.217140
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −49231.7 −1.82949
$$899$$ −37105.0 −1.37655
$$900$$ 0 0
$$901$$ −3742.22 −0.138370
$$902$$ 43198.9 1.59464
$$903$$ 0 0
$$904$$ −19182.2 −0.705742
$$905$$ −7733.86 −0.284069
$$906$$ 0 0
$$907$$ 16865.5 0.617431 0.308715 0.951154i $$-0.400101\pi$$
0.308715 + 0.951154i $$0.400101\pi$$
$$908$$ −7665.65 −0.280169
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −26754.1 −0.973000 −0.486500 0.873681i $$-0.661727\pi$$
−0.486500 + 0.873681i $$0.661727\pi$$
$$912$$ 0 0
$$913$$ −1202.15 −0.0435766
$$914$$ −53722.7 −1.94419
$$915$$ 0 0
$$916$$ 4672.02 0.168524
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −41527.5 −1.49061 −0.745303 0.666726i $$-0.767695\pi$$
−0.745303 + 0.666726i $$0.767695\pi$$
$$920$$ 1444.04 0.0517484
$$921$$ 0 0
$$922$$ −40361.5 −1.44169
$$923$$ 30272.0 1.07954
$$924$$ 0 0
$$925$$ −9315.27 −0.331118
$$926$$ −33008.6 −1.17141
$$927$$ 0 0
$$928$$ 22096.8 0.781642
$$929$$ 4584.68 0.161914 0.0809572 0.996718i $$-0.474202\pi$$
0.0809572 + 0.996718i $$0.474202\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −6473.92 −0.227532
$$933$$ 0 0
$$934$$ 12821.0 0.449159
$$935$$ 2709.48 0.0947694
$$936$$ 0 0
$$937$$ 6928.18 0.241552 0.120776 0.992680i $$-0.461462\pi$$
0.120776 + 0.992680i $$0.461462\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −2505.07 −0.0869217
$$941$$ 20944.9 0.725596 0.362798 0.931868i $$-0.381822\pi$$
0.362798 + 0.931868i $$0.381822\pi$$
$$942$$ 0 0
$$943$$ 13866.4 0.478846
$$944$$ −33893.3 −1.16857
$$945$$ 0 0
$$946$$ 25562.8 0.878559
$$947$$ −29278.9 −1.00468 −0.502342 0.864669i $$-0.667528\pi$$
−0.502342 + 0.864669i $$0.667528\pi$$
$$948$$ 0 0
$$949$$ −10652.8 −0.364387
$$950$$ −33127.1 −1.13135
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −2136.81 −0.0726316 −0.0363158 0.999340i $$-0.511562\pi$$
−0.0363158 + 0.999340i $$0.511562\pi$$
$$954$$ 0 0
$$955$$ −1993.48 −0.0675470
$$956$$ 14090.9 0.476707
$$957$$ 0 0
$$958$$ 23027.4 0.776597
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 63571.4 2.13391
$$962$$ −12225.8 −0.409745
$$963$$ 0 0
$$964$$ 1702.70 0.0568881
$$965$$ −7873.47 −0.262649
$$966$$ 0 0
$$967$$ −3921.32 −0.130405 −0.0652023 0.997872i $$-0.520769\pi$$
−0.0652023 + 0.997872i $$0.520769\pi$$
$$968$$ −13570.4 −0.450586
$$969$$ 0 0
$$970$$ 12230.9 0.404857
$$971$$ 47809.1 1.58009 0.790045 0.613049i $$-0.210057\pi$$
0.790045 + 0.613049i $$0.210057\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −12946.1 −0.425894
$$975$$ 0 0
$$976$$ 46910.5 1.53849
$$977$$ 51329.1 1.68082 0.840411 0.541949i $$-0.182314\pi$$
0.840411 + 0.541949i $$0.182314\pi$$
$$978$$ 0 0
$$979$$ 52742.4 1.72181
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 44035.8 1.43100
$$983$$ −16326.3 −0.529734 −0.264867 0.964285i $$-0.585328\pi$$
−0.264867 + 0.964285i $$0.585328\pi$$
$$984$$ 0 0
$$985$$ −10631.6 −0.343909
$$986$$ 11371.0 0.367268
$$987$$ 0 0
$$988$$ −15582.4 −0.501763
$$989$$ 8205.37 0.263818
$$990$$ 0 0
$$991$$ 33770.0 1.08248 0.541242 0.840867i $$-0.317954\pi$$
0.541242 + 0.840867i $$0.317954\pi$$
$$992$$ −55599.3 −1.77952
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 1796.56 0.0572410
$$996$$ 0 0
$$997$$ −50695.4 −1.61037 −0.805185 0.593023i $$-0.797934\pi$$
−0.805185 + 0.593023i $$0.797934\pi$$
$$998$$ −8135.48 −0.258040
$$999$$ 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.u.1.4 4
3.2 odd 2 49.4.a.e.1.1 4
7.2 even 3 441.4.e.y.361.1 8
7.3 odd 6 441.4.e.y.226.2 8
7.4 even 3 441.4.e.y.226.1 8
7.5 odd 6 441.4.e.y.361.2 8
7.6 odd 2 inner 441.4.a.u.1.3 4
12.11 even 2 784.4.a.bf.1.4 4
15.14 odd 2 1225.4.a.bb.1.4 4
21.2 odd 6 49.4.c.e.18.4 8
21.5 even 6 49.4.c.e.18.3 8
21.11 odd 6 49.4.c.e.30.4 8
21.17 even 6 49.4.c.e.30.3 8
21.20 even 2 49.4.a.e.1.2 yes 4
84.83 odd 2 784.4.a.bf.1.1 4
105.104 even 2 1225.4.a.bb.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.a.e.1.1 4 3.2 odd 2
49.4.a.e.1.2 yes 4 21.20 even 2
49.4.c.e.18.3 8 21.5 even 6
49.4.c.e.18.4 8 21.2 odd 6
49.4.c.e.30.3 8 21.17 even 6
49.4.c.e.30.4 8 21.11 odd 6
441.4.a.u.1.3 4 7.6 odd 2 inner
441.4.a.u.1.4 4 1.1 even 1 trivial
441.4.e.y.226.1 8 7.4 even 3
441.4.e.y.226.2 8 7.3 odd 6
441.4.e.y.361.1 8 7.2 even 3
441.4.e.y.361.2 8 7.5 odd 6
784.4.a.bf.1.1 4 84.83 odd 2
784.4.a.bf.1.4 4 12.11 even 2
1225.4.a.bb.1.3 4 105.104 even 2
1225.4.a.bb.1.4 4 15.14 odd 2