Properties

 Label 441.4.a.u.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: $$\Q(\sqrt{2}, \sqrt{65})$$ 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 1.1 Root $$3.11692$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-4.53113 q^{2} +12.5311 q^{4} -13.4791 q^{5} -20.5311 q^{8} +O(q^{10})$$ $$q-4.53113 q^{2} +12.5311 q^{4} -13.4791 q^{5} -20.5311 q^{8} +61.0753 q^{10} -0.813227 q^{11} -34.9564 q^{13} -7.21984 q^{16} +117.732 q^{17} +93.2913 q^{19} -168.908 q^{20} +3.68484 q^{22} -120.249 q^{23} +56.6848 q^{25} +158.392 q^{26} -8.56420 q^{29} +82.1070 q^{31} +196.963 q^{32} -533.458 q^{34} +28.8132 q^{37} -422.715 q^{38} +276.740 q^{40} -70.5291 q^{41} +417.179 q^{43} -10.1906 q^{44} +544.864 q^{46} +338.261 q^{47} -256.846 q^{50} -438.043 q^{52} -149.121 q^{53} +10.9615 q^{55} +38.8055 q^{58} -94.1828 q^{59} +120.525 q^{61} -372.037 q^{62} -834.706 q^{64} +471.179 q^{65} -792.366 q^{67} +1475.31 q^{68} -449.128 q^{71} +469.420 q^{73} -130.556 q^{74} +1169.05 q^{76} -1019.85 q^{79} +97.3166 q^{80} +319.576 q^{82} -104.253 q^{83} -1586.91 q^{85} -1890.29 q^{86} +16.6965 q^{88} -1572.92 q^{89} -1506.86 q^{92} -1532.70 q^{94} -1257.48 q^{95} -550.057 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 34q^{4} - 66q^{8} + O(q^{10})$$ $$4q - 2q^{2} + 34q^{4} - 66q^{8} - 100q^{11} - 174q^{16} - 340q^{22} - 352q^{23} - 128q^{25} - 260q^{29} + 30q^{32} + 212q^{37} + 540q^{43} - 460q^{44} + 696q^{46} - 1366q^{50} - 16q^{53} - 780q^{58} - 1678q^{64} + 756q^{65} - 1944q^{67} - 2248q^{71} + 284q^{74} - 1048q^{79} - 3284q^{85} - 4820q^{86} + 1260q^{88} - 3512q^{92} - 2192q^{95} + O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.53113 −1.60200 −0.800998 0.598667i $$-0.795697\pi$$
−0.800998 + 0.598667i $$0.795697\pi$$
$$3$$ 0 0
$$4$$ 12.5311 1.56639
$$5$$ −13.4791 −1.20560 −0.602802 0.797891i $$-0.705949\pi$$
−0.602802 + 0.797891i $$0.705949\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −20.5311 −0.907356
$$9$$ 0 0
$$10$$ 61.0753 1.93137
$$11$$ −0.813227 −0.0222906 −0.0111453 0.999938i $$-0.503548\pi$$
−0.0111453 + 0.999938i $$0.503548\pi$$
$$12$$ 0 0
$$13$$ −34.9564 −0.745781 −0.372891 0.927875i $$-0.621633\pi$$
−0.372891 + 0.927875i $$0.621633\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −7.21984 −0.112810
$$17$$ 117.732 1.67966 0.839829 0.542851i $$-0.182655\pi$$
0.839829 + 0.542851i $$0.182655\pi$$
$$18$$ 0 0
$$19$$ 93.2913 1.12645 0.563224 0.826304i $$-0.309561\pi$$
0.563224 + 0.826304i $$0.309561\pi$$
$$20$$ −168.908 −1.88845
$$21$$ 0 0
$$22$$ 3.68484 0.0357095
$$23$$ −120.249 −1.09016 −0.545079 0.838384i $$-0.683501\pi$$
−0.545079 + 0.838384i $$0.683501\pi$$
$$24$$ 0 0
$$25$$ 56.6848 0.453479
$$26$$ 158.392 1.19474
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −8.56420 −0.0548390 −0.0274195 0.999624i $$-0.508729\pi$$
−0.0274195 + 0.999624i $$0.508729\pi$$
$$30$$ 0 0
$$31$$ 82.1070 0.475705 0.237852 0.971301i $$-0.423557\pi$$
0.237852 + 0.971301i $$0.423557\pi$$
$$32$$ 196.963 1.08808
$$33$$ 0 0
$$34$$ −533.458 −2.69080
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 28.8132 0.128023 0.0640117 0.997949i $$-0.479611\pi$$
0.0640117 + 0.997949i $$0.479611\pi$$
$$38$$ −422.715 −1.80456
$$39$$ 0 0
$$40$$ 276.740 1.09391
$$41$$ −70.5291 −0.268654 −0.134327 0.990937i $$-0.542887\pi$$
−0.134327 + 0.990937i $$0.542887\pi$$
$$42$$ 0 0
$$43$$ 417.179 1.47952 0.739758 0.672873i $$-0.234940\pi$$
0.739758 + 0.672873i $$0.234940\pi$$
$$44$$ −10.1906 −0.0349159
$$45$$ 0 0
$$46$$ 544.864 1.74643
$$47$$ 338.261 1.04980 0.524899 0.851165i $$-0.324103\pi$$
0.524899 + 0.851165i $$0.324103\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −256.846 −0.726471
$$51$$ 0 0
$$52$$ −438.043 −1.16819
$$53$$ −149.121 −0.386477 −0.193239 0.981152i $$-0.561899\pi$$
−0.193239 + 0.981152i $$0.561899\pi$$
$$54$$ 0 0
$$55$$ 10.9615 0.0268737
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 38.8055 0.0878519
$$59$$ −94.1828 −0.207823 −0.103911 0.994587i $$-0.533136\pi$$
−0.103911 + 0.994587i $$0.533136\pi$$
$$60$$ 0 0
$$61$$ 120.525 0.252977 0.126488 0.991968i $$-0.459629\pi$$
0.126488 + 0.991968i $$0.459629\pi$$
$$62$$ −372.037 −0.762077
$$63$$ 0 0
$$64$$ −834.706 −1.63029
$$65$$ 471.179 0.899116
$$66$$ 0 0
$$67$$ −792.366 −1.44482 −0.722410 0.691465i $$-0.756965\pi$$
−0.722410 + 0.691465i $$0.756965\pi$$
$$68$$ 1475.31 2.63100
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −449.128 −0.750729 −0.375364 0.926877i $$-0.622482\pi$$
−0.375364 + 0.926877i $$0.622482\pi$$
$$72$$ 0 0
$$73$$ 469.420 0.752623 0.376311 0.926493i $$-0.377192\pi$$
0.376311 + 0.926493i $$0.377192\pi$$
$$74$$ −130.556 −0.205093
$$75$$ 0 0
$$76$$ 1169.05 1.76446
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1019.85 −1.45243 −0.726217 0.687465i $$-0.758723\pi$$
−0.726217 + 0.687465i $$0.758723\pi$$
$$80$$ 97.3166 0.136004
$$81$$ 0 0
$$82$$ 319.576 0.430382
$$83$$ −104.253 −0.137870 −0.0689352 0.997621i $$-0.521960\pi$$
−0.0689352 + 0.997621i $$0.521960\pi$$
$$84$$ 0 0
$$85$$ −1586.91 −2.02500
$$86$$ −1890.29 −2.37018
$$87$$ 0 0
$$88$$ 16.6965 0.0202256
$$89$$ −1572.92 −1.87336 −0.936680 0.350185i $$-0.886119\pi$$
−0.936680 + 0.350185i $$0.886119\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1506.86 −1.70762
$$93$$ 0 0
$$94$$ −1532.70 −1.68177
$$95$$ −1257.48 −1.35805
$$96$$ 0 0
$$97$$ −550.057 −0.575772 −0.287886 0.957665i $$-0.592952\pi$$
−0.287886 + 0.957665i $$0.592952\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 710.325 0.710325
$$101$$ −65.7169 −0.0647433 −0.0323717 0.999476i $$-0.510306\pi$$
−0.0323717 + 0.999476i $$0.510306\pi$$
$$102$$ 0 0
$$103$$ −1829.35 −1.75001 −0.875005 0.484113i $$-0.839142\pi$$
−0.875005 + 0.484113i $$0.839142\pi$$
$$104$$ 717.694 0.676689
$$105$$ 0 0
$$106$$ 675.685 0.619135
$$107$$ −861.377 −0.778248 −0.389124 0.921185i $$-0.627222\pi$$
−0.389124 + 0.921185i $$0.627222\pi$$
$$108$$ 0 0
$$109$$ −1620.52 −1.42401 −0.712007 0.702173i $$-0.752213\pi$$
−0.712007 + 0.702173i $$0.752213\pi$$
$$110$$ −49.6681 −0.0430515
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −380.409 −0.316689 −0.158344 0.987384i $$-0.550616\pi$$
−0.158344 + 0.987384i $$0.550616\pi$$
$$114$$ 0 0
$$115$$ 1620.84 1.31430
$$116$$ −107.319 −0.0858993
$$117$$ 0 0
$$118$$ 426.754 0.332931
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1330.34 −0.999503
$$122$$ −546.112 −0.405268
$$123$$ 0 0
$$124$$ 1028.89 0.745140
$$125$$ 920.824 0.658888
$$126$$ 0 0
$$127$$ 958.358 0.669610 0.334805 0.942287i $$-0.391329\pi$$
0.334805 + 0.942287i $$0.391329\pi$$
$$128$$ 2206.46 1.52363
$$129$$ 0 0
$$130$$ −2134.97 −1.44038
$$131$$ 1152.16 0.768431 0.384216 0.923243i $$-0.374472\pi$$
0.384216 + 0.923243i $$0.374472\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 3590.31 2.31459
$$135$$ 0 0
$$136$$ −2417.17 −1.52405
$$137$$ −357.377 −0.222867 −0.111434 0.993772i $$-0.535544\pi$$
−0.111434 + 0.993772i $$0.535544\pi$$
$$138$$ 0 0
$$139$$ 2736.29 1.66970 0.834852 0.550475i $$-0.185553\pi$$
0.834852 + 0.550475i $$0.185553\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 2035.06 1.20266
$$143$$ 28.4275 0.0166239
$$144$$ 0 0
$$145$$ 115.437 0.0661141
$$146$$ −2127.00 −1.20570
$$147$$ 0 0
$$148$$ 361.062 0.200535
$$149$$ −1409.94 −0.775212 −0.387606 0.921825i $$-0.626698\pi$$
−0.387606 + 0.921825i $$0.626698\pi$$
$$150$$ 0 0
$$151$$ 2352.35 1.26776 0.633879 0.773432i $$-0.281462\pi$$
0.633879 + 0.773432i $$0.281462\pi$$
$$152$$ −1915.38 −1.02209
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1106.72 −0.573511
$$156$$ 0 0
$$157$$ −1213.82 −0.617029 −0.308514 0.951220i $$-0.599832\pi$$
−0.308514 + 0.951220i $$0.599832\pi$$
$$158$$ 4621.08 2.32679
$$159$$ 0 0
$$160$$ −2654.88 −1.31179
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −722.774 −0.347313 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$164$$ −883.809 −0.420817
$$165$$ 0 0
$$166$$ 472.383 0.220868
$$167$$ 753.016 0.348923 0.174462 0.984664i $$-0.444182\pi$$
0.174462 + 0.984664i $$0.444182\pi$$
$$168$$ 0 0
$$169$$ −975.051 −0.443810
$$170$$ 7190.51 3.24404
$$171$$ 0 0
$$172$$ 5227.72 2.31750
$$173$$ −1859.14 −0.817038 −0.408519 0.912750i $$-0.633955\pi$$
−0.408519 + 0.912750i $$0.633955\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 5.87137 0.00251461
$$177$$ 0 0
$$178$$ 7127.10 3.00112
$$179$$ 522.825 0.218312 0.109156 0.994025i $$-0.465185\pi$$
0.109156 + 0.994025i $$0.465185\pi$$
$$180$$ 0 0
$$181$$ −2901.38 −1.19148 −0.595740 0.803177i $$-0.703141\pi$$
−0.595740 + 0.803177i $$0.703141\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 2468.85 0.989163
$$185$$ −388.375 −0.154345
$$186$$ 0 0
$$187$$ −95.7427 −0.0374407
$$188$$ 4238.79 1.64439
$$189$$ 0 0
$$190$$ 5697.80 2.17559
$$191$$ −2604.35 −0.986619 −0.493309 0.869854i $$-0.664213\pi$$
−0.493309 + 0.869854i $$0.664213\pi$$
$$192$$ 0 0
$$193$$ 676.245 0.252214 0.126107 0.992017i $$-0.459752\pi$$
0.126107 + 0.992017i $$0.459752\pi$$
$$194$$ 2492.38 0.922384
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3685.99 1.33308 0.666538 0.745471i $$-0.267775\pi$$
0.666538 + 0.745471i $$0.267775\pi$$
$$198$$ 0 0
$$199$$ 799.801 0.284907 0.142453 0.989802i $$-0.454501\pi$$
0.142453 + 0.989802i $$0.454501\pi$$
$$200$$ −1163.80 −0.411467
$$201$$ 0 0
$$202$$ 297.772 0.103719
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 950.665 0.323890
$$206$$ 8289.01 2.80351
$$207$$ 0 0
$$208$$ 252.380 0.0841316
$$209$$ −75.8670 −0.0251092
$$210$$ 0 0
$$211$$ −667.385 −0.217747 −0.108874 0.994056i $$-0.534724\pi$$
−0.108874 + 0.994056i $$0.534724\pi$$
$$212$$ −1868.65 −0.605375
$$213$$ 0 0
$$214$$ 3903.01 1.24675
$$215$$ −5623.18 −1.78371
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 7342.77 2.28126
$$219$$ 0 0
$$220$$ 137.360 0.0420947
$$221$$ −4115.48 −1.25266
$$222$$ 0 0
$$223$$ −2646.82 −0.794818 −0.397409 0.917642i $$-0.630091\pi$$
−0.397409 + 0.917642i $$0.630091\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 1723.68 0.507334
$$227$$ 4121.11 1.20497 0.602485 0.798131i $$-0.294178\pi$$
0.602485 + 0.798131i $$0.294178\pi$$
$$228$$ 0 0
$$229$$ −4066.92 −1.17358 −0.586790 0.809739i $$-0.699609\pi$$
−0.586790 + 0.809739i $$0.699609\pi$$
$$230$$ −7344.25 −2.10550
$$231$$ 0 0
$$232$$ 175.833 0.0497585
$$233$$ 3904.67 1.09787 0.548934 0.835865i $$-0.315034\pi$$
0.548934 + 0.835865i $$0.315034\pi$$
$$234$$ 0 0
$$235$$ −4559.44 −1.26564
$$236$$ −1180.22 −0.325532
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −5425.12 −1.46829 −0.734146 0.678991i $$-0.762417\pi$$
−0.734146 + 0.678991i $$0.762417\pi$$
$$240$$ 0 0
$$241$$ 1602.89 0.428429 0.214215 0.976787i $$-0.431281\pi$$
0.214215 + 0.976787i $$0.431281\pi$$
$$242$$ 6027.94 1.60120
$$243$$ 0 0
$$244$$ 1510.31 0.396261
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3261.13 −0.840084
$$248$$ −1685.75 −0.431634
$$249$$ 0 0
$$250$$ −4172.37 −1.05554
$$251$$ 3805.93 0.957085 0.478542 0.878064i $$-0.341165\pi$$
0.478542 + 0.878064i $$0.341165\pi$$
$$252$$ 0 0
$$253$$ 97.7897 0.0243003
$$254$$ −4342.44 −1.07271
$$255$$ 0 0
$$256$$ −3320.09 −0.810569
$$257$$ 4589.34 1.11391 0.556956 0.830542i $$-0.311969\pi$$
0.556956 + 0.830542i $$0.311969\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 5904.41 1.40837
$$261$$ 0 0
$$262$$ −5220.58 −1.23102
$$263$$ −877.175 −0.205661 −0.102831 0.994699i $$-0.532790\pi$$
−0.102831 + 0.994699i $$0.532790\pi$$
$$264$$ 0 0
$$265$$ 2010.00 0.465938
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −9929.24 −2.26315
$$269$$ −6123.55 −1.38795 −0.693977 0.719997i $$-0.744143\pi$$
−0.693977 + 0.719997i $$0.744143\pi$$
$$270$$ 0 0
$$271$$ 3489.76 0.782243 0.391122 0.920339i $$-0.372087\pi$$
0.391122 + 0.920339i $$0.372087\pi$$
$$272$$ −850.006 −0.189482
$$273$$ 0 0
$$274$$ 1619.32 0.357032
$$275$$ −46.0976 −0.0101083
$$276$$ 0 0
$$277$$ −4891.70 −1.06106 −0.530530 0.847666i $$-0.678007\pi$$
−0.530530 + 0.847666i $$0.678007\pi$$
$$278$$ −12398.5 −2.67486
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6914.46 −1.46791 −0.733954 0.679199i $$-0.762327\pi$$
−0.733954 + 0.679199i $$0.762327\pi$$
$$282$$ 0 0
$$283$$ −3559.85 −0.747742 −0.373871 0.927481i $$-0.621970\pi$$
−0.373871 + 0.927481i $$0.621970\pi$$
$$284$$ −5628.09 −1.17593
$$285$$ 0 0
$$286$$ −128.809 −0.0266315
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8947.80 1.82125
$$290$$ −523.061 −0.105914
$$291$$ 0 0
$$292$$ 5882.36 1.17890
$$293$$ 3285.11 0.655011 0.327505 0.944849i $$-0.393792\pi$$
0.327505 + 0.944849i $$0.393792\pi$$
$$294$$ 0 0
$$295$$ 1269.49 0.250552
$$296$$ −591.568 −0.116163
$$297$$ 0 0
$$298$$ 6388.61 1.24189
$$299$$ 4203.47 0.813020
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −10658.8 −2.03094
$$303$$ 0 0
$$304$$ −673.548 −0.127075
$$305$$ −1624.56 −0.304990
$$306$$ 0 0
$$307$$ 9094.65 1.69075 0.845373 0.534176i $$-0.179378\pi$$
0.845373 + 0.534176i $$0.179378\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 5014.71 0.918762
$$311$$ −8163.06 −1.48838 −0.744188 0.667971i $$-0.767163\pi$$
−0.744188 + 0.667971i $$0.767163\pi$$
$$312$$ 0 0
$$313$$ −2979.62 −0.538076 −0.269038 0.963130i $$-0.586706\pi$$
−0.269038 + 0.963130i $$0.586706\pi$$
$$314$$ 5499.98 0.988478
$$315$$ 0 0
$$316$$ −12779.9 −2.27508
$$317$$ 3888.11 0.688889 0.344445 0.938807i $$-0.388067\pi$$
0.344445 + 0.938807i $$0.388067\pi$$
$$318$$ 0 0
$$319$$ 6.96463 0.00122240
$$320$$ 11251.0 1.96548
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 10983.4 1.89205
$$324$$ 0 0
$$325$$ −1981.50 −0.338196
$$326$$ 3274.98 0.556394
$$327$$ 0 0
$$328$$ 1448.04 0.243764
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4893.03 −0.812524 −0.406262 0.913757i $$-0.633168\pi$$
−0.406262 + 0.913757i $$0.633168\pi$$
$$332$$ −1306.41 −0.215959
$$333$$ 0 0
$$334$$ −3412.01 −0.558973
$$335$$ 10680.3 1.74188
$$336$$ 0 0
$$337$$ −1722.10 −0.278364 −0.139182 0.990267i $$-0.544447\pi$$
−0.139182 + 0.990267i $$0.544447\pi$$
$$338$$ 4418.08 0.710982
$$339$$ 0 0
$$340$$ −19885.8 −3.17194
$$341$$ −66.7716 −0.0106038
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −8565.16 −1.34245
$$345$$ 0 0
$$346$$ 8423.99 1.30889
$$347$$ 238.058 0.0368289 0.0184145 0.999830i $$-0.494138\pi$$
0.0184145 + 0.999830i $$0.494138\pi$$
$$348$$ 0 0
$$349$$ 10053.1 1.54192 0.770959 0.636884i $$-0.219777\pi$$
0.770959 + 0.636884i $$0.219777\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −160.176 −0.0242539
$$353$$ −3470.16 −0.523224 −0.261612 0.965173i $$-0.584254\pi$$
−0.261612 + 0.965173i $$0.584254\pi$$
$$354$$ 0 0
$$355$$ 6053.82 0.905081
$$356$$ −19710.5 −2.93442
$$357$$ 0 0
$$358$$ −2368.99 −0.349734
$$359$$ −1407.54 −0.206928 −0.103464 0.994633i $$-0.532993\pi$$
−0.103464 + 0.994633i $$0.532993\pi$$
$$360$$ 0 0
$$361$$ 1844.27 0.268884
$$362$$ 13146.5 1.90875
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6327.34 −0.907364
$$366$$ 0 0
$$367$$ −11133.0 −1.58348 −0.791742 0.610855i $$-0.790826\pi$$
−0.791742 + 0.610855i $$0.790826\pi$$
$$368$$ 868.179 0.122981
$$369$$ 0 0
$$370$$ 1759.78 0.247261
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 9025.94 1.25294 0.626468 0.779447i $$-0.284500\pi$$
0.626468 + 0.779447i $$0.284500\pi$$
$$374$$ 433.823 0.0599798
$$375$$ 0 0
$$376$$ −6944.88 −0.952540
$$377$$ 299.373 0.0408979
$$378$$ 0 0
$$379$$ −5855.75 −0.793640 −0.396820 0.917896i $$-0.629886\pi$$
−0.396820 + 0.917896i $$0.629886\pi$$
$$380$$ −15757.6 −2.12723
$$381$$ 0 0
$$382$$ 11800.6 1.58056
$$383$$ −7788.03 −1.03903 −0.519517 0.854460i $$-0.673888\pi$$
−0.519517 + 0.854460i $$0.673888\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3064.16 −0.404045
$$387$$ 0 0
$$388$$ −6892.84 −0.901884
$$389$$ 3814.05 0.497121 0.248560 0.968616i $$-0.420043\pi$$
0.248560 + 0.968616i $$0.420043\pi$$
$$390$$ 0 0
$$391$$ −14157.1 −1.83109
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −16701.7 −2.13558
$$395$$ 13746.6 1.75106
$$396$$ 0 0
$$397$$ 10165.3 1.28509 0.642545 0.766248i $$-0.277879\pi$$
0.642545 + 0.766248i $$0.277879\pi$$
$$398$$ −3624.00 −0.456419
$$399$$ 0 0
$$400$$ −409.255 −0.0511569
$$401$$ −11502.5 −1.43244 −0.716222 0.697873i $$-0.754130\pi$$
−0.716222 + 0.697873i $$0.754130\pi$$
$$402$$ 0 0
$$403$$ −2870.16 −0.354772
$$404$$ −823.507 −0.101413
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −23.4317 −0.00285372
$$408$$ 0 0
$$409$$ −3266.27 −0.394882 −0.197441 0.980315i $$-0.563263\pi$$
−0.197441 + 0.980315i $$0.563263\pi$$
$$410$$ −4307.59 −0.518870
$$411$$ 0 0
$$412$$ −22923.8 −2.74120
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1405.23 0.166217
$$416$$ −6885.12 −0.811468
$$417$$ 0 0
$$418$$ 343.763 0.0402249
$$419$$ −6822.93 −0.795518 −0.397759 0.917490i $$-0.630212\pi$$
−0.397759 + 0.917490i $$0.630212\pi$$
$$420$$ 0 0
$$421$$ 1431.63 0.165733 0.0828665 0.996561i $$-0.473592\pi$$
0.0828665 + 0.996561i $$0.473592\pi$$
$$422$$ 3024.01 0.348830
$$423$$ 0 0
$$424$$ 3061.62 0.350673
$$425$$ 6673.61 0.761689
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −10794.0 −1.21904
$$429$$ 0 0
$$430$$ 25479.3 2.85750
$$431$$ −15142.2 −1.69228 −0.846141 0.532959i $$-0.821080\pi$$
−0.846141 + 0.532959i $$0.821080\pi$$
$$432$$ 0 0
$$433$$ 5475.65 0.607721 0.303860 0.952717i $$-0.401724\pi$$
0.303860 + 0.952717i $$0.401724\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −20306.9 −2.23056
$$437$$ −11218.2 −1.22801
$$438$$ 0 0
$$439$$ −1780.54 −0.193578 −0.0967890 0.995305i $$-0.530857\pi$$
−0.0967890 + 0.995305i $$0.530857\pi$$
$$440$$ −225.052 −0.0243840
$$441$$ 0 0
$$442$$ 18647.8 2.00675
$$443$$ 3259.64 0.349594 0.174797 0.984605i $$-0.444073\pi$$
0.174797 + 0.984605i $$0.444073\pi$$
$$444$$ 0 0
$$445$$ 21201.5 2.25853
$$446$$ 11993.1 1.27330
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6826.19 0.717478 0.358739 0.933438i $$-0.383207\pi$$
0.358739 + 0.933438i $$0.383207\pi$$
$$450$$ 0 0
$$451$$ 57.3562 0.00598846
$$452$$ −4766.95 −0.496059
$$453$$ 0 0
$$454$$ −18673.3 −1.93036
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3700.03 0.378731 0.189365 0.981907i $$-0.439357\pi$$
0.189365 + 0.981907i $$0.439357\pi$$
$$458$$ 18427.8 1.88007
$$459$$ 0 0
$$460$$ 20311.0 2.05871
$$461$$ −9400.80 −0.949759 −0.474880 0.880051i $$-0.657508\pi$$
−0.474880 + 0.880051i $$0.657508\pi$$
$$462$$ 0 0
$$463$$ 15483.9 1.55420 0.777102 0.629374i $$-0.216689\pi$$
0.777102 + 0.629374i $$0.216689\pi$$
$$464$$ 61.8321 0.00618639
$$465$$ 0 0
$$466$$ −17692.6 −1.75878
$$467$$ −2205.62 −0.218552 −0.109276 0.994011i $$-0.534853\pi$$
−0.109276 + 0.994011i $$0.534853\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 20659.4 2.02755
$$471$$ 0 0
$$472$$ 1933.68 0.188569
$$473$$ −339.261 −0.0329794
$$474$$ 0 0
$$475$$ 5288.20 0.510820
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 24581.9 2.35220
$$479$$ 2349.32 0.224098 0.112049 0.993703i $$-0.464259\pi$$
0.112049 + 0.993703i $$0.464259\pi$$
$$480$$ 0 0
$$481$$ −1007.21 −0.0954775
$$482$$ −7262.91 −0.686342
$$483$$ 0 0
$$484$$ −16670.6 −1.56561
$$485$$ 7414.25 0.694152
$$486$$ 0 0
$$487$$ 10394.3 0.967167 0.483583 0.875298i $$-0.339335\pi$$
0.483583 + 0.875298i $$0.339335\pi$$
$$488$$ −2474.50 −0.229540
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −12586.7 −1.15689 −0.578444 0.815722i $$-0.696340\pi$$
−0.578444 + 0.815722i $$0.696340\pi$$
$$492$$ 0 0
$$493$$ −1008.28 −0.0921108
$$494$$ 14776.6 1.34581
$$495$$ 0 0
$$496$$ −592.799 −0.0536642
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 10627.9 0.953450 0.476725 0.879053i $$-0.341824\pi$$
0.476725 + 0.879053i $$0.341824\pi$$
$$500$$ 11539.0 1.03208
$$501$$ 0 0
$$502$$ −17245.2 −1.53325
$$503$$ −6719.02 −0.595599 −0.297800 0.954628i $$-0.596253\pi$$
−0.297800 + 0.954628i $$0.596253\pi$$
$$504$$ 0 0
$$505$$ 885.802 0.0780548
$$506$$ −443.098 −0.0389291
$$507$$ 0 0
$$508$$ 12009.3 1.04887
$$509$$ 3904.33 0.339993 0.169997 0.985445i $$-0.445624\pi$$
0.169997 + 0.985445i $$0.445624\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −2607.89 −0.225105
$$513$$ 0 0
$$514$$ −20794.9 −1.78448
$$515$$ 24657.9 2.10982
$$516$$ 0 0
$$517$$ −275.083 −0.0234007
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −9673.84 −0.815819
$$521$$ −15699.7 −1.32018 −0.660092 0.751184i $$-0.729483\pi$$
−0.660092 + 0.751184i $$0.729483\pi$$
$$522$$ 0 0
$$523$$ −10152.1 −0.848794 −0.424397 0.905476i $$-0.639514\pi$$
−0.424397 + 0.905476i $$0.639514\pi$$
$$524$$ 14437.8 1.20366
$$525$$ 0 0
$$526$$ 3974.59 0.329469
$$527$$ 9666.61 0.799021
$$528$$ 0 0
$$529$$ 2292.83 0.188447
$$530$$ −9107.59 −0.746431
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 2465.44 0.200357
$$534$$ 0 0
$$535$$ 11610.6 0.938258
$$536$$ 16268.2 1.31097
$$537$$ 0 0
$$538$$ 27746.6 2.22350
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −19846.6 −1.57722 −0.788608 0.614896i $$-0.789198\pi$$
−0.788608 + 0.614896i $$0.789198\pi$$
$$542$$ −15812.6 −1.25315
$$543$$ 0 0
$$544$$ 23188.8 1.82760
$$545$$ 21843.0 1.71679
$$546$$ 0 0
$$547$$ −22798.9 −1.78210 −0.891052 0.453901i $$-0.850032\pi$$
−0.891052 + 0.453901i $$0.850032\pi$$
$$548$$ −4478.34 −0.349097
$$549$$ 0 0
$$550$$ 208.874 0.0161935
$$551$$ −798.965 −0.0617733
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 22164.9 1.69981
$$555$$ 0 0
$$556$$ 34288.8 2.61541
$$557$$ 17998.3 1.36914 0.684570 0.728947i $$-0.259990\pi$$
0.684570 + 0.728947i $$0.259990\pi$$
$$558$$ 0 0
$$559$$ −14583.1 −1.10340
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 31330.3 2.35158
$$563$$ 195.636 0.0146449 0.00732246 0.999973i $$-0.497669\pi$$
0.00732246 + 0.999973i $$0.497669\pi$$
$$564$$ 0 0
$$565$$ 5127.55 0.381801
$$566$$ 16130.1 1.19788
$$567$$ 0 0
$$568$$ 9221.11 0.681178
$$569$$ 19660.4 1.44852 0.724260 0.689527i $$-0.242182\pi$$
0.724260 + 0.689527i $$0.242182\pi$$
$$570$$ 0 0
$$571$$ −15764.5 −1.15538 −0.577691 0.816255i $$-0.696046\pi$$
−0.577691 + 0.816255i $$0.696046\pi$$
$$572$$ 356.228 0.0260396
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −6816.30 −0.494364
$$576$$ 0 0
$$577$$ −22306.4 −1.60941 −0.804704 0.593676i $$-0.797676\pi$$
−0.804704 + 0.593676i $$0.797676\pi$$
$$578$$ −40543.6 −2.91764
$$579$$ 0 0
$$580$$ 1446.56 0.103560
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 121.269 0.00861483
$$584$$ −9637.72 −0.682897
$$585$$ 0 0
$$586$$ −14885.3 −1.04932
$$587$$ −15953.2 −1.12173 −0.560866 0.827906i $$-0.689532\pi$$
−0.560866 + 0.827906i $$0.689532\pi$$
$$588$$ 0 0
$$589$$ 7659.87 0.535856
$$590$$ −5752.24 −0.401383
$$591$$ 0 0
$$592$$ −208.027 −0.0144423
$$593$$ −3155.68 −0.218530 −0.109265 0.994013i $$-0.534850\pi$$
−0.109265 + 0.994013i $$0.534850\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −17668.1 −1.21429
$$597$$ 0 0
$$598$$ −19046.5 −1.30246
$$599$$ −25456.3 −1.73642 −0.868212 0.496194i $$-0.834730\pi$$
−0.868212 + 0.496194i $$0.834730\pi$$
$$600$$ 0 0
$$601$$ −5580.96 −0.378789 −0.189395 0.981901i $$-0.560653\pi$$
−0.189395 + 0.981901i $$0.560653\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 29477.6 1.98581
$$605$$ 17931.7 1.20500
$$606$$ 0 0
$$607$$ −381.133 −0.0254855 −0.0127427 0.999919i $$-0.504056\pi$$
−0.0127427 + 0.999919i $$0.504056\pi$$
$$608$$ 18374.9 1.22566
$$609$$ 0 0
$$610$$ 7361.07 0.488592
$$611$$ −11824.4 −0.782919
$$612$$ 0 0
$$613$$ −8235.98 −0.542656 −0.271328 0.962487i $$-0.587463\pi$$
−0.271328 + 0.962487i $$0.587463\pi$$
$$614$$ −41209.0 −2.70857
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 27419.8 1.78911 0.894555 0.446958i $$-0.147493\pi$$
0.894555 + 0.446958i $$0.147493\pi$$
$$618$$ 0 0
$$619$$ 16373.4 1.06317 0.531585 0.847005i $$-0.321597\pi$$
0.531585 + 0.847005i $$0.321597\pi$$
$$620$$ −13868.5 −0.898342
$$621$$ 0 0
$$622$$ 36987.9 2.38437
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19497.4 −1.24784
$$626$$ 13501.0 0.861996
$$627$$ 0 0
$$628$$ −15210.6 −0.966508
$$629$$ 3392.24 0.215035
$$630$$ 0 0
$$631$$ 4059.60 0.256118 0.128059 0.991767i $$-0.459125\pi$$
0.128059 + 0.991767i $$0.459125\pi$$
$$632$$ 20938.7 1.31788
$$633$$ 0 0
$$634$$ −17617.5 −1.10360
$$635$$ −12917.8 −0.807284
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −31.5577 −0.00195827
$$639$$ 0 0
$$640$$ −29741.0 −1.83690
$$641$$ 6388.63 0.393660 0.196830 0.980438i $$-0.436935\pi$$
0.196830 + 0.980438i $$0.436935\pi$$
$$642$$ 0 0
$$643$$ 18308.0 1.12286 0.561428 0.827525i $$-0.310252\pi$$
0.561428 + 0.827525i $$0.310252\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −49767.0 −3.03105
$$647$$ 3303.90 0.200757 0.100379 0.994949i $$-0.467995\pi$$
0.100379 + 0.994949i $$0.467995\pi$$
$$648$$ 0 0
$$649$$ 76.5919 0.00463251
$$650$$ 8978.42 0.541789
$$651$$ 0 0
$$652$$ −9057.18 −0.544028
$$653$$ −4371.27 −0.261961 −0.130981 0.991385i $$-0.541813\pi$$
−0.130981 + 0.991385i $$0.541813\pi$$
$$654$$ 0 0
$$655$$ −15530.0 −0.926423
$$656$$ 509.209 0.0303068
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −6259.75 −0.370023 −0.185012 0.982736i $$-0.559232\pi$$
−0.185012 + 0.982736i $$0.559232\pi$$
$$660$$ 0 0
$$661$$ 14845.7 0.873574 0.436787 0.899565i $$-0.356116\pi$$
0.436787 + 0.899565i $$0.356116\pi$$
$$662$$ 22171.0 1.30166
$$663$$ 0 0
$$664$$ 2140.43 0.125098
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1029.84 0.0597832
$$668$$ 9436.14 0.546550
$$669$$ 0 0
$$670$$ −48394.0 −2.79048
$$671$$ −98.0138 −0.00563902
$$672$$ 0 0
$$673$$ 9409.13 0.538923 0.269462 0.963011i $$-0.413154\pi$$
0.269462 + 0.963011i $$0.413154\pi$$
$$674$$ 7803.05 0.445938
$$675$$ 0 0
$$676$$ −12218.5 −0.695180
$$677$$ 2950.63 0.167507 0.0837533 0.996487i $$-0.473309\pi$$
0.0837533 + 0.996487i $$0.473309\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 32581.1 1.83740
$$681$$ 0 0
$$682$$ 302.551 0.0169872
$$683$$ −6280.85 −0.351874 −0.175937 0.984401i $$-0.556296\pi$$
−0.175937 + 0.984401i $$0.556296\pi$$
$$684$$ 0 0
$$685$$ 4817.11 0.268689
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −3011.97 −0.166904
$$689$$ 5212.72 0.288228
$$690$$ 0 0
$$691$$ −32763.2 −1.80372 −0.901861 0.432027i $$-0.857798\pi$$
−0.901861 + 0.432027i $$0.857798\pi$$
$$692$$ −23297.1 −1.27980
$$693$$ 0 0
$$694$$ −1078.67 −0.0589998
$$695$$ −36882.5 −2.01300
$$696$$ 0 0
$$697$$ −8303.53 −0.451246
$$698$$ −45551.9 −2.47015
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1775.97 0.0956883 0.0478442 0.998855i $$-0.484765\pi$$
0.0478442 + 0.998855i $$0.484765\pi$$
$$702$$ 0 0
$$703$$ 2688.02 0.144212
$$704$$ 678.805 0.0363401
$$705$$ 0 0
$$706$$ 15723.7 0.838203
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 8862.43 0.469444 0.234722 0.972063i $$-0.424582\pi$$
0.234722 + 0.972063i $$0.424582\pi$$
$$710$$ −27430.7 −1.44994
$$711$$ 0 0
$$712$$ 32293.8 1.69981
$$713$$ −9873.28 −0.518594
$$714$$ 0 0
$$715$$ −383.175 −0.0200419
$$716$$ 6551.59 0.341961
$$717$$ 0 0
$$718$$ 6377.75 0.331498
$$719$$ 27499.2 1.42635 0.713177 0.700984i $$-0.247256\pi$$
0.713177 + 0.700984i $$0.247256\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −8356.64 −0.430750
$$723$$ 0 0
$$724$$ −36357.6 −1.86632
$$725$$ −485.460 −0.0248683
$$726$$ 0 0
$$727$$ −25434.9 −1.29756 −0.648781 0.760975i $$-0.724721\pi$$
−0.648781 + 0.760975i $$0.724721\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 28670.0 1.45359
$$731$$ 49115.3 2.48508
$$732$$ 0 0
$$733$$ 24155.6 1.21720 0.608600 0.793477i $$-0.291731\pi$$
0.608600 + 0.793477i $$0.291731\pi$$
$$734$$ 50445.2 2.53674
$$735$$ 0 0
$$736$$ −23684.6 −1.18618
$$737$$ 644.373 0.0322060
$$738$$ 0 0
$$739$$ 27512.9 1.36952 0.684762 0.728767i $$-0.259906\pi$$
0.684762 + 0.728767i $$0.259906\pi$$
$$740$$ −4866.78 −0.241765
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −5995.09 −0.296014 −0.148007 0.988986i $$-0.547286\pi$$
−0.148007 + 0.988986i $$0.547286\pi$$
$$744$$ 0 0
$$745$$ 19004.6 0.934598
$$746$$ −40897.7 −2.00720
$$747$$ 0 0
$$748$$ −1199.76 −0.0586467
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 1545.09 0.0750747 0.0375373 0.999295i $$-0.488049\pi$$
0.0375373 + 0.999295i $$0.488049\pi$$
$$752$$ −2442.19 −0.118428
$$753$$ 0 0
$$754$$ −1356.50 −0.0655183
$$755$$ −31707.5 −1.52841
$$756$$ 0 0
$$757$$ −5157.82 −0.247641 −0.123820 0.992305i $$-0.539515\pi$$
−0.123820 + 0.992305i $$0.539515\pi$$
$$758$$ 26533.1 1.27141
$$759$$ 0 0
$$760$$ 25817.5 1.23223
$$761$$ −3289.96 −0.156716 −0.0783581 0.996925i $$-0.524968\pi$$
−0.0783581 + 0.996925i $$0.524968\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −32635.4 −1.54543
$$765$$ 0 0
$$766$$ 35288.6 1.66453
$$767$$ 3292.29 0.154990
$$768$$ 0 0
$$769$$ 11146.5 0.522697 0.261348 0.965245i $$-0.415833\pi$$
0.261348 + 0.965245i $$0.415833\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8474.12 0.395065
$$773$$ −15830.2 −0.736576 −0.368288 0.929712i $$-0.620056\pi$$
−0.368288 + 0.929712i $$0.620056\pi$$
$$774$$ 0 0
$$775$$ 4654.22 0.215722
$$776$$ 11293.3 0.522430
$$777$$ 0 0
$$778$$ −17282.0 −0.796386
$$779$$ −6579.75 −0.302624
$$780$$ 0 0
$$781$$ 365.243 0.0167342
$$782$$ 64147.9 2.93341
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 16361.2 0.743892
$$786$$ 0 0
$$787$$ 15163.4 0.686809 0.343404 0.939188i $$-0.388420\pi$$
0.343404 + 0.939188i $$0.388420\pi$$
$$788$$ 46189.6 2.08812
$$789$$ 0 0
$$790$$ −62287.8 −2.80519
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −4213.10 −0.188665
$$794$$ −46060.2 −2.05871
$$795$$ 0 0
$$796$$ 10022.4 0.446275
$$797$$ 29398.3 1.30658 0.653289 0.757109i $$-0.273389\pi$$
0.653289 + 0.757109i $$0.273389\pi$$
$$798$$ 0 0
$$799$$ 39824.1 1.76330
$$800$$ 11164.8 0.493420
$$801$$ 0 0
$$802$$ 52119.5 2.29477
$$803$$ −381.745 −0.0167764
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 13005.1 0.568343
$$807$$ 0 0
$$808$$ 1349.24 0.0587453
$$809$$ 20712.9 0.900155 0.450078 0.892989i $$-0.351396\pi$$
0.450078 + 0.892989i $$0.351396\pi$$
$$810$$ 0 0
$$811$$ −27369.9 −1.18506 −0.592532 0.805547i $$-0.701872\pi$$
−0.592532 + 0.805547i $$0.701872\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 106.172 0.00457165
$$815$$ 9742.31 0.418722
$$816$$ 0 0
$$817$$ 38919.2 1.66660
$$818$$ 14799.9 0.632600
$$819$$ 0 0
$$820$$ 11912.9 0.507338
$$821$$ 35362.7 1.50325 0.751623 0.659592i $$-0.229271\pi$$
0.751623 + 0.659592i $$0.229271\pi$$
$$822$$ 0 0
$$823$$ −29190.4 −1.23635 −0.618174 0.786042i $$-0.712127\pi$$
−0.618174 + 0.786042i $$0.712127\pi$$
$$824$$ 37558.6 1.58788
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 7302.08 0.307035 0.153518 0.988146i $$-0.450940\pi$$
0.153518 + 0.988146i $$0.450940\pi$$
$$828$$ 0 0
$$829$$ −4250.77 −0.178088 −0.0890442 0.996028i $$-0.528381\pi$$
−0.0890442 + 0.996028i $$0.528381\pi$$
$$830$$ −6367.28 −0.266279
$$831$$ 0 0
$$832$$ 29178.3 1.21584
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −10149.9 −0.420663
$$836$$ −950.699 −0.0393309
$$837$$ 0 0
$$838$$ 30915.6 1.27442
$$839$$ −39527.7 −1.62652 −0.813258 0.581903i $$-0.802308\pi$$
−0.813258 + 0.581903i $$0.802308\pi$$
$$840$$ 0 0
$$841$$ −24315.7 −0.996993
$$842$$ −6486.92 −0.265504
$$843$$ 0 0
$$844$$ −8363.09 −0.341078
$$845$$ 13142.8 0.535059
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 1076.63 0.0435985
$$849$$ 0 0
$$850$$ −30239.0 −1.22022
$$851$$ −3464.76 −0.139566
$$852$$ 0 0
$$853$$ −31656.1 −1.27067 −0.635337 0.772235i $$-0.719139\pi$$
−0.635337 + 0.772235i $$0.719139\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 17685.1 0.706148
$$857$$ 1193.16 0.0475583 0.0237792 0.999717i $$-0.492430\pi$$
0.0237792 + 0.999717i $$0.492430\pi$$
$$858$$ 0 0
$$859$$ −29060.0 −1.15427 −0.577134 0.816650i $$-0.695829\pi$$
−0.577134 + 0.816650i $$0.695829\pi$$
$$860$$ −70464.8 −2.79399
$$861$$ 0 0
$$862$$ 68611.2 2.71103
$$863$$ −23063.0 −0.909702 −0.454851 0.890567i $$-0.650308\pi$$
−0.454851 + 0.890567i $$0.650308\pi$$
$$864$$ 0 0
$$865$$ 25059.4 0.985024
$$866$$ −24810.9 −0.973566
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 829.371 0.0323757
$$870$$ 0 0
$$871$$ 27698.2 1.07752
$$872$$ 33271.1 1.29209
$$873$$ 0 0
$$874$$ 50831.1 1.96726
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 33871.0 1.30415 0.652077 0.758153i $$-0.273898\pi$$
0.652077 + 0.758153i $$0.273898\pi$$
$$878$$ 8067.88 0.310111
$$879$$ 0 0
$$880$$ −79.1405 −0.00303162
$$881$$ −43331.1 −1.65705 −0.828525 0.559953i $$-0.810819\pi$$
−0.828525 + 0.559953i $$0.810819\pi$$
$$882$$ 0 0
$$883$$ −40897.3 −1.55867 −0.779334 0.626609i $$-0.784442\pi$$
−0.779334 + 0.626609i $$0.784442\pi$$
$$884$$ −51571.6 −1.96215
$$885$$ 0 0
$$886$$ −14769.8 −0.560047
$$887$$ −45065.8 −1.70593 −0.852965 0.521968i $$-0.825198\pi$$
−0.852965 + 0.521968i $$0.825198\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −96066.5 −3.61816
$$891$$ 0 0
$$892$$ −33167.7 −1.24500
$$893$$ 31556.8 1.18254
$$894$$ 0 0
$$895$$ −7047.19 −0.263197
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −30930.3 −1.14940
$$899$$ −703.180 −0.0260872
$$900$$ 0 0
$$901$$ −17556.3 −0.649150
$$902$$ −259.888 −0.00959349
$$903$$ 0 0
$$904$$ 7810.22 0.287350
$$905$$ 39107.9 1.43645
$$906$$ 0 0
$$907$$ 25282.5 0.925570 0.462785 0.886471i $$-0.346850\pi$$
0.462785 + 0.886471i $$0.346850\pi$$
$$908$$ 51642.2 1.88745
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 41646.1 1.51460 0.757298 0.653070i $$-0.226519\pi$$
0.757298 + 0.653070i $$0.226519\pi$$
$$912$$ 0 0
$$913$$ 84.7812 0.00307322
$$914$$ −16765.3 −0.606725
$$915$$ 0 0
$$916$$ −50963.1 −1.83829
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −26112.5 −0.937292 −0.468646 0.883386i $$-0.655258\pi$$
−0.468646 + 0.883386i $$0.655258\pi$$
$$920$$ −33277.7 −1.19254
$$921$$ 0 0
$$922$$ 42596.3 1.52151
$$923$$ 15699.9 0.559879
$$924$$ 0 0
$$925$$ 1633.27 0.0580559
$$926$$ −70159.4 −2.48983
$$927$$ 0 0
$$928$$ −1686.83 −0.0596691
$$929$$ 32357.0 1.14273 0.571366 0.820695i $$-0.306414\pi$$
0.571366 + 0.820695i $$0.306414\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 48929.9 1.71969
$$933$$ 0 0
$$934$$ 9993.95 0.350120
$$935$$ 1290.52 0.0451386
$$936$$ 0 0
$$937$$ −32947.0 −1.14870 −0.574350 0.818610i $$-0.694745\pi$$
−0.574350 + 0.818610i $$0.694745\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −57134.9 −1.98249
$$941$$ −501.602 −0.0173770 −0.00868849 0.999962i $$-0.502766\pi$$
−0.00868849 + 0.999962i $$0.502766\pi$$
$$942$$ 0 0
$$943$$ 8481.06 0.292875
$$944$$ 679.984 0.0234445
$$945$$ 0 0
$$946$$ 1537.24 0.0528328
$$947$$ 6436.90 0.220878 0.110439 0.993883i $$-0.464774\pi$$
0.110439 + 0.993883i $$0.464774\pi$$
$$948$$ 0 0
$$949$$ −16409.2 −0.561292
$$950$$ −23961.5 −0.818331
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −47511.2 −1.61494 −0.807470 0.589908i $$-0.799164\pi$$
−0.807470 + 0.589908i $$0.799164\pi$$
$$954$$ 0 0
$$955$$ 35104.2 1.18947
$$956$$ −67982.9 −2.29992
$$957$$ 0 0
$$958$$ −10645.1 −0.359004
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −23049.4 −0.773705
$$962$$ 4563.78 0.152955
$$963$$ 0 0
$$964$$ 20086.1 0.671087
$$965$$ −9115.15 −0.304069
$$966$$ 0 0
$$967$$ 7817.32 0.259967 0.129984 0.991516i $$-0.458508\pi$$
0.129984 + 0.991516i $$0.458508\pi$$
$$968$$ 27313.4 0.906905
$$969$$ 0 0
$$970$$ −33594.9 −1.11203
$$971$$ 1503.50 0.0496905 0.0248453 0.999691i $$-0.492091\pi$$
0.0248453 + 0.999691i $$0.492091\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −47097.9 −1.54940
$$975$$ 0 0
$$976$$ −870.168 −0.0285383
$$977$$ −33389.1 −1.09336 −0.546680 0.837342i $$-0.684108\pi$$
−0.546680 + 0.837342i $$0.684108\pi$$
$$978$$ 0 0
$$979$$ 1279.14 0.0417584
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 57032.2 1.85333
$$983$$ −5451.02 −0.176867 −0.0884337 0.996082i $$-0.528186\pi$$
−0.0884337 + 0.996082i $$0.528186\pi$$
$$984$$ 0 0
$$985$$ −49683.7 −1.60716
$$986$$ 4568.64 0.147561
$$987$$ 0 0
$$988$$ −40865.6 −1.31590
$$989$$ −50165.4 −1.61291
$$990$$ 0 0
$$991$$ −46530.0 −1.49150 −0.745750 0.666226i $$-0.767908\pi$$
−0.745750 + 0.666226i $$0.767908\pi$$
$$992$$ 16172.0 0.517603
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −10780.6 −0.343484
$$996$$ 0 0
$$997$$ −11410.0 −0.362444 −0.181222 0.983442i $$-0.558005\pi$$
−0.181222 + 0.983442i $$0.558005\pi$$
$$998$$ −48156.5 −1.52742
$$999$$ 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.1.1 4
3.2 odd 2 49.4.a.e.1.3 4
7.2 even 3 441.4.e.y.361.4 8
7.3 odd 6 441.4.e.y.226.3 8
7.4 even 3 441.4.e.y.226.4 8
7.5 odd 6 441.4.e.y.361.3 8
7.6 odd 2 inner 441.4.a.u.1.2 4
12.11 even 2 784.4.a.bf.1.3 4
15.14 odd 2 1225.4.a.bb.1.2 4
21.2 odd 6 49.4.c.e.18.2 8
21.5 even 6 49.4.c.e.18.1 8
21.11 odd 6 49.4.c.e.30.2 8
21.17 even 6 49.4.c.e.30.1 8
21.20 even 2 49.4.a.e.1.4 yes 4
84.83 odd 2 784.4.a.bf.1.2 4
105.104 even 2 1225.4.a.bb.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.a.e.1.3 4 3.2 odd 2
49.4.a.e.1.4 yes 4 21.20 even 2
49.4.c.e.18.1 8 21.5 even 6
49.4.c.e.18.2 8 21.2 odd 6
49.4.c.e.30.1 8 21.17 even 6
49.4.c.e.30.2 8 21.11 odd 6
441.4.a.u.1.1 4 1.1 even 1 trivial
441.4.a.u.1.2 4 7.6 odd 2 inner
441.4.e.y.226.3 8 7.3 odd 6
441.4.e.y.226.4 8 7.4 even 3
441.4.e.y.361.3 8 7.5 odd 6
441.4.e.y.361.4 8 7.2 even 3
784.4.a.bf.1.2 4 84.83 odd 2
784.4.a.bf.1.3 4 12.11 even 2
1225.4.a.bb.1.1 4 105.104 even 2
1225.4.a.bb.1.2 4 15.14 odd 2