# Properties

 Label 135.4.a.g.1.1 Level $135$ Weight $4$ Character 135.1 Self dual yes Analytic conductor $7.965$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 135.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.96525785077$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.5637.1 Defining polynomial: $$x^{3} - x^{2} - 23x + 6$$ x^3 - x^2 - 23*x + 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.45938$$ of defining polynomial Character $$\chi$$ $$=$$ 135.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.45938 q^{2} +11.8861 q^{4} -5.00000 q^{5} +5.08123 q^{7} -17.3296 q^{8} +O(q^{10})$$ $$q-4.45938 q^{2} +11.8861 q^{4} -5.00000 q^{5} +5.08123 q^{7} -17.3296 q^{8} +22.2969 q^{10} -58.3007 q^{11} +21.2119 q^{13} -22.6592 q^{14} -17.8095 q^{16} +68.8451 q^{17} -40.8133 q^{19} -59.4305 q^{20} +259.985 q^{22} +144.318 q^{23} +25.0000 q^{25} -94.5921 q^{26} +60.3960 q^{28} +220.058 q^{29} +291.545 q^{31} +218.056 q^{32} -307.006 q^{34} -25.4062 q^{35} +260.637 q^{37} +182.002 q^{38} +86.6479 q^{40} +169.766 q^{41} -438.596 q^{43} -692.967 q^{44} -643.571 q^{46} +255.481 q^{47} -317.181 q^{49} -111.485 q^{50} +252.127 q^{52} -214.714 q^{53} +291.503 q^{55} -88.0557 q^{56} -981.322 q^{58} -331.524 q^{59} +54.9647 q^{61} -1300.11 q^{62} -829.920 q^{64} -106.060 q^{65} +758.179 q^{67} +818.299 q^{68} +113.296 q^{70} +904.348 q^{71} +866.622 q^{73} -1162.28 q^{74} -485.110 q^{76} -296.239 q^{77} +206.961 q^{79} +89.0475 q^{80} -757.054 q^{82} +463.397 q^{83} -344.225 q^{85} +1955.87 q^{86} +1010.33 q^{88} -601.736 q^{89} +107.783 q^{91} +1715.38 q^{92} -1139.29 q^{94} +204.066 q^{95} +229.363 q^{97} +1414.43 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 23 q^{4} - 15 q^{5} + 44 q^{7} + 36 q^{8}+O(q^{10})$$ 3 * q + q^2 + 23 * q^4 - 15 * q^5 + 44 * q^7 + 36 * q^8 $$3 q + q^{2} + 23 q^{4} - 15 q^{5} + 44 q^{7} + 36 q^{8} - 5 q^{10} - 38 q^{11} + 28 q^{13} + 108 q^{14} + 191 q^{16} + 19 q^{17} + 187 q^{19} - 115 q^{20} + 122 q^{22} + 81 q^{23} + 75 q^{25} - 416 q^{26} + 410 q^{28} - 160 q^{29} + 227 q^{31} + 569 q^{32} + 17 q^{34} - 220 q^{35} + 78 q^{37} + 757 q^{38} - 180 q^{40} + 338 q^{41} + 22 q^{43} - 1636 q^{44} - 1425 q^{46} + 472 q^{47} - 197 q^{49} + 25 q^{50} - 1566 q^{52} - 521 q^{53} + 190 q^{55} + 1254 q^{56} - 2096 q^{58} - 140 q^{59} + 595 q^{61} - 1407 q^{62} - 918 q^{64} - 140 q^{65} + 878 q^{67} + 3053 q^{68} - 540 q^{70} + 602 q^{71} + 1294 q^{73} - 2878 q^{74} + 525 q^{76} - 288 q^{77} + 629 q^{79} - 955 q^{80} - 1682 q^{82} + 1287 q^{83} - 95 q^{85} + 3730 q^{86} - 858 q^{88} - 2154 q^{89} - 440 q^{91} - 1959 q^{92} - 1108 q^{94} - 935 q^{95} + 1392 q^{97} + 2693 q^{98}+O(q^{100})$$ 3 * q + q^2 + 23 * q^4 - 15 * q^5 + 44 * q^7 + 36 * q^8 - 5 * q^10 - 38 * q^11 + 28 * q^13 + 108 * q^14 + 191 * q^16 + 19 * q^17 + 187 * q^19 - 115 * q^20 + 122 * q^22 + 81 * q^23 + 75 * q^25 - 416 * q^26 + 410 * q^28 - 160 * q^29 + 227 * q^31 + 569 * q^32 + 17 * q^34 - 220 * q^35 + 78 * q^37 + 757 * q^38 - 180 * q^40 + 338 * q^41 + 22 * q^43 - 1636 * q^44 - 1425 * q^46 + 472 * q^47 - 197 * q^49 + 25 * q^50 - 1566 * q^52 - 521 * q^53 + 190 * q^55 + 1254 * q^56 - 2096 * q^58 - 140 * q^59 + 595 * q^61 - 1407 * q^62 - 918 * q^64 - 140 * q^65 + 878 * q^67 + 3053 * q^68 - 540 * q^70 + 602 * q^71 + 1294 * q^73 - 2878 * q^74 + 525 * q^76 - 288 * q^77 + 629 * q^79 - 955 * q^80 - 1682 * q^82 + 1287 * q^83 - 95 * q^85 + 3730 * q^86 - 858 * q^88 - 2154 * q^89 - 440 * q^91 - 1959 * q^92 - 1108 * q^94 - 935 * q^95 + 1392 * q^97 + 2693 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.45938 −1.57663 −0.788315 0.615272i $$-0.789046\pi$$
−0.788315 + 0.615272i $$0.789046\pi$$
$$3$$ 0 0
$$4$$ 11.8861 1.48576
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 5.08123 0.274361 0.137180 0.990546i $$-0.456196\pi$$
0.137180 + 0.990546i $$0.456196\pi$$
$$8$$ −17.3296 −0.765867
$$9$$ 0 0
$$10$$ 22.2969 0.705090
$$11$$ −58.3007 −1.59803 −0.799014 0.601312i $$-0.794645\pi$$
−0.799014 + 0.601312i $$0.794645\pi$$
$$12$$ 0 0
$$13$$ 21.2119 0.452548 0.226274 0.974064i $$-0.427345\pi$$
0.226274 + 0.974064i $$0.427345\pi$$
$$14$$ −22.6592 −0.432566
$$15$$ 0 0
$$16$$ −17.8095 −0.278274
$$17$$ 68.8451 0.982199 0.491099 0.871104i $$-0.336595\pi$$
0.491099 + 0.871104i $$0.336595\pi$$
$$18$$ 0 0
$$19$$ −40.8133 −0.492800 −0.246400 0.969168i $$-0.579248\pi$$
−0.246400 + 0.969168i $$0.579248\pi$$
$$20$$ −59.4305 −0.664453
$$21$$ 0 0
$$22$$ 259.985 2.51950
$$23$$ 144.318 1.30837 0.654184 0.756336i $$-0.273012\pi$$
0.654184 + 0.756336i $$0.273012\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ −94.5921 −0.713501
$$27$$ 0 0
$$28$$ 60.3960 0.407635
$$29$$ 220.058 1.40909 0.704547 0.709657i $$-0.251150\pi$$
0.704547 + 0.709657i $$0.251150\pi$$
$$30$$ 0 0
$$31$$ 291.545 1.68913 0.844566 0.535452i $$-0.179859\pi$$
0.844566 + 0.535452i $$0.179859\pi$$
$$32$$ 218.056 1.20460
$$33$$ 0 0
$$34$$ −307.006 −1.54856
$$35$$ −25.4062 −0.122698
$$36$$ 0 0
$$37$$ 260.637 1.15807 0.579033 0.815304i $$-0.303430\pi$$
0.579033 + 0.815304i $$0.303430\pi$$
$$38$$ 182.002 0.776964
$$39$$ 0 0
$$40$$ 86.6479 0.342506
$$41$$ 169.766 0.646660 0.323330 0.946286i $$-0.395198\pi$$
0.323330 + 0.946286i $$0.395198\pi$$
$$42$$ 0 0
$$43$$ −438.596 −1.55547 −0.777735 0.628592i $$-0.783631\pi$$
−0.777735 + 0.628592i $$0.783631\pi$$
$$44$$ −692.967 −2.37429
$$45$$ 0 0
$$46$$ −643.571 −2.06281
$$47$$ 255.481 0.792887 0.396444 0.918059i $$-0.370244\pi$$
0.396444 + 0.918059i $$0.370244\pi$$
$$48$$ 0 0
$$49$$ −317.181 −0.924726
$$50$$ −111.485 −0.315326
$$51$$ 0 0
$$52$$ 252.127 0.672379
$$53$$ −214.714 −0.556477 −0.278239 0.960512i $$-0.589751\pi$$
−0.278239 + 0.960512i $$0.589751\pi$$
$$54$$ 0 0
$$55$$ 291.503 0.714660
$$56$$ −88.0557 −0.210124
$$57$$ 0 0
$$58$$ −981.322 −2.22162
$$59$$ −331.524 −0.731537 −0.365769 0.930706i $$-0.619194\pi$$
−0.365769 + 0.930706i $$0.619194\pi$$
$$60$$ 0 0
$$61$$ 54.9647 0.115369 0.0576845 0.998335i $$-0.481628\pi$$
0.0576845 + 0.998335i $$0.481628\pi$$
$$62$$ −1300.11 −2.66314
$$63$$ 0 0
$$64$$ −829.920 −1.62094
$$65$$ −106.060 −0.202386
$$66$$ 0 0
$$67$$ 758.179 1.38248 0.691241 0.722624i $$-0.257064\pi$$
0.691241 + 0.722624i $$0.257064\pi$$
$$68$$ 818.299 1.45931
$$69$$ 0 0
$$70$$ 113.296 0.193449
$$71$$ 904.348 1.51164 0.755819 0.654780i $$-0.227239\pi$$
0.755819 + 0.654780i $$0.227239\pi$$
$$72$$ 0 0
$$73$$ 866.622 1.38946 0.694729 0.719271i $$-0.255524\pi$$
0.694729 + 0.719271i $$0.255524\pi$$
$$74$$ −1162.28 −1.82584
$$75$$ 0 0
$$76$$ −485.110 −0.732184
$$77$$ −296.239 −0.438437
$$78$$ 0 0
$$79$$ 206.961 0.294746 0.147373 0.989081i $$-0.452918\pi$$
0.147373 + 0.989081i $$0.452918\pi$$
$$80$$ 89.0475 0.124448
$$81$$ 0 0
$$82$$ −757.054 −1.01954
$$83$$ 463.397 0.612825 0.306412 0.951899i $$-0.400871\pi$$
0.306412 + 0.951899i $$0.400871\pi$$
$$84$$ 0 0
$$85$$ −344.225 −0.439253
$$86$$ 1955.87 2.45240
$$87$$ 0 0
$$88$$ 1010.33 1.22388
$$89$$ −601.736 −0.716673 −0.358337 0.933592i $$-0.616656\pi$$
−0.358337 + 0.933592i $$0.616656\pi$$
$$90$$ 0 0
$$91$$ 107.783 0.124162
$$92$$ 1715.38 1.94392
$$93$$ 0 0
$$94$$ −1139.29 −1.25009
$$95$$ 204.066 0.220387
$$96$$ 0 0
$$97$$ 229.363 0.240086 0.120043 0.992769i $$-0.461697\pi$$
0.120043 + 0.992769i $$0.461697\pi$$
$$98$$ 1414.43 1.45795
$$99$$ 0 0
$$100$$ 297.152 0.297152
$$101$$ −1345.66 −1.32573 −0.662863 0.748740i $$-0.730659\pi$$
−0.662863 + 0.748740i $$0.730659\pi$$
$$102$$ 0 0
$$103$$ −1596.30 −1.52707 −0.763534 0.645768i $$-0.776537\pi$$
−0.763534 + 0.645768i $$0.776537\pi$$
$$104$$ −367.594 −0.346592
$$105$$ 0 0
$$106$$ 957.494 0.877358
$$107$$ −958.786 −0.866256 −0.433128 0.901333i $$-0.642590\pi$$
−0.433128 + 0.901333i $$0.642590\pi$$
$$108$$ 0 0
$$109$$ 1690.23 1.48527 0.742635 0.669696i $$-0.233576\pi$$
0.742635 + 0.669696i $$0.233576\pi$$
$$110$$ −1299.93 −1.12675
$$111$$ 0 0
$$112$$ −90.4943 −0.0763474
$$113$$ 11.6211 0.00967456 0.00483728 0.999988i $$-0.498460\pi$$
0.00483728 + 0.999988i $$0.498460\pi$$
$$114$$ 0 0
$$115$$ −721.592 −0.585120
$$116$$ 2615.63 2.09358
$$117$$ 0 0
$$118$$ 1478.39 1.15336
$$119$$ 349.818 0.269477
$$120$$ 0 0
$$121$$ 2067.97 1.55370
$$122$$ −245.109 −0.181894
$$123$$ 0 0
$$124$$ 3465.34 2.50965
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 309.141 0.215999 0.107999 0.994151i $$-0.465556\pi$$
0.107999 + 0.994151i $$0.465556\pi$$
$$128$$ 1956.48 1.35102
$$129$$ 0 0
$$130$$ 472.960 0.319087
$$131$$ −2785.03 −1.85747 −0.928736 0.370742i $$-0.879103\pi$$
−0.928736 + 0.370742i $$0.879103\pi$$
$$132$$ 0 0
$$133$$ −207.382 −0.135205
$$134$$ −3381.01 −2.17966
$$135$$ 0 0
$$136$$ −1193.06 −0.752233
$$137$$ 2489.41 1.55244 0.776222 0.630460i $$-0.217134\pi$$
0.776222 + 0.630460i $$0.217134\pi$$
$$138$$ 0 0
$$139$$ 1786.05 1.08986 0.544931 0.838481i $$-0.316556\pi$$
0.544931 + 0.838481i $$0.316556\pi$$
$$140$$ −301.980 −0.182300
$$141$$ 0 0
$$142$$ −4032.83 −2.38329
$$143$$ −1236.67 −0.723185
$$144$$ 0 0
$$145$$ −1100.29 −0.630166
$$146$$ −3864.60 −2.19066
$$147$$ 0 0
$$148$$ 3097.95 1.72061
$$149$$ −1568.83 −0.862575 −0.431288 0.902214i $$-0.641941\pi$$
−0.431288 + 0.902214i $$0.641941\pi$$
$$150$$ 0 0
$$151$$ −438.327 −0.236229 −0.118114 0.993000i $$-0.537685\pi$$
−0.118114 + 0.993000i $$0.537685\pi$$
$$152$$ 707.277 0.377419
$$153$$ 0 0
$$154$$ 1321.04 0.691252
$$155$$ −1457.73 −0.755403
$$156$$ 0 0
$$157$$ −44.7479 −0.0227469 −0.0113735 0.999935i $$-0.503620\pi$$
−0.0113735 + 0.999935i $$0.503620\pi$$
$$158$$ −922.917 −0.464705
$$159$$ 0 0
$$160$$ −1090.28 −0.538714
$$161$$ 733.315 0.358965
$$162$$ 0 0
$$163$$ −2611.84 −1.25506 −0.627531 0.778591i $$-0.715935\pi$$
−0.627531 + 0.778591i $$0.715935\pi$$
$$164$$ 2017.86 0.960783
$$165$$ 0 0
$$166$$ −2066.47 −0.966198
$$167$$ 188.947 0.0875516 0.0437758 0.999041i $$-0.486061\pi$$
0.0437758 + 0.999041i $$0.486061\pi$$
$$168$$ 0 0
$$169$$ −1747.05 −0.795200
$$170$$ 1535.03 0.692539
$$171$$ 0 0
$$172$$ −5213.19 −2.31106
$$173$$ 1505.02 0.661413 0.330707 0.943734i $$-0.392713\pi$$
0.330707 + 0.943734i $$0.392713\pi$$
$$174$$ 0 0
$$175$$ 127.031 0.0548722
$$176$$ 1038.31 0.444689
$$177$$ 0 0
$$178$$ 2683.37 1.12993
$$179$$ 3136.62 1.30973 0.654865 0.755746i $$-0.272725\pi$$
0.654865 + 0.755746i $$0.272725\pi$$
$$180$$ 0 0
$$181$$ 4512.67 1.85317 0.926586 0.376084i $$-0.122730\pi$$
0.926586 + 0.376084i $$0.122730\pi$$
$$182$$ −480.644 −0.195757
$$183$$ 0 0
$$184$$ −2500.98 −1.00204
$$185$$ −1303.18 −0.517902
$$186$$ 0 0
$$187$$ −4013.71 −1.56958
$$188$$ 3036.67 1.17804
$$189$$ 0 0
$$190$$ −910.010 −0.347469
$$191$$ −1207.43 −0.457418 −0.228709 0.973495i $$-0.573450\pi$$
−0.228709 + 0.973495i $$0.573450\pi$$
$$192$$ 0 0
$$193$$ 923.164 0.344305 0.172152 0.985070i $$-0.444928\pi$$
0.172152 + 0.985070i $$0.444928\pi$$
$$194$$ −1022.82 −0.378526
$$195$$ 0 0
$$196$$ −3770.04 −1.37392
$$197$$ 1180.87 0.427075 0.213537 0.976935i $$-0.431501\pi$$
0.213537 + 0.976935i $$0.431501\pi$$
$$198$$ 0 0
$$199$$ −839.805 −0.299157 −0.149578 0.988750i $$-0.547792\pi$$
−0.149578 + 0.988750i $$0.547792\pi$$
$$200$$ −433.240 −0.153173
$$201$$ 0 0
$$202$$ 6000.82 2.09018
$$203$$ 1118.17 0.386600
$$204$$ 0 0
$$205$$ −848.832 −0.289195
$$206$$ 7118.51 2.40762
$$207$$ 0 0
$$208$$ −377.774 −0.125932
$$209$$ 2379.44 0.787509
$$210$$ 0 0
$$211$$ −2589.65 −0.844923 −0.422461 0.906381i $$-0.638834\pi$$
−0.422461 + 0.906381i $$0.638834\pi$$
$$212$$ −2552.12 −0.826792
$$213$$ 0 0
$$214$$ 4275.59 1.36576
$$215$$ 2192.98 0.695627
$$216$$ 0 0
$$217$$ 1481.41 0.463432
$$218$$ −7537.38 −2.34172
$$219$$ 0 0
$$220$$ 3464.84 1.06181
$$221$$ 1460.34 0.444492
$$222$$ 0 0
$$223$$ −4180.76 −1.25544 −0.627722 0.778437i $$-0.716013\pi$$
−0.627722 + 0.778437i $$0.716013\pi$$
$$224$$ 1107.99 0.330495
$$225$$ 0 0
$$226$$ −51.8232 −0.0152532
$$227$$ 2602.67 0.760993 0.380497 0.924782i $$-0.375753\pi$$
0.380497 + 0.924782i $$0.375753\pi$$
$$228$$ 0 0
$$229$$ −1845.35 −0.532508 −0.266254 0.963903i $$-0.585786\pi$$
−0.266254 + 0.963903i $$0.585786\pi$$
$$230$$ 3217.85 0.922517
$$231$$ 0 0
$$232$$ −3813.51 −1.07918
$$233$$ −240.637 −0.0676594 −0.0338297 0.999428i $$-0.510770\pi$$
−0.0338297 + 0.999428i $$0.510770\pi$$
$$234$$ 0 0
$$235$$ −1277.40 −0.354590
$$236$$ −3940.52 −1.08689
$$237$$ 0 0
$$238$$ −1559.97 −0.424865
$$239$$ −2567.64 −0.694924 −0.347462 0.937694i $$-0.612956\pi$$
−0.347462 + 0.937694i $$0.612956\pi$$
$$240$$ 0 0
$$241$$ 3987.99 1.06593 0.532965 0.846137i $$-0.321078\pi$$
0.532965 + 0.846137i $$0.321078\pi$$
$$242$$ −9221.86 −2.44960
$$243$$ 0 0
$$244$$ 653.315 0.171411
$$245$$ 1585.91 0.413550
$$246$$ 0 0
$$247$$ −865.728 −0.223016
$$248$$ −5052.36 −1.29365
$$249$$ 0 0
$$250$$ 557.423 0.141018
$$251$$ −967.393 −0.243272 −0.121636 0.992575i $$-0.538814\pi$$
−0.121636 + 0.992575i $$0.538814\pi$$
$$252$$ 0 0
$$253$$ −8413.86 −2.09081
$$254$$ −1378.58 −0.340550
$$255$$ 0 0
$$256$$ −2085.34 −0.509116
$$257$$ −3743.03 −0.908498 −0.454249 0.890875i $$-0.650092\pi$$
−0.454249 + 0.890875i $$0.650092\pi$$
$$258$$ 0 0
$$259$$ 1324.36 0.317728
$$260$$ −1260.63 −0.300697
$$261$$ 0 0
$$262$$ 12419.5 2.92855
$$263$$ 2497.47 0.585554 0.292777 0.956181i $$-0.405421\pi$$
0.292777 + 0.956181i $$0.405421\pi$$
$$264$$ 0 0
$$265$$ 1073.57 0.248864
$$266$$ 924.795 0.213168
$$267$$ 0 0
$$268$$ 9011.79 2.05404
$$269$$ 2142.91 0.485709 0.242855 0.970063i $$-0.421916\pi$$
0.242855 + 0.970063i $$0.421916\pi$$
$$270$$ 0 0
$$271$$ 1540.64 0.345341 0.172671 0.984980i $$-0.444760\pi$$
0.172671 + 0.984980i $$0.444760\pi$$
$$272$$ −1226.10 −0.273320
$$273$$ 0 0
$$274$$ −11101.2 −2.44763
$$275$$ −1457.52 −0.319606
$$276$$ 0 0
$$277$$ 6777.80 1.47018 0.735088 0.677972i $$-0.237141\pi$$
0.735088 + 0.677972i $$0.237141\pi$$
$$278$$ −7964.69 −1.71831
$$279$$ 0 0
$$280$$ 440.278 0.0939702
$$281$$ −827.653 −0.175707 −0.0878535 0.996133i $$-0.528001\pi$$
−0.0878535 + 0.996133i $$0.528001\pi$$
$$282$$ 0 0
$$283$$ −3171.98 −0.666270 −0.333135 0.942879i $$-0.608106\pi$$
−0.333135 + 0.942879i $$0.608106\pi$$
$$284$$ 10749.2 2.24593
$$285$$ 0 0
$$286$$ 5514.78 1.14020
$$287$$ 862.623 0.177418
$$288$$ 0 0
$$289$$ −173.358 −0.0352856
$$290$$ 4906.61 0.993539
$$291$$ 0 0
$$292$$ 10300.8 2.06440
$$293$$ −1376.02 −0.274362 −0.137181 0.990546i $$-0.543804\pi$$
−0.137181 + 0.990546i $$0.543804\pi$$
$$294$$ 0 0
$$295$$ 1657.62 0.327153
$$296$$ −4516.73 −0.886923
$$297$$ 0 0
$$298$$ 6996.02 1.35996
$$299$$ 3061.27 0.592099
$$300$$ 0 0
$$301$$ −2228.61 −0.426760
$$302$$ 1954.67 0.372445
$$303$$ 0 0
$$304$$ 726.864 0.137133
$$305$$ −274.823 −0.0515946
$$306$$ 0 0
$$307$$ −119.504 −0.0222165 −0.0111083 0.999938i $$-0.503536\pi$$
−0.0111083 + 0.999938i $$0.503536\pi$$
$$308$$ −3521.13 −0.651412
$$309$$ 0 0
$$310$$ 6500.56 1.19099
$$311$$ −2139.35 −0.390069 −0.195035 0.980796i $$-0.562482\pi$$
−0.195035 + 0.980796i $$0.562482\pi$$
$$312$$ 0 0
$$313$$ 5163.50 0.932455 0.466227 0.884665i $$-0.345613\pi$$
0.466227 + 0.884665i $$0.345613\pi$$
$$314$$ 199.548 0.0358635
$$315$$ 0 0
$$316$$ 2459.95 0.437922
$$317$$ 8631.69 1.52935 0.764675 0.644416i $$-0.222899\pi$$
0.764675 + 0.644416i $$0.222899\pi$$
$$318$$ 0 0
$$319$$ −12829.5 −2.25177
$$320$$ 4149.60 0.724905
$$321$$ 0 0
$$322$$ −3270.13 −0.565955
$$323$$ −2809.79 −0.484028
$$324$$ 0 0
$$325$$ 530.298 0.0905097
$$326$$ 11647.2 1.97877
$$327$$ 0 0
$$328$$ −2941.98 −0.495255
$$329$$ 1298.16 0.217537
$$330$$ 0 0
$$331$$ −2942.34 −0.488597 −0.244298 0.969700i $$-0.578558\pi$$
−0.244298 + 0.969700i $$0.578558\pi$$
$$332$$ 5507.98 0.910512
$$333$$ 0 0
$$334$$ −842.585 −0.138037
$$335$$ −3790.90 −0.618265
$$336$$ 0 0
$$337$$ 9897.46 1.59985 0.799924 0.600101i $$-0.204873\pi$$
0.799924 + 0.600101i $$0.204873\pi$$
$$338$$ 7790.79 1.25374
$$339$$ 0 0
$$340$$ −4091.49 −0.652625
$$341$$ −16997.3 −2.69928
$$342$$ 0 0
$$343$$ −3354.53 −0.528070
$$344$$ 7600.68 1.19128
$$345$$ 0 0
$$346$$ −6711.46 −1.04280
$$347$$ 12015.7 1.85890 0.929451 0.368947i $$-0.120282\pi$$
0.929451 + 0.368947i $$0.120282\pi$$
$$348$$ 0 0
$$349$$ 7894.62 1.21086 0.605428 0.795900i $$-0.293002\pi$$
0.605428 + 0.795900i $$0.293002\pi$$
$$350$$ −566.479 −0.0865131
$$351$$ 0 0
$$352$$ −12712.8 −1.92499
$$353$$ −741.154 −0.111750 −0.0558748 0.998438i $$-0.517795\pi$$
−0.0558748 + 0.998438i $$0.517795\pi$$
$$354$$ 0 0
$$355$$ −4521.74 −0.676025
$$356$$ −7152.30 −1.06481
$$357$$ 0 0
$$358$$ −13987.4 −2.06496
$$359$$ 11564.4 1.70012 0.850060 0.526685i $$-0.176565\pi$$
0.850060 + 0.526685i $$0.176565\pi$$
$$360$$ 0 0
$$361$$ −5193.28 −0.757148
$$362$$ −20123.7 −2.92177
$$363$$ 0 0
$$364$$ 1281.12 0.184474
$$365$$ −4333.11 −0.621385
$$366$$ 0 0
$$367$$ −1148.57 −0.163365 −0.0816823 0.996658i $$-0.526029\pi$$
−0.0816823 + 0.996658i $$0.526029\pi$$
$$368$$ −2570.24 −0.364084
$$369$$ 0 0
$$370$$ 5811.39 0.816540
$$371$$ −1091.01 −0.152676
$$372$$ 0 0
$$373$$ −4602.98 −0.638963 −0.319481 0.947593i $$-0.603509\pi$$
−0.319481 + 0.947593i $$0.603509\pi$$
$$374$$ 17898.7 2.47465
$$375$$ 0 0
$$376$$ −4427.38 −0.607246
$$377$$ 4667.85 0.637683
$$378$$ 0 0
$$379$$ −3988.46 −0.540563 −0.270282 0.962781i $$-0.587117\pi$$
−0.270282 + 0.962781i $$0.587117\pi$$
$$380$$ 2425.55 0.327443
$$381$$ 0 0
$$382$$ 5384.41 0.721179
$$383$$ −7788.20 −1.03906 −0.519528 0.854454i $$-0.673892\pi$$
−0.519528 + 0.854454i $$0.673892\pi$$
$$384$$ 0 0
$$385$$ 1481.20 0.196075
$$386$$ −4116.74 −0.542841
$$387$$ 0 0
$$388$$ 2726.23 0.356710
$$389$$ −8524.87 −1.11113 −0.555563 0.831474i $$-0.687497\pi$$
−0.555563 + 0.831474i $$0.687497\pi$$
$$390$$ 0 0
$$391$$ 9935.60 1.28508
$$392$$ 5496.62 0.708217
$$393$$ 0 0
$$394$$ −5265.97 −0.673339
$$395$$ −1034.80 −0.131814
$$396$$ 0 0
$$397$$ −155.729 −0.0196872 −0.00984361 0.999952i $$-0.503133\pi$$
−0.00984361 + 0.999952i $$0.503133\pi$$
$$398$$ 3745.01 0.471659
$$399$$ 0 0
$$400$$ −445.238 −0.0556547
$$401$$ −5933.96 −0.738972 −0.369486 0.929236i $$-0.620466\pi$$
−0.369486 + 0.929236i $$0.620466\pi$$
$$402$$ 0 0
$$403$$ 6184.23 0.764414
$$404$$ −15994.7 −1.96971
$$405$$ 0 0
$$406$$ −4986.33 −0.609526
$$407$$ −15195.3 −1.85062
$$408$$ 0 0
$$409$$ 14161.4 1.71207 0.856035 0.516917i $$-0.172921\pi$$
0.856035 + 0.516917i $$0.172921\pi$$
$$410$$ 3785.27 0.455954
$$411$$ 0 0
$$412$$ −18973.8 −2.26886
$$413$$ −1684.55 −0.200705
$$414$$ 0 0
$$415$$ −2316.99 −0.274064
$$416$$ 4625.39 0.545140
$$417$$ 0 0
$$418$$ −10610.8 −1.24161
$$419$$ −9624.24 −1.12214 −0.561068 0.827770i $$-0.689609\pi$$
−0.561068 + 0.827770i $$0.689609\pi$$
$$420$$ 0 0
$$421$$ −1536.26 −0.177845 −0.0889223 0.996039i $$-0.528342\pi$$
−0.0889223 + 0.996039i $$0.528342\pi$$
$$422$$ 11548.2 1.33213
$$423$$ 0 0
$$424$$ 3720.91 0.426187
$$425$$ 1721.13 0.196440
$$426$$ 0 0
$$427$$ 279.288 0.0316527
$$428$$ −11396.2 −1.28705
$$429$$ 0 0
$$430$$ −9779.33 −1.09675
$$431$$ −11582.2 −1.29441 −0.647207 0.762314i $$-0.724063\pi$$
−0.647207 + 0.762314i $$0.724063\pi$$
$$432$$ 0 0
$$433$$ 14892.6 1.65287 0.826437 0.563029i $$-0.190364\pi$$
0.826437 + 0.563029i $$0.190364\pi$$
$$434$$ −6606.17 −0.730660
$$435$$ 0 0
$$436$$ 20090.2 2.20676
$$437$$ −5890.10 −0.644764
$$438$$ 0 0
$$439$$ −1642.51 −0.178571 −0.0892853 0.996006i $$-0.528458\pi$$
−0.0892853 + 0.996006i $$0.528458\pi$$
$$440$$ −5051.63 −0.547334
$$441$$ 0 0
$$442$$ −6512.20 −0.700800
$$443$$ −3916.31 −0.420021 −0.210011 0.977699i $$-0.567350\pi$$
−0.210011 + 0.977699i $$0.567350\pi$$
$$444$$ 0 0
$$445$$ 3008.68 0.320506
$$446$$ 18643.6 1.97937
$$447$$ 0 0
$$448$$ −4217.02 −0.444722
$$449$$ 3985.25 0.418876 0.209438 0.977822i $$-0.432837\pi$$
0.209438 + 0.977822i $$0.432837\pi$$
$$450$$ 0 0
$$451$$ −9897.50 −1.03338
$$452$$ 138.130 0.0143741
$$453$$ 0 0
$$454$$ −11606.3 −1.19980
$$455$$ −538.914 −0.0555267
$$456$$ 0 0
$$457$$ −14177.8 −1.45122 −0.725611 0.688105i $$-0.758443\pi$$
−0.725611 + 0.688105i $$0.758443\pi$$
$$458$$ 8229.13 0.839567
$$459$$ 0 0
$$460$$ −8576.91 −0.869349
$$461$$ 16394.3 1.65631 0.828154 0.560500i $$-0.189391\pi$$
0.828154 + 0.560500i $$0.189391\pi$$
$$462$$ 0 0
$$463$$ 3319.60 0.333207 0.166603 0.986024i $$-0.446720\pi$$
0.166603 + 0.986024i $$0.446720\pi$$
$$464$$ −3919.12 −0.392114
$$465$$ 0 0
$$466$$ 1073.09 0.106674
$$467$$ 2529.79 0.250674 0.125337 0.992114i $$-0.459999\pi$$
0.125337 + 0.992114i $$0.459999\pi$$
$$468$$ 0 0
$$469$$ 3852.49 0.379299
$$470$$ 5696.43 0.559057
$$471$$ 0 0
$$472$$ 5745.17 0.560260
$$473$$ 25570.4 2.48569
$$474$$ 0 0
$$475$$ −1020.33 −0.0985601
$$476$$ 4157.97 0.400379
$$477$$ 0 0
$$478$$ 11450.1 1.09564
$$479$$ 8646.75 0.824802 0.412401 0.911002i $$-0.364690\pi$$
0.412401 + 0.911002i $$0.364690\pi$$
$$480$$ 0 0
$$481$$ 5528.60 0.524080
$$482$$ −17784.0 −1.68058
$$483$$ 0 0
$$484$$ 24580.1 2.30842
$$485$$ −1146.82 −0.107370
$$486$$ 0 0
$$487$$ −15251.7 −1.41914 −0.709569 0.704636i $$-0.751110\pi$$
−0.709569 + 0.704636i $$0.751110\pi$$
$$488$$ −952.515 −0.0883572
$$489$$ 0 0
$$490$$ −7072.16 −0.652015
$$491$$ 1766.87 0.162398 0.0811992 0.996698i $$-0.474125\pi$$
0.0811992 + 0.996698i $$0.474125\pi$$
$$492$$ 0 0
$$493$$ 15149.9 1.38401
$$494$$ 3860.61 0.351614
$$495$$ 0 0
$$496$$ −5192.28 −0.470041
$$497$$ 4595.20 0.414734
$$498$$ 0 0
$$499$$ 11733.8 1.05266 0.526332 0.850279i $$-0.323567\pi$$
0.526332 + 0.850279i $$0.323567\pi$$
$$500$$ −1485.76 −0.132891
$$501$$ 0 0
$$502$$ 4313.98 0.383550
$$503$$ 977.608 0.0866589 0.0433294 0.999061i $$-0.486203\pi$$
0.0433294 + 0.999061i $$0.486203\pi$$
$$504$$ 0 0
$$505$$ 6728.31 0.592883
$$506$$ 37520.6 3.29643
$$507$$ 0 0
$$508$$ 3674.48 0.320923
$$509$$ −9674.72 −0.842484 −0.421242 0.906948i $$-0.638406\pi$$
−0.421242 + 0.906948i $$0.638406\pi$$
$$510$$ 0 0
$$511$$ 4403.51 0.381213
$$512$$ −6352.52 −0.548329
$$513$$ 0 0
$$514$$ 16691.6 1.43237
$$515$$ 7981.49 0.682926
$$516$$ 0 0
$$517$$ −14894.7 −1.26706
$$518$$ −5905.81 −0.500939
$$519$$ 0 0
$$520$$ 1837.97 0.155000
$$521$$ −5178.95 −0.435497 −0.217748 0.976005i $$-0.569871\pi$$
−0.217748 + 0.976005i $$0.569871\pi$$
$$522$$ 0 0
$$523$$ 14280.7 1.19398 0.596992 0.802248i $$-0.296363\pi$$
0.596992 + 0.802248i $$0.296363\pi$$
$$524$$ −33103.1 −2.75976
$$525$$ 0 0
$$526$$ −11137.2 −0.923203
$$527$$ 20071.5 1.65906
$$528$$ 0 0
$$529$$ 8660.78 0.711826
$$530$$ −4787.47 −0.392367
$$531$$ 0 0
$$532$$ −2464.96 −0.200883
$$533$$ 3601.07 0.292645
$$534$$ 0 0
$$535$$ 4793.93 0.387401
$$536$$ −13138.9 −1.05880
$$537$$ 0 0
$$538$$ −9556.08 −0.765784
$$539$$ 18491.9 1.47774
$$540$$ 0 0
$$541$$ 12923.8 1.02706 0.513529 0.858072i $$-0.328338\pi$$
0.513529 + 0.858072i $$0.328338\pi$$
$$542$$ −6870.32 −0.544475
$$543$$ 0 0
$$544$$ 15012.1 1.18316
$$545$$ −8451.14 −0.664233
$$546$$ 0 0
$$547$$ 13653.3 1.06723 0.533614 0.845728i $$-0.320834\pi$$
0.533614 + 0.845728i $$0.320834\pi$$
$$548$$ 29589.4 2.30656
$$549$$ 0 0
$$550$$ 6499.63 0.503900
$$551$$ −8981.28 −0.694402
$$552$$ 0 0
$$553$$ 1051.62 0.0808666
$$554$$ −30224.8 −2.31792
$$555$$ 0 0
$$556$$ 21229.2 1.61928
$$557$$ 9313.63 0.708494 0.354247 0.935152i $$-0.384737\pi$$
0.354247 + 0.935152i $$0.384737\pi$$
$$558$$ 0 0
$$559$$ −9303.46 −0.703925
$$560$$ 452.471 0.0341436
$$561$$ 0 0
$$562$$ 3690.82 0.277025
$$563$$ 10625.4 0.795394 0.397697 0.917517i $$-0.369810\pi$$
0.397697 + 0.917517i $$0.369810\pi$$
$$564$$ 0 0
$$565$$ −58.1057 −0.00432660
$$566$$ 14145.1 1.05046
$$567$$ 0 0
$$568$$ −15672.0 −1.15771
$$569$$ −9060.50 −0.667550 −0.333775 0.942653i $$-0.608323\pi$$
−0.333775 + 0.942653i $$0.608323\pi$$
$$570$$ 0 0
$$571$$ −21379.1 −1.56688 −0.783440 0.621467i $$-0.786537\pi$$
−0.783440 + 0.621467i $$0.786537\pi$$
$$572$$ −14699.2 −1.07448
$$573$$ 0 0
$$574$$ −3846.77 −0.279723
$$575$$ 3607.96 0.261674
$$576$$ 0 0
$$577$$ 6347.76 0.457991 0.228996 0.973427i $$-0.426456\pi$$
0.228996 + 0.973427i $$0.426456\pi$$
$$578$$ 773.071 0.0556324
$$579$$ 0 0
$$580$$ −13078.1 −0.936277
$$581$$ 2354.63 0.168135
$$582$$ 0 0
$$583$$ 12518.0 0.889266
$$584$$ −15018.2 −1.06414
$$585$$ 0 0
$$586$$ 6136.22 0.432568
$$587$$ −14773.7 −1.03880 −0.519401 0.854531i $$-0.673845\pi$$
−0.519401 + 0.854531i $$0.673845\pi$$
$$588$$ 0 0
$$589$$ −11898.9 −0.832405
$$590$$ −7391.95 −0.515800
$$591$$ 0 0
$$592$$ −4641.81 −0.322259
$$593$$ −26868.1 −1.86061 −0.930304 0.366790i $$-0.880457\pi$$
−0.930304 + 0.366790i $$0.880457\pi$$
$$594$$ 0 0
$$595$$ −1749.09 −0.120514
$$596$$ −18647.3 −1.28158
$$597$$ 0 0
$$598$$ −13651.4 −0.933522
$$599$$ 1749.83 0.119359 0.0596794 0.998218i $$-0.480992\pi$$
0.0596794 + 0.998218i $$0.480992\pi$$
$$600$$ 0 0
$$601$$ −17964.0 −1.21924 −0.609622 0.792692i $$-0.708679\pi$$
−0.609622 + 0.792692i $$0.708679\pi$$
$$602$$ 9938.21 0.672843
$$603$$ 0 0
$$604$$ −5209.99 −0.350979
$$605$$ −10339.8 −0.694834
$$606$$ 0 0
$$607$$ −4418.22 −0.295437 −0.147718 0.989029i $$-0.547193\pi$$
−0.147718 + 0.989029i $$0.547193\pi$$
$$608$$ −8899.58 −0.593628
$$609$$ 0 0
$$610$$ 1225.54 0.0813455
$$611$$ 5419.24 0.358820
$$612$$ 0 0
$$613$$ −20179.7 −1.32961 −0.664804 0.747018i $$-0.731485\pi$$
−0.664804 + 0.747018i $$0.731485\pi$$
$$614$$ 532.915 0.0350272
$$615$$ 0 0
$$616$$ 5133.71 0.335784
$$617$$ 4673.01 0.304908 0.152454 0.988311i $$-0.451282\pi$$
0.152454 + 0.988311i $$0.451282\pi$$
$$618$$ 0 0
$$619$$ −19976.8 −1.29715 −0.648574 0.761151i $$-0.724634\pi$$
−0.648574 + 0.761151i $$0.724634\pi$$
$$620$$ −17326.7 −1.12235
$$621$$ 0 0
$$622$$ 9540.19 0.614994
$$623$$ −3057.56 −0.196627
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −23026.0 −1.47014
$$627$$ 0 0
$$628$$ −531.877 −0.0337965
$$629$$ 17943.5 1.13745
$$630$$ 0 0
$$631$$ 10457.5 0.659757 0.329879 0.944023i $$-0.392992\pi$$
0.329879 + 0.944023i $$0.392992\pi$$
$$632$$ −3586.54 −0.225736
$$633$$ 0 0
$$634$$ −38492.0 −2.41122
$$635$$ −1545.70 −0.0965975
$$636$$ 0 0
$$637$$ −6728.02 −0.418483
$$638$$ 57211.8 3.55021
$$639$$ 0 0
$$640$$ −9782.40 −0.604193
$$641$$ −4395.86 −0.270867 −0.135434 0.990786i $$-0.543243\pi$$
−0.135434 + 0.990786i $$0.543243\pi$$
$$642$$ 0 0
$$643$$ 5786.36 0.354886 0.177443 0.984131i $$-0.443217\pi$$
0.177443 + 0.984131i $$0.443217\pi$$
$$644$$ 8716.26 0.533336
$$645$$ 0 0
$$646$$ 12529.9 0.763133
$$647$$ 25367.7 1.54143 0.770717 0.637178i $$-0.219898\pi$$
0.770717 + 0.637178i $$0.219898\pi$$
$$648$$ 0 0
$$649$$ 19328.1 1.16902
$$650$$ −2364.80 −0.142700
$$651$$ 0 0
$$652$$ −31044.6 −1.86472
$$653$$ 21633.2 1.29644 0.648218 0.761454i $$-0.275514\pi$$
0.648218 + 0.761454i $$0.275514\pi$$
$$654$$ 0 0
$$655$$ 13925.1 0.830687
$$656$$ −3023.46 −0.179948
$$657$$ 0 0
$$658$$ −5788.98 −0.342976
$$659$$ −18312.4 −1.08248 −0.541238 0.840870i $$-0.682044\pi$$
−0.541238 + 0.840870i $$0.682044\pi$$
$$660$$ 0 0
$$661$$ −5526.08 −0.325174 −0.162587 0.986694i $$-0.551984\pi$$
−0.162587 + 0.986694i $$0.551984\pi$$
$$662$$ 13121.0 0.770337
$$663$$ 0 0
$$664$$ −8030.48 −0.469342
$$665$$ 1036.91 0.0604656
$$666$$ 0 0
$$667$$ 31758.4 1.84361
$$668$$ 2245.84 0.130081
$$669$$ 0 0
$$670$$ 16905.1 0.974775
$$671$$ −3204.48 −0.184363
$$672$$ 0 0
$$673$$ 1437.24 0.0823204 0.0411602 0.999153i $$-0.486895\pi$$
0.0411602 + 0.999153i $$0.486895\pi$$
$$674$$ −44136.6 −2.52237
$$675$$ 0 0
$$676$$ −20765.7 −1.18148
$$677$$ 23405.9 1.32875 0.664373 0.747401i $$-0.268699\pi$$
0.664373 + 0.747401i $$0.268699\pi$$
$$678$$ 0 0
$$679$$ 1165.45 0.0658701
$$680$$ 5965.28 0.336409
$$681$$ 0 0
$$682$$ 75797.4 4.25577
$$683$$ −19227.4 −1.07719 −0.538593 0.842566i $$-0.681044\pi$$
−0.538593 + 0.842566i $$0.681044\pi$$
$$684$$ 0 0
$$685$$ −12447.1 −0.694274
$$686$$ 14959.2 0.832570
$$687$$ 0 0
$$688$$ 7811.17 0.432846
$$689$$ −4554.50 −0.251833
$$690$$ 0 0
$$691$$ −35284.8 −1.94254 −0.971271 0.237975i $$-0.923516\pi$$
−0.971271 + 0.237975i $$0.923516\pi$$
$$692$$ 17888.8 0.982703
$$693$$ 0 0
$$694$$ −53582.8 −2.93080
$$695$$ −8930.26 −0.487401
$$696$$ 0 0
$$697$$ 11687.6 0.635149
$$698$$ −35205.1 −1.90907
$$699$$ 0 0
$$700$$ 1509.90 0.0815270
$$701$$ −9173.00 −0.494236 −0.247118 0.968985i $$-0.579484\pi$$
−0.247118 + 0.968985i $$0.579484\pi$$
$$702$$ 0 0
$$703$$ −10637.4 −0.570695
$$704$$ 48384.9 2.59030
$$705$$ 0 0
$$706$$ 3305.09 0.176188
$$707$$ −6837.63 −0.363728
$$708$$ 0 0
$$709$$ −33951.6 −1.79842 −0.899210 0.437517i $$-0.855858\pi$$
−0.899210 + 0.437517i $$0.855858\pi$$
$$710$$ 20164.2 1.06584
$$711$$ 0 0
$$712$$ 10427.8 0.548876
$$713$$ 42075.3 2.21001
$$714$$ 0 0
$$715$$ 6183.35 0.323418
$$716$$ 37282.1 1.94595
$$717$$ 0 0
$$718$$ −51569.9 −2.68046
$$719$$ 6727.90 0.348968 0.174484 0.984660i $$-0.444174\pi$$
0.174484 + 0.984660i $$0.444174\pi$$
$$720$$ 0 0
$$721$$ −8111.17 −0.418968
$$722$$ 23158.8 1.19374
$$723$$ 0 0
$$724$$ 53638.0 2.75337
$$725$$ 5501.45 0.281819
$$726$$ 0 0
$$727$$ −36726.1 −1.87359 −0.936793 0.349885i $$-0.886221\pi$$
−0.936793 + 0.349885i $$0.886221\pi$$
$$728$$ −1867.83 −0.0950912
$$729$$ 0 0
$$730$$ 19323.0 0.979694
$$731$$ −30195.1 −1.52778
$$732$$ 0 0
$$733$$ 26691.4 1.34498 0.672489 0.740108i $$-0.265225\pi$$
0.672489 + 0.740108i $$0.265225\pi$$
$$734$$ 5121.91 0.257566
$$735$$ 0 0
$$736$$ 31469.5 1.57606
$$737$$ −44202.4 −2.20925
$$738$$ 0 0
$$739$$ 12207.0 0.607634 0.303817 0.952730i $$-0.401739\pi$$
0.303817 + 0.952730i $$0.401739\pi$$
$$740$$ −15489.8 −0.769480
$$741$$ 0 0
$$742$$ 4865.25 0.240713
$$743$$ 12473.3 0.615882 0.307941 0.951405i $$-0.400360\pi$$
0.307941 + 0.951405i $$0.400360\pi$$
$$744$$ 0 0
$$745$$ 7844.16 0.385755
$$746$$ 20526.4 1.00741
$$747$$ 0 0
$$748$$ −47707.4 −2.33202
$$749$$ −4871.82 −0.237667
$$750$$ 0 0
$$751$$ −15102.6 −0.733825 −0.366913 0.930255i $$-0.619585\pi$$
−0.366913 + 0.930255i $$0.619585\pi$$
$$752$$ −4549.99 −0.220640
$$753$$ 0 0
$$754$$ −20815.7 −1.00539
$$755$$ 2191.63 0.105645
$$756$$ 0 0
$$757$$ −3418.34 −0.164124 −0.0820618 0.996627i $$-0.526150\pi$$
−0.0820618 + 0.996627i $$0.526150\pi$$
$$758$$ 17786.1 0.852268
$$759$$ 0 0
$$760$$ −3536.39 −0.168787
$$761$$ 12684.5 0.604224 0.302112 0.953272i $$-0.402308\pi$$
0.302112 + 0.953272i $$0.402308\pi$$
$$762$$ 0 0
$$763$$ 8588.45 0.407500
$$764$$ −14351.7 −0.679614
$$765$$ 0 0
$$766$$ 34730.6 1.63821
$$767$$ −7032.25 −0.331056
$$768$$ 0 0
$$769$$ −27580.2 −1.29333 −0.646663 0.762776i $$-0.723836\pi$$
−0.646663 + 0.762776i $$0.723836\pi$$
$$770$$ −6605.22 −0.309137
$$771$$ 0 0
$$772$$ 10972.8 0.511555
$$773$$ −17386.0 −0.808966 −0.404483 0.914546i $$-0.632548\pi$$
−0.404483 + 0.914546i $$0.632548\pi$$
$$774$$ 0 0
$$775$$ 7288.63 0.337826
$$776$$ −3974.77 −0.183874
$$777$$ 0 0
$$778$$ 38015.7 1.75183
$$779$$ −6928.72 −0.318674
$$780$$ 0 0
$$781$$ −52724.1 −2.41564
$$782$$ −44306.7 −2.02609
$$783$$ 0 0
$$784$$ 5648.84 0.257327
$$785$$ 223.739 0.0101727
$$786$$ 0 0
$$787$$ 4680.29 0.211988 0.105994 0.994367i $$-0.466198\pi$$
0.105994 + 0.994367i $$0.466198\pi$$
$$788$$ 14036.0 0.634531
$$789$$ 0 0
$$790$$ 4614.58 0.207822
$$791$$ 59.0498 0.00265432
$$792$$ 0 0
$$793$$ 1165.91 0.0522100
$$794$$ 694.456 0.0310395
$$795$$ 0 0
$$796$$ −9982.00 −0.444476
$$797$$ −7278.62 −0.323490 −0.161745 0.986833i $$-0.551712\pi$$
−0.161745 + 0.986833i $$0.551712\pi$$
$$798$$ 0 0
$$799$$ 17588.6 0.778773
$$800$$ 5451.40 0.240920
$$801$$ 0 0
$$802$$ 26461.8 1.16508
$$803$$ −50524.7 −2.22039
$$804$$ 0 0
$$805$$ −3666.58 −0.160534
$$806$$ −27577.9 −1.20520
$$807$$ 0 0
$$808$$ 23319.8 1.01533
$$809$$ −29212.5 −1.26954 −0.634770 0.772701i $$-0.718905\pi$$
−0.634770 + 0.772701i $$0.718905\pi$$
$$810$$ 0 0
$$811$$ 41992.4 1.81819 0.909094 0.416590i $$-0.136775\pi$$
0.909094 + 0.416590i $$0.136775\pi$$
$$812$$ 13290.6 0.574396
$$813$$ 0 0
$$814$$ 67761.6 2.91774
$$815$$ 13059.2 0.561281
$$816$$ 0 0
$$817$$ 17900.5 0.766536
$$818$$ −63151.2 −2.69930
$$819$$ 0 0
$$820$$ −10089.3 −0.429675
$$821$$ 8722.99 0.370809 0.185405 0.982662i $$-0.440640\pi$$
0.185405 + 0.982662i $$0.440640\pi$$
$$822$$ 0 0
$$823$$ 13584.8 0.575379 0.287690 0.957724i $$-0.407113\pi$$
0.287690 + 0.957724i $$0.407113\pi$$
$$824$$ 27663.2 1.16953
$$825$$ 0 0
$$826$$ 7512.05 0.316438
$$827$$ −26573.6 −1.11736 −0.558679 0.829384i $$-0.688692\pi$$
−0.558679 + 0.829384i $$0.688692\pi$$
$$828$$ 0 0
$$829$$ −43238.4 −1.81150 −0.905748 0.423816i $$-0.860690\pi$$
−0.905748 + 0.423816i $$0.860690\pi$$
$$830$$ 10332.3 0.432097
$$831$$ 0 0
$$832$$ −17604.2 −0.733552
$$833$$ −21836.3 −0.908265
$$834$$ 0 0
$$835$$ −944.733 −0.0391543
$$836$$ 28282.3 1.17005
$$837$$ 0 0
$$838$$ 42918.2 1.76919
$$839$$ 4093.21 0.168431 0.0842154 0.996448i $$-0.473162\pi$$
0.0842154 + 0.996448i $$0.473162\pi$$
$$840$$ 0 0
$$841$$ 24036.5 0.985546
$$842$$ 6850.75 0.280395
$$843$$ 0 0
$$844$$ −30780.8 −1.25535
$$845$$ 8735.27 0.355624
$$846$$ 0 0
$$847$$ 10507.8 0.426273
$$848$$ 3823.96 0.154853
$$849$$ 0 0
$$850$$ −7675.16 −0.309713
$$851$$ 37614.7 1.51517
$$852$$ 0 0
$$853$$ −20201.5 −0.810886 −0.405443 0.914120i $$-0.632883\pi$$
−0.405443 + 0.914120i $$0.632883\pi$$
$$854$$ −1245.45 −0.0499046
$$855$$ 0 0
$$856$$ 16615.4 0.663436
$$857$$ −4551.65 −0.181425 −0.0907126 0.995877i $$-0.528914\pi$$
−0.0907126 + 0.995877i $$0.528914\pi$$
$$858$$ 0 0
$$859$$ 11962.6 0.475154 0.237577 0.971369i $$-0.423647\pi$$
0.237577 + 0.971369i $$0.423647\pi$$
$$860$$ 26065.9 1.03354
$$861$$ 0 0
$$862$$ 51649.3 2.04081
$$863$$ 7164.61 0.282603 0.141301 0.989967i $$-0.454871\pi$$
0.141301 + 0.989967i $$0.454871\pi$$
$$864$$ 0 0
$$865$$ −7525.10 −0.295793
$$866$$ −66412.0 −2.60597
$$867$$ 0 0
$$868$$ 17608.2 0.688549
$$869$$ −12065.9 −0.471012
$$870$$ 0 0
$$871$$ 16082.4 0.625640
$$872$$ −29291.0 −1.13752
$$873$$ 0 0
$$874$$ 26266.2 1.01655
$$875$$ −635.154 −0.0245396
$$876$$ 0 0
$$877$$ −19218.7 −0.739987 −0.369994 0.929034i $$-0.620640\pi$$
−0.369994 + 0.929034i $$0.620640\pi$$
$$878$$ 7324.56 0.281540
$$879$$ 0 0
$$880$$ −5191.53 −0.198871
$$881$$ 45914.5 1.75585 0.877923 0.478802i $$-0.158929\pi$$
0.877923 + 0.478802i $$0.158929\pi$$
$$882$$ 0 0
$$883$$ −44656.7 −1.70194 −0.850972 0.525211i $$-0.823986\pi$$
−0.850972 + 0.525211i $$0.823986\pi$$
$$884$$ 17357.7 0.660410
$$885$$ 0 0
$$886$$ 17464.3 0.662218
$$887$$ −14975.2 −0.566873 −0.283437 0.958991i $$-0.591475\pi$$
−0.283437 + 0.958991i $$0.591475\pi$$
$$888$$ 0 0
$$889$$ 1570.82 0.0592616
$$890$$ −13416.9 −0.505319
$$891$$ 0 0
$$892$$ −49692.9 −1.86529
$$893$$ −10427.0 −0.390735
$$894$$ 0 0
$$895$$ −15683.1 −0.585729
$$896$$ 9941.33 0.370666
$$897$$ 0 0
$$898$$ −17771.7 −0.660413
$$899$$ 64156.8 2.38015
$$900$$ 0 0
$$901$$ −14782.0 −0.546571
$$902$$ 44136.7 1.62926
$$903$$ 0 0
$$904$$ −201.390 −0.00740943
$$905$$ −22563.3 −0.828763
$$906$$ 0 0
$$907$$ −14818.1 −0.542479 −0.271240 0.962512i $$-0.587434\pi$$
−0.271240 + 0.962512i $$0.587434\pi$$
$$908$$ 30935.6 1.13065
$$909$$ 0 0
$$910$$ 2403.22 0.0875451
$$911$$ −21846.5 −0.794518 −0.397259 0.917707i $$-0.630039\pi$$
−0.397259 + 0.917707i $$0.630039\pi$$
$$912$$ 0 0
$$913$$ −27016.4 −0.979312
$$914$$ 63224.2 2.28804
$$915$$ 0 0
$$916$$ −21934.0 −0.791179
$$917$$ −14151.4 −0.509618
$$918$$ 0 0
$$919$$ 28878.9 1.03659 0.518296 0.855201i $$-0.326566\pi$$
0.518296 + 0.855201i $$0.326566\pi$$
$$920$$ 12504.9 0.448124
$$921$$ 0 0
$$922$$ −73108.4 −2.61139
$$923$$ 19182.9 0.684089
$$924$$ 0 0
$$925$$ 6515.92 0.231613
$$926$$ −14803.4 −0.525344
$$927$$ 0 0
$$928$$ 47985.0 1.69740
$$929$$ −4608.79 −0.162766 −0.0813829 0.996683i $$-0.525934\pi$$
−0.0813829 + 0.996683i $$0.525934\pi$$
$$930$$ 0 0
$$931$$ 12945.2 0.455705
$$932$$ −2860.23 −0.100526
$$933$$ 0 0
$$934$$ −11281.3 −0.395220
$$935$$ 20068.6 0.701938
$$936$$ 0 0
$$937$$ −21063.8 −0.734391 −0.367195 0.930144i $$-0.619682\pi$$
−0.367195 + 0.930144i $$0.619682\pi$$
$$938$$ −17179.7 −0.598014
$$939$$ 0 0
$$940$$ −15183.3 −0.526836
$$941$$ 20244.8 0.701339 0.350670 0.936499i $$-0.385954\pi$$
0.350670 + 0.936499i $$0.385954\pi$$
$$942$$ 0 0
$$943$$ 24500.4 0.846069
$$944$$ 5904.27 0.203567
$$945$$ 0 0
$$946$$ −114028. −3.91901
$$947$$ 29978.3 1.02868 0.514342 0.857585i $$-0.328036\pi$$
0.514342 + 0.857585i $$0.328036\pi$$
$$948$$ 0 0
$$949$$ 18382.7 0.628797
$$950$$ 4550.05 0.155393
$$951$$ 0 0
$$952$$ −6062.20 −0.206383
$$953$$ −5782.65 −0.196556 −0.0982782 0.995159i $$-0.531334\pi$$
−0.0982782 + 0.995159i $$0.531334\pi$$
$$954$$ 0 0
$$955$$ 6037.17 0.204564
$$956$$ −30519.2 −1.03249
$$957$$ 0 0
$$958$$ −38559.2 −1.30041
$$959$$ 12649.3 0.425930
$$960$$ 0 0
$$961$$ 55207.7 1.85317
$$962$$ −24654.2 −0.826281
$$963$$ 0 0
$$964$$ 47401.6 1.58372
$$965$$ −4615.82 −0.153978
$$966$$ 0 0
$$967$$ −26119.2 −0.868600 −0.434300 0.900768i $$-0.643004\pi$$
−0.434300 + 0.900768i $$0.643004\pi$$
$$968$$ −35837.0 −1.18992
$$969$$ 0 0
$$970$$ 5114.09 0.169282
$$971$$ −5101.93 −0.168619 −0.0843093 0.996440i $$-0.526868\pi$$
−0.0843093 + 0.996440i $$0.526868\pi$$
$$972$$ 0 0
$$973$$ 9075.34 0.299016
$$974$$ 68013.1 2.23746
$$975$$ 0 0
$$976$$ −978.894 −0.0321041
$$977$$ 45902.4 1.50312 0.751559 0.659666i $$-0.229302\pi$$
0.751559 + 0.659666i $$0.229302\pi$$
$$978$$ 0 0
$$979$$ 35081.6 1.14526
$$980$$ 18850.2 0.614437
$$981$$ 0 0
$$982$$ −7879.14 −0.256042
$$983$$ −13416.5 −0.435321 −0.217661 0.976025i $$-0.569843\pi$$
−0.217661 + 0.976025i $$0.569843\pi$$
$$984$$ 0 0
$$985$$ −5904.37 −0.190994
$$986$$ −67559.2 −2.18207
$$987$$ 0 0
$$988$$ −10290.1 −0.331349
$$989$$ −63297.4 −2.03513
$$990$$ 0 0
$$991$$ 3806.66 0.122021 0.0610104 0.998137i $$-0.480568\pi$$
0.0610104 + 0.998137i $$0.480568\pi$$
$$992$$ 63573.2 2.03473
$$993$$ 0 0
$$994$$ −20491.8 −0.653883
$$995$$ 4199.02 0.133787
$$996$$ 0 0
$$997$$ −22523.8 −0.715483 −0.357742 0.933821i $$-0.616453\pi$$
−0.357742 + 0.933821i $$0.616453\pi$$
$$998$$ −52325.7 −1.65966
$$999$$ 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 135.4.a.g.1.1 yes 3
3.2 odd 2 135.4.a.f.1.3 3
4.3 odd 2 2160.4.a.be.1.3 3
5.2 odd 4 675.4.b.k.649.2 6
5.3 odd 4 675.4.b.k.649.5 6
5.4 even 2 675.4.a.q.1.3 3
9.2 odd 6 405.4.e.t.271.1 6
9.4 even 3 405.4.e.r.136.3 6
9.5 odd 6 405.4.e.t.136.1 6
9.7 even 3 405.4.e.r.271.3 6
12.11 even 2 2160.4.a.bm.1.3 3
15.2 even 4 675.4.b.l.649.5 6
15.8 even 4 675.4.b.l.649.2 6
15.14 odd 2 675.4.a.r.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.3 3 3.2 odd 2
135.4.a.g.1.1 yes 3 1.1 even 1 trivial
405.4.e.r.136.3 6 9.4 even 3
405.4.e.r.271.3 6 9.7 even 3
405.4.e.t.136.1 6 9.5 odd 6
405.4.e.t.271.1 6 9.2 odd 6
675.4.a.q.1.3 3 5.4 even 2
675.4.a.r.1.1 3 15.14 odd 2
675.4.b.k.649.2 6 5.2 odd 4
675.4.b.k.649.5 6 5.3 odd 4
675.4.b.l.649.2 6 15.8 even 4
675.4.b.l.649.5 6 15.2 even 4
2160.4.a.be.1.3 3 4.3 odd 2
2160.4.a.bm.1.3 3 12.11 even 2