# Properties

 Label 153.4.a.g.1.2 Level $153$ Weight $4$ Character 153.1 Self dual yes Analytic conductor $9.027$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$153 = 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 153.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.02729223088$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2636.1 Defining polynomial: $$x^{3} - 14x - 4$$ x^3 - 14*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.87707$$ of defining polynomial Character $$\chi$$ $$=$$ 153.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.36122 q^{2} -6.14708 q^{4} -3.03171 q^{5} -7.94049 q^{7} +19.2573 q^{8} +O(q^{10})$$ $$q-1.36122 q^{2} -6.14708 q^{4} -3.03171 q^{5} -7.94049 q^{7} +19.2573 q^{8} +4.12682 q^{10} -27.6161 q^{11} +58.1117 q^{13} +10.8088 q^{14} +22.9632 q^{16} +17.0000 q^{17} +89.1688 q^{19} +18.6361 q^{20} +37.5916 q^{22} +115.269 q^{23} -115.809 q^{25} -79.1029 q^{26} +48.8108 q^{28} +128.558 q^{29} +273.460 q^{31} -185.316 q^{32} -23.1408 q^{34} +24.0732 q^{35} -132.351 q^{37} -121.379 q^{38} -58.3825 q^{40} +470.559 q^{41} +352.642 q^{43} +169.758 q^{44} -156.907 q^{46} -152.598 q^{47} -279.949 q^{49} +157.641 q^{50} -357.217 q^{52} -527.614 q^{53} +83.7239 q^{55} -152.912 q^{56} -174.995 q^{58} +292.020 q^{59} -53.8962 q^{61} -372.239 q^{62} +68.5514 q^{64} -176.178 q^{65} +52.9572 q^{67} -104.500 q^{68} -32.7690 q^{70} -788.400 q^{71} +295.780 q^{73} +180.159 q^{74} -548.127 q^{76} +219.285 q^{77} -720.325 q^{79} -69.6175 q^{80} -640.535 q^{82} +116.051 q^{83} -51.5390 q^{85} -480.024 q^{86} -531.812 q^{88} +813.329 q^{89} -461.435 q^{91} -708.569 q^{92} +207.720 q^{94} -270.334 q^{95} +794.693 q^{97} +381.072 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10})$$ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 $$3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100})$$ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 - 56 * q^10 + 28 * q^11 + 30 * q^13 - 92 * q^14 + 137 * q^16 + 51 * q^17 + 80 * q^19 + 168 * q^20 + 286 * q^22 - 142 * q^23 - 223 * q^25 - 26 * q^26 + 476 * q^28 + 456 * q^29 + 230 * q^31 + 71 * q^32 - 17 * q^34 + 332 * q^35 + 356 * q^37 - 724 * q^38 - 424 * q^40 + 294 * q^41 + 556 * q^43 + 1122 * q^44 - 704 * q^46 - 640 * q^47 - 269 * q^49 - 547 * q^50 - 774 * q^52 - 302 * q^53 + 76 * q^55 - 684 * q^56 - 1304 * q^58 - 636 * q^59 - 84 * q^61 - 508 * q^62 - 919 * q^64 - 408 * q^65 + 1008 * q^67 + 425 * q^68 - 1504 * q^70 + 402 * q^71 + 838 * q^73 - 836 * q^74 - 908 * q^76 + 504 * q^77 - 594 * q^79 + 40 * q^80 + 358 * q^82 + 2396 * q^83 + 136 * q^85 + 1264 * q^86 + 1838 * q^88 + 170 * q^89 - 1016 * q^91 - 4896 * q^92 - 2016 * q^94 + 472 * q^95 - 270 * q^97 - 2857 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.36122 −0.481264 −0.240632 0.970616i $$-0.577355\pi$$
−0.240632 + 0.970616i $$0.577355\pi$$
$$3$$ 0 0
$$4$$ −6.14708 −0.768385
$$5$$ −3.03171 −0.271164 −0.135582 0.990766i $$-0.543290\pi$$
−0.135582 + 0.990766i $$0.543290\pi$$
$$6$$ 0 0
$$7$$ −7.94049 −0.428746 −0.214373 0.976752i $$-0.568771\pi$$
−0.214373 + 0.976752i $$0.568771\pi$$
$$8$$ 19.2573 0.851061
$$9$$ 0 0
$$10$$ 4.12682 0.130502
$$11$$ −27.6161 −0.756961 −0.378481 0.925609i $$-0.623553\pi$$
−0.378481 + 0.925609i $$0.623553\pi$$
$$12$$ 0 0
$$13$$ 58.1117 1.23979 0.619896 0.784684i $$-0.287175\pi$$
0.619896 + 0.784684i $$0.287175\pi$$
$$14$$ 10.8088 0.206340
$$15$$ 0 0
$$16$$ 22.9632 0.358799
$$17$$ 17.0000 0.242536
$$18$$ 0 0
$$19$$ 89.1688 1.07667 0.538335 0.842731i $$-0.319054\pi$$
0.538335 + 0.842731i $$0.319054\pi$$
$$20$$ 18.6361 0.208358
$$21$$ 0 0
$$22$$ 37.5916 0.364298
$$23$$ 115.269 1.04501 0.522507 0.852635i $$-0.324997\pi$$
0.522507 + 0.852635i $$0.324997\pi$$
$$24$$ 0 0
$$25$$ −115.809 −0.926470
$$26$$ −79.1029 −0.596668
$$27$$ 0 0
$$28$$ 48.8108 0.329442
$$29$$ 128.558 0.823191 0.411596 0.911367i $$-0.364972\pi$$
0.411596 + 0.911367i $$0.364972\pi$$
$$30$$ 0 0
$$31$$ 273.460 1.58435 0.792174 0.610295i $$-0.208949\pi$$
0.792174 + 0.610295i $$0.208949\pi$$
$$32$$ −185.316 −1.02374
$$33$$ 0 0
$$34$$ −23.1408 −0.116724
$$35$$ 24.0732 0.116260
$$36$$ 0 0
$$37$$ −132.351 −0.588063 −0.294031 0.955796i $$-0.594997\pi$$
−0.294031 + 0.955796i $$0.594997\pi$$
$$38$$ −121.379 −0.518163
$$39$$ 0 0
$$40$$ −58.3825 −0.230777
$$41$$ 470.559 1.79241 0.896207 0.443636i $$-0.146312\pi$$
0.896207 + 0.443636i $$0.146312\pi$$
$$42$$ 0 0
$$43$$ 352.642 1.25064 0.625318 0.780370i $$-0.284969\pi$$
0.625318 + 0.780370i $$0.284969\pi$$
$$44$$ 169.758 0.581637
$$45$$ 0 0
$$46$$ −156.907 −0.502928
$$47$$ −152.598 −0.473589 −0.236795 0.971560i $$-0.576097\pi$$
−0.236795 + 0.971560i $$0.576097\pi$$
$$48$$ 0 0
$$49$$ −279.949 −0.816177
$$50$$ 157.641 0.445877
$$51$$ 0 0
$$52$$ −357.217 −0.952637
$$53$$ −527.614 −1.36742 −0.683711 0.729753i $$-0.739635\pi$$
−0.683711 + 0.729753i $$0.739635\pi$$
$$54$$ 0 0
$$55$$ 83.7239 0.205261
$$56$$ −152.912 −0.364889
$$57$$ 0 0
$$58$$ −174.995 −0.396173
$$59$$ 292.020 0.644368 0.322184 0.946677i $$-0.395583\pi$$
0.322184 + 0.946677i $$0.395583\pi$$
$$60$$ 0 0
$$61$$ −53.8962 −0.113126 −0.0565632 0.998399i $$-0.518014\pi$$
−0.0565632 + 0.998399i $$0.518014\pi$$
$$62$$ −372.239 −0.762490
$$63$$ 0 0
$$64$$ 68.5514 0.133889
$$65$$ −176.178 −0.336187
$$66$$ 0 0
$$67$$ 52.9572 0.0965635 0.0482817 0.998834i $$-0.484625\pi$$
0.0482817 + 0.998834i $$0.484625\pi$$
$$68$$ −104.500 −0.186361
$$69$$ 0 0
$$70$$ −32.7690 −0.0559520
$$71$$ −788.400 −1.31783 −0.658915 0.752218i $$-0.728984\pi$$
−0.658915 + 0.752218i $$0.728984\pi$$
$$72$$ 0 0
$$73$$ 295.780 0.474224 0.237112 0.971482i $$-0.423799\pi$$
0.237112 + 0.971482i $$0.423799\pi$$
$$74$$ 180.159 0.283014
$$75$$ 0 0
$$76$$ −548.127 −0.827296
$$77$$ 219.285 0.324544
$$78$$ 0 0
$$79$$ −720.325 −1.02586 −0.512930 0.858430i $$-0.671440\pi$$
−0.512930 + 0.858430i $$0.671440\pi$$
$$80$$ −69.6175 −0.0972934
$$81$$ 0 0
$$82$$ −640.535 −0.862625
$$83$$ 116.051 0.153473 0.0767363 0.997051i $$-0.475550\pi$$
0.0767363 + 0.997051i $$0.475550\pi$$
$$84$$ 0 0
$$85$$ −51.5390 −0.0657669
$$86$$ −480.024 −0.601887
$$87$$ 0 0
$$88$$ −531.812 −0.644220
$$89$$ 813.329 0.968682 0.484341 0.874879i $$-0.339059\pi$$
0.484341 + 0.874879i $$0.339059\pi$$
$$90$$ 0 0
$$91$$ −461.435 −0.531556
$$92$$ −708.569 −0.802972
$$93$$ 0 0
$$94$$ 207.720 0.227922
$$95$$ −270.334 −0.291954
$$96$$ 0 0
$$97$$ 794.693 0.831844 0.415922 0.909400i $$-0.363459\pi$$
0.415922 + 0.909400i $$0.363459\pi$$
$$98$$ 381.072 0.392797
$$99$$ 0 0
$$100$$ 711.885 0.711885
$$101$$ −265.513 −0.261579 −0.130790 0.991410i $$-0.541751\pi$$
−0.130790 + 0.991410i $$0.541751\pi$$
$$102$$ 0 0
$$103$$ 523.107 0.500420 0.250210 0.968192i $$-0.419500\pi$$
0.250210 + 0.968192i $$0.419500\pi$$
$$104$$ 1119.07 1.05514
$$105$$ 0 0
$$106$$ 718.199 0.658091
$$107$$ 986.039 0.890878 0.445439 0.895312i $$-0.353048\pi$$
0.445439 + 0.895312i $$0.353048\pi$$
$$108$$ 0 0
$$109$$ 1814.39 1.59438 0.797188 0.603732i $$-0.206320\pi$$
0.797188 + 0.603732i $$0.206320\pi$$
$$110$$ −113.967 −0.0987846
$$111$$ 0 0
$$112$$ −182.339 −0.153834
$$113$$ 707.339 0.588857 0.294429 0.955673i $$-0.404871\pi$$
0.294429 + 0.955673i $$0.404871\pi$$
$$114$$ 0 0
$$115$$ −349.463 −0.283370
$$116$$ −790.253 −0.632527
$$117$$ 0 0
$$118$$ −397.503 −0.310112
$$119$$ −134.988 −0.103986
$$120$$ 0 0
$$121$$ −568.350 −0.427010
$$122$$ 73.3647 0.0544437
$$123$$ 0 0
$$124$$ −1680.98 −1.21739
$$125$$ 730.061 0.522389
$$126$$ 0 0
$$127$$ 2648.18 1.85030 0.925151 0.379600i $$-0.123938\pi$$
0.925151 + 0.379600i $$0.123938\pi$$
$$128$$ 1389.22 0.959302
$$129$$ 0 0
$$130$$ 239.817 0.161795
$$131$$ 1979.08 1.31995 0.659974 0.751289i $$-0.270567\pi$$
0.659974 + 0.751289i $$0.270567\pi$$
$$132$$ 0 0
$$133$$ −708.044 −0.461618
$$134$$ −72.0865 −0.0464726
$$135$$ 0 0
$$136$$ 327.374 0.206413
$$137$$ −3141.92 −1.95936 −0.979679 0.200570i $$-0.935721\pi$$
−0.979679 + 0.200570i $$0.935721\pi$$
$$138$$ 0 0
$$139$$ 1468.07 0.895830 0.447915 0.894076i $$-0.352167\pi$$
0.447915 + 0.894076i $$0.352167\pi$$
$$140$$ −147.980 −0.0893327
$$141$$ 0 0
$$142$$ 1073.19 0.634224
$$143$$ −1604.82 −0.938474
$$144$$ 0 0
$$145$$ −389.749 −0.223220
$$146$$ −402.621 −0.228227
$$147$$ 0 0
$$148$$ 813.570 0.451858
$$149$$ 286.027 0.157263 0.0786316 0.996904i $$-0.474945\pi$$
0.0786316 + 0.996904i $$0.474945\pi$$
$$150$$ 0 0
$$151$$ −669.626 −0.360883 −0.180442 0.983586i $$-0.557753\pi$$
−0.180442 + 0.983586i $$0.557753\pi$$
$$152$$ 1717.15 0.916311
$$153$$ 0 0
$$154$$ −298.496 −0.156191
$$155$$ −829.049 −0.429618
$$156$$ 0 0
$$157$$ 720.809 0.366413 0.183206 0.983074i $$-0.441352\pi$$
0.183206 + 0.983074i $$0.441352\pi$$
$$158$$ 980.522 0.493710
$$159$$ 0 0
$$160$$ 561.825 0.277601
$$161$$ −915.294 −0.448045
$$162$$ 0 0
$$163$$ −676.599 −0.325125 −0.162562 0.986698i $$-0.551976\pi$$
−0.162562 + 0.986698i $$0.551976\pi$$
$$164$$ −2892.56 −1.37726
$$165$$ 0 0
$$166$$ −157.971 −0.0738609
$$167$$ 2835.67 1.31396 0.656979 0.753909i $$-0.271834\pi$$
0.656979 + 0.753909i $$0.271834\pi$$
$$168$$ 0 0
$$169$$ 1179.97 0.537083
$$170$$ 70.1560 0.0316513
$$171$$ 0 0
$$172$$ −2167.72 −0.960970
$$173$$ 177.314 0.0779243 0.0389621 0.999241i $$-0.487595\pi$$
0.0389621 + 0.999241i $$0.487595\pi$$
$$174$$ 0 0
$$175$$ 919.578 0.397220
$$176$$ −634.153 −0.271597
$$177$$ 0 0
$$178$$ −1107.12 −0.466192
$$179$$ 1023.76 0.427483 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$180$$ 0 0
$$181$$ −3450.21 −1.41686 −0.708432 0.705779i $$-0.750597\pi$$
−0.708432 + 0.705779i $$0.750597\pi$$
$$182$$ 628.116 0.255819
$$183$$ 0 0
$$184$$ 2219.78 0.889370
$$185$$ 401.248 0.159461
$$186$$ 0 0
$$187$$ −469.474 −0.183590
$$188$$ 938.031 0.363899
$$189$$ 0 0
$$190$$ 367.984 0.140507
$$191$$ 490.894 0.185968 0.0929839 0.995668i $$-0.470360\pi$$
0.0929839 + 0.995668i $$0.470360\pi$$
$$192$$ 0 0
$$193$$ −3548.80 −1.32357 −0.661783 0.749696i $$-0.730200\pi$$
−0.661783 + 0.749696i $$0.730200\pi$$
$$194$$ −1081.75 −0.400337
$$195$$ 0 0
$$196$$ 1720.87 0.627138
$$197$$ −1363.15 −0.492996 −0.246498 0.969143i $$-0.579280\pi$$
−0.246498 + 0.969143i $$0.579280\pi$$
$$198$$ 0 0
$$199$$ 3737.46 1.33137 0.665683 0.746235i $$-0.268140\pi$$
0.665683 + 0.746235i $$0.268140\pi$$
$$200$$ −2230.16 −0.788482
$$201$$ 0 0
$$202$$ 361.422 0.125889
$$203$$ −1020.81 −0.352940
$$204$$ 0 0
$$205$$ −1426.60 −0.486038
$$206$$ −712.064 −0.240834
$$207$$ 0 0
$$208$$ 1334.43 0.444836
$$209$$ −2462.50 −0.814997
$$210$$ 0 0
$$211$$ −5266.12 −1.71817 −0.859087 0.511829i $$-0.828968\pi$$
−0.859087 + 0.511829i $$0.828968\pi$$
$$212$$ 3243.28 1.05071
$$213$$ 0 0
$$214$$ −1342.22 −0.428748
$$215$$ −1069.11 −0.339128
$$216$$ 0 0
$$217$$ −2171.40 −0.679283
$$218$$ −2469.78 −0.767316
$$219$$ 0 0
$$220$$ −514.657 −0.157719
$$221$$ 987.899 0.300694
$$222$$ 0 0
$$223$$ 704.546 0.211569 0.105785 0.994389i $$-0.466265\pi$$
0.105785 + 0.994389i $$0.466265\pi$$
$$224$$ 1471.50 0.438923
$$225$$ 0 0
$$226$$ −962.845 −0.283396
$$227$$ 2151.26 0.629006 0.314503 0.949256i $$-0.398162\pi$$
0.314503 + 0.949256i $$0.398162\pi$$
$$228$$ 0 0
$$229$$ −3916.94 −1.13030 −0.565149 0.824989i $$-0.691181\pi$$
−0.565149 + 0.824989i $$0.691181\pi$$
$$230$$ 475.696 0.136376
$$231$$ 0 0
$$232$$ 2475.67 0.700586
$$233$$ 5192.74 1.46003 0.730017 0.683429i $$-0.239512\pi$$
0.730017 + 0.683429i $$0.239512\pi$$
$$234$$ 0 0
$$235$$ 462.632 0.128420
$$236$$ −1795.07 −0.495123
$$237$$ 0 0
$$238$$ 183.749 0.0500448
$$239$$ −334.305 −0.0904786 −0.0452393 0.998976i $$-0.514405\pi$$
−0.0452393 + 0.998976i $$0.514405\pi$$
$$240$$ 0 0
$$241$$ −1918.45 −0.512773 −0.256386 0.966574i $$-0.582532\pi$$
−0.256386 + 0.966574i $$0.582532\pi$$
$$242$$ 773.651 0.205505
$$243$$ 0 0
$$244$$ 331.304 0.0869245
$$245$$ 848.722 0.221318
$$246$$ 0 0
$$247$$ 5181.75 1.33485
$$248$$ 5266.09 1.34838
$$249$$ 0 0
$$250$$ −993.775 −0.251407
$$251$$ −7695.71 −1.93525 −0.967627 0.252385i $$-0.918785\pi$$
−0.967627 + 0.252385i $$0.918785\pi$$
$$252$$ 0 0
$$253$$ −3183.29 −0.791035
$$254$$ −3604.76 −0.890484
$$255$$ 0 0
$$256$$ −2439.44 −0.595567
$$257$$ −5335.10 −1.29492 −0.647460 0.762099i $$-0.724169\pi$$
−0.647460 + 0.762099i $$0.724169\pi$$
$$258$$ 0 0
$$259$$ 1050.93 0.252130
$$260$$ 1082.98 0.258321
$$261$$ 0 0
$$262$$ −2693.97 −0.635244
$$263$$ −3934.15 −0.922396 −0.461198 0.887297i $$-0.652580\pi$$
−0.461198 + 0.887297i $$0.652580\pi$$
$$264$$ 0 0
$$265$$ 1599.57 0.370795
$$266$$ 963.804 0.222160
$$267$$ 0 0
$$268$$ −325.532 −0.0741979
$$269$$ −3424.04 −0.776088 −0.388044 0.921641i $$-0.626849\pi$$
−0.388044 + 0.921641i $$0.626849\pi$$
$$270$$ 0 0
$$271$$ 549.034 0.123068 0.0615340 0.998105i $$-0.480401\pi$$
0.0615340 + 0.998105i $$0.480401\pi$$
$$272$$ 390.374 0.0870216
$$273$$ 0 0
$$274$$ 4276.85 0.942970
$$275$$ 3198.19 0.701302
$$276$$ 0 0
$$277$$ 5203.65 1.12873 0.564363 0.825527i $$-0.309122\pi$$
0.564363 + 0.825527i $$0.309122\pi$$
$$278$$ −1998.37 −0.431131
$$279$$ 0 0
$$280$$ 463.585 0.0989447
$$281$$ 1986.73 0.421774 0.210887 0.977510i $$-0.432365\pi$$
0.210887 + 0.977510i $$0.432365\pi$$
$$282$$ 0 0
$$283$$ 753.696 0.158313 0.0791565 0.996862i $$-0.474777\pi$$
0.0791565 + 0.996862i $$0.474777\pi$$
$$284$$ 4846.36 1.01260
$$285$$ 0 0
$$286$$ 2184.51 0.451654
$$287$$ −3736.47 −0.768490
$$288$$ 0 0
$$289$$ 289.000 0.0588235
$$290$$ 530.534 0.107428
$$291$$ 0 0
$$292$$ −1818.18 −0.364387
$$293$$ 7202.22 1.43603 0.718017 0.696025i $$-0.245050\pi$$
0.718017 + 0.696025i $$0.245050\pi$$
$$294$$ 0 0
$$295$$ −885.318 −0.174729
$$296$$ −2548.72 −0.500477
$$297$$ 0 0
$$298$$ −389.345 −0.0756852
$$299$$ 6698.50 1.29560
$$300$$ 0 0
$$301$$ −2800.15 −0.536205
$$302$$ 911.509 0.173680
$$303$$ 0 0
$$304$$ 2047.60 0.386308
$$305$$ 163.398 0.0306758
$$306$$ 0 0
$$307$$ 2425.71 0.450953 0.225477 0.974249i $$-0.427606\pi$$
0.225477 + 0.974249i $$0.427606\pi$$
$$308$$ −1347.96 −0.249375
$$309$$ 0 0
$$310$$ 1128.52 0.206760
$$311$$ 9544.94 1.74033 0.870167 0.492757i $$-0.164011\pi$$
0.870167 + 0.492757i $$0.164011\pi$$
$$312$$ 0 0
$$313$$ 588.379 0.106253 0.0531264 0.998588i $$-0.483081\pi$$
0.0531264 + 0.998588i $$0.483081\pi$$
$$314$$ −981.180 −0.176341
$$315$$ 0 0
$$316$$ 4427.89 0.788255
$$317$$ −7653.31 −1.35600 −0.678001 0.735061i $$-0.737154\pi$$
−0.678001 + 0.735061i $$0.737154\pi$$
$$318$$ 0 0
$$319$$ −3550.26 −0.623124
$$320$$ −207.828 −0.0363060
$$321$$ 0 0
$$322$$ 1245.92 0.215628
$$323$$ 1515.87 0.261131
$$324$$ 0 0
$$325$$ −6729.85 −1.14863
$$326$$ 921.001 0.156471
$$327$$ 0 0
$$328$$ 9061.70 1.52545
$$329$$ 1211.70 0.203050
$$330$$ 0 0
$$331$$ 752.266 0.124919 0.0624597 0.998047i $$-0.480106\pi$$
0.0624597 + 0.998047i $$0.480106\pi$$
$$332$$ −713.373 −0.117926
$$333$$ 0 0
$$334$$ −3859.98 −0.632361
$$335$$ −160.551 −0.0261845
$$336$$ 0 0
$$337$$ −1968.57 −0.318204 −0.159102 0.987262i $$-0.550860\pi$$
−0.159102 + 0.987262i $$0.550860\pi$$
$$338$$ −1606.20 −0.258479
$$339$$ 0 0
$$340$$ 316.814 0.0505343
$$341$$ −7551.89 −1.19929
$$342$$ 0 0
$$343$$ 4946.52 0.778678
$$344$$ 6790.93 1.06437
$$345$$ 0 0
$$346$$ −241.363 −0.0375022
$$347$$ −3983.10 −0.616207 −0.308104 0.951353i $$-0.599694\pi$$
−0.308104 + 0.951353i $$0.599694\pi$$
$$348$$ 0 0
$$349$$ 1495.61 0.229393 0.114697 0.993401i $$-0.463410\pi$$
0.114697 + 0.993401i $$0.463410\pi$$
$$350$$ −1251.75 −0.191168
$$351$$ 0 0
$$352$$ 5117.72 0.774930
$$353$$ −6482.49 −0.977417 −0.488708 0.872447i $$-0.662532\pi$$
−0.488708 + 0.872447i $$0.662532\pi$$
$$354$$ 0 0
$$355$$ 2390.20 0.357348
$$356$$ −4999.59 −0.744320
$$357$$ 0 0
$$358$$ −1393.56 −0.205732
$$359$$ 4943.42 0.726751 0.363376 0.931643i $$-0.381624\pi$$
0.363376 + 0.931643i $$0.381624\pi$$
$$360$$ 0 0
$$361$$ 1092.08 0.159218
$$362$$ 4696.50 0.681886
$$363$$ 0 0
$$364$$ 2836.48 0.408439
$$365$$ −896.717 −0.128593
$$366$$ 0 0
$$367$$ −14.8871 −0.00211743 −0.00105872 0.999999i $$-0.500337\pi$$
−0.00105872 + 0.999999i $$0.500337\pi$$
$$368$$ 2646.95 0.374950
$$369$$ 0 0
$$370$$ −546.188 −0.0767431
$$371$$ 4189.51 0.586276
$$372$$ 0 0
$$373$$ 1923.18 0.266966 0.133483 0.991051i $$-0.457384\pi$$
0.133483 + 0.991051i $$0.457384\pi$$
$$374$$ 639.058 0.0883554
$$375$$ 0 0
$$376$$ −2938.63 −0.403053
$$377$$ 7470.70 1.02059
$$378$$ 0 0
$$379$$ −9592.87 −1.30014 −0.650069 0.759875i $$-0.725260\pi$$
−0.650069 + 0.759875i $$0.725260\pi$$
$$380$$ 1661.76 0.224333
$$381$$ 0 0
$$382$$ −668.215 −0.0894996
$$383$$ 9083.77 1.21190 0.605951 0.795502i $$-0.292793\pi$$
0.605951 + 0.795502i $$0.292793\pi$$
$$384$$ 0 0
$$385$$ −664.809 −0.0880046
$$386$$ 4830.70 0.636985
$$387$$ 0 0
$$388$$ −4885.04 −0.639176
$$389$$ 1143.78 0.149079 0.0745396 0.997218i $$-0.476251\pi$$
0.0745396 + 0.997218i $$0.476251\pi$$
$$390$$ 0 0
$$391$$ 1959.58 0.253453
$$392$$ −5391.06 −0.694616
$$393$$ 0 0
$$394$$ 1855.55 0.237262
$$395$$ 2183.81 0.278176
$$396$$ 0 0
$$397$$ 10604.5 1.34061 0.670307 0.742084i $$-0.266162\pi$$
0.670307 + 0.742084i $$0.266162\pi$$
$$398$$ −5087.51 −0.640739
$$399$$ 0 0
$$400$$ −2659.33 −0.332417
$$401$$ −13785.4 −1.71674 −0.858368 0.513035i $$-0.828521\pi$$
−0.858368 + 0.513035i $$0.828521\pi$$
$$402$$ 0 0
$$403$$ 15891.2 1.96426
$$404$$ 1632.13 0.200993
$$405$$ 0 0
$$406$$ 1389.55 0.169857
$$407$$ 3655.01 0.445141
$$408$$ 0 0
$$409$$ −9505.94 −1.14924 −0.574619 0.818421i $$-0.694850\pi$$
−0.574619 + 0.818421i $$0.694850\pi$$
$$410$$ 1941.91 0.233913
$$411$$ 0 0
$$412$$ −3215.58 −0.384515
$$413$$ −2318.78 −0.276270
$$414$$ 0 0
$$415$$ −351.832 −0.0416162
$$416$$ −10769.1 −1.26922
$$417$$ 0 0
$$418$$ 3352.00 0.392229
$$419$$ −9680.86 −1.12874 −0.564369 0.825523i $$-0.690880\pi$$
−0.564369 + 0.825523i $$0.690880\pi$$
$$420$$ 0 0
$$421$$ −12360.3 −1.43089 −0.715444 0.698671i $$-0.753775\pi$$
−0.715444 + 0.698671i $$0.753775\pi$$
$$422$$ 7168.36 0.826897
$$423$$ 0 0
$$424$$ −10160.4 −1.16376
$$425$$ −1968.75 −0.224702
$$426$$ 0 0
$$427$$ 427.962 0.0485025
$$428$$ −6061.25 −0.684537
$$429$$ 0 0
$$430$$ 1455.29 0.163210
$$431$$ −2970.58 −0.331990 −0.165995 0.986127i $$-0.553084\pi$$
−0.165995 + 0.986127i $$0.553084\pi$$
$$432$$ 0 0
$$433$$ 6131.50 0.680510 0.340255 0.940333i $$-0.389487\pi$$
0.340255 + 0.940333i $$0.389487\pi$$
$$434$$ 2955.76 0.326915
$$435$$ 0 0
$$436$$ −11153.2 −1.22509
$$437$$ 10278.4 1.12513
$$438$$ 0 0
$$439$$ −2544.91 −0.276679 −0.138339 0.990385i $$-0.544176\pi$$
−0.138339 + 0.990385i $$0.544176\pi$$
$$440$$ 1612.30 0.174689
$$441$$ 0 0
$$442$$ −1344.75 −0.144713
$$443$$ −8529.82 −0.914817 −0.457408 0.889257i $$-0.651222\pi$$
−0.457408 + 0.889257i $$0.651222\pi$$
$$444$$ 0 0
$$445$$ −2465.77 −0.262672
$$446$$ −959.043 −0.101821
$$447$$ 0 0
$$448$$ −544.331 −0.0574046
$$449$$ −8855.74 −0.930798 −0.465399 0.885101i $$-0.654089\pi$$
−0.465399 + 0.885101i $$0.654089\pi$$
$$450$$ 0 0
$$451$$ −12995.0 −1.35679
$$452$$ −4348.07 −0.452469
$$453$$ 0 0
$$454$$ −2928.35 −0.302718
$$455$$ 1398.94 0.144139
$$456$$ 0 0
$$457$$ −7154.78 −0.732356 −0.366178 0.930545i $$-0.619334\pi$$
−0.366178 + 0.930545i $$0.619334\pi$$
$$458$$ 5331.82 0.543973
$$459$$ 0 0
$$460$$ 2148.17 0.217737
$$461$$ 7263.06 0.733784 0.366892 0.930264i $$-0.380422\pi$$
0.366892 + 0.930264i $$0.380422\pi$$
$$462$$ 0 0
$$463$$ 352.898 0.0354224 0.0177112 0.999843i $$-0.494362\pi$$
0.0177112 + 0.999843i $$0.494362\pi$$
$$464$$ 2952.09 0.295360
$$465$$ 0 0
$$466$$ −7068.47 −0.702662
$$467$$ −1483.02 −0.146951 −0.0734753 0.997297i $$-0.523409\pi$$
−0.0734753 + 0.997297i $$0.523409\pi$$
$$468$$ 0 0
$$469$$ −420.506 −0.0414012
$$470$$ −629.745 −0.0618042
$$471$$ 0 0
$$472$$ 5623.51 0.548396
$$473$$ −9738.60 −0.946683
$$474$$ 0 0
$$475$$ −10326.5 −0.997502
$$476$$ 829.783 0.0799014
$$477$$ 0 0
$$478$$ 455.063 0.0435441
$$479$$ 9990.10 0.952942 0.476471 0.879190i $$-0.341916\pi$$
0.476471 + 0.879190i $$0.341916\pi$$
$$480$$ 0 0
$$481$$ −7691.13 −0.729075
$$482$$ 2611.44 0.246779
$$483$$ 0 0
$$484$$ 3493.69 0.328108
$$485$$ −2409.27 −0.225566
$$486$$ 0 0
$$487$$ −1129.88 −0.105133 −0.0525663 0.998617i $$-0.516740\pi$$
−0.0525663 + 0.998617i $$0.516740\pi$$
$$488$$ −1037.90 −0.0962774
$$489$$ 0 0
$$490$$ −1155.30 −0.106512
$$491$$ −18774.9 −1.72566 −0.862832 0.505491i $$-0.831311\pi$$
−0.862832 + 0.505491i $$0.831311\pi$$
$$492$$ 0 0
$$493$$ 2185.48 0.199653
$$494$$ −7053.51 −0.642414
$$495$$ 0 0
$$496$$ 6279.49 0.568463
$$497$$ 6260.28 0.565014
$$498$$ 0 0
$$499$$ 17329.1 1.55462 0.777310 0.629118i $$-0.216584\pi$$
0.777310 + 0.629118i $$0.216584\pi$$
$$500$$ −4487.74 −0.401396
$$501$$ 0 0
$$502$$ 10475.6 0.931369
$$503$$ 20837.0 1.84707 0.923533 0.383518i $$-0.125288\pi$$
0.923533 + 0.383518i $$0.125288\pi$$
$$504$$ 0 0
$$505$$ 804.957 0.0709309
$$506$$ 4333.16 0.380697
$$507$$ 0 0
$$508$$ −16278.6 −1.42174
$$509$$ −11835.0 −1.03060 −0.515301 0.857009i $$-0.672320\pi$$
−0.515301 + 0.857009i $$0.672320\pi$$
$$510$$ 0 0
$$511$$ −2348.63 −0.203322
$$512$$ −7793.12 −0.672676
$$513$$ 0 0
$$514$$ 7262.26 0.623199
$$515$$ −1585.91 −0.135696
$$516$$ 0 0
$$517$$ 4214.16 0.358489
$$518$$ −1430.55 −0.121341
$$519$$ 0 0
$$520$$ −3392.71 −0.286115
$$521$$ −7686.37 −0.646346 −0.323173 0.946340i $$-0.604750\pi$$
−0.323173 + 0.946340i $$0.604750\pi$$
$$522$$ 0 0
$$523$$ 11476.4 0.959518 0.479759 0.877400i $$-0.340724\pi$$
0.479759 + 0.877400i $$0.340724\pi$$
$$524$$ −12165.6 −1.01423
$$525$$ 0 0
$$526$$ 5355.25 0.443916
$$527$$ 4648.81 0.384261
$$528$$ 0 0
$$529$$ 1120.01 0.0920535
$$530$$ −2177.37 −0.178451
$$531$$ 0 0
$$532$$ 4352.40 0.354700
$$533$$ 27345.0 2.22222
$$534$$ 0 0
$$535$$ −2989.38 −0.241574
$$536$$ 1019.81 0.0821814
$$537$$ 0 0
$$538$$ 4660.88 0.373504
$$539$$ 7731.09 0.617814
$$540$$ 0 0
$$541$$ −546.481 −0.0434289 −0.0217145 0.999764i $$-0.506912\pi$$
−0.0217145 + 0.999764i $$0.506912\pi$$
$$542$$ −747.357 −0.0592283
$$543$$ 0 0
$$544$$ −3150.38 −0.248293
$$545$$ −5500.69 −0.432337
$$546$$ 0 0
$$547$$ 8397.33 0.656388 0.328194 0.944610i $$-0.393560\pi$$
0.328194 + 0.944610i $$0.393560\pi$$
$$548$$ 19313.6 1.50554
$$549$$ 0 0
$$550$$ −4353.44 −0.337512
$$551$$ 11463.3 0.886305
$$552$$ 0 0
$$553$$ 5719.73 0.439833
$$554$$ −7083.32 −0.543215
$$555$$ 0 0
$$556$$ −9024.36 −0.688342
$$557$$ 4881.65 0.371350 0.185675 0.982611i $$-0.440553\pi$$
0.185675 + 0.982611i $$0.440553\pi$$
$$558$$ 0 0
$$559$$ 20492.6 1.55053
$$560$$ 552.797 0.0417142
$$561$$ 0 0
$$562$$ −2704.38 −0.202985
$$563$$ −7198.57 −0.538870 −0.269435 0.963019i $$-0.586837\pi$$
−0.269435 + 0.963019i $$0.586837\pi$$
$$564$$ 0 0
$$565$$ −2144.44 −0.159677
$$566$$ −1025.95 −0.0761904
$$567$$ 0 0
$$568$$ −15182.5 −1.12155
$$569$$ 23946.9 1.76433 0.882167 0.470937i $$-0.156084\pi$$
0.882167 + 0.470937i $$0.156084\pi$$
$$570$$ 0 0
$$571$$ 1593.15 0.116763 0.0583813 0.998294i $$-0.481406\pi$$
0.0583813 + 0.998294i $$0.481406\pi$$
$$572$$ 9864.95 0.721109
$$573$$ 0 0
$$574$$ 5086.16 0.369847
$$575$$ −13349.2 −0.968174
$$576$$ 0 0
$$577$$ 12937.4 0.933435 0.466717 0.884406i $$-0.345436\pi$$
0.466717 + 0.884406i $$0.345436\pi$$
$$578$$ −393.393 −0.0283097
$$579$$ 0 0
$$580$$ 2395.82 0.171519
$$581$$ −921.499 −0.0658007
$$582$$ 0 0
$$583$$ 14570.6 1.03508
$$584$$ 5695.92 0.403594
$$585$$ 0 0
$$586$$ −9803.82 −0.691112
$$587$$ 12899.2 0.906998 0.453499 0.891257i $$-0.350176\pi$$
0.453499 + 0.891257i $$0.350176\pi$$
$$588$$ 0 0
$$589$$ 24384.1 1.70582
$$590$$ 1205.11 0.0840911
$$591$$ 0 0
$$592$$ −3039.19 −0.210997
$$593$$ −4357.13 −0.301730 −0.150865 0.988554i $$-0.548206\pi$$
−0.150865 + 0.988554i $$0.548206\pi$$
$$594$$ 0 0
$$595$$ 409.245 0.0281973
$$596$$ −1758.23 −0.120839
$$597$$ 0 0
$$598$$ −9118.14 −0.623526
$$599$$ −13726.8 −0.936328 −0.468164 0.883642i $$-0.655084\pi$$
−0.468164 + 0.883642i $$0.655084\pi$$
$$600$$ 0 0
$$601$$ 2531.41 0.171811 0.0859056 0.996303i $$-0.472622\pi$$
0.0859056 + 0.996303i $$0.472622\pi$$
$$602$$ 3811.62 0.258057
$$603$$ 0 0
$$604$$ 4116.24 0.277297
$$605$$ 1723.07 0.115790
$$606$$ 0 0
$$607$$ 185.004 0.0123708 0.00618540 0.999981i $$-0.498031\pi$$
0.00618540 + 0.999981i $$0.498031\pi$$
$$608$$ −16524.4 −1.10223
$$609$$ 0 0
$$610$$ −222.420 −0.0147632
$$611$$ −8867.73 −0.587152
$$612$$ 0 0
$$613$$ −17706.9 −1.16668 −0.583339 0.812228i $$-0.698254\pi$$
−0.583339 + 0.812228i $$0.698254\pi$$
$$614$$ −3301.93 −0.217028
$$615$$ 0 0
$$616$$ 4222.84 0.276207
$$617$$ 6183.89 0.403491 0.201746 0.979438i $$-0.435339\pi$$
0.201746 + 0.979438i $$0.435339\pi$$
$$618$$ 0 0
$$619$$ −1247.51 −0.0810046 −0.0405023 0.999179i $$-0.512896\pi$$
−0.0405023 + 0.999179i $$0.512896\pi$$
$$620$$ 5096.23 0.330112
$$621$$ 0 0
$$622$$ −12992.8 −0.837561
$$623$$ −6458.23 −0.415318
$$624$$ 0 0
$$625$$ 12262.8 0.784817
$$626$$ −800.914 −0.0511357
$$627$$ 0 0
$$628$$ −4430.87 −0.281546
$$629$$ −2249.96 −0.142626
$$630$$ 0 0
$$631$$ 24053.3 1.51750 0.758752 0.651379i $$-0.225809\pi$$
0.758752 + 0.651379i $$0.225809\pi$$
$$632$$ −13871.5 −0.873069
$$633$$ 0 0
$$634$$ 10417.8 0.652596
$$635$$ −8028.51 −0.501735
$$636$$ 0 0
$$637$$ −16268.3 −1.01189
$$638$$ 4832.69 0.299887
$$639$$ 0 0
$$640$$ −4211.70 −0.260128
$$641$$ 21286.8 1.31167 0.655834 0.754905i $$-0.272317\pi$$
0.655834 + 0.754905i $$0.272317\pi$$
$$642$$ 0 0
$$643$$ −1789.41 −0.109747 −0.0548736 0.998493i $$-0.517476\pi$$
−0.0548736 + 0.998493i $$0.517476\pi$$
$$644$$ 5626.38 0.344271
$$645$$ 0 0
$$646$$ −2063.43 −0.125673
$$647$$ 4378.61 0.266060 0.133030 0.991112i $$-0.457529\pi$$
0.133030 + 0.991112i $$0.457529\pi$$
$$648$$ 0 0
$$649$$ −8064.45 −0.487762
$$650$$ 9160.81 0.552795
$$651$$ 0 0
$$652$$ 4159.11 0.249821
$$653$$ 7665.15 0.459358 0.229679 0.973266i $$-0.426232\pi$$
0.229679 + 0.973266i $$0.426232\pi$$
$$654$$ 0 0
$$655$$ −5999.99 −0.357922
$$656$$ 10805.5 0.643117
$$657$$ 0 0
$$658$$ −1649.39 −0.0977205
$$659$$ 4710.22 0.278428 0.139214 0.990262i $$-0.455542\pi$$
0.139214 + 0.990262i $$0.455542\pi$$
$$660$$ 0 0
$$661$$ −31266.6 −1.83983 −0.919916 0.392116i $$-0.871743\pi$$
−0.919916 + 0.392116i $$0.871743\pi$$
$$662$$ −1024.00 −0.0601192
$$663$$ 0 0
$$664$$ 2234.82 0.130614
$$665$$ 2146.58 0.125174
$$666$$ 0 0
$$667$$ 14818.7 0.860246
$$668$$ −17431.1 −1.00962
$$669$$ 0 0
$$670$$ 218.545 0.0126017
$$671$$ 1488.40 0.0856322
$$672$$ 0 0
$$673$$ 11723.0 0.671454 0.335727 0.941959i $$-0.391018\pi$$
0.335727 + 0.941959i $$0.391018\pi$$
$$674$$ 2679.65 0.153140
$$675$$ 0 0
$$676$$ −7253.37 −0.412686
$$677$$ 289.531 0.0164366 0.00821829 0.999966i $$-0.497384\pi$$
0.00821829 + 0.999966i $$0.497384\pi$$
$$678$$ 0 0
$$679$$ −6310.25 −0.356650
$$680$$ −992.502 −0.0559716
$$681$$ 0 0
$$682$$ 10279.8 0.577175
$$683$$ −1720.10 −0.0963660 −0.0481830 0.998839i $$-0.515343\pi$$
−0.0481830 + 0.998839i $$0.515343\pi$$
$$684$$ 0 0
$$685$$ 9525.37 0.531308
$$686$$ −6733.30 −0.374750
$$687$$ 0 0
$$688$$ 8097.77 0.448728
$$689$$ −30660.5 −1.69532
$$690$$ 0 0
$$691$$ −16777.7 −0.923665 −0.461832 0.886967i $$-0.652808\pi$$
−0.461832 + 0.886967i $$0.652808\pi$$
$$692$$ −1089.96 −0.0598758
$$693$$ 0 0
$$694$$ 5421.88 0.296559
$$695$$ −4450.77 −0.242917
$$696$$ 0 0
$$697$$ 7999.50 0.434724
$$698$$ −2035.86 −0.110399
$$699$$ 0 0
$$700$$ −5652.71 −0.305218
$$701$$ −23981.1 −1.29209 −0.646043 0.763301i $$-0.723577\pi$$
−0.646043 + 0.763301i $$0.723577\pi$$
$$702$$ 0 0
$$703$$ −11801.6 −0.633150
$$704$$ −1893.12 −0.101349
$$705$$ 0 0
$$706$$ 8824.10 0.470396
$$707$$ 2108.30 0.112151
$$708$$ 0 0
$$709$$ −7709.28 −0.408361 −0.204181 0.978933i $$-0.565453\pi$$
−0.204181 + 0.978933i $$0.565453\pi$$
$$710$$ −3253.59 −0.171979
$$711$$ 0 0
$$712$$ 15662.5 0.824407
$$713$$ 31521.5 1.65567
$$714$$ 0 0
$$715$$ 4865.34 0.254480
$$716$$ −6293.13 −0.328471
$$717$$ 0 0
$$718$$ −6729.09 −0.349760
$$719$$ 11976.5 0.621209 0.310605 0.950539i $$-0.399468\pi$$
0.310605 + 0.950539i $$0.399468\pi$$
$$720$$ 0 0
$$721$$ −4153.72 −0.214553
$$722$$ −1486.56 −0.0766260
$$723$$ 0 0
$$724$$ 21208.7 1.08870
$$725$$ −14888.1 −0.762662
$$726$$ 0 0
$$727$$ −18597.3 −0.948745 −0.474372 0.880324i $$-0.657325\pi$$
−0.474372 + 0.880324i $$0.657325\pi$$
$$728$$ −8886.00 −0.452386
$$729$$ 0 0
$$730$$ 1220.63 0.0618870
$$731$$ 5994.91 0.303324
$$732$$ 0 0
$$733$$ −23569.5 −1.18767 −0.593833 0.804588i $$-0.702386\pi$$
−0.593833 + 0.804588i $$0.702386\pi$$
$$734$$ 20.2646 0.00101905
$$735$$ 0 0
$$736$$ −21361.3 −1.06982
$$737$$ −1462.47 −0.0730948
$$738$$ 0 0
$$739$$ 10149.1 0.505199 0.252599 0.967571i $$-0.418715\pi$$
0.252599 + 0.967571i $$0.418715\pi$$
$$740$$ −2466.50 −0.122528
$$741$$ 0 0
$$742$$ −5702.85 −0.282154
$$743$$ −27758.0 −1.37058 −0.685291 0.728269i $$-0.740325\pi$$
−0.685291 + 0.728269i $$0.740325\pi$$
$$744$$ 0 0
$$745$$ −867.148 −0.0426441
$$746$$ −2617.87 −0.128481
$$747$$ 0 0
$$748$$ 2885.89 0.141068
$$749$$ −7829.63 −0.381960
$$750$$ 0 0
$$751$$ −815.225 −0.0396112 −0.0198056 0.999804i $$-0.506305\pi$$
−0.0198056 + 0.999804i $$0.506305\pi$$
$$752$$ −3504.13 −0.169924
$$753$$ 0 0
$$754$$ −10169.3 −0.491172
$$755$$ 2030.11 0.0978585
$$756$$ 0 0
$$757$$ −13239.4 −0.635659 −0.317829 0.948148i $$-0.602954\pi$$
−0.317829 + 0.948148i $$0.602954\pi$$
$$758$$ 13058.0 0.625710
$$759$$ 0 0
$$760$$ −5205.90 −0.248471
$$761$$ 11028.2 0.525324 0.262662 0.964888i $$-0.415400\pi$$
0.262662 + 0.964888i $$0.415400\pi$$
$$762$$ 0 0
$$763$$ −14407.1 −0.683582
$$764$$ −3017.56 −0.142895
$$765$$ 0 0
$$766$$ −12365.0 −0.583246
$$767$$ 16969.8 0.798882
$$768$$ 0 0
$$769$$ −18921.2 −0.887277 −0.443639 0.896206i $$-0.646313\pi$$
−0.443639 + 0.896206i $$0.646313\pi$$
$$770$$ 904.952 0.0423535
$$771$$ 0 0
$$772$$ 21814.7 1.01701
$$773$$ −38728.6 −1.80203 −0.901016 0.433786i $$-0.857177\pi$$
−0.901016 + 0.433786i $$0.857177\pi$$
$$774$$ 0 0
$$775$$ −31669.0 −1.46785
$$776$$ 15303.6 0.707949
$$777$$ 0 0
$$778$$ −1556.93 −0.0717465
$$779$$ 41959.2 1.92984
$$780$$ 0 0
$$781$$ 21772.5 0.997545
$$782$$ −2667.42 −0.121978
$$783$$ 0 0
$$784$$ −6428.50 −0.292844
$$785$$ −2185.28 −0.0993579
$$786$$ 0 0
$$787$$ −20587.3 −0.932477 −0.466239 0.884659i $$-0.654391\pi$$
−0.466239 + 0.884659i $$0.654391\pi$$
$$788$$ 8379.37 0.378811
$$789$$ 0 0
$$790$$ −2972.66 −0.133876
$$791$$ −5616.62 −0.252470
$$792$$ 0 0
$$793$$ −3132.00 −0.140253
$$794$$ −14435.0 −0.645190
$$795$$ 0 0
$$796$$ −22974.5 −1.02300
$$797$$ 15871.4 0.705385 0.352693 0.935739i $$-0.385266\pi$$
0.352693 + 0.935739i $$0.385266\pi$$
$$798$$ 0 0
$$799$$ −2594.17 −0.114862
$$800$$ 21461.3 0.948463
$$801$$ 0 0
$$802$$ 18765.0 0.826204
$$803$$ −8168.28 −0.358969
$$804$$ 0 0
$$805$$ 2774.90 0.121494
$$806$$ −21631.4 −0.945329
$$807$$ 0 0
$$808$$ −5113.06 −0.222620
$$809$$ −39667.1 −1.72388 −0.861942 0.507007i $$-0.830752\pi$$
−0.861942 + 0.507007i $$0.830752\pi$$
$$810$$ 0 0
$$811$$ 8003.87 0.346552 0.173276 0.984873i $$-0.444565\pi$$
0.173276 + 0.984873i $$0.444565\pi$$
$$812$$ 6275.00 0.271194
$$813$$ 0 0
$$814$$ −4975.28 −0.214230
$$815$$ 2051.25 0.0881621
$$816$$ 0 0
$$817$$ 31444.7 1.34652
$$818$$ 12939.7 0.553088
$$819$$ 0 0
$$820$$ 8769.40 0.373464
$$821$$ 13279.1 0.564489 0.282244 0.959343i $$-0.408921\pi$$
0.282244 + 0.959343i $$0.408921\pi$$
$$822$$ 0 0
$$823$$ 28934.0 1.22549 0.612745 0.790281i $$-0.290065\pi$$
0.612745 + 0.790281i $$0.290065\pi$$
$$824$$ 10073.6 0.425887
$$825$$ 0 0
$$826$$ 3156.37 0.132959
$$827$$ 13679.6 0.575193 0.287597 0.957752i $$-0.407144\pi$$
0.287597 + 0.957752i $$0.407144\pi$$
$$828$$ 0 0
$$829$$ 16514.5 0.691886 0.345943 0.938256i $$-0.387559\pi$$
0.345943 + 0.938256i $$0.387559\pi$$
$$830$$ 478.921 0.0200284
$$831$$ 0 0
$$832$$ 3983.64 0.165995
$$833$$ −4759.13 −0.197952
$$834$$ 0 0
$$835$$ −8596.93 −0.356298
$$836$$ 15137.1 0.626231
$$837$$ 0 0
$$838$$ 13177.8 0.543221
$$839$$ 87.9839 0.00362043 0.00181022 0.999998i $$-0.499424\pi$$
0.00181022 + 0.999998i $$0.499424\pi$$
$$840$$ 0 0
$$841$$ −7861.94 −0.322356
$$842$$ 16825.1 0.688635
$$843$$ 0 0
$$844$$ 32371.3 1.32022
$$845$$ −3577.33 −0.145638
$$846$$ 0 0
$$847$$ 4512.98 0.183079
$$848$$ −12115.7 −0.490630
$$849$$ 0 0
$$850$$ 2679.90 0.108141
$$851$$ −15256.0 −0.614534
$$852$$ 0 0
$$853$$ 8162.96 0.327660 0.163830 0.986489i $$-0.447615\pi$$
0.163830 + 0.986489i $$0.447615\pi$$
$$854$$ −582.551 −0.0233425
$$855$$ 0 0
$$856$$ 18988.4 0.758191
$$857$$ −18724.9 −0.746361 −0.373181 0.927759i $$-0.621733\pi$$
−0.373181 + 0.927759i $$0.621733\pi$$
$$858$$ 0 0
$$859$$ −46422.5 −1.84391 −0.921953 0.387301i $$-0.873407\pi$$
−0.921953 + 0.387301i $$0.873407\pi$$
$$860$$ 6571.88 0.260580
$$861$$ 0 0
$$862$$ 4043.61 0.159775
$$863$$ −29112.3 −1.14831 −0.574157 0.818746i $$-0.694670\pi$$
−0.574157 + 0.818746i $$0.694670\pi$$
$$864$$ 0 0
$$865$$ −537.563 −0.0211303
$$866$$ −8346.33 −0.327505
$$867$$ 0 0
$$868$$ 13347.8 0.521950
$$869$$ 19892.6 0.776536
$$870$$ 0 0
$$871$$ 3077.43 0.119719
$$872$$ 34940.2 1.35691
$$873$$ 0 0
$$874$$ −13991.2 −0.541487
$$875$$ −5797.04 −0.223972
$$876$$ 0 0
$$877$$ 39163.0 1.50791 0.753957 0.656924i $$-0.228143\pi$$
0.753957 + 0.656924i $$0.228143\pi$$
$$878$$ 3464.19 0.133156
$$879$$ 0 0
$$880$$ 1922.57 0.0736473
$$881$$ 35073.2 1.34125 0.670627 0.741795i $$-0.266025\pi$$
0.670627 + 0.741795i $$0.266025\pi$$
$$882$$ 0 0
$$883$$ −48775.7 −1.85893 −0.929463 0.368915i $$-0.879729\pi$$
−0.929463 + 0.368915i $$0.879729\pi$$
$$884$$ −6072.69 −0.231048
$$885$$ 0 0
$$886$$ 11611.0 0.440269
$$887$$ −13296.0 −0.503309 −0.251654 0.967817i $$-0.580975\pi$$
−0.251654 + 0.967817i $$0.580975\pi$$
$$888$$ 0 0
$$889$$ −21027.9 −0.793309
$$890$$ 3356.46 0.126415
$$891$$ 0 0
$$892$$ −4330.90 −0.162566
$$893$$ −13607.0 −0.509899
$$894$$ 0 0
$$895$$ −3103.74 −0.115918
$$896$$ −11031.1 −0.411297
$$897$$ 0 0
$$898$$ 12054.6 0.447960
$$899$$ 35155.3 1.30422
$$900$$ 0 0
$$901$$ −8969.43 −0.331648
$$902$$ 17689.1 0.652974
$$903$$ 0 0
$$904$$ 13621.4 0.501153
$$905$$ 10460.0 0.384202
$$906$$ 0 0
$$907$$ −11675.0 −0.427410 −0.213705 0.976898i $$-0.568553\pi$$
−0.213705 + 0.976898i $$0.568553\pi$$
$$908$$ −13224.0 −0.483319
$$909$$ 0 0
$$910$$ −1904.26 −0.0693689
$$911$$ −18552.9 −0.674738 −0.337369 0.941372i $$-0.609537\pi$$
−0.337369 + 0.941372i $$0.609537\pi$$
$$912$$ 0 0
$$913$$ −3204.87 −0.116173
$$914$$ 9739.24 0.352457
$$915$$ 0 0
$$916$$ 24077.7 0.868504
$$917$$ −15714.9 −0.565922
$$918$$ 0 0
$$919$$ 33956.8 1.21886 0.609429 0.792841i $$-0.291399\pi$$
0.609429 + 0.792841i $$0.291399\pi$$
$$920$$ −6729.71 −0.241165
$$921$$ 0 0
$$922$$ −9886.63 −0.353144
$$923$$ −45815.3 −1.63383
$$924$$ 0 0
$$925$$ 15327.4 0.544823
$$926$$ −480.372 −0.0170475
$$927$$ 0 0
$$928$$ −23823.8 −0.842732
$$929$$ 23695.3 0.836832 0.418416 0.908256i $$-0.362585\pi$$
0.418416 + 0.908256i $$0.362585\pi$$
$$930$$ 0 0
$$931$$ −24962.7 −0.878753
$$932$$ −31920.2 −1.12187
$$933$$ 0 0
$$934$$ 2018.72 0.0707221
$$935$$ 1423.31 0.0497830
$$936$$ 0 0
$$937$$ 7990.62 0.278593 0.139297 0.990251i $$-0.455516\pi$$
0.139297 + 0.990251i $$0.455516\pi$$
$$938$$ 572.402 0.0199249
$$939$$ 0 0
$$940$$ −2843.83 −0.0986762
$$941$$ −24385.9 −0.844799 −0.422400 0.906410i $$-0.638812\pi$$
−0.422400 + 0.906410i $$0.638812\pi$$
$$942$$ 0 0
$$943$$ 54241.0 1.87310
$$944$$ 6705.69 0.231199
$$945$$ 0 0
$$946$$ 13256.4 0.455605
$$947$$ 1174.62 0.0403064 0.0201532 0.999797i $$-0.493585\pi$$
0.0201532 + 0.999797i $$0.493585\pi$$
$$948$$ 0 0
$$949$$ 17188.3 0.587939
$$950$$ 14056.7 0.480063
$$951$$ 0 0
$$952$$ −2599.51 −0.0884985
$$953$$ 33546.9 1.14029 0.570143 0.821546i $$-0.306888\pi$$
0.570143 + 0.821546i $$0.306888\pi$$
$$954$$ 0 0
$$955$$ −1488.25 −0.0504277
$$956$$ 2055.00 0.0695224
$$957$$ 0 0
$$958$$ −13598.7 −0.458617
$$959$$ 24948.4 0.840067
$$960$$ 0 0
$$961$$ 44989.1 1.51016
$$962$$ 10469.3 0.350878
$$963$$ 0 0
$$964$$ 11792.9 0.394007
$$965$$ 10758.9 0.358903
$$966$$ 0 0
$$967$$ 24766.8 0.823625 0.411813 0.911269i $$-0.364896\pi$$
0.411813 + 0.911269i $$0.364896\pi$$
$$968$$ −10944.9 −0.363411
$$969$$ 0 0
$$970$$ 3279.56 0.108557
$$971$$ −42324.3 −1.39882 −0.699409 0.714721i $$-0.746553\pi$$
−0.699409 + 0.714721i $$0.746553\pi$$
$$972$$ 0 0
$$973$$ −11657.2 −0.384083
$$974$$ 1538.01 0.0505966
$$975$$ 0 0
$$976$$ −1237.63 −0.0405896
$$977$$ 11320.4 0.370698 0.185349 0.982673i $$-0.440658\pi$$
0.185349 + 0.982673i $$0.440658\pi$$
$$978$$ 0 0
$$979$$ −22461.0 −0.733254
$$980$$ −5217.16 −0.170057
$$981$$ 0 0
$$982$$ 25556.8 0.830500
$$983$$ 11311.9 0.367032 0.183516 0.983017i $$-0.441252\pi$$
0.183516 + 0.983017i $$0.441252\pi$$
$$984$$ 0 0
$$985$$ 4132.66 0.133683
$$986$$ −2974.92 −0.0960860
$$987$$ 0 0
$$988$$ −31852.6 −1.02568
$$989$$ 40648.8 1.30693
$$990$$ 0 0
$$991$$ −29405.5 −0.942580 −0.471290 0.881978i $$-0.656212\pi$$
−0.471290 + 0.881978i $$0.656212\pi$$
$$992$$ −50676.5 −1.62196
$$993$$ 0 0
$$994$$ −8521.63 −0.271921
$$995$$ −11330.9 −0.361018
$$996$$ 0 0
$$997$$ −54905.9 −1.74412 −0.872060 0.489398i $$-0.837216\pi$$
−0.872060 + 0.489398i $$0.837216\pi$$
$$998$$ −23588.7 −0.748183
$$999$$ 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 153.4.a.g.1.2 3
3.2 odd 2 17.4.a.b.1.2 3
4.3 odd 2 2448.4.a.bi.1.1 3
12.11 even 2 272.4.a.h.1.2 3
15.2 even 4 425.4.b.f.324.4 6
15.8 even 4 425.4.b.f.324.3 6
15.14 odd 2 425.4.a.g.1.2 3
21.20 even 2 833.4.a.d.1.2 3
24.5 odd 2 1088.4.a.v.1.2 3
24.11 even 2 1088.4.a.x.1.2 3
33.32 even 2 2057.4.a.e.1.2 3
51.38 odd 4 289.4.b.b.288.3 6
51.47 odd 4 289.4.b.b.288.4 6
51.50 odd 2 289.4.a.b.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.2 3 3.2 odd 2
153.4.a.g.1.2 3 1.1 even 1 trivial
272.4.a.h.1.2 3 12.11 even 2
289.4.a.b.1.2 3 51.50 odd 2
289.4.b.b.288.3 6 51.38 odd 4
289.4.b.b.288.4 6 51.47 odd 4
425.4.a.g.1.2 3 15.14 odd 2
425.4.b.f.324.3 6 15.8 even 4
425.4.b.f.324.4 6 15.2 even 4
833.4.a.d.1.2 3 21.20 even 2
1088.4.a.v.1.2 3 24.5 odd 2
1088.4.a.x.1.2 3 24.11 even 2
2057.4.a.e.1.2 3 33.32 even 2
2448.4.a.bi.1.1 3 4.3 odd 2