Properties

 Label 825.4.a.m.1.1 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.56155 q^{2} -3.00000 q^{3} -5.56155 q^{4} +4.68466 q^{6} +10.2462 q^{7} +21.1771 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-1.56155 q^{2} -3.00000 q^{3} -5.56155 q^{4} +4.68466 q^{6} +10.2462 q^{7} +21.1771 q^{8} +9.00000 q^{9} -11.0000 q^{11} +16.6847 q^{12} +40.8769 q^{13} -16.0000 q^{14} +11.4233 q^{16} +98.7083 q^{17} -14.0540 q^{18} -39.6458 q^{19} -30.7386 q^{21} +17.1771 q^{22} -61.6932 q^{23} -63.5312 q^{24} -63.8314 q^{26} -27.0000 q^{27} -56.9848 q^{28} -149.093 q^{29} +54.7386 q^{31} -187.255 q^{32} +33.0000 q^{33} -154.138 q^{34} -50.0540 q^{36} -44.8939 q^{37} +61.9091 q^{38} -122.631 q^{39} +336.479 q^{41} +48.0000 q^{42} +2.36745 q^{43} +61.1771 q^{44} +96.3371 q^{46} +333.295 q^{47} -34.2699 q^{48} -238.015 q^{49} -296.125 q^{51} -227.339 q^{52} -640.064 q^{53} +42.1619 q^{54} +216.985 q^{56} +118.938 q^{57} +232.816 q^{58} -370.773 q^{59} -714.405 q^{61} -85.4773 q^{62} +92.2159 q^{63} +201.022 q^{64} -51.5312 q^{66} +404.985 q^{67} -548.972 q^{68} +185.080 q^{69} +939.292 q^{71} +190.594 q^{72} +362.570 q^{73} +70.1042 q^{74} +220.492 q^{76} -112.708 q^{77} +191.494 q^{78} +951.835 q^{79} +81.0000 q^{81} -525.430 q^{82} -735.221 q^{83} +170.955 q^{84} -3.69690 q^{86} +447.278 q^{87} -232.948 q^{88} +385.879 q^{89} +418.833 q^{91} +343.110 q^{92} -164.216 q^{93} -520.458 q^{94} +561.764 q^{96} +966.345 q^{97} +371.673 q^{98} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 6 q^{3} - 7 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 - 6 * q^3 - 7 * q^4 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9 $$2 q + q^{2} - 6 q^{3} - 7 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9} - 22 q^{11} + 21 q^{12} + 90 q^{13} - 32 q^{14} - 39 q^{16} + 16 q^{17} + 9 q^{18} - 170 q^{19} - 12 q^{21} - 11 q^{22} + 124 q^{23} + 9 q^{24} + 62 q^{26} - 54 q^{27} - 48 q^{28} - 158 q^{29} + 60 q^{31} - 123 q^{32} + 66 q^{33} - 366 q^{34} - 63 q^{36} + 372 q^{37} - 272 q^{38} - 270 q^{39} + 38 q^{41} + 96 q^{42} + 516 q^{43} + 77 q^{44} + 572 q^{46} - 224 q^{47} + 117 q^{48} - 542 q^{49} - 48 q^{51} - 298 q^{52} - 472 q^{53} - 27 q^{54} + 368 q^{56} + 510 q^{57} + 210 q^{58} + 248 q^{59} + 72 q^{61} - 72 q^{62} + 36 q^{63} + 769 q^{64} + 33 q^{66} + 744 q^{67} - 430 q^{68} - 372 q^{69} + 2060 q^{71} - 27 q^{72} + 486 q^{73} + 1138 q^{74} + 408 q^{76} - 44 q^{77} - 186 q^{78} + 642 q^{79} + 162 q^{81} - 1290 q^{82} + 286 q^{83} + 144 q^{84} + 1312 q^{86} + 474 q^{87} + 33 q^{88} + 244 q^{89} + 112 q^{91} + 76 q^{92} - 180 q^{93} - 1948 q^{94} + 369 q^{96} + 168 q^{97} - 407 q^{98} - 198 q^{99}+O(q^{100})$$ 2 * q + q^2 - 6 * q^3 - 7 * q^4 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9 - 22 * q^11 + 21 * q^12 + 90 * q^13 - 32 * q^14 - 39 * q^16 + 16 * q^17 + 9 * q^18 - 170 * q^19 - 12 * q^21 - 11 * q^22 + 124 * q^23 + 9 * q^24 + 62 * q^26 - 54 * q^27 - 48 * q^28 - 158 * q^29 + 60 * q^31 - 123 * q^32 + 66 * q^33 - 366 * q^34 - 63 * q^36 + 372 * q^37 - 272 * q^38 - 270 * q^39 + 38 * q^41 + 96 * q^42 + 516 * q^43 + 77 * q^44 + 572 * q^46 - 224 * q^47 + 117 * q^48 - 542 * q^49 - 48 * q^51 - 298 * q^52 - 472 * q^53 - 27 * q^54 + 368 * q^56 + 510 * q^57 + 210 * q^58 + 248 * q^59 + 72 * q^61 - 72 * q^62 + 36 * q^63 + 769 * q^64 + 33 * q^66 + 744 * q^67 - 430 * q^68 - 372 * q^69 + 2060 * q^71 - 27 * q^72 + 486 * q^73 + 1138 * q^74 + 408 * q^76 - 44 * q^77 - 186 * q^78 + 642 * q^79 + 162 * q^81 - 1290 * q^82 + 286 * q^83 + 144 * q^84 + 1312 * q^86 + 474 * q^87 + 33 * q^88 + 244 * q^89 + 112 * q^91 + 76 * q^92 - 180 * q^93 - 1948 * q^94 + 369 * q^96 + 168 * q^97 - 407 * q^98 - 198 * q^99

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.56155 −0.552092 −0.276046 0.961144i $$-0.589024\pi$$
−0.276046 + 0.961144i $$0.589024\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ −5.56155 −0.695194
$$5$$ 0 0
$$6$$ 4.68466 0.318751
$$7$$ 10.2462 0.553243 0.276622 0.960979i $$-0.410785\pi$$
0.276622 + 0.960979i $$0.410785\pi$$
$$8$$ 21.1771 0.935904
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 16.6847 0.401371
$$13$$ 40.8769 0.872093 0.436047 0.899924i $$-0.356378\pi$$
0.436047 + 0.899924i $$0.356378\pi$$
$$14$$ −16.0000 −0.305441
$$15$$ 0 0
$$16$$ 11.4233 0.178489
$$17$$ 98.7083 1.40825 0.704126 0.710075i $$-0.251339\pi$$
0.704126 + 0.710075i $$0.251339\pi$$
$$18$$ −14.0540 −0.184031
$$19$$ −39.6458 −0.478704 −0.239352 0.970933i $$-0.576935\pi$$
−0.239352 + 0.970933i $$0.576935\pi$$
$$20$$ 0 0
$$21$$ −30.7386 −0.319415
$$22$$ 17.1771 0.166462
$$23$$ −61.6932 −0.559301 −0.279650 0.960102i $$-0.590219\pi$$
−0.279650 + 0.960102i $$0.590219\pi$$
$$24$$ −63.5312 −0.540344
$$25$$ 0 0
$$26$$ −63.8314 −0.481476
$$27$$ −27.0000 −0.192450
$$28$$ −56.9848 −0.384612
$$29$$ −149.093 −0.954684 −0.477342 0.878718i $$-0.658400\pi$$
−0.477342 + 0.878718i $$0.658400\pi$$
$$30$$ 0 0
$$31$$ 54.7386 0.317140 0.158570 0.987348i $$-0.449312\pi$$
0.158570 + 0.987348i $$0.449312\pi$$
$$32$$ −187.255 −1.03445
$$33$$ 33.0000 0.174078
$$34$$ −154.138 −0.777485
$$35$$ 0 0
$$36$$ −50.0540 −0.231731
$$37$$ −44.8939 −0.199473 −0.0997367 0.995014i $$-0.531800\pi$$
−0.0997367 + 0.995014i $$0.531800\pi$$
$$38$$ 61.9091 0.264289
$$39$$ −122.631 −0.503503
$$40$$ 0 0
$$41$$ 336.479 1.28169 0.640844 0.767671i $$-0.278585\pi$$
0.640844 + 0.767671i $$0.278585\pi$$
$$42$$ 48.0000 0.176347
$$43$$ 2.36745 0.00839611 0.00419806 0.999991i $$-0.498664\pi$$
0.00419806 + 0.999991i $$0.498664\pi$$
$$44$$ 61.1771 0.209609
$$45$$ 0 0
$$46$$ 96.3371 0.308786
$$47$$ 333.295 1.03439 0.517193 0.855869i $$-0.326977\pi$$
0.517193 + 0.855869i $$0.326977\pi$$
$$48$$ −34.2699 −0.103051
$$49$$ −238.015 −0.693922
$$50$$ 0 0
$$51$$ −296.125 −0.813055
$$52$$ −227.339 −0.606274
$$53$$ −640.064 −1.65886 −0.829430 0.558610i $$-0.811335\pi$$
−0.829430 + 0.558610i $$0.811335\pi$$
$$54$$ 42.1619 0.106250
$$55$$ 0 0
$$56$$ 216.985 0.517782
$$57$$ 118.938 0.276380
$$58$$ 232.816 0.527074
$$59$$ −370.773 −0.818144 −0.409072 0.912502i $$-0.634147\pi$$
−0.409072 + 0.912502i $$0.634147\pi$$
$$60$$ 0 0
$$61$$ −714.405 −1.49951 −0.749756 0.661715i $$-0.769829\pi$$
−0.749756 + 0.661715i $$0.769829\pi$$
$$62$$ −85.4773 −0.175091
$$63$$ 92.2159 0.184414
$$64$$ 201.022 0.392621
$$65$$ 0 0
$$66$$ −51.5312 −0.0961069
$$67$$ 404.985 0.738459 0.369230 0.929338i $$-0.379622\pi$$
0.369230 + 0.929338i $$0.379622\pi$$
$$68$$ −548.972 −0.979009
$$69$$ 185.080 0.322912
$$70$$ 0 0
$$71$$ 939.292 1.57005 0.785024 0.619465i $$-0.212651\pi$$
0.785024 + 0.619465i $$0.212651\pi$$
$$72$$ 190.594 0.311968
$$73$$ 362.570 0.581310 0.290655 0.956828i $$-0.406127\pi$$
0.290655 + 0.956828i $$0.406127\pi$$
$$74$$ 70.1042 0.110128
$$75$$ 0 0
$$76$$ 220.492 0.332792
$$77$$ −112.708 −0.166809
$$78$$ 191.494 0.277980
$$79$$ 951.835 1.35557 0.677784 0.735261i $$-0.262941\pi$$
0.677784 + 0.735261i $$0.262941\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −525.430 −0.707610
$$83$$ −735.221 −0.972302 −0.486151 0.873875i $$-0.661599\pi$$
−0.486151 + 0.873875i $$0.661599\pi$$
$$84$$ 170.955 0.222056
$$85$$ 0 0
$$86$$ −3.69690 −0.00463543
$$87$$ 447.278 0.551187
$$88$$ −232.948 −0.282186
$$89$$ 385.879 0.459585 0.229793 0.973240i $$-0.426195\pi$$
0.229793 + 0.973240i $$0.426195\pi$$
$$90$$ 0 0
$$91$$ 418.833 0.482480
$$92$$ 343.110 0.388823
$$93$$ −164.216 −0.183101
$$94$$ −520.458 −0.571076
$$95$$ 0 0
$$96$$ 561.764 0.597238
$$97$$ 966.345 1.01152 0.505760 0.862674i $$-0.331212\pi$$
0.505760 + 0.862674i $$0.331212\pi$$
$$98$$ 371.673 0.383109
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ 348.600 0.343436 0.171718 0.985146i $$-0.445068\pi$$
0.171718 + 0.985146i $$0.445068\pi$$
$$102$$ 462.415 0.448881
$$103$$ 1536.38 1.46975 0.734873 0.678204i $$-0.237242\pi$$
0.734873 + 0.678204i $$0.237242\pi$$
$$104$$ 865.653 0.816195
$$105$$ 0 0
$$106$$ 999.494 0.915844
$$107$$ 779.180 0.703983 0.351991 0.936003i $$-0.385505\pi$$
0.351991 + 0.936003i $$0.385505\pi$$
$$108$$ 150.162 0.133790
$$109$$ −1501.79 −1.31968 −0.659842 0.751404i $$-0.729377\pi$$
−0.659842 + 0.751404i $$0.729377\pi$$
$$110$$ 0 0
$$111$$ 134.682 0.115166
$$112$$ 117.045 0.0987478
$$113$$ −170.000 −0.141524 −0.0707622 0.997493i $$-0.522543\pi$$
−0.0707622 + 0.997493i $$0.522543\pi$$
$$114$$ −185.727 −0.152587
$$115$$ 0 0
$$116$$ 829.187 0.663691
$$117$$ 367.892 0.290698
$$118$$ 578.981 0.451691
$$119$$ 1011.39 0.779106
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 1115.58 0.827869
$$123$$ −1009.44 −0.739983
$$124$$ −304.432 −0.220474
$$125$$ 0 0
$$126$$ −144.000 −0.101814
$$127$$ 1739.82 1.21562 0.607811 0.794082i $$-0.292048\pi$$
0.607811 + 0.794082i $$0.292048\pi$$
$$128$$ 1184.13 0.817683
$$129$$ −7.10235 −0.00484750
$$130$$ 0 0
$$131$$ 312.837 0.208647 0.104323 0.994543i $$-0.466732\pi$$
0.104323 + 0.994543i $$0.466732\pi$$
$$132$$ −183.531 −0.121018
$$133$$ −406.220 −0.264840
$$134$$ −632.405 −0.407698
$$135$$ 0 0
$$136$$ 2090.35 1.31799
$$137$$ 716.928 0.447090 0.223545 0.974694i $$-0.428237\pi$$
0.223545 + 0.974694i $$0.428237\pi$$
$$138$$ −289.011 −0.178277
$$139$$ −876.483 −0.534837 −0.267418 0.963581i $$-0.586171\pi$$
−0.267418 + 0.963581i $$0.586171\pi$$
$$140$$ 0 0
$$141$$ −999.886 −0.597203
$$142$$ −1466.75 −0.866811
$$143$$ −449.646 −0.262946
$$144$$ 102.810 0.0594963
$$145$$ 0 0
$$146$$ −566.172 −0.320937
$$147$$ 714.045 0.400636
$$148$$ 249.680 0.138673
$$149$$ −2376.36 −1.30657 −0.653285 0.757112i $$-0.726610\pi$$
−0.653285 + 0.757112i $$0.726610\pi$$
$$150$$ 0 0
$$151$$ −92.8466 −0.0500381 −0.0250190 0.999687i $$-0.507965\pi$$
−0.0250190 + 0.999687i $$0.507965\pi$$
$$152$$ −839.583 −0.448021
$$153$$ 888.375 0.469417
$$154$$ 176.000 0.0920941
$$155$$ 0 0
$$156$$ 682.017 0.350032
$$157$$ 1881.24 0.956301 0.478150 0.878278i $$-0.341307\pi$$
0.478150 + 0.878278i $$0.341307\pi$$
$$158$$ −1486.34 −0.748398
$$159$$ 1920.19 0.957743
$$160$$ 0 0
$$161$$ −632.121 −0.309429
$$162$$ −126.486 −0.0613436
$$163$$ 2465.49 1.18474 0.592369 0.805667i $$-0.298193\pi$$
0.592369 + 0.805667i $$0.298193\pi$$
$$164$$ −1871.35 −0.891022
$$165$$ 0 0
$$166$$ 1148.09 0.536800
$$167$$ 1254.30 0.581200 0.290600 0.956845i $$-0.406145\pi$$
0.290600 + 0.956845i $$0.406145\pi$$
$$168$$ −650.955 −0.298942
$$169$$ −526.080 −0.239454
$$170$$ 0 0
$$171$$ −356.813 −0.159568
$$172$$ −13.1667 −0.00583693
$$173$$ 1206.71 0.530314 0.265157 0.964205i $$-0.414576\pi$$
0.265157 + 0.964205i $$0.414576\pi$$
$$174$$ −698.449 −0.304306
$$175$$ 0 0
$$176$$ −125.656 −0.0538164
$$177$$ 1112.32 0.472356
$$178$$ −602.570 −0.253733
$$179$$ −1442.29 −0.602244 −0.301122 0.953586i $$-0.597361\pi$$
−0.301122 + 0.953586i $$0.597361\pi$$
$$180$$ 0 0
$$181$$ 4261.81 1.75015 0.875076 0.483985i $$-0.160811\pi$$
0.875076 + 0.483985i $$0.160811\pi$$
$$182$$ −654.030 −0.266373
$$183$$ 2143.22 0.865744
$$184$$ −1306.48 −0.523451
$$185$$ 0 0
$$186$$ 256.432 0.101089
$$187$$ −1085.79 −0.424604
$$188$$ −1853.64 −0.719099
$$189$$ −276.648 −0.106472
$$190$$ 0 0
$$191$$ −852.223 −0.322852 −0.161426 0.986885i $$-0.551609\pi$$
−0.161426 + 0.986885i $$0.551609\pi$$
$$192$$ −603.065 −0.226680
$$193$$ 2459.95 0.917468 0.458734 0.888574i $$-0.348303\pi$$
0.458734 + 0.888574i $$0.348303\pi$$
$$194$$ −1509.00 −0.558452
$$195$$ 0 0
$$196$$ 1323.73 0.482410
$$197$$ −3477.06 −1.25751 −0.628756 0.777602i $$-0.716436\pi$$
−0.628756 + 0.777602i $$0.716436\pi$$
$$198$$ 154.594 0.0554874
$$199$$ 3995.04 1.42312 0.711560 0.702626i $$-0.247989\pi$$
0.711560 + 0.702626i $$0.247989\pi$$
$$200$$ 0 0
$$201$$ −1214.95 −0.426350
$$202$$ −544.358 −0.189608
$$203$$ −1527.64 −0.528173
$$204$$ 1646.91 0.565231
$$205$$ 0 0
$$206$$ −2399.14 −0.811436
$$207$$ −555.239 −0.186434
$$208$$ 466.949 0.155659
$$209$$ 436.104 0.144335
$$210$$ 0 0
$$211$$ 1046.13 0.341319 0.170660 0.985330i $$-0.445410\pi$$
0.170660 + 0.985330i $$0.445410\pi$$
$$212$$ 3559.75 1.15323
$$213$$ −2817.88 −0.906468
$$214$$ −1216.73 −0.388664
$$215$$ 0 0
$$216$$ −571.781 −0.180115
$$217$$ 560.864 0.175456
$$218$$ 2345.13 0.728587
$$219$$ −1087.71 −0.335619
$$220$$ 0 0
$$221$$ 4034.89 1.22813
$$222$$ −210.313 −0.0635823
$$223$$ 506.265 0.152027 0.0760135 0.997107i $$-0.475781\pi$$
0.0760135 + 0.997107i $$0.475781\pi$$
$$224$$ −1918.65 −0.572300
$$225$$ 0 0
$$226$$ 265.464 0.0781345
$$227$$ 4286.29 1.25326 0.626632 0.779315i $$-0.284433\pi$$
0.626632 + 0.779315i $$0.284433\pi$$
$$228$$ −661.477 −0.192138
$$229$$ 5709.37 1.64754 0.823769 0.566926i $$-0.191867\pi$$
0.823769 + 0.566926i $$0.191867\pi$$
$$230$$ 0 0
$$231$$ 338.125 0.0963073
$$232$$ −3157.35 −0.893492
$$233$$ −2946.09 −0.828348 −0.414174 0.910198i $$-0.635929\pi$$
−0.414174 + 0.910198i $$0.635929\pi$$
$$234$$ −574.483 −0.160492
$$235$$ 0 0
$$236$$ 2062.07 0.568769
$$237$$ −2855.51 −0.782637
$$238$$ −1579.33 −0.430139
$$239$$ −2078.89 −0.562646 −0.281323 0.959613i $$-0.590773\pi$$
−0.281323 + 0.959613i $$0.590773\pi$$
$$240$$ 0 0
$$241$$ 1853.37 0.495378 0.247689 0.968840i $$-0.420329\pi$$
0.247689 + 0.968840i $$0.420329\pi$$
$$242$$ −188.948 −0.0501902
$$243$$ −243.000 −0.0641500
$$244$$ 3973.20 1.04245
$$245$$ 0 0
$$246$$ 1576.29 0.408539
$$247$$ −1620.60 −0.417475
$$248$$ 1159.20 0.296813
$$249$$ 2205.66 0.561359
$$250$$ 0 0
$$251$$ −2358.39 −0.593068 −0.296534 0.955022i $$-0.595831\pi$$
−0.296534 + 0.955022i $$0.595831\pi$$
$$252$$ −512.864 −0.128204
$$253$$ 678.625 0.168635
$$254$$ −2716.82 −0.671135
$$255$$ 0 0
$$256$$ −3457.26 −0.844057
$$257$$ −5519.25 −1.33962 −0.669809 0.742534i $$-0.733624\pi$$
−0.669809 + 0.742534i $$0.733624\pi$$
$$258$$ 11.0907 0.00267627
$$259$$ −459.993 −0.110357
$$260$$ 0 0
$$261$$ −1341.84 −0.318228
$$262$$ −488.512 −0.115192
$$263$$ −2259.65 −0.529795 −0.264898 0.964277i $$-0.585338\pi$$
−0.264898 + 0.964277i $$0.585338\pi$$
$$264$$ 698.844 0.162920
$$265$$ 0 0
$$266$$ 634.333 0.146216
$$267$$ −1157.64 −0.265342
$$268$$ −2252.34 −0.513373
$$269$$ 7039.53 1.59557 0.797783 0.602944i $$-0.206006\pi$$
0.797783 + 0.602944i $$0.206006\pi$$
$$270$$ 0 0
$$271$$ 5155.08 1.15553 0.577765 0.816203i $$-0.303925\pi$$
0.577765 + 0.816203i $$0.303925\pi$$
$$272$$ 1127.57 0.251357
$$273$$ −1256.50 −0.278560
$$274$$ −1119.52 −0.246835
$$275$$ 0 0
$$276$$ −1029.33 −0.224487
$$277$$ −9074.52 −1.96836 −0.984179 0.177175i $$-0.943304\pi$$
−0.984179 + 0.177175i $$0.943304\pi$$
$$278$$ 1368.67 0.295279
$$279$$ 492.648 0.105713
$$280$$ 0 0
$$281$$ −3407.79 −0.723459 −0.361729 0.932283i $$-0.617814\pi$$
−0.361729 + 0.932283i $$0.617814\pi$$
$$282$$ 1561.38 0.329711
$$283$$ 8827.73 1.85425 0.927127 0.374746i $$-0.122270\pi$$
0.927127 + 0.374746i $$0.122270\pi$$
$$284$$ −5223.92 −1.09149
$$285$$ 0 0
$$286$$ 702.146 0.145170
$$287$$ 3447.64 0.709085
$$288$$ −1685.29 −0.344815
$$289$$ 4830.33 0.983174
$$290$$ 0 0
$$291$$ −2899.03 −0.584001
$$292$$ −2016.45 −0.404123
$$293$$ 4528.29 0.902886 0.451443 0.892300i $$-0.350909\pi$$
0.451443 + 0.892300i $$0.350909\pi$$
$$294$$ −1115.02 −0.221188
$$295$$ 0 0
$$296$$ −950.722 −0.186688
$$297$$ 297.000 0.0580259
$$298$$ 3710.81 0.721347
$$299$$ −2521.83 −0.487762
$$300$$ 0 0
$$301$$ 24.2574 0.00464509
$$302$$ 144.985 0.0276256
$$303$$ −1045.80 −0.198283
$$304$$ −452.886 −0.0854434
$$305$$ 0 0
$$306$$ −1387.24 −0.259162
$$307$$ −568.106 −0.105614 −0.0528071 0.998605i $$-0.516817\pi$$
−0.0528071 + 0.998605i $$0.516817\pi$$
$$308$$ 626.833 0.115965
$$309$$ −4609.14 −0.848559
$$310$$ 0 0
$$311$$ −6853.59 −1.24962 −0.624809 0.780778i $$-0.714823\pi$$
−0.624809 + 0.780778i $$0.714823\pi$$
$$312$$ −2596.96 −0.471230
$$313$$ 1138.92 0.205673 0.102837 0.994698i $$-0.467208\pi$$
0.102837 + 0.994698i $$0.467208\pi$$
$$314$$ −2937.65 −0.527966
$$315$$ 0 0
$$316$$ −5293.68 −0.942382
$$317$$ −3207.48 −0.568297 −0.284148 0.958780i $$-0.591711\pi$$
−0.284148 + 0.958780i $$0.591711\pi$$
$$318$$ −2998.48 −0.528763
$$319$$ 1640.02 0.287848
$$320$$ 0 0
$$321$$ −2337.54 −0.406445
$$322$$ 987.091 0.170834
$$323$$ −3913.37 −0.674136
$$324$$ −450.486 −0.0772438
$$325$$ 0 0
$$326$$ −3850.00 −0.654085
$$327$$ 4505.37 0.761920
$$328$$ 7125.65 1.19954
$$329$$ 3415.02 0.572267
$$330$$ 0 0
$$331$$ −9135.12 −1.51695 −0.758477 0.651700i $$-0.774056\pi$$
−0.758477 + 0.651700i $$0.774056\pi$$
$$332$$ 4088.97 0.675938
$$333$$ −404.045 −0.0664911
$$334$$ −1958.65 −0.320876
$$335$$ 0 0
$$336$$ −351.136 −0.0570121
$$337$$ 3470.05 0.560907 0.280453 0.959868i $$-0.409515\pi$$
0.280453 + 0.959868i $$0.409515\pi$$
$$338$$ 821.501 0.132200
$$339$$ 510.000 0.0817091
$$340$$ 0 0
$$341$$ −602.125 −0.0956214
$$342$$ 557.182 0.0880963
$$343$$ −5953.20 −0.937151
$$344$$ 50.1357 0.00785795
$$345$$ 0 0
$$346$$ −1884.34 −0.292782
$$347$$ 89.3315 0.0138201 0.00691004 0.999976i $$-0.497800\pi$$
0.00691004 + 0.999976i $$0.497800\pi$$
$$348$$ −2487.56 −0.383182
$$349$$ −149.375 −0.0229107 −0.0114554 0.999934i $$-0.503646\pi$$
−0.0114554 + 0.999934i $$0.503646\pi$$
$$350$$ 0 0
$$351$$ −1103.68 −0.167834
$$352$$ 2059.80 0.311897
$$353$$ −7867.64 −1.18627 −0.593133 0.805104i $$-0.702109\pi$$
−0.593133 + 0.805104i $$0.702109\pi$$
$$354$$ −1736.94 −0.260784
$$355$$ 0 0
$$356$$ −2146.09 −0.319501
$$357$$ −3034.16 −0.449817
$$358$$ 2252.21 0.332494
$$359$$ 4974.22 0.731279 0.365639 0.930757i $$-0.380850\pi$$
0.365639 + 0.930757i $$0.380850\pi$$
$$360$$ 0 0
$$361$$ −5287.21 −0.770842
$$362$$ −6655.04 −0.966246
$$363$$ −363.000 −0.0524864
$$364$$ −2329.36 −0.335417
$$365$$ 0 0
$$366$$ −3346.74 −0.477970
$$367$$ 13266.7 1.88696 0.943479 0.331433i $$-0.107532\pi$$
0.943479 + 0.331433i $$0.107532\pi$$
$$368$$ −704.739 −0.0998290
$$369$$ 3028.31 0.427229
$$370$$ 0 0
$$371$$ −6558.23 −0.917754
$$372$$ 913.295 0.127291
$$373$$ 4632.77 0.643099 0.321549 0.946893i $$-0.395796\pi$$
0.321549 + 0.946893i $$0.395796\pi$$
$$374$$ 1695.52 0.234421
$$375$$ 0 0
$$376$$ 7058.22 0.968085
$$377$$ −6094.45 −0.832573
$$378$$ 432.000 0.0587822
$$379$$ 6503.31 0.881406 0.440703 0.897653i $$-0.354729\pi$$
0.440703 + 0.897653i $$0.354729\pi$$
$$380$$ 0 0
$$381$$ −5219.45 −0.701839
$$382$$ 1330.79 0.178244
$$383$$ −12734.5 −1.69897 −0.849484 0.527614i $$-0.823087\pi$$
−0.849484 + 0.527614i $$0.823087\pi$$
$$384$$ −3552.39 −0.472090
$$385$$ 0 0
$$386$$ −3841.35 −0.506527
$$387$$ 21.3071 0.00279870
$$388$$ −5374.38 −0.703203
$$389$$ 12024.6 1.56728 0.783639 0.621216i $$-0.213361\pi$$
0.783639 + 0.621216i $$0.213361\pi$$
$$390$$ 0 0
$$391$$ −6089.63 −0.787636
$$392$$ −5040.47 −0.649444
$$393$$ −938.511 −0.120462
$$394$$ 5429.61 0.694263
$$395$$ 0 0
$$396$$ 550.594 0.0698696
$$397$$ 5223.65 0.660371 0.330186 0.943916i $$-0.392889\pi$$
0.330186 + 0.943916i $$0.392889\pi$$
$$398$$ −6238.46 −0.785693
$$399$$ 1218.66 0.152905
$$400$$ 0 0
$$401$$ 9648.18 1.20151 0.600757 0.799432i $$-0.294866\pi$$
0.600757 + 0.799432i $$0.294866\pi$$
$$402$$ 1897.22 0.235384
$$403$$ 2237.55 0.276576
$$404$$ −1938.76 −0.238755
$$405$$ 0 0
$$406$$ 2385.48 0.291600
$$407$$ 493.833 0.0601435
$$408$$ −6271.06 −0.760941
$$409$$ −2010.47 −0.243060 −0.121530 0.992588i $$-0.538780\pi$$
−0.121530 + 0.992588i $$0.538780\pi$$
$$410$$ 0 0
$$411$$ −2150.78 −0.258127
$$412$$ −8544.65 −1.02176
$$413$$ −3799.02 −0.452633
$$414$$ 867.034 0.102929
$$415$$ 0 0
$$416$$ −7654.39 −0.902133
$$417$$ 2629.45 0.308788
$$418$$ −681.000 −0.0796861
$$419$$ 4435.27 0.517129 0.258565 0.965994i $$-0.416750\pi$$
0.258565 + 0.965994i $$0.416750\pi$$
$$420$$ 0 0
$$421$$ 15217.9 1.76170 0.880852 0.473392i $$-0.156971\pi$$
0.880852 + 0.473392i $$0.156971\pi$$
$$422$$ −1633.58 −0.188440
$$423$$ 2999.66 0.344795
$$424$$ −13554.7 −1.55253
$$425$$ 0 0
$$426$$ 4400.26 0.500454
$$427$$ −7319.95 −0.829595
$$428$$ −4333.45 −0.489405
$$429$$ 1348.94 0.151812
$$430$$ 0 0
$$431$$ −5622.11 −0.628324 −0.314162 0.949369i $$-0.601723\pi$$
−0.314162 + 0.949369i $$0.601723\pi$$
$$432$$ −308.429 −0.0343502
$$433$$ 14306.3 1.58780 0.793898 0.608051i $$-0.208049\pi$$
0.793898 + 0.608051i $$0.208049\pi$$
$$434$$ −875.818 −0.0968678
$$435$$ 0 0
$$436$$ 8352.29 0.917436
$$437$$ 2445.88 0.267740
$$438$$ 1698.52 0.185293
$$439$$ 4384.20 0.476643 0.238322 0.971186i $$-0.423403\pi$$
0.238322 + 0.971186i $$0.423403\pi$$
$$440$$ 0 0
$$441$$ −2142.14 −0.231307
$$442$$ −6300.69 −0.678039
$$443$$ 10090.0 1.08214 0.541071 0.840977i $$-0.318019\pi$$
0.541071 + 0.840977i $$0.318019\pi$$
$$444$$ −749.040 −0.0800627
$$445$$ 0 0
$$446$$ −790.560 −0.0839330
$$447$$ 7129.07 0.754348
$$448$$ 2059.71 0.217215
$$449$$ 9582.52 1.00719 0.503594 0.863941i $$-0.332011\pi$$
0.503594 + 0.863941i $$0.332011\pi$$
$$450$$ 0 0
$$451$$ −3701.27 −0.386444
$$452$$ 945.464 0.0983869
$$453$$ 278.540 0.0288895
$$454$$ −6693.27 −0.691918
$$455$$ 0 0
$$456$$ 2518.75 0.258665
$$457$$ −9999.34 −1.02352 −0.511761 0.859128i $$-0.671007\pi$$
−0.511761 + 0.859128i $$0.671007\pi$$
$$458$$ −8915.49 −0.909593
$$459$$ −2665.12 −0.271018
$$460$$ 0 0
$$461$$ −11115.8 −1.12302 −0.561512 0.827468i $$-0.689780\pi$$
−0.561512 + 0.827468i $$0.689780\pi$$
$$462$$ −528.000 −0.0531705
$$463$$ 1567.16 0.157305 0.0786524 0.996902i $$-0.474938\pi$$
0.0786524 + 0.996902i $$0.474938\pi$$
$$464$$ −1703.13 −0.170401
$$465$$ 0 0
$$466$$ 4600.48 0.457325
$$467$$ −12648.8 −1.25335 −0.626675 0.779281i $$-0.715585\pi$$
−0.626675 + 0.779281i $$0.715585\pi$$
$$468$$ −2046.05 −0.202091
$$469$$ 4149.56 0.408548
$$470$$ 0 0
$$471$$ −5643.72 −0.552120
$$472$$ −7851.88 −0.765704
$$473$$ −26.0420 −0.00253152
$$474$$ 4459.02 0.432088
$$475$$ 0 0
$$476$$ −5624.88 −0.541630
$$477$$ −5760.58 −0.552953
$$478$$ 3246.30 0.310633
$$479$$ −10719.2 −1.02249 −0.511247 0.859434i $$-0.670816\pi$$
−0.511247 + 0.859434i $$0.670816\pi$$
$$480$$ 0 0
$$481$$ −1835.12 −0.173959
$$482$$ −2894.14 −0.273494
$$483$$ 1896.36 0.178649
$$484$$ −672.948 −0.0631995
$$485$$ 0 0
$$486$$ 379.457 0.0354167
$$487$$ −7161.20 −0.666335 −0.333167 0.942868i $$-0.608117\pi$$
−0.333167 + 0.942868i $$0.608117\pi$$
$$488$$ −15129.0 −1.40340
$$489$$ −7396.48 −0.684009
$$490$$ 0 0
$$491$$ 14567.3 1.33893 0.669463 0.742845i $$-0.266524\pi$$
0.669463 + 0.742845i $$0.266524\pi$$
$$492$$ 5614.04 0.514432
$$493$$ −14716.7 −1.34444
$$494$$ 2530.65 0.230485
$$495$$ 0 0
$$496$$ 625.295 0.0566060
$$497$$ 9624.18 0.868619
$$498$$ −3444.26 −0.309922
$$499$$ −4638.99 −0.416172 −0.208086 0.978111i $$-0.566723\pi$$
−0.208086 + 0.978111i $$0.566723\pi$$
$$500$$ 0 0
$$501$$ −3762.89 −0.335556
$$502$$ 3682.75 0.327428
$$503$$ 12206.3 1.08201 0.541006 0.841019i $$-0.318044\pi$$
0.541006 + 0.841019i $$0.318044\pi$$
$$504$$ 1952.86 0.172594
$$505$$ 0 0
$$506$$ −1059.71 −0.0931024
$$507$$ 1578.24 0.138249
$$508$$ −9676.09 −0.845093
$$509$$ 10018.6 0.872427 0.436214 0.899843i $$-0.356319\pi$$
0.436214 + 0.899843i $$0.356319\pi$$
$$510$$ 0 0
$$511$$ 3714.97 0.321606
$$512$$ −4074.36 −0.351686
$$513$$ 1070.44 0.0921267
$$514$$ 8618.61 0.739592
$$515$$ 0 0
$$516$$ 39.5001 0.00336995
$$517$$ −3666.25 −0.311879
$$518$$ 718.303 0.0609274
$$519$$ −3620.12 −0.306177
$$520$$ 0 0
$$521$$ 1054.72 0.0886916 0.0443458 0.999016i $$-0.485880\pi$$
0.0443458 + 0.999016i $$0.485880\pi$$
$$522$$ 2095.35 0.175691
$$523$$ 16234.2 1.35730 0.678652 0.734460i $$-0.262564\pi$$
0.678652 + 0.734460i $$0.262564\pi$$
$$524$$ −1739.86 −0.145050
$$525$$ 0 0
$$526$$ 3528.57 0.292496
$$527$$ 5403.16 0.446613
$$528$$ 376.969 0.0310709
$$529$$ −8360.95 −0.687183
$$530$$ 0 0
$$531$$ −3336.95 −0.272715
$$532$$ 2259.21 0.184115
$$533$$ 13754.2 1.11775
$$534$$ 1807.71 0.146493
$$535$$ 0 0
$$536$$ 8576.40 0.691127
$$537$$ 4326.86 0.347706
$$538$$ −10992.6 −0.880900
$$539$$ 2618.17 0.209225
$$540$$ 0 0
$$541$$ 675.936 0.0537167 0.0268584 0.999639i $$-0.491450\pi$$
0.0268584 + 0.999639i $$0.491450\pi$$
$$542$$ −8049.93 −0.637960
$$543$$ −12785.4 −1.01045
$$544$$ −18483.6 −1.45676
$$545$$ 0 0
$$546$$ 1962.09 0.153791
$$547$$ −13058.2 −1.02071 −0.510355 0.859964i $$-0.670486\pi$$
−0.510355 + 0.859964i $$0.670486\pi$$
$$548$$ −3987.23 −0.310814
$$549$$ −6429.65 −0.499837
$$550$$ 0 0
$$551$$ 5910.91 0.457011
$$552$$ 3919.44 0.302215
$$553$$ 9752.70 0.749959
$$554$$ 14170.3 1.08672
$$555$$ 0 0
$$556$$ 4874.61 0.371815
$$557$$ 6710.48 0.510471 0.255236 0.966879i $$-0.417847\pi$$
0.255236 + 0.966879i $$0.417847\pi$$
$$558$$ −769.295 −0.0583636
$$559$$ 96.7741 0.00732219
$$560$$ 0 0
$$561$$ 3257.37 0.245145
$$562$$ 5321.45 0.399416
$$563$$ 20820.5 1.55858 0.779288 0.626666i $$-0.215581\pi$$
0.779288 + 0.626666i $$0.215581\pi$$
$$564$$ 5560.92 0.415172
$$565$$ 0 0
$$566$$ −13785.0 −1.02372
$$567$$ 829.943 0.0614715
$$568$$ 19891.5 1.46941
$$569$$ 3251.08 0.239530 0.119765 0.992802i $$-0.461786\pi$$
0.119765 + 0.992802i $$0.461786\pi$$
$$570$$ 0 0
$$571$$ −4637.50 −0.339883 −0.169941 0.985454i $$-0.554358\pi$$
−0.169941 + 0.985454i $$0.554358\pi$$
$$572$$ 2500.73 0.182798
$$573$$ 2556.67 0.186399
$$574$$ −5383.67 −0.391481
$$575$$ 0 0
$$576$$ 1809.20 0.130874
$$577$$ −14462.4 −1.04346 −0.521730 0.853111i $$-0.674713\pi$$
−0.521730 + 0.853111i $$0.674713\pi$$
$$578$$ −7542.82 −0.542803
$$579$$ −7379.86 −0.529700
$$580$$ 0 0
$$581$$ −7533.23 −0.537920
$$582$$ 4526.99 0.322423
$$583$$ 7040.71 0.500165
$$584$$ 7678.18 0.544050
$$585$$ 0 0
$$586$$ −7071.17 −0.498476
$$587$$ 22759.7 1.60033 0.800166 0.599779i $$-0.204745\pi$$
0.800166 + 0.599779i $$0.204745\pi$$
$$588$$ −3971.20 −0.278520
$$589$$ −2170.16 −0.151816
$$590$$ 0 0
$$591$$ 10431.2 0.726025
$$592$$ −512.836 −0.0356038
$$593$$ 14956.4 1.03573 0.517864 0.855463i $$-0.326727\pi$$
0.517864 + 0.855463i $$0.326727\pi$$
$$594$$ −463.781 −0.0320356
$$595$$ 0 0
$$596$$ 13216.2 0.908319
$$597$$ −11985.1 −0.821638
$$598$$ 3937.96 0.269290
$$599$$ 2150.77 0.146708 0.0733539 0.997306i $$-0.476630\pi$$
0.0733539 + 0.997306i $$0.476630\pi$$
$$600$$ 0 0
$$601$$ 27759.8 1.88410 0.942050 0.335472i $$-0.108896\pi$$
0.942050 + 0.335472i $$0.108896\pi$$
$$602$$ −37.8792 −0.00256452
$$603$$ 3644.86 0.246153
$$604$$ 516.371 0.0347862
$$605$$ 0 0
$$606$$ 1633.07 0.109470
$$607$$ 10991.5 0.734974 0.367487 0.930029i $$-0.380218\pi$$
0.367487 + 0.930029i $$0.380218\pi$$
$$608$$ 7423.87 0.495194
$$609$$ 4582.91 0.304941
$$610$$ 0 0
$$611$$ 13624.1 0.902081
$$612$$ −4940.74 −0.326336
$$613$$ −10646.1 −0.701457 −0.350728 0.936477i $$-0.614066\pi$$
−0.350728 + 0.936477i $$0.614066\pi$$
$$614$$ 887.128 0.0583088
$$615$$ 0 0
$$616$$ −2386.83 −0.156117
$$617$$ −7199.92 −0.469786 −0.234893 0.972021i $$-0.575474\pi$$
−0.234893 + 0.972021i $$0.575474\pi$$
$$618$$ 7197.41 0.468483
$$619$$ 12186.9 0.791332 0.395666 0.918395i $$-0.370514\pi$$
0.395666 + 0.918395i $$0.370514\pi$$
$$620$$ 0 0
$$621$$ 1665.72 0.107637
$$622$$ 10702.2 0.689905
$$623$$ 3953.80 0.254262
$$624$$ −1400.85 −0.0898698
$$625$$ 0 0
$$626$$ −1778.49 −0.113551
$$627$$ −1308.31 −0.0833317
$$628$$ −10462.6 −0.664814
$$629$$ −4431.40 −0.280909
$$630$$ 0 0
$$631$$ 7370.64 0.465009 0.232505 0.972595i $$-0.425308\pi$$
0.232505 + 0.972595i $$0.425308\pi$$
$$632$$ 20157.1 1.26868
$$633$$ −3138.38 −0.197061
$$634$$ 5008.65 0.313752
$$635$$ 0 0
$$636$$ −10679.3 −0.665818
$$637$$ −9729.32 −0.605164
$$638$$ −2560.98 −0.158919
$$639$$ 8453.63 0.523349
$$640$$ 0 0
$$641$$ −25014.9 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$642$$ 3650.19 0.224395
$$643$$ 21668.2 1.32894 0.664472 0.747313i $$-0.268657\pi$$
0.664472 + 0.747313i $$0.268657\pi$$
$$644$$ 3515.58 0.215113
$$645$$ 0 0
$$646$$ 6110.94 0.372185
$$647$$ 27625.3 1.67861 0.839305 0.543661i $$-0.182962\pi$$
0.839305 + 0.543661i $$0.182962\pi$$
$$648$$ 1715.34 0.103989
$$649$$ 4078.50 0.246680
$$650$$ 0 0
$$651$$ −1682.59 −0.101299
$$652$$ −13712.0 −0.823623
$$653$$ 14314.0 0.857810 0.428905 0.903350i $$-0.358899\pi$$
0.428905 + 0.903350i $$0.358899\pi$$
$$654$$ −7035.38 −0.420650
$$655$$ 0 0
$$656$$ 3843.70 0.228767
$$657$$ 3263.13 0.193770
$$658$$ −5332.73 −0.315944
$$659$$ −28327.8 −1.67450 −0.837249 0.546822i $$-0.815837\pi$$
−0.837249 + 0.546822i $$0.815837\pi$$
$$660$$ 0 0
$$661$$ −32190.9 −1.89422 −0.947112 0.320905i $$-0.896013\pi$$
−0.947112 + 0.320905i $$0.896013\pi$$
$$662$$ 14265.0 0.837498
$$663$$ −12104.7 −0.709059
$$664$$ −15569.8 −0.909981
$$665$$ 0 0
$$666$$ 630.938 0.0367092
$$667$$ 9198.01 0.533955
$$668$$ −6975.84 −0.404047
$$669$$ −1518.80 −0.0877729
$$670$$ 0 0
$$671$$ 7858.46 0.452120
$$672$$ 5755.95 0.330418
$$673$$ 6207.38 0.355538 0.177769 0.984072i $$-0.443112\pi$$
0.177769 + 0.984072i $$0.443112\pi$$
$$674$$ −5418.66 −0.309672
$$675$$ 0 0
$$676$$ 2925.82 0.166467
$$677$$ −28831.1 −1.63674 −0.818368 0.574695i $$-0.805121\pi$$
−0.818368 + 0.574695i $$0.805121\pi$$
$$678$$ −796.392 −0.0451110
$$679$$ 9901.37 0.559617
$$680$$ 0 0
$$681$$ −12858.9 −0.723573
$$682$$ 940.250 0.0527918
$$683$$ −3193.10 −0.178888 −0.0894441 0.995992i $$-0.528509\pi$$
−0.0894441 + 0.995992i $$0.528509\pi$$
$$684$$ 1984.43 0.110931
$$685$$ 0 0
$$686$$ 9296.24 0.517394
$$687$$ −17128.1 −0.951206
$$688$$ 27.0441 0.00149861
$$689$$ −26163.8 −1.44668
$$690$$ 0 0
$$691$$ 7682.49 0.422946 0.211473 0.977384i $$-0.432174\pi$$
0.211473 + 0.977384i $$0.432174\pi$$
$$692$$ −6711.17 −0.368671
$$693$$ −1014.37 −0.0556031
$$694$$ −139.496 −0.00762996
$$695$$ 0 0
$$696$$ 9472.05 0.515858
$$697$$ 33213.3 1.80494
$$698$$ 233.256 0.0126488
$$699$$ 8838.28 0.478247
$$700$$ 0 0
$$701$$ 26551.6 1.43058 0.715292 0.698825i $$-0.246293\pi$$
0.715292 + 0.698825i $$0.246293\pi$$
$$702$$ 1723.45 0.0926601
$$703$$ 1779.86 0.0954887
$$704$$ −2211.24 −0.118380
$$705$$ 0 0
$$706$$ 12285.7 0.654928
$$707$$ 3571.83 0.190004
$$708$$ −6186.22 −0.328379
$$709$$ −16304.6 −0.863655 −0.431828 0.901956i $$-0.642131\pi$$
−0.431828 + 0.901956i $$0.642131\pi$$
$$710$$ 0 0
$$711$$ 8566.52 0.451856
$$712$$ 8171.79 0.430127
$$713$$ −3377.00 −0.177377
$$714$$ 4738.00 0.248341
$$715$$ 0 0
$$716$$ 8021.36 0.418676
$$717$$ 6236.68 0.324844
$$718$$ −7767.50 −0.403733
$$719$$ −3973.62 −0.206107 −0.103053 0.994676i $$-0.532861\pi$$
−0.103053 + 0.994676i $$0.532861\pi$$
$$720$$ 0 0
$$721$$ 15742.1 0.813128
$$722$$ 8256.25 0.425576
$$723$$ −5560.11 −0.286007
$$724$$ −23702.3 −1.21670
$$725$$ 0 0
$$726$$ 566.844 0.0289773
$$727$$ 10780.4 0.549961 0.274980 0.961450i $$-0.411329\pi$$
0.274980 + 0.961450i $$0.411329\pi$$
$$728$$ 8869.67 0.451555
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 233.687 0.0118238
$$732$$ −11919.6 −0.601860
$$733$$ 9211.46 0.464165 0.232083 0.972696i $$-0.425446\pi$$
0.232083 + 0.972696i $$0.425446\pi$$
$$734$$ −20716.6 −1.04177
$$735$$ 0 0
$$736$$ 11552.3 0.578566
$$737$$ −4454.83 −0.222654
$$738$$ −4728.87 −0.235870
$$739$$ 11084.7 0.551768 0.275884 0.961191i $$-0.411029\pi$$
0.275884 + 0.961191i $$0.411029\pi$$
$$740$$ 0 0
$$741$$ 4861.80 0.241029
$$742$$ 10241.0 0.506685
$$743$$ 27420.4 1.35391 0.676955 0.736024i $$-0.263299\pi$$
0.676955 + 0.736024i $$0.263299\pi$$
$$744$$ −3477.61 −0.171365
$$745$$ 0 0
$$746$$ −7234.32 −0.355050
$$747$$ −6616.99 −0.324101
$$748$$ 6038.69 0.295182
$$749$$ 7983.64 0.389474
$$750$$ 0 0
$$751$$ −11290.8 −0.548614 −0.274307 0.961642i $$-0.588448\pi$$
−0.274307 + 0.961642i $$0.588448\pi$$
$$752$$ 3807.33 0.184626
$$753$$ 7075.16 0.342408
$$754$$ 9516.81 0.459657
$$755$$ 0 0
$$756$$ 1538.59 0.0740185
$$757$$ −3739.19 −0.179528 −0.0897642 0.995963i $$-0.528611\pi$$
−0.0897642 + 0.995963i $$0.528611\pi$$
$$758$$ −10155.3 −0.486617
$$759$$ −2035.87 −0.0973617
$$760$$ 0 0
$$761$$ −15621.1 −0.744107 −0.372053 0.928211i $$-0.621346\pi$$
−0.372053 + 0.928211i $$0.621346\pi$$
$$762$$ 8150.45 0.387480
$$763$$ −15387.7 −0.730106
$$764$$ 4739.69 0.224445
$$765$$ 0 0
$$766$$ 19885.7 0.937987
$$767$$ −15156.0 −0.713498
$$768$$ 10371.8 0.487317
$$769$$ 40241.7 1.88706 0.943531 0.331284i $$-0.107482\pi$$
0.943531 + 0.331284i $$0.107482\pi$$
$$770$$ 0 0
$$771$$ 16557.8 0.773428
$$772$$ −13681.2 −0.637818
$$773$$ −22821.4 −1.06187 −0.530936 0.847412i $$-0.678160\pi$$
−0.530936 + 0.847412i $$0.678160\pi$$
$$774$$ −33.2721 −0.00154514
$$775$$ 0 0
$$776$$ 20464.4 0.946685
$$777$$ 1379.98 0.0637148
$$778$$ −18777.0 −0.865282
$$779$$ −13340.0 −0.613549
$$780$$ 0 0
$$781$$ −10332.2 −0.473387
$$782$$ 9509.28 0.434848
$$783$$ 4025.51 0.183729
$$784$$ −2718.92 −0.123857
$$785$$ 0 0
$$786$$ 1465.53 0.0665062
$$787$$ 29454.3 1.33410 0.667048 0.745015i $$-0.267558\pi$$
0.667048 + 0.745015i $$0.267558\pi$$
$$788$$ 19337.8 0.874216
$$789$$ 6778.96 0.305878
$$790$$ 0 0
$$791$$ −1741.86 −0.0782974
$$792$$ −2096.53 −0.0940619
$$793$$ −29202.7 −1.30771
$$794$$ −8157.00 −0.364586
$$795$$ 0 0
$$796$$ −22218.6 −0.989344
$$797$$ 27440.3 1.21955 0.609777 0.792573i $$-0.291259\pi$$
0.609777 + 0.792573i $$0.291259\pi$$
$$798$$ −1903.00 −0.0844179
$$799$$ 32899.0 1.45668
$$800$$ 0 0
$$801$$ 3472.91 0.153195
$$802$$ −15066.1 −0.663346
$$803$$ −3988.27 −0.175272
$$804$$ 6757.03 0.296396
$$805$$ 0 0
$$806$$ −3494.05 −0.152695
$$807$$ −21118.6 −0.921201
$$808$$ 7382.34 0.321423
$$809$$ −5060.18 −0.219909 −0.109954 0.993937i $$-0.535070\pi$$
−0.109954 + 0.993937i $$0.535070\pi$$
$$810$$ 0 0
$$811$$ 30480.1 1.31973 0.659865 0.751384i $$-0.270613\pi$$
0.659865 + 0.751384i $$0.270613\pi$$
$$812$$ 8496.03 0.367183
$$813$$ −15465.2 −0.667146
$$814$$ −771.146 −0.0332048
$$815$$ 0 0
$$816$$ −3382.72 −0.145121
$$817$$ −93.8596 −0.00401925
$$818$$ 3139.46 0.134192
$$819$$ 3769.50 0.160827
$$820$$ 0 0
$$821$$ 37909.0 1.61149 0.805745 0.592263i $$-0.201765\pi$$
0.805745 + 0.592263i $$0.201765\pi$$
$$822$$ 3358.56 0.142510
$$823$$ −23636.0 −1.00109 −0.500546 0.865710i $$-0.666867\pi$$
−0.500546 + 0.865710i $$0.666867\pi$$
$$824$$ 32536.0 1.37554
$$825$$ 0 0
$$826$$ 5932.36 0.249895
$$827$$ 42634.3 1.79267 0.896336 0.443376i $$-0.146219\pi$$
0.896336 + 0.443376i $$0.146219\pi$$
$$828$$ 3087.99 0.129608
$$829$$ −45152.5 −1.89169 −0.945845 0.324619i $$-0.894764\pi$$
−0.945845 + 0.324619i $$0.894764\pi$$
$$830$$ 0 0
$$831$$ 27223.6 1.13643
$$832$$ 8217.15 0.342402
$$833$$ −23494.1 −0.977217
$$834$$ −4106.02 −0.170480
$$835$$ 0 0
$$836$$ −2425.42 −0.100341
$$837$$ −1477.94 −0.0610337
$$838$$ −6925.91 −0.285503
$$839$$ 30431.5 1.25222 0.626110 0.779734i $$-0.284646\pi$$
0.626110 + 0.779734i $$0.284646\pi$$
$$840$$ 0 0
$$841$$ −2160.34 −0.0885784
$$842$$ −23763.6 −0.972623
$$843$$ 10223.4 0.417689
$$844$$ −5818.09 −0.237283
$$845$$ 0 0
$$846$$ −4684.13 −0.190359
$$847$$ 1239.79 0.0502949
$$848$$ −7311.64 −0.296088
$$849$$ −26483.2 −1.07055
$$850$$ 0 0
$$851$$ 2769.65 0.111566
$$852$$ 15671.8 0.630171
$$853$$ −10367.2 −0.416139 −0.208070 0.978114i $$-0.566718\pi$$
−0.208070 + 0.978114i $$0.566718\pi$$
$$854$$ 11430.5 0.458013
$$855$$ 0 0
$$856$$ 16500.8 0.658860
$$857$$ −12947.1 −0.516063 −0.258032 0.966136i $$-0.583074\pi$$
−0.258032 + 0.966136i $$0.583074\pi$$
$$858$$ −2106.44 −0.0838142
$$859$$ −20383.5 −0.809636 −0.404818 0.914397i $$-0.632665\pi$$
−0.404818 + 0.914397i $$0.632665\pi$$
$$860$$ 0 0
$$861$$ −10342.9 −0.409391
$$862$$ 8779.22 0.346893
$$863$$ −9056.42 −0.357224 −0.178612 0.983920i $$-0.557161\pi$$
−0.178612 + 0.983920i $$0.557161\pi$$
$$864$$ 5055.88 0.199079
$$865$$ 0 0
$$866$$ −22340.0 −0.876610
$$867$$ −14491.0 −0.567636
$$868$$ −3119.27 −0.121976
$$869$$ −10470.2 −0.408719
$$870$$ 0 0
$$871$$ 16554.5 0.644005
$$872$$ −31803.6 −1.23510
$$873$$ 8697.10 0.337173
$$874$$ −3819.37 −0.147817
$$875$$ 0 0
$$876$$ 6049.36 0.233321
$$877$$ 2867.88 0.110424 0.0552118 0.998475i $$-0.482417\pi$$
0.0552118 + 0.998475i $$0.482417\pi$$
$$878$$ −6846.16 −0.263151
$$879$$ −13584.9 −0.521281
$$880$$ 0 0
$$881$$ −11862.5 −0.453640 −0.226820 0.973937i $$-0.572833\pi$$
−0.226820 + 0.973937i $$0.572833\pi$$
$$882$$ 3345.06 0.127703
$$883$$ −33463.8 −1.27537 −0.637683 0.770299i $$-0.720107\pi$$
−0.637683 + 0.770299i $$0.720107\pi$$
$$884$$ −22440.3 −0.853787
$$885$$ 0 0
$$886$$ −15756.0 −0.597442
$$887$$ −2420.75 −0.0916357 −0.0458178 0.998950i $$-0.514589\pi$$
−0.0458178 + 0.998950i $$0.514589\pi$$
$$888$$ 2852.17 0.107784
$$889$$ 17826.5 0.672534
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ −2815.62 −0.105688
$$893$$ −13213.8 −0.495165
$$894$$ −11132.4 −0.416470
$$895$$ 0 0
$$896$$ 12132.9 0.452378
$$897$$ 7565.48 0.281610
$$898$$ −14963.6 −0.556060
$$899$$ −8161.14 −0.302769
$$900$$ 0 0
$$901$$ −63179.7 −2.33609
$$902$$ 5779.73 0.213352
$$903$$ −72.7722 −0.00268185
$$904$$ −3600.10 −0.132453
$$905$$ 0 0
$$906$$ −434.955 −0.0159497
$$907$$ 38154.1 1.39679 0.698393 0.715714i $$-0.253899\pi$$
0.698393 + 0.715714i $$0.253899\pi$$
$$908$$ −23838.4 −0.871262
$$909$$ 3137.40 0.114479
$$910$$ 0 0
$$911$$ −35758.0 −1.30045 −0.650227 0.759740i $$-0.725326\pi$$
−0.650227 + 0.759740i $$0.725326\pi$$
$$912$$ 1358.66 0.0493308
$$913$$ 8087.44 0.293160
$$914$$ 15614.5 0.565079
$$915$$ 0 0
$$916$$ −31753.0 −1.14536
$$917$$ 3205.39 0.115432
$$918$$ 4161.73 0.149627
$$919$$ 17387.5 0.624115 0.312058 0.950063i $$-0.398982\pi$$
0.312058 + 0.950063i $$0.398982\pi$$
$$920$$ 0 0
$$921$$ 1704.32 0.0609764
$$922$$ 17357.9 0.620013
$$923$$ 38395.3 1.36923
$$924$$ −1880.50 −0.0669523
$$925$$ 0 0
$$926$$ −2447.20 −0.0868467
$$927$$ 13827.4 0.489915
$$928$$ 27918.3 0.987569
$$929$$ 6955.93 0.245658 0.122829 0.992428i $$-0.460803\pi$$
0.122829 + 0.992428i $$0.460803\pi$$
$$930$$ 0 0
$$931$$ 9436.31 0.332183
$$932$$ 16384.9 0.575863
$$933$$ 20560.8 0.721467
$$934$$ 19751.7 0.691965
$$935$$ 0 0
$$936$$ 7790.88 0.272065
$$937$$ 16074.5 0.560438 0.280219 0.959936i $$-0.409593\pi$$
0.280219 + 0.959936i $$0.409593\pi$$
$$938$$ −6479.76 −0.225556
$$939$$ −3416.77 −0.118746
$$940$$ 0 0
$$941$$ −687.126 −0.0238041 −0.0119021 0.999929i $$-0.503789\pi$$
−0.0119021 + 0.999929i $$0.503789\pi$$
$$942$$ 8812.96 0.304821
$$943$$ −20758.5 −0.716849
$$944$$ −4235.44 −0.146030
$$945$$ 0 0
$$946$$ 40.6659 0.00139763
$$947$$ −35352.0 −1.21308 −0.606540 0.795053i $$-0.707443\pi$$
−0.606540 + 0.795053i $$0.707443\pi$$
$$948$$ 15881.0 0.544085
$$949$$ 14820.7 0.506956
$$950$$ 0 0
$$951$$ 9622.44 0.328106
$$952$$ 21418.2 0.729168
$$953$$ 19390.7 0.659103 0.329552 0.944138i $$-0.393102\pi$$
0.329552 + 0.944138i $$0.393102\pi$$
$$954$$ 8995.45 0.305281
$$955$$ 0 0
$$956$$ 11561.9 0.391148
$$957$$ −4920.06 −0.166189
$$958$$ 16738.7 0.564511
$$959$$ 7345.80 0.247349
$$960$$ 0 0
$$961$$ −26794.7 −0.899422
$$962$$ 2865.64 0.0960416
$$963$$ 7012.62 0.234661
$$964$$ −10307.6 −0.344384
$$965$$ 0 0
$$966$$ −2961.27 −0.0986308
$$967$$ −28643.6 −0.952551 −0.476275 0.879296i $$-0.658013\pi$$
−0.476275 + 0.879296i $$0.658013\pi$$
$$968$$ 2562.43 0.0850821
$$969$$ 11740.1 0.389213
$$970$$ 0 0
$$971$$ −19574.8 −0.646946 −0.323473 0.946237i $$-0.604851\pi$$
−0.323473 + 0.946237i $$0.604851\pi$$
$$972$$ 1351.46 0.0445967
$$973$$ −8980.63 −0.295895
$$974$$ 11182.6 0.367878
$$975$$ 0 0
$$976$$ −8160.86 −0.267646
$$977$$ −50095.5 −1.64043 −0.820213 0.572058i $$-0.806145\pi$$
−0.820213 + 0.572058i $$0.806145\pi$$
$$978$$ 11550.0 0.377636
$$979$$ −4244.67 −0.138570
$$980$$ 0 0
$$981$$ −13516.1 −0.439895
$$982$$ −22747.6 −0.739211
$$983$$ −14445.4 −0.468706 −0.234353 0.972152i $$-0.575297\pi$$
−0.234353 + 0.972152i $$0.575297\pi$$
$$984$$ −21376.9 −0.692553
$$985$$ 0 0
$$986$$ 22980.9 0.742253
$$987$$ −10245.0 −0.330399
$$988$$ 9013.05 0.290226
$$989$$ −146.056 −0.00469595
$$990$$ 0 0
$$991$$ 29120.1 0.933430 0.466715 0.884408i $$-0.345437\pi$$
0.466715 + 0.884408i $$0.345437\pi$$
$$992$$ −10250.1 −0.328064
$$993$$ 27405.4 0.875814
$$994$$ −15028.7 −0.479558
$$995$$ 0 0
$$996$$ −12266.9 −0.390253
$$997$$ 9137.45 0.290257 0.145128 0.989413i $$-0.453640\pi$$
0.145128 + 0.989413i $$0.453640\pi$$
$$998$$ 7244.03 0.229765
$$999$$ 1212.14 0.0383887
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 825.4.a.m.1.1 2
3.2 odd 2 2475.4.a.n.1.2 2
5.2 odd 4 825.4.c.j.199.2 4
5.3 odd 4 825.4.c.j.199.3 4
5.4 even 2 165.4.a.c.1.2 2
15.14 odd 2 495.4.a.d.1.1 2
55.54 odd 2 1815.4.a.n.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.2 2 5.4 even 2
495.4.a.d.1.1 2 15.14 odd 2
825.4.a.m.1.1 2 1.1 even 1 trivial
825.4.c.j.199.2 4 5.2 odd 4
825.4.c.j.199.3 4 5.3 odd 4
1815.4.a.n.1.1 2 55.54 odd 2
2475.4.a.n.1.2 2 3.2 odd 2