# Properties

 Label 2475.4.a.n.1.2 Level $2475$ Weight $4$ Character 2475.1 Self dual yes Analytic conductor $146.030$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$146.029727264$$ 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 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.56155 q^{2} -5.56155 q^{4} +10.2462 q^{7} -21.1771 q^{8} +O(q^{10})$$ $$q+1.56155 q^{2} -5.56155 q^{4} +10.2462 q^{7} -21.1771 q^{8} +11.0000 q^{11} +40.8769 q^{13} +16.0000 q^{14} +11.4233 q^{16} -98.7083 q^{17} -39.6458 q^{19} +17.1771 q^{22} +61.6932 q^{23} +63.8314 q^{26} -56.9848 q^{28} +149.093 q^{29} +54.7386 q^{31} +187.255 q^{32} -154.138 q^{34} -44.8939 q^{37} -61.9091 q^{38} -336.479 q^{41} +2.36745 q^{43} -61.1771 q^{44} +96.3371 q^{46} -333.295 q^{47} -238.015 q^{49} -227.339 q^{52} +640.064 q^{53} -216.985 q^{56} +232.816 q^{58} +370.773 q^{59} -714.405 q^{61} +85.4773 q^{62} +201.022 q^{64} +404.985 q^{67} +548.972 q^{68} -939.292 q^{71} +362.570 q^{73} -70.1042 q^{74} +220.492 q^{76} +112.708 q^{77} +951.835 q^{79} -525.430 q^{82} +735.221 q^{83} +3.69690 q^{86} -232.948 q^{88} -385.879 q^{89} +418.833 q^{91} -343.110 q^{92} -520.458 q^{94} +966.345 q^{97} -371.673 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8}+O(q^{10})$$ 2 * q - q^2 - 7 * q^4 + 4 * q^7 + 3 * q^8 $$2 q - q^{2} - 7 q^{4} + 4 q^{7} + 3 q^{8} + 22 q^{11} + 90 q^{13} + 32 q^{14} - 39 q^{16} - 16 q^{17} - 170 q^{19} - 11 q^{22} - 124 q^{23} - 62 q^{26} - 48 q^{28} + 158 q^{29} + 60 q^{31} + 123 q^{32} - 366 q^{34} + 372 q^{37} + 272 q^{38} - 38 q^{41} + 516 q^{43} - 77 q^{44} + 572 q^{46} + 224 q^{47} - 542 q^{49} - 298 q^{52} + 472 q^{53} - 368 q^{56} + 210 q^{58} - 248 q^{59} + 72 q^{61} + 72 q^{62} + 769 q^{64} + 744 q^{67} + 430 q^{68} - 2060 q^{71} + 486 q^{73} - 1138 q^{74} + 408 q^{76} + 44 q^{77} + 642 q^{79} - 1290 q^{82} - 286 q^{83} - 1312 q^{86} + 33 q^{88} - 244 q^{89} + 112 q^{91} - 76 q^{92} - 1948 q^{94} + 168 q^{97} + 407 q^{98}+O(q^{100})$$ 2 * q - q^2 - 7 * q^4 + 4 * q^7 + 3 * q^8 + 22 * q^11 + 90 * q^13 + 32 * q^14 - 39 * q^16 - 16 * q^17 - 170 * q^19 - 11 * q^22 - 124 * q^23 - 62 * q^26 - 48 * q^28 + 158 * q^29 + 60 * q^31 + 123 * q^32 - 366 * q^34 + 372 * q^37 + 272 * q^38 - 38 * q^41 + 516 * q^43 - 77 * q^44 + 572 * q^46 + 224 * q^47 - 542 * q^49 - 298 * q^52 + 472 * q^53 - 368 * q^56 + 210 * q^58 - 248 * q^59 + 72 * q^61 + 72 * q^62 + 769 * q^64 + 744 * q^67 + 430 * q^68 - 2060 * q^71 + 486 * q^73 - 1138 * q^74 + 408 * q^76 + 44 * q^77 + 642 * q^79 - 1290 * q^82 - 286 * q^83 - 1312 * q^86 + 33 * q^88 - 244 * q^89 + 112 * q^91 - 76 * q^92 - 1948 * q^94 + 168 * q^97 + 407 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.56155 0.552092 0.276046 0.961144i $$-0.410976\pi$$
0.276046 + 0.961144i $$0.410976\pi$$
$$3$$ 0 0
$$4$$ −5.56155 −0.695194
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 10.2462 0.553243 0.276622 0.960979i $$-0.410785\pi$$
0.276622 + 0.960979i $$0.410785\pi$$
$$8$$ −21.1771 −0.935904
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ 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 $$-0.748661\pi$$
−0.704126 + 0.710075i $$0.748661\pi$$
$$18$$ 0 0
$$19$$ −39.6458 −0.478704 −0.239352 0.970933i $$-0.576935\pi$$
−0.239352 + 0.970933i $$0.576935\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 17.1771 0.166462
$$23$$ 61.6932 0.559301 0.279650 0.960102i $$-0.409781\pi$$
0.279650 + 0.960102i $$0.409781\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 63.8314 0.481476
$$27$$ 0 0
$$28$$ −56.9848 −0.384612
$$29$$ 149.093 0.954684 0.477342 0.878718i $$-0.341600\pi$$
0.477342 + 0.878718i $$0.341600\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ −154.138 −0.777485
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −44.8939 −0.199473 −0.0997367 0.995014i $$-0.531800\pi$$
−0.0997367 + 0.995014i $$0.531800\pi$$
$$38$$ −61.9091 −0.264289
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −336.479 −1.28169 −0.640844 0.767671i $$-0.721415\pi$$
−0.640844 + 0.767671i $$0.721415\pi$$
$$42$$ 0 0
$$43$$ 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 $$-0.673023\pi$$
−0.517193 + 0.855869i $$0.673023\pi$$
$$48$$ 0 0
$$49$$ −238.015 −0.693922
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −227.339 −0.606274
$$53$$ 640.064 1.65886 0.829430 0.558610i $$-0.188665\pi$$
0.829430 + 0.558610i $$0.188665\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −216.985 −0.517782
$$57$$ 0 0
$$58$$ 232.816 0.527074
$$59$$ 370.773 0.818144 0.409072 0.912502i $$-0.365853\pi$$
0.409072 + 0.912502i $$0.365853\pi$$
$$60$$ 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
$$63$$ 0 0
$$64$$ 201.022 0.392621
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 404.985 0.738459 0.369230 0.929338i $$-0.379622\pi$$
0.369230 + 0.929338i $$0.379622\pi$$
$$68$$ 548.972 0.979009
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −939.292 −1.57005 −0.785024 0.619465i $$-0.787349\pi$$
−0.785024 + 0.619465i $$0.787349\pi$$
$$72$$ 0 0
$$73$$ 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
$$78$$ 0 0
$$79$$ 951.835 1.35557 0.677784 0.735261i $$-0.262941\pi$$
0.677784 + 0.735261i $$0.262941\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −525.430 −0.707610
$$83$$ 735.221 0.972302 0.486151 0.873875i $$-0.338401\pi$$
0.486151 + 0.873875i $$0.338401\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 3.69690 0.00463543
$$87$$ 0 0
$$88$$ −232.948 −0.282186
$$89$$ −385.879 −0.459585 −0.229793 0.973240i $$-0.573805\pi$$
−0.229793 + 0.973240i $$0.573805\pi$$
$$90$$ 0 0
$$91$$ 418.833 0.482480
$$92$$ −343.110 −0.388823
$$93$$ 0 0
$$94$$ −520.458 −0.571076
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 966.345 1.01152 0.505760 0.862674i $$-0.331212\pi$$
0.505760 + 0.862674i $$0.331212\pi$$
$$98$$ −371.673 −0.383109
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −348.600 −0.343436 −0.171718 0.985146i $$-0.554932\pi$$
−0.171718 + 0.985146i $$0.554932\pi$$
$$102$$ 0 0
$$103$$ 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 $$-0.614495\pi$$
−0.351991 + 0.936003i $$0.614495\pi$$
$$108$$ 0 0
$$109$$ −1501.79 −1.31968 −0.659842 0.751404i $$-0.729377\pi$$
−0.659842 + 0.751404i $$0.729377\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 117.045 0.0987478
$$113$$ 170.000 0.141524 0.0707622 0.997493i $$-0.477457\pi$$
0.0707622 + 0.997493i $$0.477457\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −829.187 −0.663691
$$117$$ 0 0
$$118$$ 578.981 0.451691
$$119$$ −1011.39 −0.779106
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −1115.58 −0.827869
$$123$$ 0 0
$$124$$ −304.432 −0.220474
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1739.82 1.21562 0.607811 0.794082i $$-0.292048\pi$$
0.607811 + 0.794082i $$0.292048\pi$$
$$128$$ −1184.13 −0.817683
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −312.837 −0.208647 −0.104323 0.994543i $$-0.533268\pi$$
−0.104323 + 0.994543i $$0.533268\pi$$
$$132$$ 0 0
$$133$$ −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 $$-0.571763\pi$$
−0.223545 + 0.974694i $$0.571763\pi$$
$$138$$ 0 0
$$139$$ −876.483 −0.534837 −0.267418 0.963581i $$-0.586171\pi$$
−0.267418 + 0.963581i $$0.586171\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1466.75 −0.866811
$$143$$ 449.646 0.262946
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 566.172 0.320937
$$147$$ 0 0
$$148$$ 249.680 0.138673
$$149$$ 2376.36 1.30657 0.653285 0.757112i $$-0.273390\pi$$
0.653285 + 0.757112i $$0.273390\pi$$
$$150$$ 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
$$153$$ 0 0
$$154$$ 176.000 0.0920941
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1881.24 0.956301 0.478150 0.878278i $$-0.341307\pi$$
0.478150 + 0.878278i $$0.341307\pi$$
$$158$$ 1486.34 0.748398
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 632.121 0.309429
$$162$$ 0 0
$$163$$ 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 $$-0.593855\pi$$
−0.290600 + 0.956845i $$0.593855\pi$$
$$168$$ 0 0
$$169$$ −526.080 −0.239454
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −13.1667 −0.00583693
$$173$$ −1206.71 −0.530314 −0.265157 0.964205i $$-0.585424\pi$$
−0.265157 + 0.964205i $$0.585424\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 125.656 0.0538164
$$177$$ 0 0
$$178$$ −602.570 −0.253733
$$179$$ 1442.29 0.602244 0.301122 0.953586i $$-0.402639\pi$$
0.301122 + 0.953586i $$0.402639\pi$$
$$180$$ 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
$$183$$ 0 0
$$184$$ −1306.48 −0.523451
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1085.79 −0.424604
$$188$$ 1853.64 0.719099
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 852.223 0.322852 0.161426 0.986885i $$-0.448391\pi$$
0.161426 + 0.986885i $$0.448391\pi$$
$$192$$ 0 0
$$193$$ 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 $$-0.283564\pi$$
0.628756 + 0.777602i $$0.283564\pi$$
$$198$$ 0 0
$$199$$ 3995.04 1.42312 0.711560 0.702626i $$-0.247989\pi$$
0.711560 + 0.702626i $$0.247989\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −544.358 −0.189608
$$203$$ 1527.64 0.528173
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2399.14 0.811436
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ −1216.73 −0.388664
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 560.864 0.175456
$$218$$ −2345.13 −0.728587
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4034.89 −1.22813
$$222$$ 0 0
$$223$$ 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 $$-0.715567\pi$$
−0.626632 + 0.779315i $$0.715567\pi$$
$$228$$ 0 0
$$229$$ 5709.37 1.64754 0.823769 0.566926i $$-0.191867\pi$$
0.823769 + 0.566926i $$0.191867\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −3157.35 −0.893492
$$233$$ 2946.09 0.828348 0.414174 0.910198i $$-0.364071\pi$$
0.414174 + 0.910198i $$0.364071\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −2062.07 −0.568769
$$237$$ 0 0
$$238$$ −1579.33 −0.430139
$$239$$ 2078.89 0.562646 0.281323 0.959613i $$-0.409227\pi$$
0.281323 + 0.959613i $$0.409227\pi$$
$$240$$ 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
$$243$$ 0 0
$$244$$ 3973.20 1.04245
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1620.60 −0.417475
$$248$$ −1159.20 −0.296813
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2358.39 0.593068 0.296534 0.955022i $$-0.404169\pi$$
0.296534 + 0.955022i $$0.404169\pi$$
$$252$$ 0 0
$$253$$ 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 $$-0.266376\pi$$
0.669809 + 0.742534i $$0.266376\pi$$
$$258$$ 0 0
$$259$$ −459.993 −0.110357
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −488.512 −0.115192
$$263$$ 2259.65 0.529795 0.264898 0.964277i $$-0.414662\pi$$
0.264898 + 0.964277i $$0.414662\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −634.333 −0.146216
$$267$$ 0 0
$$268$$ −2252.34 −0.513373
$$269$$ −7039.53 −1.59557 −0.797783 0.602944i $$-0.793994\pi$$
−0.797783 + 0.602944i $$0.793994\pi$$
$$270$$ 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
$$273$$ 0 0
$$274$$ −1119.52 −0.246835
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −9074.52 −1.96836 −0.984179 0.177175i $$-0.943304\pi$$
−0.984179 + 0.177175i $$0.943304\pi$$
$$278$$ −1368.67 −0.295279
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 3407.79 0.723459 0.361729 0.932283i $$-0.382186\pi$$
0.361729 + 0.932283i $$0.382186\pi$$
$$282$$ 0 0
$$283$$ 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
$$288$$ 0 0
$$289$$ 4830.33 0.983174
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −2016.45 −0.404123
$$293$$ −4528.29 −0.902886 −0.451443 0.892300i $$-0.649091\pi$$
−0.451443 + 0.892300i $$0.649091\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 950.722 0.186688
$$297$$ 0 0
$$298$$ 3710.81 0.721347
$$299$$ 2521.83 0.487762
$$300$$ 0 0
$$301$$ 24.2574 0.00464509
$$302$$ −144.985 −0.0276256
$$303$$ 0 0
$$304$$ −452.886 −0.0854434
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −568.106 −0.105614 −0.0528071 0.998605i $$-0.516817\pi$$
−0.0528071 + 0.998605i $$0.516817\pi$$
$$308$$ −626.833 −0.115965
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 6853.59 1.24962 0.624809 0.780778i $$-0.285177\pi$$
0.624809 + 0.780778i $$0.285177\pi$$
$$312$$ 0 0
$$313$$ 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 $$-0.408289\pi$$
0.284148 + 0.958780i $$0.408289\pi$$
$$318$$ 0 0
$$319$$ 1640.02 0.287848
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 987.091 0.170834
$$323$$ 3913.37 0.674136
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 3850.00 0.654085
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ −1958.65 −0.320876
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 3470.05 0.560907 0.280453 0.959868i $$-0.409515\pi$$
0.280453 + 0.959868i $$0.409515\pi$$
$$338$$ −821.501 −0.132200
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 602.125 0.0956214
$$342$$ 0 0
$$343$$ −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 $$-0.502200\pi$$
−0.00691004 + 0.999976i $$0.502200\pi$$
$$348$$ 0 0
$$349$$ −149.375 −0.0229107 −0.0114554 0.999934i $$-0.503646\pi$$
−0.0114554 + 0.999934i $$0.503646\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2059.80 0.311897
$$353$$ 7867.64 1.18627 0.593133 0.805104i $$-0.297891\pi$$
0.593133 + 0.805104i $$0.297891\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2146.09 0.319501
$$357$$ 0 0
$$358$$ 2252.21 0.332494
$$359$$ −4974.22 −0.731279 −0.365639 0.930757i $$-0.619150\pi$$
−0.365639 + 0.930757i $$0.619150\pi$$
$$360$$ 0 0
$$361$$ −5287.21 −0.770842
$$362$$ 6655.04 0.966246
$$363$$ 0 0
$$364$$ −2329.36 −0.335417
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13266.7 1.88696 0.943479 0.331433i $$-0.107532\pi$$
0.943479 + 0.331433i $$0.107532\pi$$
$$368$$ 704.739 0.0998290
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6558.23 0.917754
$$372$$ 0 0
$$373$$ 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
$$378$$ 0 0
$$379$$ 6503.31 0.881406 0.440703 0.897653i $$-0.354729\pi$$
0.440703 + 0.897653i $$0.354729\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 1330.79 0.178244
$$383$$ 12734.5 1.69897 0.849484 0.527614i $$-0.176913\pi$$
0.849484 + 0.527614i $$0.176913\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 3841.35 0.506527
$$387$$ 0 0
$$388$$ −5374.38 −0.703203
$$389$$ −12024.6 −1.56728 −0.783639 0.621216i $$-0.786639\pi$$
−0.783639 + 0.621216i $$0.786639\pi$$
$$390$$ 0 0
$$391$$ −6089.63 −0.787636
$$392$$ 5040.47 0.649444
$$393$$ 0 0
$$394$$ 5429.61 0.694263
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 5223.65 0.660371 0.330186 0.943916i $$-0.392889\pi$$
0.330186 + 0.943916i $$0.392889\pi$$
$$398$$ 6238.46 0.785693
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −9648.18 −1.20151 −0.600757 0.799432i $$-0.705134\pi$$
−0.600757 + 0.799432i $$0.705134\pi$$
$$402$$ 0 0
$$403$$ 2237.55 0.276576
$$404$$ 1938.76 0.238755
$$405$$ 0 0
$$406$$ 2385.48 0.291600
$$407$$ −493.833 −0.0601435
$$408$$ 0 0
$$409$$ −2010.47 −0.243060 −0.121530 0.992588i $$-0.538780\pi$$
−0.121530 + 0.992588i $$0.538780\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −8544.65 −1.02176
$$413$$ 3799.02 0.452633
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 7654.39 0.902133
$$417$$ 0 0
$$418$$ −681.000 −0.0796861
$$419$$ −4435.27 −0.517129 −0.258565 0.965994i $$-0.583250\pi$$
−0.258565 + 0.965994i $$0.583250\pi$$
$$420$$ 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
$$423$$ 0 0
$$424$$ −13554.7 −1.55253
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −7319.95 −0.829595
$$428$$ 4333.45 0.489405
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5622.11 0.628324 0.314162 0.949369i $$-0.398277\pi$$
0.314162 + 0.949369i $$0.398277\pi$$
$$432$$ 0 0
$$433$$ 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
$$438$$ 0 0
$$439$$ 4384.20 0.476643 0.238322 0.971186i $$-0.423403\pi$$
0.238322 + 0.971186i $$0.423403\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −6300.69 −0.678039
$$443$$ −10090.0 −1.08214 −0.541071 0.840977i $$-0.681981\pi$$
−0.541071 + 0.840977i $$0.681981\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 790.560 0.0839330
$$447$$ 0 0
$$448$$ 2059.71 0.217215
$$449$$ −9582.52 −1.00719 −0.503594 0.863941i $$-0.667989\pi$$
−0.503594 + 0.863941i $$0.667989\pi$$
$$450$$ 0 0
$$451$$ −3701.27 −0.386444
$$452$$ −945.464 −0.0983869
$$453$$ 0 0
$$454$$ −6693.27 −0.691918
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9999.34 −1.02352 −0.511761 0.859128i $$-0.671007\pi$$
−0.511761 + 0.859128i $$0.671007\pi$$
$$458$$ 8915.49 0.909593
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 11115.8 1.12302 0.561512 0.827468i $$-0.310220\pi$$
0.561512 + 0.827468i $$0.310220\pi$$
$$462$$ 0 0
$$463$$ 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 $$-0.284415\pi$$
0.626675 + 0.779281i $$0.284415\pi$$
$$468$$ 0 0
$$469$$ 4149.56 0.408548
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −7851.88 −0.765704
$$473$$ 26.0420 0.00253152
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 5624.88 0.541630
$$477$$ 0 0
$$478$$ 3246.30 0.310633
$$479$$ 10719.2 1.02249 0.511247 0.859434i $$-0.329184\pi$$
0.511247 + 0.859434i $$0.329184\pi$$
$$480$$ 0 0
$$481$$ −1835.12 −0.173959
$$482$$ 2894.14 0.273494
$$483$$ 0 0
$$484$$ −672.948 −0.0631995
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −7161.20 −0.666335 −0.333167 0.942868i $$-0.608117\pi$$
−0.333167 + 0.942868i $$0.608117\pi$$
$$488$$ 15129.0 1.40340
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −14567.3 −1.33893 −0.669463 0.742845i $$-0.733476\pi$$
−0.669463 + 0.742845i $$0.733476\pi$$
$$492$$ 0 0
$$493$$ −14716.7 −1.34444
$$494$$ −2530.65 −0.230485
$$495$$ 0 0
$$496$$ 625.295 0.0566060
$$497$$ −9624.18 −0.868619
$$498$$ 0 0
$$499$$ −4638.99 −0.416172 −0.208086 0.978111i $$-0.566723\pi$$
−0.208086 + 0.978111i $$0.566723\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 3682.75 0.327428
$$503$$ −12206.3 −1.08201 −0.541006 0.841019i $$-0.681956\pi$$
−0.541006 + 0.841019i $$0.681956\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1059.71 0.0931024
$$507$$ 0 0
$$508$$ −9676.09 −0.845093
$$509$$ −10018.6 −0.872427 −0.436214 0.899843i $$-0.643681\pi$$
−0.436214 + 0.899843i $$0.643681\pi$$
$$510$$ 0 0
$$511$$ 3714.97 0.321606
$$512$$ 4074.36 0.351686
$$513$$ 0 0
$$514$$ 8618.61 0.739592
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3666.25 −0.311879
$$518$$ −718.303 −0.0609274
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1054.72 −0.0886916 −0.0443458 0.999016i $$-0.514120\pi$$
−0.0443458 + 0.999016i $$0.514120\pi$$
$$522$$ 0 0
$$523$$ 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
$$528$$ 0 0
$$529$$ −8360.95 −0.687183
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2259.21 0.184115
$$533$$ −13754.2 −1.11775
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −8576.40 −0.691127
$$537$$ 0 0
$$538$$ −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
$$543$$ 0 0
$$544$$ −18483.6 −1.45676
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −13058.2 −1.02071 −0.510355 0.859964i $$-0.670486\pi$$
−0.510355 + 0.859964i $$0.670486\pi$$
$$548$$ 3987.23 0.310814
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5910.91 −0.457011
$$552$$ 0 0
$$553$$ 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 $$-0.582153\pi$$
−0.255236 + 0.966879i $$0.582153\pi$$
$$558$$ 0 0
$$559$$ 96.7741 0.00732219
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 5321.45 0.399416
$$563$$ −20820.5 −1.55858 −0.779288 0.626666i $$-0.784419\pi$$
−0.779288 + 0.626666i $$0.784419\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 13785.0 1.02372
$$567$$ 0 0
$$568$$ 19891.5 1.46941
$$569$$ −3251.08 −0.239530 −0.119765 0.992802i $$-0.538214\pi$$
−0.119765 + 0.992802i $$0.538214\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ −5383.67 −0.391481
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −14462.4 −1.04346 −0.521730 0.853111i $$-0.674713\pi$$
−0.521730 + 0.853111i $$0.674713\pi$$
$$578$$ 7542.82 0.542803
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 7533.23 0.537920
$$582$$ 0 0
$$583$$ 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 $$-0.795255\pi$$
−0.800166 + 0.599779i $$0.795255\pi$$
$$588$$ 0 0
$$589$$ −2170.16 −0.151816
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −512.836 −0.0356038
$$593$$ −14956.4 −1.03573 −0.517864 0.855463i $$-0.673273\pi$$
−0.517864 + 0.855463i $$0.673273\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −13216.2 −0.908319
$$597$$ 0 0
$$598$$ 3937.96 0.269290
$$599$$ −2150.77 −0.146708 −0.0733539 0.997306i $$-0.523370\pi$$
−0.0733539 + 0.997306i $$0.523370\pi$$
$$600$$ 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
$$603$$ 0 0
$$604$$ 516.371 0.0347862
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 10991.5 0.734974 0.367487 0.930029i $$-0.380218\pi$$
0.367487 + 0.930029i $$0.380218\pi$$
$$608$$ −7423.87 −0.495194
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −13624.1 −0.902081
$$612$$ 0 0
$$613$$ −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 $$-0.424526\pi$$
0.234893 + 0.972021i $$0.424526\pi$$
$$618$$ 0 0
$$619$$ 12186.9 0.791332 0.395666 0.918395i $$-0.370514\pi$$
0.395666 + 0.918395i $$0.370514\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 10702.2 0.689905
$$623$$ −3953.80 −0.254262
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1778.49 0.113551
$$627$$ 0 0
$$628$$ −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
$$633$$ 0 0
$$634$$ 5008.65 0.313752
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −9729.32 −0.605164
$$638$$ 2560.98 0.158919
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 25014.9 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$642$$ 0 0
$$643$$ 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 $$-0.817038\pi$$
−0.839305 + 0.543661i $$0.817038\pi$$
$$648$$ 0 0
$$649$$ 4078.50 0.246680
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −13712.0 −0.823623
$$653$$ −14314.0 −0.857810 −0.428905 0.903350i $$-0.641101\pi$$
−0.428905 + 0.903350i $$0.641101\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −3843.70 −0.228767
$$657$$ 0 0
$$658$$ −5332.73 −0.315944
$$659$$ 28327.8 1.67450 0.837249 0.546822i $$-0.184163\pi$$
0.837249 + 0.546822i $$0.184163\pi$$
$$660$$ 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
$$663$$ 0 0
$$664$$ −15569.8 −0.909981
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9198.01 0.533955
$$668$$ 6975.84 0.404047
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −7858.46 −0.452120
$$672$$ 0 0
$$673$$ 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 $$-0.194879\pi$$
0.818368 + 0.574695i $$0.194879\pi$$
$$678$$ 0 0
$$679$$ 9901.37 0.559617
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 940.250 0.0527918
$$683$$ 3193.10 0.178888 0.0894441 0.995992i $$-0.471491\pi$$
0.0894441 + 0.995992i $$0.471491\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −9296.24 −0.517394
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ −139.496 −0.00762996
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 33213.3 1.80494
$$698$$ −233.256 −0.0126488
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −26551.6 −1.43058 −0.715292 0.698825i $$-0.753707\pi$$
−0.715292 + 0.698825i $$0.753707\pi$$
$$702$$ 0 0
$$703$$ 1779.86 0.0954887
$$704$$ 2211.24 0.118380
$$705$$ 0 0
$$706$$ 12285.7 0.654928
$$707$$ −3571.83 −0.190004
$$708$$ 0 0
$$709$$ −16304.6 −0.863655 −0.431828 0.901956i $$-0.642131\pi$$
−0.431828 + 0.901956i $$0.642131\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 8171.79 0.430127
$$713$$ 3377.00 0.177377
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −8021.36 −0.418676
$$717$$ 0 0
$$718$$ −7767.50 −0.403733
$$719$$ 3973.62 0.206107 0.103053 0.994676i $$-0.467139\pi$$
0.103053 + 0.994676i $$0.467139\pi$$
$$720$$ 0 0
$$721$$ 15742.1 0.813128
$$722$$ −8256.25 −0.425576
$$723$$ 0 0
$$724$$ −23702.3 −1.21670
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 10780.4 0.549961 0.274980 0.961450i $$-0.411329\pi$$
0.274980 + 0.961450i $$0.411329\pi$$
$$728$$ −8869.67 −0.451555
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −233.687 −0.0118238
$$732$$ 0 0
$$733$$ 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
$$738$$ 0 0
$$739$$ 11084.7 0.551768 0.275884 0.961191i $$-0.411029\pi$$
0.275884 + 0.961191i $$0.411029\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 10241.0 0.506685
$$743$$ −27420.4 −1.35391 −0.676955 0.736024i $$-0.736701\pi$$
−0.676955 + 0.736024i $$0.736701\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 7234.32 0.355050
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ 9516.81 0.459657
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −3739.19 −0.179528 −0.0897642 0.995963i $$-0.528611\pi$$
−0.0897642 + 0.995963i $$0.528611\pi$$
$$758$$ 10155.3 0.486617
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 15621.1 0.744107 0.372053 0.928211i $$-0.378654\pi$$
0.372053 + 0.928211i $$0.378654\pi$$
$$762$$ 0 0
$$763$$ −15387.7 −0.730106
$$764$$ −4739.69 −0.224445
$$765$$ 0 0
$$766$$ 19885.7 0.937987
$$767$$ 15156.0 0.713498
$$768$$ 0 0
$$769$$ 40241.7 1.88706 0.943531 0.331284i $$-0.107482\pi$$
0.943531 + 0.331284i $$0.107482\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −13681.2 −0.637818
$$773$$ 22821.4 1.06187 0.530936 0.847412i $$-0.321840\pi$$
0.530936 + 0.847412i $$0.321840\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −20464.4 −0.946685
$$777$$ 0 0
$$778$$ −18777.0 −0.865282
$$779$$ 13340.0 0.613549
$$780$$ 0 0
$$781$$ −10332.2 −0.473387
$$782$$ −9509.28 −0.434848
$$783$$ 0 0
$$784$$ −2718.92 −0.123857
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 29454.3 1.33410 0.667048 0.745015i $$-0.267558\pi$$
0.667048 + 0.745015i $$0.267558\pi$$
$$788$$ −19337.8 −0.874216
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1741.86 0.0782974
$$792$$ 0 0
$$793$$ −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 $$-0.708741\pi$$
−0.609777 + 0.792573i $$0.708741\pi$$
$$798$$ 0 0
$$799$$ 32899.0 1.45668
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −15066.1 −0.663346
$$803$$ 3988.27 0.175272
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 3494.05 0.152695
$$807$$ 0 0
$$808$$ 7382.34 0.321423
$$809$$ 5060.18 0.219909 0.109954 0.993937i $$-0.464930\pi$$
0.109954 + 0.993937i $$0.464930\pi$$
$$810$$ 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
$$813$$ 0 0
$$814$$ −771.146 −0.0332048
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −93.8596 −0.00401925
$$818$$ −3139.46 −0.134192
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −37909.0 −1.61149 −0.805745 0.592263i $$-0.798235\pi$$
−0.805745 + 0.592263i $$0.798235\pi$$
$$822$$ 0 0
$$823$$ −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 $$-0.853781\pi$$
−0.896336 + 0.443376i $$0.853781\pi$$
$$828$$ 0 0
$$829$$ −45152.5 −1.89169 −0.945845 0.324619i $$-0.894764\pi$$
−0.945845 + 0.324619i $$0.894764\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 8217.15 0.342402
$$833$$ 23494.1 0.977217
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2425.42 0.100341
$$837$$ 0 0
$$838$$ −6925.91 −0.285503
$$839$$ −30431.5 −1.25222 −0.626110 0.779734i $$-0.715354\pi$$
−0.626110 + 0.779734i $$0.715354\pi$$
$$840$$ 0 0
$$841$$ −2160.34 −0.0885784
$$842$$ 23763.6 0.972623
$$843$$ 0 0
$$844$$ −5818.09 −0.237283
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1239.79 0.0502949
$$848$$ 7311.64 0.296088
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2769.65 −0.111566
$$852$$ 0 0
$$853$$ −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 $$-0.416926\pi$$
0.258032 + 0.966136i $$0.416926\pi$$
$$858$$ 0 0
$$859$$ −20383.5 −0.809636 −0.404818 0.914397i $$-0.632665\pi$$
−0.404818 + 0.914397i $$0.632665\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8779.22 0.346893
$$863$$ 9056.42 0.357224 0.178612 0.983920i $$-0.442839\pi$$
0.178612 + 0.983920i $$0.442839\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 22340.0 0.876610
$$867$$ 0 0
$$868$$ −3119.27 −0.121976
$$869$$ 10470.2 0.408719
$$870$$ 0 0
$$871$$ 16554.5 0.644005
$$872$$ 31803.6 1.23510
$$873$$ 0 0
$$874$$ −3819.37 −0.147817
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2867.88 0.110424 0.0552118 0.998475i $$-0.482417\pi$$
0.0552118 + 0.998475i $$0.482417\pi$$
$$878$$ 6846.16 0.263151
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 11862.5 0.453640 0.226820 0.973937i $$-0.427167\pi$$
0.226820 + 0.973937i $$0.427167\pi$$
$$882$$ 0 0
$$883$$ −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 $$-0.485411\pi$$
0.0458178 + 0.998950i $$0.485411\pi$$
$$888$$ 0 0
$$889$$ 17826.5 0.672534
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −2815.62 −0.105688
$$893$$ 13213.8 0.495165
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −12132.9 −0.452378
$$897$$ 0 0
$$898$$ −14963.6 −0.556060
$$899$$ 8161.14 0.302769
$$900$$ 0 0
$$901$$ −63179.7 −2.33609
$$902$$ −5779.73 −0.213352
$$903$$ 0 0
$$904$$ −3600.10 −0.132453
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 38154.1 1.39679 0.698393 0.715714i $$-0.253899\pi$$
0.698393 + 0.715714i $$0.253899\pi$$
$$908$$ 23838.4 0.871262
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 35758.0 1.30045 0.650227 0.759740i $$-0.274674\pi$$
0.650227 + 0.759740i $$0.274674\pi$$
$$912$$ 0 0
$$913$$ 8087.44 0.293160
$$914$$ −15614.5 −0.565079
$$915$$ 0 0
$$916$$ −31753.0 −1.14536
$$917$$ −3205.39 −0.115432
$$918$$ 0 0
$$919$$ 17387.5 0.624115 0.312058 0.950063i $$-0.398982\pi$$
0.312058 + 0.950063i $$0.398982\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 17357.9 0.620013
$$923$$ −38395.3 −1.36923
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 2447.20 0.0868467
$$927$$ 0 0
$$928$$ 27918.3 0.987569
$$929$$ −6955.93 −0.245658 −0.122829 0.992428i $$-0.539197\pi$$
−0.122829 + 0.992428i $$0.539197\pi$$
$$930$$ 0 0
$$931$$ 9436.31 0.332183
$$932$$ −16384.9 −0.575863
$$933$$ 0 0
$$934$$ 19751.7 0.691965
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 16074.5 0.560438 0.280219 0.959936i $$-0.409593\pi$$
0.280219 + 0.959936i $$0.409593\pi$$
$$938$$ 6479.76 0.225556
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 687.126 0.0238041 0.0119021 0.999929i $$-0.496211\pi$$
0.0119021 + 0.999929i $$0.496211\pi$$
$$942$$ 0 0
$$943$$ −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 $$-0.292557\pi$$
0.606540 + 0.795053i $$0.292557\pi$$
$$948$$ 0 0
$$949$$ 14820.7 0.506956
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 21418.2 0.729168
$$953$$ −19390.7 −0.659103 −0.329552 0.944138i $$-0.606898\pi$$
−0.329552 + 0.944138i $$0.606898\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −11561.9 −0.391148
$$957$$ 0 0
$$958$$ 16738.7 0.564511
$$959$$ −7345.80 −0.247349
$$960$$ 0 0
$$961$$ −26794.7 −0.899422
$$962$$ −2865.64 −0.0960416
$$963$$ 0 0
$$964$$ −10307.6 −0.344384
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −28643.6 −0.952551 −0.476275 0.879296i $$-0.658013\pi$$
−0.476275 + 0.879296i $$0.658013\pi$$
$$968$$ −2562.43 −0.0850821
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 19574.8 0.646946 0.323473 0.946237i $$-0.395149\pi$$
0.323473 + 0.946237i $$0.395149\pi$$
$$972$$ 0 0
$$973$$ −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 $$-0.193855\pi$$
0.820213 + 0.572058i $$0.193855\pi$$
$$978$$ 0 0
$$979$$ −4244.67 −0.138570
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −22747.6 −0.739211
$$983$$ 14445.4 0.468706 0.234353 0.972152i $$-0.424703\pi$$
0.234353 + 0.972152i $$0.424703\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −22980.9 −0.742253
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ −15028.7 −0.479558
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 9137.45 0.290257 0.145128 0.989413i $$-0.453640\pi$$
0.145128 + 0.989413i $$0.453640\pi$$
$$998$$ −7244.03 −0.229765
$$999$$ 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 2475.4.a.n.1.2 2
3.2 odd 2 825.4.a.m.1.1 2
5.4 even 2 495.4.a.d.1.1 2
15.2 even 4 825.4.c.j.199.2 4
15.8 even 4 825.4.c.j.199.3 4
15.14 odd 2 165.4.a.c.1.2 2
165.164 even 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 15.14 odd 2
495.4.a.d.1.1 2 5.4 even 2
825.4.a.m.1.1 2 3.2 odd 2
825.4.c.j.199.2 4 15.2 even 4
825.4.c.j.199.3 4 15.8 even 4
1815.4.a.n.1.1 2 165.164 even 2
2475.4.a.n.1.2 2 1.1 even 1 trivial