# Properties

 Label 825.4.c.j.199.3 Level $825$ Weight $4$ Character 825.199 Analytic conductor $48.677$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.3 Root $$1.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.4.c.j.199.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.56155i q^{2} -3.00000i q^{3} +5.56155 q^{4} +4.68466 q^{6} -10.2462i q^{7} +21.1771i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+1.56155i q^{2} -3.00000i q^{3} +5.56155 q^{4} +4.68466 q^{6} -10.2462i q^{7} +21.1771i q^{8} -9.00000 q^{9} -11.0000 q^{11} -16.6847i q^{12} +40.8769i q^{13} +16.0000 q^{14} +11.4233 q^{16} -98.7083i q^{17} -14.0540i q^{18} +39.6458 q^{19} -30.7386 q^{21} -17.1771i q^{22} -61.6932i q^{23} +63.5312 q^{24} -63.8314 q^{26} +27.0000i q^{27} -56.9848i q^{28} +149.093 q^{29} +54.7386 q^{31} +187.255i q^{32} +33.0000i q^{33} +154.138 q^{34} -50.0540 q^{36} +44.8939i q^{37} +61.9091i q^{38} +122.631 q^{39} +336.479 q^{41} -48.0000i q^{42} +2.36745i q^{43} -61.1771 q^{44} +96.3371 q^{46} -333.295i q^{47} -34.2699i q^{48} +238.015 q^{49} -296.125 q^{51} +227.339i q^{52} -640.064i q^{53} -42.1619 q^{54} +216.985 q^{56} -118.938i q^{57} +232.816i q^{58} +370.773 q^{59} -714.405 q^{61} +85.4773i q^{62} +92.2159i q^{63} -201.022 q^{64} -51.5312 q^{66} -404.985i q^{67} -548.972i q^{68} -185.080 q^{69} +939.292 q^{71} -190.594i q^{72} +362.570i q^{73} -70.1042 q^{74} +220.492 q^{76} +112.708i q^{77} +191.494i q^{78} -951.835 q^{79} +81.0000 q^{81} +525.430i q^{82} -735.221i q^{83} -170.955 q^{84} -3.69690 q^{86} -447.278i q^{87} -232.948i q^{88} -385.879 q^{89} +418.833 q^{91} -343.110i q^{92} -164.216i q^{93} +520.458 q^{94} +561.764 q^{96} -966.345i q^{97} +371.673i q^{98} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 14 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q + 14 * q^4 - 6 * q^6 - 36 * q^9 $$4 q + 14 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 64 q^{14} - 78 q^{16} + 340 q^{19} - 24 q^{21} - 18 q^{24} + 124 q^{26} + 316 q^{29} + 120 q^{31} + 732 q^{34} - 126 q^{36} + 540 q^{39} + 76 q^{41} - 154 q^{44} + 1144 q^{46} + 1084 q^{49} - 96 q^{51} + 54 q^{54} + 736 q^{56} - 496 q^{59} + 144 q^{61} - 1538 q^{64} + 66 q^{66} + 744 q^{69} + 4120 q^{71} - 2276 q^{74} + 816 q^{76} - 1284 q^{79} + 324 q^{81} - 288 q^{84} + 2624 q^{86} - 488 q^{89} + 224 q^{91} + 3896 q^{94} + 738 q^{96} + 396 q^{99}+O(q^{100})$$ 4 * q + 14 * q^4 - 6 * q^6 - 36 * q^9 - 44 * q^11 + 64 * q^14 - 78 * q^16 + 340 * q^19 - 24 * q^21 - 18 * q^24 + 124 * q^26 + 316 * q^29 + 120 * q^31 + 732 * q^34 - 126 * q^36 + 540 * q^39 + 76 * q^41 - 154 * q^44 + 1144 * q^46 + 1084 * q^49 - 96 * q^51 + 54 * q^54 + 736 * q^56 - 496 * q^59 + 144 * q^61 - 1538 * q^64 + 66 * q^66 + 744 * q^69 + 4120 * q^71 - 2276 * q^74 + 816 * q^76 - 1284 * q^79 + 324 * q^81 - 288 * q^84 + 2624 * q^86 - 488 * q^89 + 224 * q^91 + 3896 * q^94 + 738 * q^96 + 396 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.56155i 0.552092i 0.961144 + 0.276046i $$0.0890243\pi$$
−0.961144 + 0.276046i $$0.910976\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ 5.56155 0.695194
$$5$$ 0 0
$$6$$ 4.68466 0.318751
$$7$$ − 10.2462i − 0.553243i −0.960979 0.276622i $$-0.910785\pi$$
0.960979 0.276622i $$-0.0892149\pi$$
$$8$$ 21.1771i 0.935904i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ − 16.6847i − 0.401371i
$$13$$ 40.8769i 0.872093i 0.899924 + 0.436047i $$0.143622\pi$$
−0.899924 + 0.436047i $$0.856378\pi$$
$$14$$ 16.0000 0.305441
$$15$$ 0 0
$$16$$ 11.4233 0.178489
$$17$$ − 98.7083i − 1.40825i −0.710075 0.704126i $$-0.751339\pi$$
0.710075 0.704126i $$-0.248661\pi$$
$$18$$ − 14.0540i − 0.184031i
$$19$$ 39.6458 0.478704 0.239352 0.970933i $$-0.423065\pi$$
0.239352 + 0.970933i $$0.423065\pi$$
$$20$$ 0 0
$$21$$ −30.7386 −0.319415
$$22$$ − 17.1771i − 0.166462i
$$23$$ − 61.6932i − 0.559301i −0.960102 0.279650i $$-0.909781\pi$$
0.960102 0.279650i $$-0.0902185\pi$$
$$24$$ 63.5312 0.540344
$$25$$ 0 0
$$26$$ −63.8314 −0.481476
$$27$$ 27.0000i 0.192450i
$$28$$ − 56.9848i − 0.384612i
$$29$$ 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 $$0.449312\pi$$
$$32$$ 187.255i 1.03445i
$$33$$ 33.0000i 0.174078i
$$34$$ 154.138 0.777485
$$35$$ 0 0
$$36$$ −50.0540 −0.231731
$$37$$ 44.8939i 0.199473i 0.995014 + 0.0997367i $$0.0318000\pi$$
−0.995014 + 0.0997367i $$0.968200\pi$$
$$38$$ 61.9091i 0.264289i
$$39$$ 122.631 0.503503
$$40$$ 0 0
$$41$$ 336.479 1.28169 0.640844 0.767671i $$-0.278585\pi$$
0.640844 + 0.767671i $$0.278585\pi$$
$$42$$ − 48.0000i − 0.176347i
$$43$$ 2.36745i 0.00839611i 0.999991 + 0.00419806i $$0.00133629\pi$$
−0.999991 + 0.00419806i $$0.998664\pi$$
$$44$$ −61.1771 −0.209609
$$45$$ 0 0
$$46$$ 96.3371 0.308786
$$47$$ − 333.295i − 1.03439i −0.855869 0.517193i $$-0.826977\pi$$
0.855869 0.517193i $$-0.173023\pi$$
$$48$$ − 34.2699i − 0.103051i
$$49$$ 238.015 0.693922
$$50$$ 0 0
$$51$$ −296.125 −0.813055
$$52$$ 227.339i 0.606274i
$$53$$ − 640.064i − 1.65886i −0.558610 0.829430i $$-0.688665\pi$$
0.558610 0.829430i $$-0.311335\pi$$
$$54$$ −42.1619 −0.106250
$$55$$ 0 0
$$56$$ 216.985 0.517782
$$57$$ − 118.938i − 0.276380i
$$58$$ 232.816i 0.527074i
$$59$$ 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 $$0.769829\pi$$
$$62$$ 85.4773i 0.175091i
$$63$$ 92.2159i 0.184414i
$$64$$ −201.022 −0.392621
$$65$$ 0 0
$$66$$ −51.5312 −0.0961069
$$67$$ − 404.985i − 0.738459i −0.929338 0.369230i $$-0.879622\pi$$
0.929338 0.369230i $$-0.120378\pi$$
$$68$$ − 548.972i − 0.979009i
$$69$$ −185.080 −0.322912
$$70$$ 0 0
$$71$$ 939.292 1.57005 0.785024 0.619465i $$-0.212651\pi$$
0.785024 + 0.619465i $$0.212651\pi$$
$$72$$ − 190.594i − 0.311968i
$$73$$ 362.570i 0.581310i 0.956828 + 0.290655i $$0.0938732\pi$$
−0.956828 + 0.290655i $$0.906127\pi$$
$$74$$ −70.1042 −0.110128
$$75$$ 0 0
$$76$$ 220.492 0.332792
$$77$$ 112.708i 0.166809i
$$78$$ 191.494i 0.277980i
$$79$$ −951.835 −1.35557 −0.677784 0.735261i $$-0.737059\pi$$
−0.677784 + 0.735261i $$0.737059\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 525.430i 0.707610i
$$83$$ − 735.221i − 0.972302i −0.873875 0.486151i $$-0.838401\pi$$
0.873875 0.486151i $$-0.161599\pi$$
$$84$$ −170.955 −0.222056
$$85$$ 0 0
$$86$$ −3.69690 −0.00463543
$$87$$ − 447.278i − 0.551187i
$$88$$ − 232.948i − 0.282186i
$$89$$ −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
$$92$$ − 343.110i − 0.388823i
$$93$$ − 164.216i − 0.183101i
$$94$$ 520.458 0.571076
$$95$$ 0 0
$$96$$ 561.764 0.597238
$$97$$ − 966.345i − 1.01152i −0.862674 0.505760i $$-0.831212\pi$$
0.862674 0.505760i $$-0.168788\pi$$
$$98$$ 371.673i 0.383109i
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ 348.600 0.343436 0.171718 0.985146i $$-0.445068\pi$$
0.171718 + 0.985146i $$0.445068\pi$$
$$102$$ − 462.415i − 0.448881i
$$103$$ 1536.38i 1.46975i 0.678204 + 0.734873i $$0.262758\pi$$
−0.678204 + 0.734873i $$0.737242\pi$$
$$104$$ −865.653 −0.816195
$$105$$ 0 0
$$106$$ 999.494 0.915844
$$107$$ − 779.180i − 0.703983i −0.936003 0.351991i $$-0.885505\pi$$
0.936003 0.351991i $$-0.114495\pi$$
$$108$$ 150.162i 0.133790i
$$109$$ 1501.79 1.31968 0.659842 0.751404i $$-0.270623\pi$$
0.659842 + 0.751404i $$0.270623\pi$$
$$110$$ 0 0
$$111$$ 134.682 0.115166
$$112$$ − 117.045i − 0.0987478i
$$113$$ − 170.000i − 0.141524i −0.997493 0.0707622i $$-0.977457\pi$$
0.997493 0.0707622i $$-0.0225431\pi$$
$$114$$ 185.727 0.152587
$$115$$ 0 0
$$116$$ 829.187 0.663691
$$117$$ − 367.892i − 0.290698i
$$118$$ 578.981i 0.451691i
$$119$$ −1011.39 −0.779106
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ − 1115.58i − 0.827869i
$$123$$ − 1009.44i − 0.739983i
$$124$$ 304.432 0.220474
$$125$$ 0 0
$$126$$ −144.000 −0.101814
$$127$$ − 1739.82i − 1.21562i −0.794082 0.607811i $$-0.792048\pi$$
0.794082 0.607811i $$-0.207952\pi$$
$$128$$ 1184.13i 0.817683i
$$129$$ 7.10235 0.00484750
$$130$$ 0 0
$$131$$ 312.837 0.208647 0.104323 0.994543i $$-0.466732\pi$$
0.104323 + 0.994543i $$0.466732\pi$$
$$132$$ 183.531i 0.121018i
$$133$$ − 406.220i − 0.264840i
$$134$$ 632.405 0.407698
$$135$$ 0 0
$$136$$ 2090.35 1.31799
$$137$$ − 716.928i − 0.447090i −0.974694 0.223545i $$-0.928237\pi$$
0.974694 0.223545i $$-0.0717629\pi$$
$$138$$ − 289.011i − 0.178277i
$$139$$ 876.483 0.534837 0.267418 0.963581i $$-0.413829\pi$$
0.267418 + 0.963581i $$0.413829\pi$$
$$140$$ 0 0
$$141$$ −999.886 −0.597203
$$142$$ 1466.75i 0.866811i
$$143$$ − 449.646i − 0.262946i
$$144$$ −102.810 −0.0594963
$$145$$ 0 0
$$146$$ −566.172 −0.320937
$$147$$ − 714.045i − 0.400636i
$$148$$ 249.680i 0.138673i
$$149$$ 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 $$0.507965\pi$$
$$152$$ 839.583i 0.448021i
$$153$$ 888.375i 0.469417i
$$154$$ −176.000 −0.0920941
$$155$$ 0 0
$$156$$ 682.017 0.350032
$$157$$ − 1881.24i − 0.956301i −0.878278 0.478150i $$-0.841307\pi$$
0.878278 0.478150i $$-0.158693\pi$$
$$158$$ − 1486.34i − 0.748398i
$$159$$ −1920.19 −0.957743
$$160$$ 0 0
$$161$$ −632.121 −0.309429
$$162$$ 126.486i 0.0613436i
$$163$$ 2465.49i 1.18474i 0.805667 + 0.592369i $$0.201807\pi$$
−0.805667 + 0.592369i $$0.798193\pi$$
$$164$$ 1871.35 0.891022
$$165$$ 0 0
$$166$$ 1148.09 0.536800
$$167$$ − 1254.30i − 0.581200i −0.956845 0.290600i $$-0.906145\pi$$
0.956845 0.290600i $$-0.0938549\pi$$
$$168$$ − 650.955i − 0.298942i
$$169$$ 526.080 0.239454
$$170$$ 0 0
$$171$$ −356.813 −0.159568
$$172$$ 13.1667i 0.00583693i
$$173$$ 1206.71i 0.530314i 0.964205 + 0.265157i $$0.0854238\pi$$
−0.964205 + 0.265157i $$0.914576\pi$$
$$174$$ 698.449 0.304306
$$175$$ 0 0
$$176$$ −125.656 −0.0538164
$$177$$ − 1112.32i − 0.472356i
$$178$$ − 602.570i − 0.253733i
$$179$$ 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 $$0.160811\pi$$
$$182$$ 654.030i 0.266373i
$$183$$ 2143.22i 0.865744i
$$184$$ 1306.48 0.523451
$$185$$ 0 0
$$186$$ 256.432 0.101089
$$187$$ 1085.79i 0.424604i
$$188$$ − 1853.64i − 0.719099i
$$189$$ 276.648 0.106472
$$190$$ 0 0
$$191$$ −852.223 −0.322852 −0.161426 0.986885i $$-0.551609\pi$$
−0.161426 + 0.986885i $$0.551609\pi$$
$$192$$ 603.065i 0.226680i
$$193$$ 2459.95i 0.917468i 0.888574 + 0.458734i $$0.151697\pi$$
−0.888574 + 0.458734i $$0.848303\pi$$
$$194$$ 1509.00 0.558452
$$195$$ 0 0
$$196$$ 1323.73 0.482410
$$197$$ 3477.06i 1.25751i 0.777602 + 0.628756i $$0.216436\pi$$
−0.777602 + 0.628756i $$0.783564\pi$$
$$198$$ 154.594i 0.0554874i
$$199$$ −3995.04 −1.42312 −0.711560 0.702626i $$-0.752011\pi$$
−0.711560 + 0.702626i $$0.752011\pi$$
$$200$$ 0 0
$$201$$ −1214.95 −0.426350
$$202$$ 544.358i 0.189608i
$$203$$ − 1527.64i − 0.528173i
$$204$$ −1646.91 −0.565231
$$205$$ 0 0
$$206$$ −2399.14 −0.811436
$$207$$ 555.239i 0.186434i
$$208$$ 466.949i 0.155659i
$$209$$ −436.104 −0.144335
$$210$$ 0 0
$$211$$ 1046.13 0.341319 0.170660 0.985330i $$-0.445410\pi$$
0.170660 + 0.985330i $$0.445410\pi$$
$$212$$ − 3559.75i − 1.15323i
$$213$$ − 2817.88i − 0.906468i
$$214$$ 1216.73 0.388664
$$215$$ 0 0
$$216$$ −571.781 −0.180115
$$217$$ − 560.864i − 0.175456i
$$218$$ 2345.13i 0.728587i
$$219$$ 1087.71 0.335619
$$220$$ 0 0
$$221$$ 4034.89 1.22813
$$222$$ 210.313i 0.0635823i
$$223$$ 506.265i 0.152027i 0.997107 + 0.0760135i $$0.0242192\pi$$
−0.997107 + 0.0760135i $$0.975781\pi$$
$$224$$ 1918.65 0.572300
$$225$$ 0 0
$$226$$ 265.464 0.0781345
$$227$$ − 4286.29i − 1.25326i −0.779315 0.626632i $$-0.784433\pi$$
0.779315 0.626632i $$-0.215567\pi$$
$$228$$ − 661.477i − 0.192138i
$$229$$ −5709.37 −1.64754 −0.823769 0.566926i $$-0.808133\pi$$
−0.823769 + 0.566926i $$0.808133\pi$$
$$230$$ 0 0
$$231$$ 338.125 0.0963073
$$232$$ 3157.35i 0.893492i
$$233$$ − 2946.09i − 0.828348i −0.910198 0.414174i $$-0.864071\pi$$
0.910198 0.414174i $$-0.135929\pi$$
$$234$$ 574.483 0.160492
$$235$$ 0 0
$$236$$ 2062.07 0.568769
$$237$$ 2855.51i 0.782637i
$$238$$ − 1579.33i − 0.430139i
$$239$$ 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 $$0.420329\pi$$
$$242$$ 188.948i 0.0501902i
$$243$$ − 243.000i − 0.0641500i
$$244$$ −3973.20 −1.04245
$$245$$ 0 0
$$246$$ 1576.29 0.408539
$$247$$ 1620.60i 0.417475i
$$248$$ 1159.20i 0.296813i
$$249$$ −2205.66 −0.561359
$$250$$ 0 0
$$251$$ −2358.39 −0.593068 −0.296534 0.955022i $$-0.595831\pi$$
−0.296534 + 0.955022i $$0.595831\pi$$
$$252$$ 512.864i 0.128204i
$$253$$ 678.625i 0.168635i
$$254$$ 2716.82 0.671135
$$255$$ 0 0
$$256$$ −3457.26 −0.844057
$$257$$ 5519.25i 1.33962i 0.742534 + 0.669809i $$0.233624\pi$$
−0.742534 + 0.669809i $$0.766376\pi$$
$$258$$ 11.0907i 0.00267627i
$$259$$ 459.993 0.110357
$$260$$ 0 0
$$261$$ −1341.84 −0.318228
$$262$$ 488.512i 0.115192i
$$263$$ − 2259.65i − 0.529795i −0.964277 0.264898i $$-0.914662\pi$$
0.964277 0.264898i $$-0.0853382\pi$$
$$264$$ −698.844 −0.162920
$$265$$ 0 0
$$266$$ 634.333 0.146216
$$267$$ 1157.64i 0.265342i
$$268$$ − 2252.34i − 0.513373i
$$269$$ −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 $$0.303925\pi$$
$$272$$ − 1127.57i − 0.251357i
$$273$$ − 1256.50i − 0.278560i
$$274$$ 1119.52 0.246835
$$275$$ 0 0
$$276$$ −1029.33 −0.224487
$$277$$ 9074.52i 1.96836i 0.177175 + 0.984179i $$0.443304\pi$$
−0.177175 + 0.984179i $$0.556696\pi$$
$$278$$ 1368.67i 0.295279i
$$279$$ −492.648 −0.105713
$$280$$ 0 0
$$281$$ −3407.79 −0.723459 −0.361729 0.932283i $$-0.617814\pi$$
−0.361729 + 0.932283i $$0.617814\pi$$
$$282$$ − 1561.38i − 0.329711i
$$283$$ 8827.73i 1.85425i 0.374746 + 0.927127i $$0.377730\pi$$
−0.374746 + 0.927127i $$0.622270\pi$$
$$284$$ 5223.92 1.09149
$$285$$ 0 0
$$286$$ 702.146 0.145170
$$287$$ − 3447.64i − 0.709085i
$$288$$ − 1685.29i − 0.344815i
$$289$$ −4830.33 −0.983174
$$290$$ 0 0
$$291$$ −2899.03 −0.584001
$$292$$ 2016.45i 0.404123i
$$293$$ 4528.29i 0.902886i 0.892300 + 0.451443i $$0.149091\pi$$
−0.892300 + 0.451443i $$0.850909\pi$$
$$294$$ 1115.02 0.221188
$$295$$ 0 0
$$296$$ −950.722 −0.186688
$$297$$ − 297.000i − 0.0580259i
$$298$$ 3710.81i 0.721347i
$$299$$ 2521.83 0.487762
$$300$$ 0 0
$$301$$ 24.2574 0.00464509
$$302$$ − 144.985i − 0.0276256i
$$303$$ − 1045.80i − 0.198283i
$$304$$ 452.886 0.0854434
$$305$$ 0 0
$$306$$ −1387.24 −0.259162
$$307$$ 568.106i 0.105614i 0.998605 + 0.0528071i $$0.0168168\pi$$
−0.998605 + 0.0528071i $$0.983183\pi$$
$$308$$ 626.833i 0.115965i
$$309$$ 4609.14 0.848559
$$310$$ 0 0
$$311$$ −6853.59 −1.24962 −0.624809 0.780778i $$-0.714823\pi$$
−0.624809 + 0.780778i $$0.714823\pi$$
$$312$$ 2596.96i 0.471230i
$$313$$ 1138.92i 0.205673i 0.994698 + 0.102837i $$0.0327919\pi$$
−0.994698 + 0.102837i $$0.967208\pi$$
$$314$$ 2937.65 0.527966
$$315$$ 0 0
$$316$$ −5293.68 −0.942382
$$317$$ 3207.48i 0.568297i 0.958780 + 0.284148i $$0.0917109\pi$$
−0.958780 + 0.284148i $$0.908289\pi$$
$$318$$ − 2998.48i − 0.528763i
$$319$$ −1640.02 −0.287848
$$320$$ 0 0
$$321$$ −2337.54 −0.406445
$$322$$ − 987.091i − 0.170834i
$$323$$ − 3913.37i − 0.674136i
$$324$$ 450.486 0.0772438
$$325$$ 0 0
$$326$$ −3850.00 −0.654085
$$327$$ − 4505.37i − 0.761920i
$$328$$ 7125.65i 1.19954i
$$329$$ −3415.02 −0.572267
$$330$$ 0 0
$$331$$ −9135.12 −1.51695 −0.758477 0.651700i $$-0.774056\pi$$
−0.758477 + 0.651700i $$0.774056\pi$$
$$332$$ − 4088.97i − 0.675938i
$$333$$ − 404.045i − 0.0664911i
$$334$$ 1958.65 0.320876
$$335$$ 0 0
$$336$$ −351.136 −0.0570121
$$337$$ − 3470.05i − 0.560907i −0.959868 0.280453i $$-0.909515\pi$$
0.959868 0.280453i $$-0.0904848\pi$$
$$338$$ 821.501i 0.132200i
$$339$$ −510.000 −0.0817091
$$340$$ 0 0
$$341$$ −602.125 −0.0956214
$$342$$ − 557.182i − 0.0880963i
$$343$$ − 5953.20i − 0.937151i
$$344$$ −50.1357 −0.00785795
$$345$$ 0 0
$$346$$ −1884.34 −0.292782
$$347$$ − 89.3315i − 0.0138201i −0.999976 0.00691004i $$-0.997800\pi$$
0.999976 0.00691004i $$-0.00219955\pi$$
$$348$$ − 2487.56i − 0.383182i
$$349$$ 149.375 0.0229107 0.0114554 0.999934i $$-0.496354\pi$$
0.0114554 + 0.999934i $$0.496354\pi$$
$$350$$ 0 0
$$351$$ −1103.68 −0.167834
$$352$$ − 2059.80i − 0.311897i
$$353$$ − 7867.64i − 1.18627i −0.805104 0.593133i $$-0.797891\pi$$
0.805104 0.593133i $$-0.202109\pi$$
$$354$$ 1736.94 0.260784
$$355$$ 0 0
$$356$$ −2146.09 −0.319501
$$357$$ 3034.16i 0.449817i
$$358$$ 2252.21i 0.332494i
$$359$$ −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
$$362$$ 6655.04i 0.966246i
$$363$$ − 363.000i − 0.0524864i
$$364$$ 2329.36 0.335417
$$365$$ 0 0
$$366$$ −3346.74 −0.477970
$$367$$ − 13266.7i − 1.88696i −0.331433 0.943479i $$-0.607532\pi$$
0.331433 0.943479i $$-0.392468\pi$$
$$368$$ − 704.739i − 0.0998290i
$$369$$ −3028.31 −0.427229
$$370$$ 0 0
$$371$$ −6558.23 −0.917754
$$372$$ − 913.295i − 0.127291i
$$373$$ 4632.77i 0.643099i 0.946893 + 0.321549i $$0.104204\pi$$
−0.946893 + 0.321549i $$0.895796\pi$$
$$374$$ −1695.52 −0.234421
$$375$$ 0 0
$$376$$ 7058.22 0.968085
$$377$$ 6094.45i 0.832573i
$$378$$ 432.000i 0.0587822i
$$379$$ −6503.31 −0.881406 −0.440703 0.897653i $$-0.645271\pi$$
−0.440703 + 0.897653i $$0.645271\pi$$
$$380$$ 0 0
$$381$$ −5219.45 −0.701839
$$382$$ − 1330.79i − 0.178244i
$$383$$ − 12734.5i − 1.69897i −0.527614 0.849484i $$-0.676913\pi$$
0.527614 0.849484i $$-0.323087\pi$$
$$384$$ 3552.39 0.472090
$$385$$ 0 0
$$386$$ −3841.35 −0.506527
$$387$$ − 21.3071i − 0.00279870i
$$388$$ − 5374.38i − 0.703203i
$$389$$ −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
$$392$$ 5040.47i 0.649444i
$$393$$ − 938.511i − 0.120462i
$$394$$ −5429.61 −0.694263
$$395$$ 0 0
$$396$$ 550.594 0.0698696
$$397$$ − 5223.65i − 0.660371i −0.943916 0.330186i $$-0.892889\pi$$
0.943916 0.330186i $$-0.107111\pi$$
$$398$$ − 6238.46i − 0.785693i
$$399$$ −1218.66 −0.152905
$$400$$ 0 0
$$401$$ 9648.18 1.20151 0.600757 0.799432i $$-0.294866\pi$$
0.600757 + 0.799432i $$0.294866\pi$$
$$402$$ − 1897.22i − 0.235384i
$$403$$ 2237.55i 0.276576i
$$404$$ 1938.76 0.238755
$$405$$ 0 0
$$406$$ 2385.48 0.291600
$$407$$ − 493.833i − 0.0601435i
$$408$$ − 6271.06i − 0.760941i
$$409$$ 2010.47 0.243060 0.121530 0.992588i $$-0.461220\pi$$
0.121530 + 0.992588i $$0.461220\pi$$
$$410$$ 0 0
$$411$$ −2150.78 −0.258127
$$412$$ 8544.65i 1.02176i
$$413$$ − 3799.02i − 0.452633i
$$414$$ −867.034 −0.102929
$$415$$ 0 0
$$416$$ −7654.39 −0.902133
$$417$$ − 2629.45i − 0.308788i
$$418$$ − 681.000i − 0.0796861i
$$419$$ −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 $$0.156971\pi$$
$$422$$ 1633.58i 0.188440i
$$423$$ 2999.66i 0.344795i
$$424$$ 13554.7 1.55253
$$425$$ 0 0
$$426$$ 4400.26 0.500454
$$427$$ 7319.95i 0.829595i
$$428$$ − 4333.45i − 0.489405i
$$429$$ −1348.94 −0.151812
$$430$$ 0 0
$$431$$ −5622.11 −0.628324 −0.314162 0.949369i $$-0.601723\pi$$
−0.314162 + 0.949369i $$0.601723\pi$$
$$432$$ 308.429i 0.0343502i
$$433$$ 14306.3i 1.58780i 0.608051 + 0.793898i $$0.291951\pi$$
−0.608051 + 0.793898i $$0.708049\pi$$
$$434$$ 875.818 0.0968678
$$435$$ 0 0
$$436$$ 8352.29 0.917436
$$437$$ − 2445.88i − 0.267740i
$$438$$ 1698.52i 0.185293i
$$439$$ −4384.20 −0.476643 −0.238322 0.971186i $$-0.576597\pi$$
−0.238322 + 0.971186i $$0.576597\pi$$
$$440$$ 0 0
$$441$$ −2142.14 −0.231307
$$442$$ 6300.69i 0.678039i
$$443$$ 10090.0i 1.08214i 0.840977 + 0.541071i $$0.181981\pi$$
−0.840977 + 0.541071i $$0.818019\pi$$
$$444$$ 749.040 0.0800627
$$445$$ 0 0
$$446$$ −790.560 −0.0839330
$$447$$ − 7129.07i − 0.754348i
$$448$$ 2059.71i 0.217215i
$$449$$ −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
$$452$$ − 945.464i − 0.0983869i
$$453$$ 278.540i 0.0288895i
$$454$$ 6693.27 0.691918
$$455$$ 0 0
$$456$$ 2518.75 0.258665
$$457$$ 9999.34i 1.02352i 0.859128 + 0.511761i $$0.171007\pi$$
−0.859128 + 0.511761i $$0.828993\pi$$
$$458$$ − 8915.49i − 0.909593i
$$459$$ 2665.12 0.271018
$$460$$ 0 0
$$461$$ −11115.8 −1.12302 −0.561512 0.827468i $$-0.689780\pi$$
−0.561512 + 0.827468i $$0.689780\pi$$
$$462$$ 528.000i 0.0531705i
$$463$$ 1567.16i 0.157305i 0.996902 + 0.0786524i $$0.0250617\pi$$
−0.996902 + 0.0786524i $$0.974938\pi$$
$$464$$ 1703.13 0.170401
$$465$$ 0 0
$$466$$ 4600.48 0.457325
$$467$$ 12648.8i 1.25335i 0.779281 + 0.626675i $$0.215585\pi$$
−0.779281 + 0.626675i $$0.784415\pi$$
$$468$$ − 2046.05i − 0.202091i
$$469$$ −4149.56 −0.408548
$$470$$ 0 0
$$471$$ −5643.72 −0.552120
$$472$$ 7851.88i 0.765704i
$$473$$ − 26.0420i − 0.00253152i
$$474$$ −4459.02 −0.432088
$$475$$ 0 0
$$476$$ −5624.88 −0.541630
$$477$$ 5760.58i 0.552953i
$$478$$ 3246.30i 0.310633i
$$479$$ 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
$$482$$ 2894.14i 0.273494i
$$483$$ 1896.36i 0.178649i
$$484$$ 672.948 0.0631995
$$485$$ 0 0
$$486$$ 379.457 0.0354167
$$487$$ 7161.20i 0.666335i 0.942868 + 0.333167i $$0.108117\pi$$
−0.942868 + 0.333167i $$0.891883\pi$$
$$488$$ − 15129.0i − 1.40340i
$$489$$ 7396.48 0.684009
$$490$$ 0 0
$$491$$ 14567.3 1.33893 0.669463 0.742845i $$-0.266524\pi$$
0.669463 + 0.742845i $$0.266524\pi$$
$$492$$ − 5614.04i − 0.514432i
$$493$$ − 14716.7i − 1.34444i
$$494$$ −2530.65 −0.230485
$$495$$ 0 0
$$496$$ 625.295 0.0566060
$$497$$ − 9624.18i − 0.868619i
$$498$$ − 3444.26i − 0.309922i
$$499$$ 4638.99 0.416172 0.208086 0.978111i $$-0.433277\pi$$
0.208086 + 0.978111i $$0.433277\pi$$
$$500$$ 0 0
$$501$$ −3762.89 −0.335556
$$502$$ − 3682.75i − 0.327428i
$$503$$ 12206.3i 1.08201i 0.841019 + 0.541006i $$0.181956\pi$$
−0.841019 + 0.541006i $$0.818044\pi$$
$$504$$ −1952.86 −0.172594
$$505$$ 0 0
$$506$$ −1059.71 −0.0931024
$$507$$ − 1578.24i − 0.138249i
$$508$$ − 9676.09i − 0.845093i
$$509$$ −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
$$512$$ 4074.36i 0.351686i
$$513$$ 1070.44i 0.0921267i
$$514$$ −8618.61 −0.739592
$$515$$ 0 0
$$516$$ 39.5001 0.00336995
$$517$$ 3666.25i 0.311879i
$$518$$ 718.303i 0.0609274i
$$519$$ 3620.12 0.306177
$$520$$ 0 0
$$521$$ 1054.72 0.0886916 0.0443458 0.999016i $$-0.485880\pi$$
0.0443458 + 0.999016i $$0.485880\pi$$
$$522$$ − 2095.35i − 0.175691i
$$523$$ 16234.2i 1.35730i 0.734460 + 0.678652i $$0.237436\pi$$
−0.734460 + 0.678652i $$0.762564\pi$$
$$524$$ 1739.86 0.145050
$$525$$ 0 0
$$526$$ 3528.57 0.292496
$$527$$ − 5403.16i − 0.446613i
$$528$$ 376.969i 0.0310709i
$$529$$ 8360.95 0.687183
$$530$$ 0 0
$$531$$ −3336.95 −0.272715
$$532$$ − 2259.21i − 0.184115i
$$533$$ 13754.2i 1.11775i
$$534$$ −1807.71 −0.146493
$$535$$ 0 0
$$536$$ 8576.40 0.691127
$$537$$ − 4326.86i − 0.347706i
$$538$$ − 10992.6i − 0.880900i
$$539$$ −2618.17 −0.209225
$$540$$ 0 0
$$541$$ 675.936 0.0537167 0.0268584 0.999639i $$-0.491450\pi$$
0.0268584 + 0.999639i $$0.491450\pi$$
$$542$$ 8049.93i 0.637960i
$$543$$ − 12785.4i − 1.01045i
$$544$$ 18483.6 1.45676
$$545$$ 0 0
$$546$$ 1962.09 0.153791
$$547$$ 13058.2i 1.02071i 0.859964 + 0.510355i $$0.170486\pi$$
−0.859964 + 0.510355i $$0.829514\pi$$
$$548$$ − 3987.23i − 0.310814i
$$549$$ 6429.65 0.499837
$$550$$ 0 0
$$551$$ 5910.91 0.457011
$$552$$ − 3919.44i − 0.302215i
$$553$$ 9752.70i 0.749959i
$$554$$ −14170.3 −1.08672
$$555$$ 0 0
$$556$$ 4874.61 0.371815
$$557$$ − 6710.48i − 0.510471i −0.966879 0.255236i $$-0.917847\pi$$
0.966879 0.255236i $$-0.0821530\pi$$
$$558$$ − 769.295i − 0.0583636i
$$559$$ −96.7741 −0.00732219
$$560$$ 0 0
$$561$$ 3257.37 0.245145
$$562$$ − 5321.45i − 0.399416i
$$563$$ 20820.5i 1.55858i 0.626666 + 0.779288i $$0.284419\pi$$
−0.626666 + 0.779288i $$0.715581\pi$$
$$564$$ −5560.92 −0.415172
$$565$$ 0 0
$$566$$ −13785.0 −1.02372
$$567$$ − 829.943i − 0.0614715i
$$568$$ 19891.5i 1.46941i
$$569$$ −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 $$0.554358\pi$$
$$572$$ − 2500.73i − 0.182798i
$$573$$ 2556.67i 0.186399i
$$574$$ 5383.67 0.391481
$$575$$ 0 0
$$576$$ 1809.20 0.130874
$$577$$ 14462.4i 1.04346i 0.853111 + 0.521730i $$0.174713\pi$$
−0.853111 + 0.521730i $$0.825287\pi$$
$$578$$ − 7542.82i − 0.542803i
$$579$$ 7379.86 0.529700
$$580$$ 0 0
$$581$$ −7533.23 −0.537920
$$582$$ − 4526.99i − 0.322423i
$$583$$ 7040.71i 0.500165i
$$584$$ −7678.18 −0.544050
$$585$$ 0 0
$$586$$ −7071.17 −0.498476
$$587$$ − 22759.7i − 1.60033i −0.599779 0.800166i $$-0.704745\pi$$
0.599779 0.800166i $$-0.295255\pi$$
$$588$$ − 3971.20i − 0.278520i
$$589$$ 2170.16 0.151816
$$590$$ 0 0
$$591$$ 10431.2 0.726025
$$592$$ 512.836i 0.0356038i
$$593$$ 14956.4i 1.03573i 0.855463 + 0.517864i $$0.173273\pi$$
−0.855463 + 0.517864i $$0.826727\pi$$
$$594$$ 463.781 0.0320356
$$595$$ 0 0
$$596$$ 13216.2 0.908319
$$597$$ 11985.1i 0.821638i
$$598$$ 3937.96i 0.269290i
$$599$$ −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 $$0.108896\pi$$
$$602$$ 37.8792i 0.00256452i
$$603$$ 3644.86i 0.246153i
$$604$$ −516.371 −0.0347862
$$605$$ 0 0
$$606$$ 1633.07 0.109470
$$607$$ − 10991.5i − 0.734974i −0.930029 0.367487i $$-0.880218\pi$$
0.930029 0.367487i $$-0.119782\pi$$
$$608$$ 7423.87i 0.495194i
$$609$$ −4582.91 −0.304941
$$610$$ 0 0
$$611$$ 13624.1 0.902081
$$612$$ 4940.74i 0.326336i
$$613$$ − 10646.1i − 0.701457i −0.936477 0.350728i $$-0.885934\pi$$
0.936477 0.350728i $$-0.114066\pi$$
$$614$$ −887.128 −0.0583088
$$615$$ 0 0
$$616$$ −2386.83 −0.156117
$$617$$ 7199.92i 0.469786i 0.972021 + 0.234893i $$0.0754739\pi$$
−0.972021 + 0.234893i $$0.924526\pi$$
$$618$$ 7197.41i 0.468483i
$$619$$ −12186.9 −0.791332 −0.395666 0.918395i $$-0.629486\pi$$
−0.395666 + 0.918395i $$0.629486\pi$$
$$620$$ 0 0
$$621$$ 1665.72 0.107637
$$622$$ − 10702.2i − 0.689905i
$$623$$ 3953.80i 0.254262i
$$624$$ 1400.85 0.0898698
$$625$$ 0 0
$$626$$ −1778.49 −0.113551
$$627$$ 1308.31i 0.0833317i
$$628$$ − 10462.6i − 0.664814i
$$629$$ 4431.40 0.280909
$$630$$ 0 0
$$631$$ 7370.64 0.465009 0.232505 0.972595i $$-0.425308\pi$$
0.232505 + 0.972595i $$0.425308\pi$$
$$632$$ − 20157.1i − 1.26868i
$$633$$ − 3138.38i − 0.197061i
$$634$$ −5008.65 −0.313752
$$635$$ 0 0
$$636$$ −10679.3 −0.665818
$$637$$ 9729.32i 0.605164i
$$638$$ − 2560.98i − 0.158919i
$$639$$ −8453.63 −0.523349
$$640$$ 0 0
$$641$$ −25014.9 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$642$$ − 3650.19i − 0.224395i
$$643$$ 21668.2i 1.32894i 0.747313 + 0.664472i $$0.231343\pi$$
−0.747313 + 0.664472i $$0.768657\pi$$
$$644$$ −3515.58 −0.215113
$$645$$ 0 0
$$646$$ 6110.94 0.372185
$$647$$ − 27625.3i − 1.67861i −0.543661 0.839305i $$-0.682962\pi$$
0.543661 0.839305i $$-0.317038\pi$$
$$648$$ 1715.34i 0.103989i
$$649$$ −4078.50 −0.246680
$$650$$ 0 0
$$651$$ −1682.59 −0.101299
$$652$$ 13712.0i 0.823623i
$$653$$ 14314.0i 0.857810i 0.903350 + 0.428905i $$0.141101\pi$$
−0.903350 + 0.428905i $$0.858899\pi$$
$$654$$ 7035.38 0.420650
$$655$$ 0 0
$$656$$ 3843.70 0.228767
$$657$$ − 3263.13i − 0.193770i
$$658$$ − 5332.73i − 0.315944i
$$659$$ 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 $$0.896013\pi$$
$$662$$ − 14265.0i − 0.837498i
$$663$$ − 12104.7i − 0.709059i
$$664$$ 15569.8 0.909981
$$665$$ 0 0
$$666$$ 630.938 0.0367092
$$667$$ − 9198.01i − 0.533955i
$$668$$ − 6975.84i − 0.404047i
$$669$$ 1518.80 0.0877729
$$670$$ 0 0
$$671$$ 7858.46 0.452120
$$672$$ − 5755.95i − 0.330418i
$$673$$ 6207.38i 0.355538i 0.984072 + 0.177769i $$0.0568879\pi$$
−0.984072 + 0.177769i $$0.943112\pi$$
$$674$$ 5418.66 0.309672
$$675$$ 0 0
$$676$$ 2925.82 0.166467
$$677$$ 28831.1i 1.63674i 0.574695 + 0.818368i $$0.305121\pi$$
−0.574695 + 0.818368i $$0.694879\pi$$
$$678$$ − 796.392i − 0.0451110i
$$679$$ −9901.37 −0.559617
$$680$$ 0 0
$$681$$ −12858.9 −0.723573
$$682$$ − 940.250i − 0.0527918i
$$683$$ − 3193.10i − 0.178888i −0.995992 0.0894441i $$-0.971491\pi$$
0.995992 0.0894441i $$-0.0285090\pi$$
$$684$$ −1984.43 −0.110931
$$685$$ 0 0
$$686$$ 9296.24 0.517394
$$687$$ 17128.1i 0.951206i
$$688$$ 27.0441i 0.00149861i
$$689$$ 26163.8 1.44668
$$690$$ 0 0
$$691$$ 7682.49 0.422946 0.211473 0.977384i $$-0.432174\pi$$
0.211473 + 0.977384i $$0.432174\pi$$
$$692$$ 6711.17i 0.368671i
$$693$$ − 1014.37i − 0.0556031i
$$694$$ 139.496 0.00762996
$$695$$ 0 0
$$696$$ 9472.05 0.515858
$$697$$ − 33213.3i − 1.80494i
$$698$$ 233.256i 0.0126488i
$$699$$ −8838.28 −0.478247
$$700$$ 0 0
$$701$$ 26551.6 1.43058 0.715292 0.698825i $$-0.246293\pi$$
0.715292 + 0.698825i $$0.246293\pi$$
$$702$$ − 1723.45i − 0.0926601i
$$703$$ 1779.86i 0.0954887i
$$704$$ 2211.24 0.118380
$$705$$ 0 0
$$706$$ 12285.7 0.654928
$$707$$ − 3571.83i − 0.190004i
$$708$$ − 6186.22i − 0.328379i
$$709$$ 16304.6 0.863655 0.431828 0.901956i $$-0.357869\pi$$
0.431828 + 0.901956i $$0.357869\pi$$
$$710$$ 0 0
$$711$$ 8566.52 0.451856
$$712$$ − 8171.79i − 0.430127i
$$713$$ − 3377.00i − 0.177377i
$$714$$ −4738.00 −0.248341
$$715$$ 0 0
$$716$$ 8021.36 0.418676
$$717$$ − 6236.68i − 0.324844i
$$718$$ − 7767.50i − 0.403733i
$$719$$ 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
$$722$$ − 8256.25i − 0.425576i
$$723$$ − 5560.11i − 0.286007i
$$724$$ 23702.3 1.21670
$$725$$ 0 0
$$726$$ 566.844 0.0289773
$$727$$ − 10780.4i − 0.549961i −0.961450 0.274980i $$-0.911329\pi$$
0.961450 0.274980i $$-0.0886714\pi$$
$$728$$ 8869.67i 0.451555i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 233.687 0.0118238
$$732$$ 11919.6i 0.601860i
$$733$$ 9211.46i 0.464165i 0.972696 + 0.232083i $$0.0745540\pi$$
−0.972696 + 0.232083i $$0.925446\pi$$
$$734$$ 20716.6 1.04177
$$735$$ 0 0
$$736$$ 11552.3 0.578566
$$737$$ 4454.83i 0.222654i
$$738$$ − 4728.87i − 0.235870i
$$739$$ −11084.7 −0.551768 −0.275884 0.961191i $$-0.588971\pi$$
−0.275884 + 0.961191i $$0.588971\pi$$
$$740$$ 0 0
$$741$$ 4861.80 0.241029
$$742$$ − 10241.0i − 0.506685i
$$743$$ 27420.4i 1.35391i 0.736024 + 0.676955i $$0.236701\pi$$
−0.736024 + 0.676955i $$0.763299\pi$$
$$744$$ 3477.61 0.171365
$$745$$ 0 0
$$746$$ −7234.32 −0.355050
$$747$$ 6616.99i 0.324101i
$$748$$ 6038.69i 0.295182i
$$749$$ −7983.64 −0.389474
$$750$$ 0 0
$$751$$ −11290.8 −0.548614 −0.274307 0.961642i $$-0.588448\pi$$
−0.274307 + 0.961642i $$0.588448\pi$$
$$752$$ − 3807.33i − 0.184626i
$$753$$ 7075.16i 0.342408i
$$754$$ −9516.81 −0.459657
$$755$$ 0 0
$$756$$ 1538.59 0.0740185
$$757$$ 3739.19i 0.179528i 0.995963 + 0.0897642i $$0.0286113\pi$$
−0.995963 + 0.0897642i $$0.971389\pi$$
$$758$$ − 10155.3i − 0.486617i
$$759$$ 2035.87 0.0973617
$$760$$ 0 0
$$761$$ −15621.1 −0.744107 −0.372053 0.928211i $$-0.621346\pi$$
−0.372053 + 0.928211i $$0.621346\pi$$
$$762$$ − 8150.45i − 0.387480i
$$763$$ − 15387.7i − 0.730106i
$$764$$ −4739.69 −0.224445
$$765$$ 0 0
$$766$$ 19885.7 0.937987
$$767$$ 15156.0i 0.713498i
$$768$$ 10371.8i 0.487317i
$$769$$ −40241.7 −1.88706 −0.943531 0.331284i $$-0.892518\pi$$
−0.943531 + 0.331284i $$0.892518\pi$$
$$770$$ 0 0
$$771$$ 16557.8 0.773428
$$772$$ 13681.2i 0.637818i
$$773$$ − 22821.4i − 1.06187i −0.847412 0.530936i $$-0.821840\pi$$
0.847412 0.530936i $$-0.178160\pi$$
$$774$$ 33.2721 0.00154514
$$775$$ 0 0
$$776$$ 20464.4 0.946685
$$777$$ − 1379.98i − 0.0637148i
$$778$$ − 18777.0i − 0.865282i
$$779$$ 13340.0 0.613549
$$780$$ 0 0
$$781$$ −10332.2 −0.473387
$$782$$ − 9509.28i − 0.434848i
$$783$$ 4025.51i 0.183729i
$$784$$ 2718.92 0.123857
$$785$$ 0 0
$$786$$ 1465.53 0.0665062
$$787$$ − 29454.3i − 1.33410i −0.745015 0.667048i $$-0.767558\pi$$
0.745015 0.667048i $$-0.232442\pi$$
$$788$$ 19337.8i 0.874216i
$$789$$ −6778.96 −0.305878
$$790$$ 0 0
$$791$$ −1741.86 −0.0782974
$$792$$ 2096.53i 0.0940619i
$$793$$ − 29202.7i − 1.30771i
$$794$$ 8157.00 0.364586
$$795$$ 0 0
$$796$$ −22218.6 −0.989344
$$797$$ − 27440.3i − 1.21955i −0.792573 0.609777i $$-0.791259\pi$$
0.792573 0.609777i $$-0.208741\pi$$
$$798$$ − 1903.00i − 0.0844179i
$$799$$ −32899.0 −1.45668
$$800$$ 0 0
$$801$$ 3472.91 0.153195
$$802$$ 15066.1i 0.663346i
$$803$$ − 3988.27i − 0.175272i
$$804$$ −6757.03 −0.296396
$$805$$ 0 0
$$806$$ −3494.05 −0.152695
$$807$$ 21118.6i 0.921201i
$$808$$ 7382.34i 0.321423i
$$809$$ 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 $$0.270613\pi$$
$$812$$ − 8496.03i − 0.367183i
$$813$$ − 15465.2i − 0.667146i
$$814$$ 771.146 0.0332048
$$815$$ 0 0
$$816$$ −3382.72 −0.145121
$$817$$ 93.8596i 0.00401925i
$$818$$ 3139.46i 0.134192i
$$819$$ −3769.50 −0.160827
$$820$$ 0 0
$$821$$ 37909.0 1.61149 0.805745 0.592263i $$-0.201765\pi$$
0.805745 + 0.592263i $$0.201765\pi$$
$$822$$ − 3358.56i − 0.142510i
$$823$$ − 23636.0i − 1.00109i −0.865710 0.500546i $$-0.833133\pi$$
0.865710 0.500546i $$-0.166867\pi$$
$$824$$ −32536.0 −1.37554
$$825$$ 0 0
$$826$$ 5932.36 0.249895
$$827$$ − 42634.3i − 1.79267i −0.443376 0.896336i $$-0.646219\pi$$
0.443376 0.896336i $$-0.353781\pi$$
$$828$$ 3087.99i 0.129608i
$$829$$ 45152.5 1.89169 0.945845 0.324619i $$-0.105236\pi$$
0.945845 + 0.324619i $$0.105236\pi$$
$$830$$ 0 0
$$831$$ 27223.6 1.13643
$$832$$ − 8217.15i − 0.342402i
$$833$$ − 23494.1i − 0.977217i
$$834$$ 4106.02 0.170480
$$835$$ 0 0
$$836$$ −2425.42 −0.100341
$$837$$ 1477.94i 0.0610337i
$$838$$ − 6925.91i − 0.285503i
$$839$$ −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
$$842$$ 23763.6i 0.972623i
$$843$$ 10223.4i 0.417689i
$$844$$ 5818.09 0.237283
$$845$$ 0 0
$$846$$ −4684.13 −0.190359
$$847$$ − 1239.79i − 0.0502949i
$$848$$ − 7311.64i − 0.296088i
$$849$$ 26483.2 1.07055
$$850$$ 0 0
$$851$$ 2769.65 0.111566
$$852$$ − 15671.8i − 0.630171i
$$853$$ − 10367.2i − 0.416139i −0.978114 0.208070i $$-0.933282\pi$$
0.978114 0.208070i $$-0.0667180\pi$$
$$854$$ −11430.5 −0.458013
$$855$$ 0 0
$$856$$ 16500.8 0.658860
$$857$$ 12947.1i 0.516063i 0.966136 + 0.258032i $$0.0830739\pi$$
−0.966136 + 0.258032i $$0.916926\pi$$
$$858$$ − 2106.44i − 0.0838142i
$$859$$ 20383.5 0.809636 0.404818 0.914397i $$-0.367335\pi$$
0.404818 + 0.914397i $$0.367335\pi$$
$$860$$ 0 0
$$861$$ −10342.9 −0.409391
$$862$$ − 8779.22i − 0.346893i
$$863$$ − 9056.42i − 0.357224i −0.983920 0.178612i $$-0.942839\pi$$
0.983920 0.178612i $$-0.0571606\pi$$
$$864$$ −5055.88 −0.199079
$$865$$ 0 0
$$866$$ −22340.0 −0.876610
$$867$$ 14491.0i 0.567636i
$$868$$ − 3119.27i − 0.121976i
$$869$$ 10470.2 0.408719
$$870$$ 0 0
$$871$$ 16554.5 0.644005
$$872$$ 31803.6i 1.23510i
$$873$$ 8697.10i 0.337173i
$$874$$ 3819.37 0.147817
$$875$$ 0 0
$$876$$ 6049.36 0.233321
$$877$$ − 2867.88i − 0.110424i −0.998475 0.0552118i $$-0.982417\pi$$
0.998475 0.0552118i $$-0.0175834\pi$$
$$878$$ − 6846.16i − 0.263151i
$$879$$ 13584.9 0.521281
$$880$$ 0 0
$$881$$ −11862.5 −0.453640 −0.226820 0.973937i $$-0.572833\pi$$
−0.226820 + 0.973937i $$0.572833\pi$$
$$882$$ − 3345.06i − 0.127703i
$$883$$ − 33463.8i − 1.27537i −0.770299 0.637683i $$-0.779893\pi$$
0.770299 0.637683i $$-0.220107\pi$$
$$884$$ 22440.3 0.853787
$$885$$ 0 0
$$886$$ −15756.0 −0.597442
$$887$$ 2420.75i 0.0916357i 0.998950 + 0.0458178i $$0.0145894\pi$$
−0.998950 + 0.0458178i $$0.985411\pi$$
$$888$$ 2852.17i 0.107784i
$$889$$ −17826.5 −0.672534
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 2815.62i 0.105688i
$$893$$ − 13213.8i − 0.495165i
$$894$$ 11132.4 0.416470
$$895$$ 0 0
$$896$$ 12132.9 0.452378
$$897$$ − 7565.48i − 0.281610i
$$898$$ − 14963.6i − 0.556060i
$$899$$ 8161.14 0.302769
$$900$$ 0 0
$$901$$ −63179.7 −2.33609
$$902$$ − 5779.73i − 0.213352i
$$903$$ − 72.7722i − 0.00268185i
$$904$$ 3600.10 0.132453
$$905$$ 0 0
$$906$$ −434.955 −0.0159497
$$907$$ − 38154.1i − 1.39679i −0.715714 0.698393i $$-0.753899\pi$$
0.715714 0.698393i $$-0.246101\pi$$
$$908$$ − 23838.4i − 0.871262i
$$909$$ −3137.40 −0.114479
$$910$$ 0 0
$$911$$ −35758.0 −1.30045 −0.650227 0.759740i $$-0.725326\pi$$
−0.650227 + 0.759740i $$0.725326\pi$$
$$912$$ − 1358.66i − 0.0493308i
$$913$$ 8087.44i 0.293160i
$$914$$ −15614.5 −0.565079
$$915$$ 0 0
$$916$$ −31753.0 −1.14536
$$917$$ − 3205.39i − 0.115432i
$$918$$ 4161.73i 0.149627i
$$919$$ −17387.5 −0.624115 −0.312058 0.950063i $$-0.601018\pi$$
−0.312058 + 0.950063i $$0.601018\pi$$
$$920$$ 0 0
$$921$$ 1704.32 0.0609764
$$922$$ − 17357.9i − 0.620013i
$$923$$ 38395.3i 1.36923i
$$924$$ 1880.50 0.0669523
$$925$$ 0 0
$$926$$ −2447.20 −0.0868467
$$927$$ − 13827.4i − 0.489915i
$$928$$ 27918.3i 0.987569i
$$929$$ −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
$$932$$ − 16384.9i − 0.575863i
$$933$$ 20560.8i 0.721467i
$$934$$ −19751.7 −0.691965
$$935$$ 0 0
$$936$$ 7790.88 0.272065
$$937$$ − 16074.5i − 0.560438i −0.959936 0.280219i $$-0.909593\pi$$
0.959936 0.280219i $$-0.0904070\pi$$
$$938$$ − 6479.76i − 0.225556i
$$939$$ 3416.77 0.118746
$$940$$ 0 0
$$941$$ −687.126 −0.0238041 −0.0119021 0.999929i $$-0.503789\pi$$
−0.0119021 + 0.999929i $$0.503789\pi$$
$$942$$ − 8812.96i − 0.304821i
$$943$$ − 20758.5i − 0.716849i
$$944$$ 4235.44 0.146030
$$945$$ 0 0
$$946$$ 40.6659 0.00139763
$$947$$ 35352.0i 1.21308i 0.795053 + 0.606540i $$0.207443\pi$$
−0.795053 + 0.606540i $$0.792557\pi$$
$$948$$ 15881.0i 0.544085i
$$949$$ −14820.7 −0.506956
$$950$$ 0 0
$$951$$ 9622.44 0.328106
$$952$$ − 21418.2i − 0.729168i
$$953$$ 19390.7i 0.659103i 0.944138 + 0.329552i $$0.106898\pi$$
−0.944138 + 0.329552i $$0.893102\pi$$
$$954$$ −8995.45 −0.305281
$$955$$ 0 0
$$956$$ 11561.9 0.391148
$$957$$ 4920.06i 0.166189i
$$958$$ 16738.7i 0.564511i
$$959$$ −7345.80 −0.247349
$$960$$ 0 0
$$961$$ −26794.7 −0.899422
$$962$$ − 2865.64i − 0.0960416i
$$963$$ 7012.62i 0.234661i
$$964$$ 10307.6 0.344384
$$965$$ 0 0
$$966$$ −2961.27 −0.0986308
$$967$$ 28643.6i 0.952551i 0.879296 + 0.476275i $$0.158013\pi$$
−0.879296 + 0.476275i $$0.841987\pi$$
$$968$$ 2562.43i 0.0850821i
$$969$$ −11740.1 −0.389213
$$970$$ 0 0
$$971$$ −19574.8 −0.646946 −0.323473 0.946237i $$-0.604851\pi$$
−0.323473 + 0.946237i $$0.604851\pi$$
$$972$$ − 1351.46i − 0.0445967i
$$973$$ − 8980.63i − 0.295895i
$$974$$ −11182.6 −0.367878
$$975$$ 0 0
$$976$$ −8160.86 −0.267646
$$977$$ 50095.5i 1.64043i 0.572058 + 0.820213i $$0.306145\pi$$
−0.572058 + 0.820213i $$0.693855\pi$$
$$978$$ 11550.0i 0.377636i
$$979$$ 4244.67 0.138570
$$980$$ 0 0
$$981$$ −13516.1 −0.439895
$$982$$ 22747.6i 0.739211i
$$983$$ − 14445.4i − 0.468706i −0.972152 0.234353i $$-0.924703\pi$$
0.972152 0.234353i $$-0.0752971\pi$$
$$984$$ 21376.9 0.692553
$$985$$ 0 0
$$986$$ 22980.9 0.742253
$$987$$ 10245.0i 0.330399i
$$988$$ 9013.05i 0.290226i
$$989$$ 146.056 0.00469595
$$990$$ 0 0
$$991$$ 29120.1 0.933430 0.466715 0.884408i $$-0.345437\pi$$
0.466715 + 0.884408i $$0.345437\pi$$
$$992$$ 10250.1i 0.328064i
$$993$$ 27405.4i 0.875814i
$$994$$ 15028.7 0.479558
$$995$$ 0 0
$$996$$ −12266.9 −0.390253
$$997$$ − 9137.45i − 0.290257i −0.989413 0.145128i $$-0.953640\pi$$
0.989413 0.145128i $$-0.0463595\pi$$
$$998$$ 7244.03i 0.229765i
$$999$$ −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.c.j.199.3 4
5.2 odd 4 825.4.a.m.1.1 2
5.3 odd 4 165.4.a.c.1.2 2
5.4 even 2 inner 825.4.c.j.199.2 4
15.2 even 4 2475.4.a.n.1.2 2
15.8 even 4 495.4.a.d.1.1 2
55.43 even 4 1815.4.a.n.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.2 2 5.3 odd 4
495.4.a.d.1.1 2 15.8 even 4
825.4.a.m.1.1 2 5.2 odd 4
825.4.c.j.199.2 4 5.4 even 2 inner
825.4.c.j.199.3 4 1.1 even 1 trivial
1815.4.a.n.1.1 2 55.43 even 4
2475.4.a.n.1.2 2 15.2 even 4