# Properties

 Label 1089.3.b.j.485.3 Level $1089$ Weight $3$ Character 1089.485 Analytic conductor $29.673$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1089.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.6731007888$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 48 x^{14} + 921 x^{12} + 8986 x^{10} + 46812 x^{8} + 125072 x^{6} + 152129 x^{4} + 65614 x^{2} + 5041$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 99) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 485.3 Root $$-2.99491i$$ of defining polynomial Character $$\chi$$ $$=$$ 1089.485 Dual form 1089.3.b.j.485.14

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.99491i q^{2} -4.96950 q^{4} -1.61173i q^{5} +3.32981 q^{7} +2.90358i q^{8} +O(q^{10})$$ $$q-2.99491i q^{2} -4.96950 q^{4} -1.61173i q^{5} +3.32981 q^{7} +2.90358i q^{8} -4.82698 q^{10} -7.56350 q^{13} -9.97250i q^{14} -11.1821 q^{16} -28.3526i q^{17} +26.0667 q^{19} +8.00948i q^{20} -19.5656i q^{23} +22.4023 q^{25} +22.6520i q^{26} -16.5475 q^{28} +2.15288i q^{29} -9.22840 q^{31} +45.1036i q^{32} -84.9135 q^{34} -5.36675i q^{35} -67.5342 q^{37} -78.0674i q^{38} +4.67977 q^{40} +29.0597i q^{41} -0.719802 q^{43} -58.5973 q^{46} -24.5249i q^{47} -37.9123 q^{49} -67.0930i q^{50} +37.5868 q^{52} -5.79176i q^{53} +9.66837i q^{56} +6.44768 q^{58} -73.9896i q^{59} -72.1664 q^{61} +27.6383i q^{62} +90.3531 q^{64} +12.1903i q^{65} -79.8310 q^{67} +140.898i q^{68} -16.0730 q^{70} -107.780i q^{71} +90.3799 q^{73} +202.259i q^{74} -129.538 q^{76} -114.802 q^{79} +18.0224i q^{80} +87.0314 q^{82} +125.752i q^{83} -45.6966 q^{85} +2.15574i q^{86} +83.3041i q^{89} -25.1850 q^{91} +97.2313i q^{92} -73.4499 q^{94} -42.0123i q^{95} -78.0275 q^{97} +113.544i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 32 q^{4} + 8 q^{7} + O(q^{10})$$ $$16 q - 32 q^{4} + 8 q^{7} - 24 q^{10} - 4 q^{13} + 28 q^{16} + 20 q^{19} - 44 q^{25} - 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} - 224 q^{40} - 272 q^{43} - 208 q^{46} + 348 q^{49} - 520 q^{52} - 44 q^{58} - 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} + 4 q^{73} - 1052 q^{76} - 216 q^{79} + 348 q^{82} - 416 q^{85} - 168 q^{91} - 1140 q^{94} - 44 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.99491i − 1.49746i −0.662877 0.748728i $$-0.730665\pi$$
0.662877 0.748728i $$-0.269335\pi$$
$$3$$ 0 0
$$4$$ −4.96950 −1.24238
$$5$$ − 1.61173i − 0.322345i −0.986926 0.161173i $$-0.948472\pi$$
0.986926 0.161173i $$-0.0515276\pi$$
$$6$$ 0 0
$$7$$ 3.32981 0.475688 0.237844 0.971303i $$-0.423559\pi$$
0.237844 + 0.971303i $$0.423559\pi$$
$$8$$ 2.90358i 0.362947i
$$9$$ 0 0
$$10$$ −4.82698 −0.482698
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −7.56350 −0.581807 −0.290904 0.956752i $$-0.593956\pi$$
−0.290904 + 0.956752i $$0.593956\pi$$
$$14$$ − 9.97250i − 0.712322i
$$15$$ 0 0
$$16$$ −11.1821 −0.698878
$$17$$ − 28.3526i − 1.66780i −0.551917 0.833899i $$-0.686104\pi$$
0.551917 0.833899i $$-0.313896\pi$$
$$18$$ 0 0
$$19$$ 26.0667 1.37193 0.685965 0.727635i $$-0.259380\pi$$
0.685965 + 0.727635i $$0.259380\pi$$
$$20$$ 8.00948i 0.400474i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 19.5656i − 0.850679i −0.905034 0.425339i $$-0.860155\pi$$
0.905034 0.425339i $$-0.139845\pi$$
$$24$$ 0 0
$$25$$ 22.4023 0.896093
$$26$$ 22.6520i 0.871231i
$$27$$ 0 0
$$28$$ −16.5475 −0.590983
$$29$$ 2.15288i 0.0742372i 0.999311 + 0.0371186i $$0.0118179\pi$$
−0.999311 + 0.0371186i $$0.988182\pi$$
$$30$$ 0 0
$$31$$ −9.22840 −0.297690 −0.148845 0.988861i $$-0.547556\pi$$
−0.148845 + 0.988861i $$0.547556\pi$$
$$32$$ 45.1036i 1.40949i
$$33$$ 0 0
$$34$$ −84.9135 −2.49745
$$35$$ − 5.36675i − 0.153336i
$$36$$ 0 0
$$37$$ −67.5342 −1.82525 −0.912624 0.408800i $$-0.865948\pi$$
−0.912624 + 0.408800i $$0.865948\pi$$
$$38$$ − 78.0674i − 2.05440i
$$39$$ 0 0
$$40$$ 4.67977 0.116994
$$41$$ 29.0597i 0.708774i 0.935099 + 0.354387i $$0.115310\pi$$
−0.935099 + 0.354387i $$0.884690\pi$$
$$42$$ 0 0
$$43$$ −0.719802 −0.0167396 −0.00836979 0.999965i $$-0.502664\pi$$
−0.00836979 + 0.999965i $$0.502664\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −58.5973 −1.27385
$$47$$ − 24.5249i − 0.521806i −0.965365 0.260903i $$-0.915980\pi$$
0.965365 0.260903i $$-0.0840203\pi$$
$$48$$ 0 0
$$49$$ −37.9123 −0.773721
$$50$$ − 67.0930i − 1.34186i
$$51$$ 0 0
$$52$$ 37.5868 0.722823
$$53$$ − 5.79176i − 0.109278i −0.998506 0.0546392i $$-0.982599\pi$$
0.998506 0.0546392i $$-0.0174009\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 9.66837i 0.172650i
$$57$$ 0 0
$$58$$ 6.44768 0.111167
$$59$$ − 73.9896i − 1.25406i −0.778994 0.627031i $$-0.784270\pi$$
0.778994 0.627031i $$-0.215730\pi$$
$$60$$ 0 0
$$61$$ −72.1664 −1.18306 −0.591528 0.806284i $$-0.701475\pi$$
−0.591528 + 0.806284i $$0.701475\pi$$
$$62$$ 27.6383i 0.445778i
$$63$$ 0 0
$$64$$ 90.3531 1.41177
$$65$$ 12.1903i 0.187543i
$$66$$ 0 0
$$67$$ −79.8310 −1.19151 −0.595754 0.803167i $$-0.703146\pi$$
−0.595754 + 0.803167i $$0.703146\pi$$
$$68$$ 140.898i 2.07203i
$$69$$ 0 0
$$70$$ −16.0730 −0.229614
$$71$$ − 107.780i − 1.51803i −0.651072 0.759016i $$-0.725681\pi$$
0.651072 0.759016i $$-0.274319\pi$$
$$72$$ 0 0
$$73$$ 90.3799 1.23808 0.619041 0.785359i $$-0.287522\pi$$
0.619041 + 0.785359i $$0.287522\pi$$
$$74$$ 202.259i 2.73323i
$$75$$ 0 0
$$76$$ −129.538 −1.70445
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −114.802 −1.45319 −0.726595 0.687066i $$-0.758898\pi$$
−0.726595 + 0.687066i $$0.758898\pi$$
$$80$$ 18.0224i 0.225280i
$$81$$ 0 0
$$82$$ 87.0314 1.06136
$$83$$ 125.752i 1.51509i 0.652784 + 0.757544i $$0.273601\pi$$
−0.652784 + 0.757544i $$0.726399\pi$$
$$84$$ 0 0
$$85$$ −45.6966 −0.537607
$$86$$ 2.15574i 0.0250668i
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 83.3041i 0.936001i 0.883728 + 0.468000i $$0.155025\pi$$
−0.883728 + 0.468000i $$0.844975\pi$$
$$90$$ 0 0
$$91$$ −25.1850 −0.276759
$$92$$ 97.2313i 1.05686i
$$93$$ 0 0
$$94$$ −73.4499 −0.781382
$$95$$ − 42.0123i − 0.442235i
$$96$$ 0 0
$$97$$ −78.0275 −0.804407 −0.402203 0.915550i $$-0.631756\pi$$
−0.402203 + 0.915550i $$0.631756\pi$$
$$98$$ 113.544i 1.15861i
$$99$$ 0 0
$$100$$ −111.328 −1.11328
$$101$$ 38.9858i 0.385998i 0.981199 + 0.192999i $$0.0618215\pi$$
−0.981199 + 0.192999i $$0.938179\pi$$
$$102$$ 0 0
$$103$$ 30.1474 0.292694 0.146347 0.989233i $$-0.453248\pi$$
0.146347 + 0.989233i $$0.453248\pi$$
$$104$$ − 21.9612i − 0.211165i
$$105$$ 0 0
$$106$$ −17.3458 −0.163640
$$107$$ − 49.7544i − 0.464994i −0.972597 0.232497i $$-0.925310\pi$$
0.972597 0.232497i $$-0.0746897\pi$$
$$108$$ 0 0
$$109$$ 149.823 1.37453 0.687263 0.726409i $$-0.258812\pi$$
0.687263 + 0.726409i $$0.258812\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −37.2342 −0.332448
$$113$$ 127.469i 1.12805i 0.825759 + 0.564024i $$0.190747\pi$$
−0.825759 + 0.564024i $$0.809253\pi$$
$$114$$ 0 0
$$115$$ −31.5344 −0.274212
$$116$$ − 10.6987i − 0.0922305i
$$117$$ 0 0
$$118$$ −221.593 −1.87790
$$119$$ − 94.4088i − 0.793351i
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 216.132i 1.77157i
$$123$$ 0 0
$$124$$ 45.8606 0.369843
$$125$$ − 76.3996i − 0.611197i
$$126$$ 0 0
$$127$$ 237.975 1.87382 0.936910 0.349572i $$-0.113673\pi$$
0.936910 + 0.349572i $$0.113673\pi$$
$$128$$ − 90.1853i − 0.704573i
$$129$$ 0 0
$$130$$ 36.5089 0.280837
$$131$$ 148.535i 1.13386i 0.823767 + 0.566929i $$0.191869\pi$$
−0.823767 + 0.566929i $$0.808131\pi$$
$$132$$ 0 0
$$133$$ 86.7972 0.652610
$$134$$ 239.087i 1.78423i
$$135$$ 0 0
$$136$$ 82.3239 0.605322
$$137$$ − 20.1586i − 0.147143i −0.997290 0.0735716i $$-0.976560\pi$$
0.997290 0.0735716i $$-0.0234397\pi$$
$$138$$ 0 0
$$139$$ −148.878 −1.07106 −0.535532 0.844515i $$-0.679889\pi$$
−0.535532 + 0.844515i $$0.679889\pi$$
$$140$$ 26.6701i 0.190501i
$$141$$ 0 0
$$142$$ −322.792 −2.27319
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 3.46985 0.0239300
$$146$$ − 270.680i − 1.85397i
$$147$$ 0 0
$$148$$ 335.611 2.26764
$$149$$ − 44.8984i − 0.301332i −0.988585 0.150666i $$-0.951858\pi$$
0.988585 0.150666i $$-0.0481418\pi$$
$$150$$ 0 0
$$151$$ −46.5427 −0.308230 −0.154115 0.988053i $$-0.549253\pi$$
−0.154115 + 0.988053i $$0.549253\pi$$
$$152$$ 75.6866i 0.497938i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 14.8737i 0.0959591i
$$156$$ 0 0
$$157$$ 18.1734 0.115754 0.0578769 0.998324i $$-0.481567\pi$$
0.0578769 + 0.998324i $$0.481567\pi$$
$$158$$ 343.822i 2.17609i
$$159$$ 0 0
$$160$$ 72.6947 0.454342
$$161$$ − 65.1498i − 0.404657i
$$162$$ 0 0
$$163$$ −139.162 −0.853753 −0.426876 0.904310i $$-0.640386\pi$$
−0.426876 + 0.904310i $$0.640386\pi$$
$$164$$ − 144.412i − 0.880564i
$$165$$ 0 0
$$166$$ 376.617 2.26878
$$167$$ − 145.214i − 0.869544i −0.900541 0.434772i $$-0.856829\pi$$
0.900541 0.434772i $$-0.143171\pi$$
$$168$$ 0 0
$$169$$ −111.794 −0.661500
$$170$$ 136.857i 0.805043i
$$171$$ 0 0
$$172$$ 3.57706 0.0207968
$$173$$ − 290.416i − 1.67871i −0.543586 0.839353i $$-0.682934\pi$$
0.543586 0.839353i $$-0.317066\pi$$
$$174$$ 0 0
$$175$$ 74.5956 0.426261
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 249.488 1.40162
$$179$$ − 146.411i − 0.817940i −0.912548 0.408970i $$-0.865888\pi$$
0.912548 0.408970i $$-0.134112\pi$$
$$180$$ 0 0
$$181$$ 223.461 1.23459 0.617295 0.786732i $$-0.288228\pi$$
0.617295 + 0.786732i $$0.288228\pi$$
$$182$$ 75.4270i 0.414434i
$$183$$ 0 0
$$184$$ 56.8102 0.308751
$$185$$ 108.847i 0.588361i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 121.876i 0.648279i
$$189$$ 0 0
$$190$$ −125.823 −0.662228
$$191$$ 23.2178i 0.121559i 0.998151 + 0.0607795i $$0.0193586\pi$$
−0.998151 + 0.0607795i $$0.980641\pi$$
$$192$$ 0 0
$$193$$ 125.437 0.649934 0.324967 0.945725i $$-0.394647\pi$$
0.324967 + 0.945725i $$0.394647\pi$$
$$194$$ 233.685i 1.20456i
$$195$$ 0 0
$$196$$ 188.405 0.961252
$$197$$ − 12.3847i − 0.0628665i −0.999506 0.0314333i $$-0.989993\pi$$
0.999506 0.0314333i $$-0.0100072\pi$$
$$198$$ 0 0
$$199$$ 112.611 0.565885 0.282943 0.959137i $$-0.408689\pi$$
0.282943 + 0.959137i $$0.408689\pi$$
$$200$$ 65.0469i 0.325235i
$$201$$ 0 0
$$202$$ 116.759 0.578016
$$203$$ 7.16868i 0.0353137i
$$204$$ 0 0
$$205$$ 46.8364 0.228470
$$206$$ − 90.2890i − 0.438296i
$$207$$ 0 0
$$208$$ 84.5754 0.406613
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −153.910 −0.729433 −0.364716 0.931119i $$-0.618834\pi$$
−0.364716 + 0.931119i $$0.618834\pi$$
$$212$$ 28.7822i 0.135765i
$$213$$ 0 0
$$214$$ −149.010 −0.696309
$$215$$ 1.16012i 0.00539593i
$$216$$ 0 0
$$217$$ −30.7289 −0.141608
$$218$$ − 448.708i − 2.05829i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 214.445i 0.970337i
$$222$$ 0 0
$$223$$ 282.384 1.26630 0.633148 0.774031i $$-0.281763\pi$$
0.633148 + 0.774031i $$0.281763\pi$$
$$224$$ 150.187i 0.670476i
$$225$$ 0 0
$$226$$ 381.760 1.68920
$$227$$ − 202.865i − 0.893677i −0.894615 0.446839i $$-0.852550\pi$$
0.894615 0.446839i $$-0.147450\pi$$
$$228$$ 0 0
$$229$$ −419.120 −1.83022 −0.915108 0.403208i $$-0.867895\pi$$
−0.915108 + 0.403208i $$0.867895\pi$$
$$230$$ 94.4428i 0.410621i
$$231$$ 0 0
$$232$$ −6.25105 −0.0269442
$$233$$ − 111.121i − 0.476916i −0.971153 0.238458i $$-0.923358\pi$$
0.971153 0.238458i $$-0.0766420\pi$$
$$234$$ 0 0
$$235$$ −39.5274 −0.168202
$$236$$ 367.692i 1.55802i
$$237$$ 0 0
$$238$$ −282.746 −1.18801
$$239$$ − 314.056i − 1.31404i −0.753872 0.657021i $$-0.771816\pi$$
0.753872 0.657021i $$-0.228184\pi$$
$$240$$ 0 0
$$241$$ 104.293 0.432753 0.216376 0.976310i $$-0.430576\pi$$
0.216376 + 0.976310i $$0.430576\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 358.631 1.46980
$$245$$ 61.1043i 0.249405i
$$246$$ 0 0
$$247$$ −197.155 −0.798199
$$248$$ − 26.7954i − 0.108046i
$$249$$ 0 0
$$250$$ −228.810 −0.915241
$$251$$ 333.537i 1.32883i 0.747363 + 0.664416i $$0.231320\pi$$
−0.747363 + 0.664416i $$0.768680\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ − 712.714i − 2.80596i
$$255$$ 0 0
$$256$$ 91.3152 0.356700
$$257$$ − 306.989i − 1.19451i −0.802052 0.597254i $$-0.796258\pi$$
0.802052 0.597254i $$-0.203742\pi$$
$$258$$ 0 0
$$259$$ −224.876 −0.868248
$$260$$ − 60.5797i − 0.232999i
$$261$$ 0 0
$$262$$ 444.851 1.69790
$$263$$ − 249.060i − 0.946997i −0.880794 0.473499i $$-0.842991\pi$$
0.880794 0.473499i $$-0.157009\pi$$
$$264$$ 0 0
$$265$$ −9.33473 −0.0352254
$$266$$ − 259.950i − 0.977255i
$$267$$ 0 0
$$268$$ 396.720 1.48030
$$269$$ − 196.927i − 0.732071i −0.930601 0.366035i $$-0.880715\pi$$
0.930601 0.366035i $$-0.119285\pi$$
$$270$$ 0 0
$$271$$ 463.464 1.71020 0.855099 0.518465i $$-0.173496\pi$$
0.855099 + 0.518465i $$0.173496\pi$$
$$272$$ 317.040i 1.16559i
$$273$$ 0 0
$$274$$ −60.3733 −0.220340
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 23.4550 0.0846751 0.0423376 0.999103i $$-0.486520\pi$$
0.0423376 + 0.999103i $$0.486520\pi$$
$$278$$ 445.876i 1.60387i
$$279$$ 0 0
$$280$$ 15.5828 0.0556528
$$281$$ 263.447i 0.937534i 0.883322 + 0.468767i $$0.155302\pi$$
−0.883322 + 0.468767i $$0.844698\pi$$
$$282$$ 0 0
$$283$$ 35.8237 0.126585 0.0632927 0.997995i $$-0.479840\pi$$
0.0632927 + 0.997995i $$0.479840\pi$$
$$284$$ 535.614i 1.88597i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 96.7636i 0.337155i
$$288$$ 0 0
$$289$$ −514.868 −1.78155
$$290$$ − 10.3919i − 0.0358342i
$$291$$ 0 0
$$292$$ −449.143 −1.53816
$$293$$ 114.195i 0.389745i 0.980829 + 0.194873i $$0.0624294\pi$$
−0.980829 + 0.194873i $$0.937571\pi$$
$$294$$ 0 0
$$295$$ −119.251 −0.404241
$$296$$ − 196.091i − 0.662469i
$$297$$ 0 0
$$298$$ −134.467 −0.451231
$$299$$ 147.984i 0.494931i
$$300$$ 0 0
$$301$$ −2.39681 −0.00796281
$$302$$ 139.391i 0.461560i
$$303$$ 0 0
$$304$$ −291.479 −0.958812
$$305$$ 116.313i 0.381353i
$$306$$ 0 0
$$307$$ 224.438 0.731070 0.365535 0.930798i $$-0.380886\pi$$
0.365535 + 0.930798i $$0.380886\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 44.5453 0.143695
$$311$$ 171.175i 0.550401i 0.961387 + 0.275201i $$0.0887442\pi$$
−0.961387 + 0.275201i $$0.911256\pi$$
$$312$$ 0 0
$$313$$ −274.110 −0.875749 −0.437875 0.899036i $$-0.644269\pi$$
−0.437875 + 0.899036i $$0.644269\pi$$
$$314$$ − 54.4276i − 0.173336i
$$315$$ 0 0
$$316$$ 570.509 1.80541
$$317$$ − 618.549i − 1.95126i −0.219427 0.975629i $$-0.570419\pi$$
0.219427 0.975629i $$-0.429581\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ − 145.625i − 0.455077i
$$321$$ 0 0
$$322$$ −195.118 −0.605957
$$323$$ − 739.057i − 2.28810i
$$324$$ 0 0
$$325$$ −169.440 −0.521354
$$326$$ 416.777i 1.27846i
$$327$$ 0 0
$$328$$ −84.3772 −0.257248
$$329$$ − 81.6633i − 0.248217i
$$330$$ 0 0
$$331$$ −160.328 −0.484376 −0.242188 0.970229i $$-0.577865\pi$$
−0.242188 + 0.970229i $$0.577865\pi$$
$$332$$ − 624.926i − 1.88231i
$$333$$ 0 0
$$334$$ −434.903 −1.30210
$$335$$ 128.666i 0.384077i
$$336$$ 0 0
$$337$$ 110.004 0.326422 0.163211 0.986591i $$-0.447815\pi$$
0.163211 + 0.986591i $$0.447815\pi$$
$$338$$ 334.812i 0.990568i
$$339$$ 0 0
$$340$$ 227.089 0.667910
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −289.402 −0.843738
$$344$$ − 2.09000i − 0.00607558i
$$345$$ 0 0
$$346$$ −869.771 −2.51379
$$347$$ − 221.736i − 0.639008i −0.947585 0.319504i $$-0.896484\pi$$
0.947585 0.319504i $$-0.103516\pi$$
$$348$$ 0 0
$$349$$ 466.352 1.33625 0.668126 0.744049i $$-0.267097\pi$$
0.668126 + 0.744049i $$0.267097\pi$$
$$350$$ − 223.407i − 0.638307i
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 494.404i − 1.40058i −0.713860 0.700289i $$-0.753055\pi$$
0.713860 0.700289i $$-0.246945\pi$$
$$354$$ 0 0
$$355$$ −173.712 −0.489330
$$356$$ − 413.980i − 1.16286i
$$357$$ 0 0
$$358$$ −438.489 −1.22483
$$359$$ − 28.1698i − 0.0784673i −0.999230 0.0392336i $$-0.987508\pi$$
0.999230 0.0392336i $$-0.0124917\pi$$
$$360$$ 0 0
$$361$$ 318.471 0.882191
$$362$$ − 669.246i − 1.84875i
$$363$$ 0 0
$$364$$ 125.157 0.343838
$$365$$ − 145.668i − 0.399090i
$$366$$ 0 0
$$367$$ 29.7976 0.0811925 0.0405962 0.999176i $$-0.487074\pi$$
0.0405962 + 0.999176i $$0.487074\pi$$
$$368$$ 218.784i 0.594521i
$$369$$ 0 0
$$370$$ 325.986 0.881044
$$371$$ − 19.2855i − 0.0519824i
$$372$$ 0 0
$$373$$ 313.887 0.841520 0.420760 0.907172i $$-0.361763\pi$$
0.420760 + 0.907172i $$0.361763\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 71.2099 0.189388
$$377$$ − 16.2833i − 0.0431917i
$$378$$ 0 0
$$379$$ −31.4155 −0.0828904 −0.0414452 0.999141i $$-0.513196\pi$$
−0.0414452 + 0.999141i $$0.513196\pi$$
$$380$$ 208.780i 0.549422i
$$381$$ 0 0
$$382$$ 69.5352 0.182029
$$383$$ 474.897i 1.23994i 0.784625 + 0.619970i $$0.212855\pi$$
−0.784625 + 0.619970i $$0.787145\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ − 375.674i − 0.973249i
$$387$$ 0 0
$$388$$ 387.758 0.999376
$$389$$ − 42.3470i − 0.108861i −0.998518 0.0544306i $$-0.982666\pi$$
0.998518 0.0544306i $$-0.0173344\pi$$
$$390$$ 0 0
$$391$$ −554.735 −1.41876
$$392$$ − 110.081i − 0.280820i
$$393$$ 0 0
$$394$$ −37.0911 −0.0941399
$$395$$ 185.029i 0.468429i
$$396$$ 0 0
$$397$$ −560.270 −1.41126 −0.705630 0.708580i $$-0.749336\pi$$
−0.705630 + 0.708580i $$0.749336\pi$$
$$398$$ − 337.261i − 0.847389i
$$399$$ 0 0
$$400$$ −250.504 −0.626260
$$401$$ − 565.289i − 1.40970i −0.709357 0.704850i $$-0.751014\pi$$
0.709357 0.704850i $$-0.248986\pi$$
$$402$$ 0 0
$$403$$ 69.7990 0.173198
$$404$$ − 193.740i − 0.479555i
$$405$$ 0 0
$$406$$ 21.4696 0.0528807
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −174.254 −0.426048 −0.213024 0.977047i $$-0.568331\pi$$
−0.213024 + 0.977047i $$0.568331\pi$$
$$410$$ − 140.271i − 0.342124i
$$411$$ 0 0
$$412$$ −149.818 −0.363635
$$413$$ − 246.372i − 0.596542i
$$414$$ 0 0
$$415$$ 202.678 0.488382
$$416$$ − 341.141i − 0.820050i
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 279.267i − 0.666508i −0.942837 0.333254i $$-0.891853\pi$$
0.942837 0.333254i $$-0.108147\pi$$
$$420$$ 0 0
$$421$$ 387.008 0.919259 0.459630 0.888111i $$-0.347982\pi$$
0.459630 + 0.888111i $$0.347982\pi$$
$$422$$ 460.948i 1.09229i
$$423$$ 0 0
$$424$$ 16.8168 0.0396623
$$425$$ − 635.164i − 1.49450i
$$426$$ 0 0
$$427$$ −240.301 −0.562765
$$428$$ 247.255i 0.577698i
$$429$$ 0 0
$$430$$ 3.47447 0.00808017
$$431$$ − 319.975i − 0.742402i −0.928552 0.371201i $$-0.878946\pi$$
0.928552 0.371201i $$-0.121054\pi$$
$$432$$ 0 0
$$433$$ 526.145 1.21512 0.607558 0.794276i $$-0.292149\pi$$
0.607558 + 0.794276i $$0.292149\pi$$
$$434$$ 92.0303i 0.212051i
$$435$$ 0 0
$$436$$ −744.547 −1.70768
$$437$$ − 510.010i − 1.16707i
$$438$$ 0 0
$$439$$ −22.7817 −0.0518946 −0.0259473 0.999663i $$-0.508260\pi$$
−0.0259473 + 0.999663i $$0.508260\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 642.243 1.45304
$$443$$ − 60.0022i − 0.135445i −0.997704 0.0677226i $$-0.978427\pi$$
0.997704 0.0677226i $$-0.0215733\pi$$
$$444$$ 0 0
$$445$$ 134.263 0.301716
$$446$$ − 845.715i − 1.89622i
$$447$$ 0 0
$$448$$ 300.859 0.671560
$$449$$ 761.224i 1.69538i 0.530495 + 0.847688i $$0.322006\pi$$
−0.530495 + 0.847688i $$0.677994\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 633.459i − 1.40146i
$$453$$ 0 0
$$454$$ −607.562 −1.33824
$$455$$ 40.5914i 0.0892119i
$$456$$ 0 0
$$457$$ 788.938 1.72634 0.863171 0.504912i $$-0.168475\pi$$
0.863171 + 0.504912i $$0.168475\pi$$
$$458$$ 1255.23i 2.74067i
$$459$$ 0 0
$$460$$ 156.710 0.340675
$$461$$ − 726.842i − 1.57666i −0.615250 0.788332i $$-0.710945\pi$$
0.615250 0.788332i $$-0.289055\pi$$
$$462$$ 0 0
$$463$$ 301.548 0.651291 0.325646 0.945492i $$-0.394418\pi$$
0.325646 + 0.945492i $$0.394418\pi$$
$$464$$ − 24.0736i − 0.0518827i
$$465$$ 0 0
$$466$$ −332.799 −0.714161
$$467$$ 292.485i 0.626306i 0.949703 + 0.313153i $$0.101385\pi$$
−0.949703 + 0.313153i $$0.898615\pi$$
$$468$$ 0 0
$$469$$ −265.822 −0.566785
$$470$$ 118.381i 0.251875i
$$471$$ 0 0
$$472$$ 214.835 0.455158
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 583.954 1.22938
$$476$$ 469.165i 0.985640i
$$477$$ 0 0
$$478$$ −940.571 −1.96772
$$479$$ 496.880i 1.03733i 0.854978 + 0.518664i $$0.173570\pi$$
−0.854978 + 0.518664i $$0.826430\pi$$
$$480$$ 0 0
$$481$$ 510.795 1.06194
$$482$$ − 312.350i − 0.648029i
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 125.759i 0.259297i
$$486$$ 0 0
$$487$$ 691.718 1.42037 0.710183 0.704017i $$-0.248612\pi$$
0.710183 + 0.704017i $$0.248612\pi$$
$$488$$ − 209.541i − 0.429387i
$$489$$ 0 0
$$490$$ 183.002 0.373474
$$491$$ 296.243i 0.603346i 0.953412 + 0.301673i $$0.0975450\pi$$
−0.953412 + 0.301673i $$0.902455\pi$$
$$492$$ 0 0
$$493$$ 61.0396 0.123813
$$494$$ 590.462i 1.19527i
$$495$$ 0 0
$$496$$ 103.192 0.208049
$$497$$ − 358.888i − 0.722109i
$$498$$ 0 0
$$499$$ −134.004 −0.268544 −0.134272 0.990944i $$-0.542870\pi$$
−0.134272 + 0.990944i $$0.542870\pi$$
$$500$$ 379.668i 0.759336i
$$501$$ 0 0
$$502$$ 998.914 1.98987
$$503$$ − 190.003i − 0.377740i −0.982002 0.188870i $$-0.939518\pi$$
0.982002 0.188870i $$-0.0604824\pi$$
$$504$$ 0 0
$$505$$ 62.8345 0.124425
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −1182.62 −2.32799
$$509$$ 119.100i 0.233989i 0.993133 + 0.116995i $$0.0373260\pi$$
−0.993133 + 0.116995i $$0.962674\pi$$
$$510$$ 0 0
$$511$$ 300.948 0.588940
$$512$$ − 634.222i − 1.23872i
$$513$$ 0 0
$$514$$ −919.404 −1.78872
$$515$$ − 48.5895i − 0.0943485i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 673.485i 1.30016i
$$519$$ 0 0
$$520$$ −35.3955 −0.0680682
$$521$$ 637.297i 1.22322i 0.791160 + 0.611609i $$0.209478\pi$$
−0.791160 + 0.611609i $$0.790522\pi$$
$$522$$ 0 0
$$523$$ −129.698 −0.247988 −0.123994 0.992283i $$-0.539570\pi$$
−0.123994 + 0.992283i $$0.539570\pi$$
$$524$$ − 738.147i − 1.40868i
$$525$$ 0 0
$$526$$ −745.914 −1.41809
$$527$$ 261.649i 0.496487i
$$528$$ 0 0
$$529$$ 146.187 0.276346
$$530$$ 27.9567i 0.0527485i
$$531$$ 0 0
$$532$$ −431.339 −0.810787
$$533$$ − 219.793i − 0.412370i
$$534$$ 0 0
$$535$$ −80.1905 −0.149889
$$536$$ − 231.795i − 0.432454i
$$537$$ 0 0
$$538$$ −589.779 −1.09624
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 493.991 0.913108 0.456554 0.889696i $$-0.349084\pi$$
0.456554 + 0.889696i $$0.349084\pi$$
$$542$$ − 1388.03i − 2.56095i
$$543$$ 0 0
$$544$$ 1278.80 2.35074
$$545$$ − 241.474i − 0.443072i
$$546$$ 0 0
$$547$$ 106.128 0.194018 0.0970089 0.995284i $$-0.469072\pi$$
0.0970089 + 0.995284i $$0.469072\pi$$
$$548$$ 100.178i 0.182807i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 56.1183i 0.101848i
$$552$$ 0 0
$$553$$ −382.269 −0.691264
$$554$$ − 70.2457i − 0.126797i
$$555$$ 0 0
$$556$$ 739.849 1.33066
$$557$$ − 827.637i − 1.48588i −0.669356 0.742942i $$-0.733430\pi$$
0.669356 0.742942i $$-0.266570\pi$$
$$558$$ 0 0
$$559$$ 5.44422 0.00973921
$$560$$ 60.0113i 0.107163i
$$561$$ 0 0
$$562$$ 789.001 1.40392
$$563$$ 470.917i 0.836442i 0.908345 + 0.418221i $$0.137346\pi$$
−0.908345 + 0.418221i $$0.862654\pi$$
$$564$$ 0 0
$$565$$ 205.446 0.363621
$$566$$ − 107.289i − 0.189556i
$$567$$ 0 0
$$568$$ 312.948 0.550965
$$569$$ 200.074i 0.351624i 0.984424 + 0.175812i $$0.0562551\pi$$
−0.984424 + 0.175812i $$0.943745\pi$$
$$570$$ 0 0
$$571$$ −658.692 −1.15358 −0.576788 0.816894i $$-0.695694\pi$$
−0.576788 + 0.816894i $$0.695694\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 289.798 0.504875
$$575$$ − 438.315i − 0.762287i
$$576$$ 0 0
$$577$$ −100.977 −0.175004 −0.0875020 0.996164i $$-0.527888\pi$$
−0.0875020 + 0.996164i $$0.527888\pi$$
$$578$$ 1541.98i 2.66779i
$$579$$ 0 0
$$580$$ −17.2434 −0.0297301
$$581$$ 418.732i 0.720709i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 262.425i 0.449358i
$$585$$ 0 0
$$586$$ 342.005 0.583627
$$587$$ 349.390i 0.595213i 0.954689 + 0.297606i $$0.0961883\pi$$
−0.954689 + 0.297606i $$0.903812\pi$$
$$588$$ 0 0
$$589$$ −240.554 −0.408410
$$590$$ 357.147i 0.605333i
$$591$$ 0 0
$$592$$ 755.171 1.27563
$$593$$ − 670.238i − 1.13025i −0.825005 0.565125i $$-0.808828\pi$$
0.825005 0.565125i $$-0.191172\pi$$
$$594$$ 0 0
$$595$$ −152.161 −0.255733
$$596$$ 223.123i 0.374367i
$$597$$ 0 0
$$598$$ 443.200 0.741138
$$599$$ 771.328i 1.28769i 0.765155 + 0.643847i $$0.222662\pi$$
−0.765155 + 0.643847i $$0.777338\pi$$
$$600$$ 0 0
$$601$$ 185.722 0.309021 0.154511 0.987991i $$-0.450620\pi$$
0.154511 + 0.987991i $$0.450620\pi$$
$$602$$ 7.17823i 0.0119240i
$$603$$ 0 0
$$604$$ 231.294 0.382937
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 488.185 0.804259 0.402129 0.915583i $$-0.368270\pi$$
0.402129 + 0.915583i $$0.368270\pi$$
$$608$$ 1175.70i 1.93372i
$$609$$ 0 0
$$610$$ 348.346 0.571059
$$611$$ 185.494i 0.303591i
$$612$$ 0 0
$$613$$ −510.461 −0.832726 −0.416363 0.909199i $$-0.636695\pi$$
−0.416363 + 0.909199i $$0.636695\pi$$
$$614$$ − 672.174i − 1.09475i
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1142.14i 1.85111i 0.378608 + 0.925557i $$0.376403\pi$$
−0.378608 + 0.925557i $$0.623597\pi$$
$$618$$ 0 0
$$619$$ −908.213 −1.46723 −0.733613 0.679567i $$-0.762168\pi$$
−0.733613 + 0.679567i $$0.762168\pi$$
$$620$$ − 73.9147i − 0.119217i
$$621$$ 0 0
$$622$$ 512.653 0.824202
$$623$$ 277.387i 0.445244i
$$624$$ 0 0
$$625$$ 436.923 0.699077
$$626$$ 820.934i 1.31140i
$$627$$ 0 0
$$628$$ −90.3126 −0.143810
$$629$$ 1914.77i 3.04415i
$$630$$ 0 0
$$631$$ 944.402 1.49668 0.748338 0.663318i $$-0.230852\pi$$
0.748338 + 0.663318i $$0.230852\pi$$
$$632$$ − 333.336i − 0.527431i
$$633$$ 0 0
$$634$$ −1852.50 −2.92192
$$635$$ − 383.551i − 0.604017i
$$636$$ 0 0
$$637$$ 286.750 0.450157
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −145.354 −0.227116
$$641$$ 359.081i 0.560188i 0.959973 + 0.280094i $$0.0903657\pi$$
−0.959973 + 0.280094i $$0.909634\pi$$
$$642$$ 0 0
$$643$$ 928.146 1.44346 0.721731 0.692174i $$-0.243347\pi$$
0.721731 + 0.692174i $$0.243347\pi$$
$$644$$ 323.762i 0.502737i
$$645$$ 0 0
$$646$$ −2213.41 −3.42633
$$647$$ − 436.289i − 0.674326i −0.941446 0.337163i $$-0.890533\pi$$
0.941446 0.337163i $$-0.109467\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 507.458i 0.780705i
$$651$$ 0 0
$$652$$ 691.565 1.06068
$$653$$ 243.417i 0.372767i 0.982477 + 0.186383i $$0.0596767\pi$$
−0.982477 + 0.186383i $$0.940323\pi$$
$$654$$ 0 0
$$655$$ 239.399 0.365494
$$656$$ − 324.948i − 0.495347i
$$657$$ 0 0
$$658$$ −244.574 −0.371694
$$659$$ − 348.053i − 0.528153i −0.964502 0.264076i $$-0.914933\pi$$
0.964502 0.264076i $$-0.0850671\pi$$
$$660$$ 0 0
$$661$$ 601.901 0.910592 0.455296 0.890340i $$-0.349533\pi$$
0.455296 + 0.890340i $$0.349533\pi$$
$$662$$ 480.170i 0.725332i
$$663$$ 0 0
$$664$$ −365.131 −0.549897
$$665$$ − 139.893i − 0.210366i
$$666$$ 0 0
$$667$$ 42.1224 0.0631520
$$668$$ 721.640i 1.08030i
$$669$$ 0 0
$$670$$ 385.343 0.575138
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −738.885 −1.09790 −0.548949 0.835856i $$-0.684972\pi$$
−0.548949 + 0.835856i $$0.684972\pi$$
$$674$$ − 329.453i − 0.488803i
$$675$$ 0 0
$$676$$ 555.558 0.821832
$$677$$ 42.1168i 0.0622109i 0.999516 + 0.0311055i $$0.00990277\pi$$
−0.999516 + 0.0311055i $$0.990097\pi$$
$$678$$ 0 0
$$679$$ −259.817 −0.382647
$$680$$ − 132.684i − 0.195123i
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 543.110i 0.795183i 0.917562 + 0.397592i $$0.130154\pi$$
−0.917562 + 0.397592i $$0.869846\pi$$
$$684$$ 0 0
$$685$$ −32.4902 −0.0474309
$$686$$ 866.734i 1.26346i
$$687$$ 0 0
$$688$$ 8.04886 0.0116989
$$689$$ 43.8059i 0.0635790i
$$690$$ 0 0
$$691$$ −428.173 −0.619643 −0.309821 0.950795i $$-0.600269\pi$$
−0.309821 + 0.950795i $$0.600269\pi$$
$$692$$ 1443.22i 2.08558i
$$693$$ 0 0
$$694$$ −664.080 −0.956887
$$695$$ 239.951i 0.345253i
$$696$$ 0 0
$$697$$ 823.918 1.18209
$$698$$ − 1396.68i − 2.00098i
$$699$$ 0 0
$$700$$ −370.703 −0.529576
$$701$$ 2.78056i 0.00396656i 0.999998 + 0.00198328i $$0.000631299\pi$$
−0.999998 + 0.00198328i $$0.999369\pi$$
$$702$$ 0 0
$$703$$ −1760.39 −2.50411
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −1480.70 −2.09730
$$707$$ 129.816i 0.183615i
$$708$$ 0 0
$$709$$ 1077.51 1.51976 0.759878 0.650065i $$-0.225258\pi$$
0.759878 + 0.650065i $$0.225258\pi$$
$$710$$ 520.253i 0.732751i
$$711$$ 0 0
$$712$$ −241.880 −0.339719
$$713$$ 180.559i 0.253239i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 727.591i 1.01619i
$$717$$ 0 0
$$718$$ −84.3660 −0.117501
$$719$$ − 1073.42i − 1.49293i −0.665424 0.746465i $$-0.731749\pi$$
0.665424 0.746465i $$-0.268251\pi$$
$$720$$ 0 0
$$721$$ 100.385 0.139231
$$722$$ − 953.792i − 1.32104i
$$723$$ 0 0
$$724$$ −1110.49 −1.53383
$$725$$ 48.2295i 0.0665234i
$$726$$ 0 0
$$727$$ −212.371 −0.292119 −0.146060 0.989276i $$-0.546659\pi$$
−0.146060 + 0.989276i $$0.546659\pi$$
$$728$$ − 73.1267i − 0.100449i
$$729$$ 0 0
$$730$$ −436.262 −0.597620
$$731$$ 20.4082i 0.0279182i
$$732$$ 0 0
$$733$$ 942.223 1.28543 0.642717 0.766104i $$-0.277807\pi$$
0.642717 + 0.766104i $$0.277807\pi$$
$$734$$ − 89.2414i − 0.121582i
$$735$$ 0 0
$$736$$ 882.479 1.19902
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 720.862 0.975456 0.487728 0.872996i $$-0.337826\pi$$
0.487728 + 0.872996i $$0.337826\pi$$
$$740$$ − 540.914i − 0.730965i
$$741$$ 0 0
$$742$$ −57.7583 −0.0778414
$$743$$ 836.564i 1.12593i 0.826482 + 0.562963i $$0.190339\pi$$
−0.826482 + 0.562963i $$0.809661\pi$$
$$744$$ 0 0
$$745$$ −72.3640 −0.0971329
$$746$$ − 940.064i − 1.26014i
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 165.673i − 0.221192i
$$750$$ 0 0
$$751$$ −35.2668 −0.0469598 −0.0234799 0.999724i $$-0.507475\pi$$
−0.0234799 + 0.999724i $$0.507475\pi$$
$$752$$ 274.238i 0.364679i
$$753$$ 0 0
$$754$$ −48.7670 −0.0646777
$$755$$ 75.0141i 0.0993564i
$$756$$ 0 0
$$757$$ 454.700 0.600660 0.300330 0.953835i $$-0.402903\pi$$
0.300330 + 0.953835i $$0.402903\pi$$
$$758$$ 94.0866i 0.124125i
$$759$$ 0 0
$$760$$ 121.986 0.160508
$$761$$ − 104.689i − 0.137568i −0.997632 0.0687840i $$-0.978088\pi$$
0.997632 0.0687840i $$-0.0219119\pi$$
$$762$$ 0 0
$$763$$ 498.884 0.653845
$$764$$ − 115.381i − 0.151022i
$$765$$ 0 0
$$766$$ 1422.28 1.85676
$$767$$ 559.620i 0.729622i
$$768$$ 0 0
$$769$$ 93.7806 0.121951 0.0609757 0.998139i $$-0.480579\pi$$
0.0609757 + 0.998139i $$0.480579\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −623.361 −0.807463
$$773$$ 358.726i 0.464070i 0.972707 + 0.232035i $$0.0745384\pi$$
−0.972707 + 0.232035i $$0.925462\pi$$
$$774$$ 0 0
$$775$$ −206.738 −0.266758
$$776$$ − 226.559i − 0.291957i
$$777$$ 0 0
$$778$$ −126.826 −0.163015
$$779$$ 757.491i 0.972388i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 1661.38i 2.12453i
$$783$$ 0 0
$$784$$ 423.938 0.540737
$$785$$ − 29.2905i − 0.0373127i
$$786$$ 0 0
$$787$$ −276.916 −0.351863 −0.175931 0.984402i $$-0.556294\pi$$
−0.175931 + 0.984402i $$0.556294\pi$$
$$788$$ 61.5458i 0.0781038i
$$789$$ 0 0
$$790$$ 554.147 0.701452
$$791$$ 424.449i 0.536598i
$$792$$ 0 0
$$793$$ 545.830 0.688311
$$794$$ 1677.96i 2.11330i
$$795$$ 0 0
$$796$$ −559.622 −0.703042
$$797$$ 528.612i 0.663252i 0.943411 + 0.331626i $$0.107597\pi$$
−0.943411 + 0.331626i $$0.892403\pi$$
$$798$$ 0 0
$$799$$ −695.343 −0.870267
$$800$$ 1010.43i 1.26303i
$$801$$ 0 0
$$802$$ −1692.99 −2.11096
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −105.004 −0.130439
$$806$$ − 209.042i − 0.259357i
$$807$$ 0 0
$$808$$ −113.198 −0.140097
$$809$$ − 1257.86i − 1.55483i −0.628985 0.777417i $$-0.716529\pi$$
0.628985 0.777417i $$-0.283471\pi$$
$$810$$ 0 0
$$811$$ 1041.19 1.28383 0.641916 0.766775i $$-0.278140\pi$$
0.641916 + 0.766775i $$0.278140\pi$$
$$812$$ − 35.6248i − 0.0438729i
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 224.291i 0.275203i
$$816$$ 0 0
$$817$$ −18.7628 −0.0229655
$$818$$ 521.875i 0.637989i
$$819$$ 0 0
$$820$$ −232.754 −0.283846
$$821$$ 435.664i 0.530650i 0.964159 + 0.265325i $$0.0854793\pi$$
−0.964159 + 0.265325i $$0.914521\pi$$
$$822$$ 0 0
$$823$$ 587.220 0.713512 0.356756 0.934198i $$-0.383883\pi$$
0.356756 + 0.934198i $$0.383883\pi$$
$$824$$ 87.5354i 0.106232i
$$825$$ 0 0
$$826$$ −737.862 −0.893296
$$827$$ − 274.955i − 0.332473i −0.986086 0.166236i $$-0.946839\pi$$
0.986086 0.166236i $$-0.0531615\pi$$
$$828$$ 0 0
$$829$$ −916.540 −1.10560 −0.552799 0.833315i $$-0.686440\pi$$
−0.552799 + 0.833315i $$0.686440\pi$$
$$830$$ − 607.004i − 0.731330i
$$831$$ 0 0
$$832$$ −683.385 −0.821376
$$833$$ 1074.91i 1.29041i
$$834$$ 0 0
$$835$$ −234.045 −0.280293
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −836.380 −0.998067
$$839$$ − 1058.83i − 1.26202i −0.775777 0.631008i $$-0.782642\pi$$
0.775777 0.631008i $$-0.217358\pi$$
$$840$$ 0 0
$$841$$ 836.365 0.994489
$$842$$ − 1159.06i − 1.37655i
$$843$$ 0 0
$$844$$ 764.858 0.906229
$$845$$ 180.181i 0.213232i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 64.7637i 0.0763723i
$$849$$ 0 0
$$850$$ −1902.26 −2.23795
$$851$$ 1321.35i 1.55270i
$$852$$ 0 0
$$853$$ 262.534 0.307777 0.153888 0.988088i $$-0.450820\pi$$
0.153888 + 0.988088i $$0.450820\pi$$
$$854$$ 719.680i 0.842716i
$$855$$ 0 0
$$856$$ 144.466 0.168768
$$857$$ − 139.937i − 0.163287i −0.996662 0.0816437i $$-0.973983\pi$$
0.996662 0.0816437i $$-0.0260169\pi$$
$$858$$ 0 0
$$859$$ −466.124 −0.542635 −0.271318 0.962490i $$-0.587459\pi$$
−0.271318 + 0.962490i $$0.587459\pi$$
$$860$$ − 5.76524i − 0.00670377i
$$861$$ 0 0
$$862$$ −958.299 −1.11172
$$863$$ − 223.415i − 0.258882i −0.991587 0.129441i $$-0.958682\pi$$
0.991587 0.129441i $$-0.0413183\pi$$
$$864$$ 0 0
$$865$$ −468.072 −0.541123
$$866$$ − 1575.76i − 1.81958i
$$867$$ 0 0
$$868$$ 152.707 0.175930
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 603.801 0.693228
$$872$$ 435.023i 0.498880i
$$873$$ 0 0
$$874$$ −1527.44 −1.74764
$$875$$ − 254.397i − 0.290739i
$$876$$ 0 0
$$877$$ −446.074 −0.508637 −0.254318 0.967121i $$-0.581851\pi$$
−0.254318 + 0.967121i $$0.581851\pi$$
$$878$$ 68.2292i 0.0777099i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 1631.61i − 1.85200i −0.377522 0.926001i $$-0.623224\pi$$
0.377522 0.926001i $$-0.376776\pi$$
$$882$$ 0 0
$$883$$ 1491.22 1.68881 0.844404 0.535707i $$-0.179955\pi$$
0.844404 + 0.535707i $$0.179955\pi$$
$$884$$ − 1065.68i − 1.20552i
$$885$$ 0 0
$$886$$ −179.701 −0.202823
$$887$$ − 623.153i − 0.702540i −0.936274 0.351270i $$-0.885750\pi$$
0.936274 0.351270i $$-0.114250\pi$$
$$888$$ 0 0
$$889$$ 792.413 0.891353
$$890$$ − 402.107i − 0.451806i
$$891$$ 0 0
$$892$$ −1403.31 −1.57321
$$893$$ − 639.282i − 0.715881i
$$894$$ 0 0
$$895$$ −235.975 −0.263659
$$896$$ − 300.300i − 0.335157i
$$897$$ 0 0
$$898$$ 2279.80 2.53875
$$899$$ − 19.8676i − 0.0220997i
$$900$$ 0 0
$$901$$ −164.211 −0.182254
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −370.117 −0.409422
$$905$$ − 360.158i − 0.397965i
$$906$$ 0 0
$$907$$ −966.557 −1.06566 −0.532832 0.846221i $$-0.678872\pi$$
−0.532832 + 0.846221i $$0.678872\pi$$
$$908$$ 1008.14i 1.11028i
$$909$$ 0 0
$$910$$ 121.568 0.133591
$$911$$ 617.042i 0.677324i 0.940908 + 0.338662i $$0.109974\pi$$
−0.940908 + 0.338662i $$0.890026\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ − 2362.80i − 2.58512i
$$915$$ 0 0
$$916$$ 2082.82 2.27382
$$917$$ 494.595i 0.539362i
$$918$$ 0 0
$$919$$ −1104.16 −1.20148 −0.600742 0.799443i $$-0.705128\pi$$
−0.600742 + 0.799443i $$0.705128\pi$$
$$920$$ − 91.5626i − 0.0995246i
$$921$$ 0 0
$$922$$ −2176.83 −2.36099
$$923$$ 815.195i 0.883202i
$$924$$ 0 0
$$925$$ −1512.92 −1.63559
$$926$$ − 903.109i − 0.975280i
$$927$$ 0 0
$$928$$ −97.1025 −0.104636
$$929$$ − 1078.61i − 1.16104i −0.814245 0.580521i $$-0.802849\pi$$
0.814245 0.580521i $$-0.197151\pi$$
$$930$$ 0 0
$$931$$ −988.248 −1.06149
$$932$$ 552.218i 0.592509i
$$933$$ 0 0
$$934$$ 875.967 0.937866
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 587.691 0.627205 0.313602 0.949554i $$-0.398464\pi$$
0.313602 + 0.949554i $$0.398464\pi$$
$$938$$ 796.115i 0.848736i
$$939$$ 0 0
$$940$$ 196.432 0.208970
$$941$$ − 1069.27i − 1.13631i −0.822921 0.568155i $$-0.807657\pi$$
0.822921 0.568155i $$-0.192343\pi$$
$$942$$ 0 0
$$943$$ 568.572 0.602939
$$944$$ 827.356i 0.876437i
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 435.924i − 0.460321i −0.973153 0.230160i $$-0.926075\pi$$
0.973153 0.230160i $$-0.0739251\pi$$
$$948$$ 0 0
$$949$$ −683.588 −0.720325
$$950$$ − 1748.89i − 1.84094i
$$951$$ 0 0
$$952$$ 274.123 0.287945
$$953$$ − 670.977i − 0.704068i −0.935987 0.352034i $$-0.885490\pi$$
0.935987 0.352034i $$-0.114510\pi$$
$$954$$ 0 0
$$955$$ 37.4207 0.0391840
$$956$$ 1560.70i 1.63253i
$$957$$ 0 0
$$958$$ 1488.11 1.55335
$$959$$ − 67.1244i − 0.0699942i
$$960$$ 0 0
$$961$$ −875.837 −0.911380
$$962$$ − 1529.79i − 1.59021i
$$963$$ 0 0
$$964$$ −518.287 −0.537642
$$965$$ − 202.171i − 0.209503i
$$966$$ 0 0
$$967$$ 940.682 0.972784 0.486392 0.873741i $$-0.338313\pi$$
0.486392 + 0.873741i $$0.338313\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 376.637 0.388286
$$971$$ − 1077.62i − 1.10980i −0.831916 0.554901i $$-0.812756\pi$$
0.831916 0.554901i $$-0.187244\pi$$
$$972$$ 0 0
$$973$$ −495.736 −0.509492
$$974$$ − 2071.64i − 2.12694i
$$975$$ 0 0
$$976$$ 806.969 0.826812
$$977$$ − 1006.05i − 1.02973i −0.857270 0.514866i $$-0.827842\pi$$
0.857270 0.514866i $$-0.172158\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ − 303.658i − 0.309855i
$$981$$ 0 0
$$982$$ 887.221 0.903484
$$983$$ 1173.82i 1.19412i 0.802197 + 0.597060i $$0.203665\pi$$
−0.802197 + 0.597060i $$0.796335\pi$$
$$984$$ 0 0
$$985$$ −19.9608 −0.0202647
$$986$$ − 182.808i − 0.185404i
$$987$$ 0 0
$$988$$ 979.763 0.991663
$$989$$ 14.0834i 0.0142400i
$$990$$ 0 0
$$991$$ −989.386 −0.998372 −0.499186 0.866495i $$-0.666368\pi$$
−0.499186 + 0.866495i $$0.666368\pi$$
$$992$$ − 416.234i − 0.419591i
$$993$$ 0 0
$$994$$ −1074.84 −1.08133
$$995$$ − 181.499i − 0.182411i
$$996$$ 0 0
$$997$$ 1465.59 1.47000 0.735000 0.678067i $$-0.237182\pi$$
0.735000 + 0.678067i $$0.237182\pi$$
$$998$$ 401.329i 0.402133i
$$999$$ 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 1089.3.b.j.485.3 16
3.2 odd 2 inner 1089.3.b.j.485.14 16
11.7 odd 10 99.3.l.a.71.2 yes 32
11.8 odd 10 99.3.l.a.53.7 yes 32
11.10 odd 2 1089.3.b.i.485.14 16
33.8 even 10 99.3.l.a.53.2 32
33.29 even 10 99.3.l.a.71.7 yes 32
33.32 even 2 1089.3.b.i.485.3 16

By twisted newform
Twist Min Dim Char Parity Ord Type
99.3.l.a.53.2 32 33.8 even 10
99.3.l.a.53.7 yes 32 11.8 odd 10
99.3.l.a.71.2 yes 32 11.7 odd 10
99.3.l.a.71.7 yes 32 33.29 even 10
1089.3.b.i.485.3 16 33.32 even 2
1089.3.b.i.485.14 16 11.10 odd 2
1089.3.b.j.485.3 16 1.1 even 1 trivial
1089.3.b.j.485.14 16 3.2 odd 2 inner