# Properties

 Label 1089.3.b.h.485.4 Level $1089$ Weight $3$ Character 1089.485 Analytic conductor $29.673$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Coefficient field: 8.0.3317760000.7 Defining polynomial: $$x^{8} + 12x^{6} + 23x^{4} + 12x^{2} + 1$$ x^8 + 12*x^6 + 23*x^4 + 12*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 485.4 Root $$-1.19395i$$ of defining polynomial Character $$\chi$$ $$=$$ 1089.485 Dual form 1089.3.b.h.485.5

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.356394i q^{2} +3.87298 q^{4} +3.44572i q^{5} +3.96812 q^{7} -2.80588i q^{8} +O(q^{10})$$ $$q-0.356394i q^{2} +3.87298 q^{4} +3.44572i q^{5} +3.96812 q^{7} -2.80588i q^{8} +1.22803 q^{10} -4.75615 q^{13} -1.41421i q^{14} +14.4919 q^{16} +29.1280i q^{17} -14.8644 q^{19} +13.3452i q^{20} +22.9639i q^{23} +13.1270 q^{25} +1.69506i q^{26} +15.3685 q^{28} +4.63312i q^{29} +27.4919 q^{31} -16.3884i q^{32} +10.3810 q^{34} +13.6730i q^{35} +36.8569 q^{37} +5.29760i q^{38} +9.66829 q^{40} -68.3199i q^{41} -26.8968 q^{43} +8.18418 q^{46} +42.9425i q^{47} -33.2540 q^{49} -4.67839i q^{50} -18.4205 q^{52} +78.2194i q^{53} -11.1341i q^{56} +1.65122 q^{58} +92.7992i q^{59} +40.5612 q^{61} -9.79796i q^{62} +52.1270 q^{64} -16.3884i q^{65} -54.3327 q^{67} +112.812i q^{68} +4.87298 q^{70} -91.0029i q^{71} -109.347 q^{73} -13.1356i q^{74} -57.5697 q^{76} +143.292 q^{79} +49.9351i q^{80} -24.3488 q^{82} +44.0003i q^{83} -100.367 q^{85} +9.58587i q^{86} -64.7730i q^{89} -18.8730 q^{91} +88.9386i q^{92} +15.3044 q^{94} -51.2187i q^{95} +16.8891 q^{97} +11.8515i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 8 q^{16} + 136 q^{25} + 96 q^{31} + 176 q^{34} + 16 q^{37} - 328 q^{49} + 416 q^{58} + 448 q^{64} + 216 q^{67} + 8 q^{70} + 208 q^{82} - 120 q^{91} + 352 q^{97}+O(q^{100})$$ 8 * q - 8 * q^16 + 136 * q^25 + 96 * q^31 + 176 * q^34 + 16 * q^37 - 328 * q^49 + 416 * q^58 + 448 * q^64 + 216 * q^67 + 8 * q^70 + 208 * q^82 - 120 * q^91 + 352 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 0.356394i − 0.178197i −0.996023 0.0890985i $$-0.971601\pi$$
0.996023 0.0890985i $$-0.0283986\pi$$
$$3$$ 0 0
$$4$$ 3.87298 0.968246
$$5$$ 3.44572i 0.689144i 0.938760 + 0.344572i $$0.111976\pi$$
−0.938760 + 0.344572i $$0.888024\pi$$
$$6$$ 0 0
$$7$$ 3.96812 0.566874 0.283437 0.958991i $$-0.408525\pi$$
0.283437 + 0.958991i $$0.408525\pi$$
$$8$$ − 2.80588i − 0.350735i
$$9$$ 0 0
$$10$$ 1.22803 0.122803
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −4.75615 −0.365858 −0.182929 0.983126i $$-0.558558\pi$$
−0.182929 + 0.983126i $$0.558558\pi$$
$$14$$ − 1.41421i − 0.101015i
$$15$$ 0 0
$$16$$ 14.4919 0.905746
$$17$$ 29.1280i 1.71341i 0.515804 + 0.856706i $$0.327493\pi$$
−0.515804 + 0.856706i $$0.672507\pi$$
$$18$$ 0 0
$$19$$ −14.8644 −0.782339 −0.391169 0.920319i $$-0.627929\pi$$
−0.391169 + 0.920319i $$0.627929\pi$$
$$20$$ 13.3452i 0.667261i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 22.9639i 0.998429i 0.866479 + 0.499214i $$0.166378\pi$$
−0.866479 + 0.499214i $$0.833622\pi$$
$$24$$ 0 0
$$25$$ 13.1270 0.525081
$$26$$ 1.69506i 0.0651948i
$$27$$ 0 0
$$28$$ 15.3685 0.548873
$$29$$ 4.63312i 0.159763i 0.996804 + 0.0798814i $$0.0254542\pi$$
−0.996804 + 0.0798814i $$0.974546\pi$$
$$30$$ 0 0
$$31$$ 27.4919 0.886837 0.443418 0.896315i $$-0.353766\pi$$
0.443418 + 0.896315i $$0.353766\pi$$
$$32$$ − 16.3884i − 0.512137i
$$33$$ 0 0
$$34$$ 10.3810 0.305325
$$35$$ 13.6730i 0.390658i
$$36$$ 0 0
$$37$$ 36.8569 0.996131 0.498066 0.867139i $$-0.334044\pi$$
0.498066 + 0.867139i $$0.334044\pi$$
$$38$$ 5.29760i 0.139410i
$$39$$ 0 0
$$40$$ 9.66829 0.241707
$$41$$ − 68.3199i − 1.66634i −0.553019 0.833169i $$-0.686524\pi$$
0.553019 0.833169i $$-0.313476\pi$$
$$42$$ 0 0
$$43$$ −26.8968 −0.625508 −0.312754 0.949834i $$-0.601252\pi$$
−0.312754 + 0.949834i $$0.601252\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 8.18418 0.177917
$$47$$ 42.9425i 0.913670i 0.889552 + 0.456835i $$0.151017\pi$$
−0.889552 + 0.456835i $$0.848983\pi$$
$$48$$ 0 0
$$49$$ −33.2540 −0.678654
$$50$$ − 4.67839i − 0.0935678i
$$51$$ 0 0
$$52$$ −18.4205 −0.354240
$$53$$ 78.2194i 1.47584i 0.674889 + 0.737919i $$0.264191\pi$$
−0.674889 + 0.737919i $$0.735809\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ − 11.1341i − 0.198823i
$$57$$ 0 0
$$58$$ 1.65122 0.0284692
$$59$$ 92.7992i 1.57287i 0.617674 + 0.786434i $$0.288075\pi$$
−0.617674 + 0.786434i $$0.711925\pi$$
$$60$$ 0 0
$$61$$ 40.5612 0.664937 0.332469 0.943114i $$-0.392118\pi$$
0.332469 + 0.943114i $$0.392118\pi$$
$$62$$ − 9.79796i − 0.158032i
$$63$$ 0 0
$$64$$ 52.1270 0.814485
$$65$$ − 16.3884i − 0.252129i
$$66$$ 0 0
$$67$$ −54.3327 −0.810935 −0.405468 0.914109i $$-0.632891\pi$$
−0.405468 + 0.914109i $$0.632891\pi$$
$$68$$ 112.812i 1.65900i
$$69$$ 0 0
$$70$$ 4.87298 0.0696140
$$71$$ − 91.0029i − 1.28173i −0.767653 0.640866i $$-0.778576\pi$$
0.767653 0.640866i $$-0.221424\pi$$
$$72$$ 0 0
$$73$$ −109.347 −1.49791 −0.748954 0.662622i $$-0.769444\pi$$
−0.748954 + 0.662622i $$0.769444\pi$$
$$74$$ − 13.1356i − 0.177508i
$$75$$ 0 0
$$76$$ −57.5697 −0.757496
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 143.292 1.81383 0.906913 0.421318i $$-0.138432\pi$$
0.906913 + 0.421318i $$0.138432\pi$$
$$80$$ 49.9351i 0.624189i
$$81$$ 0 0
$$82$$ −24.3488 −0.296936
$$83$$ 44.0003i 0.530124i 0.964231 + 0.265062i $$0.0853924\pi$$
−0.964231 + 0.265062i $$0.914608\pi$$
$$84$$ 0 0
$$85$$ −100.367 −1.18079
$$86$$ 9.58587i 0.111464i
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 64.7730i − 0.727786i −0.931441 0.363893i $$-0.881447\pi$$
0.931441 0.363893i $$-0.118553\pi$$
$$90$$ 0 0
$$91$$ −18.8730 −0.207395
$$92$$ 88.9386i 0.966724i
$$93$$ 0 0
$$94$$ 15.3044 0.162813
$$95$$ − 51.2187i − 0.539144i
$$96$$ 0 0
$$97$$ 16.8891 0.174115 0.0870573 0.996203i $$-0.472254\pi$$
0.0870573 + 0.996203i $$0.472254\pi$$
$$98$$ 11.8515i 0.120934i
$$99$$ 0 0
$$100$$ 50.8407 0.508407
$$101$$ 164.568i 1.62939i 0.579889 + 0.814696i $$0.303096\pi$$
−0.579889 + 0.814696i $$0.696904\pi$$
$$102$$ 0 0
$$103$$ 145.381 1.41147 0.705733 0.708478i $$-0.250618\pi$$
0.705733 + 0.708478i $$0.250618\pi$$
$$104$$ 13.3452i 0.128319i
$$105$$ 0 0
$$106$$ 27.8769 0.262990
$$107$$ − 189.414i − 1.77022i −0.465377 0.885112i $$-0.654081\pi$$
0.465377 0.885112i $$-0.345919\pi$$
$$108$$ 0 0
$$109$$ 57.5418 0.527906 0.263953 0.964536i $$-0.414974\pi$$
0.263953 + 0.964536i $$0.414974\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 57.5057 0.513444
$$113$$ 29.7769i 0.263512i 0.991282 + 0.131756i $$0.0420616\pi$$
−0.991282 + 0.131756i $$0.957938\pi$$
$$114$$ 0 0
$$115$$ −79.1270 −0.688061
$$116$$ 17.9440i 0.154690i
$$117$$ 0 0
$$118$$ 33.0731 0.280280
$$119$$ 115.583i 0.971289i
$$120$$ 0 0
$$121$$ 0 0
$$122$$ − 14.4558i − 0.118490i
$$123$$ 0 0
$$124$$ 106.476 0.858676
$$125$$ 131.375i 1.05100i
$$126$$ 0 0
$$127$$ 12.5923 0.0991520 0.0495760 0.998770i $$-0.484213\pi$$
0.0495760 + 0.998770i $$0.484213\pi$$
$$128$$ − 84.1312i − 0.657275i
$$129$$ 0 0
$$130$$ −5.84072 −0.0449286
$$131$$ − 20.3597i − 0.155418i −0.996976 0.0777089i $$-0.975240\pi$$
0.996976 0.0777089i $$-0.0247605\pi$$
$$132$$ 0 0
$$133$$ −58.9839 −0.443488
$$134$$ 19.3638i 0.144506i
$$135$$ 0 0
$$136$$ 81.7298 0.600955
$$137$$ − 35.9170i − 0.262168i −0.991371 0.131084i $$-0.958154\pi$$
0.991371 0.131084i $$-0.0418458\pi$$
$$138$$ 0 0
$$139$$ 101.091 0.727273 0.363637 0.931541i $$-0.381535\pi$$
0.363637 + 0.931541i $$0.381535\pi$$
$$140$$ 52.9554i 0.378253i
$$141$$ 0 0
$$142$$ −32.4329 −0.228401
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −15.9644 −0.110100
$$146$$ 38.9707i 0.266923i
$$147$$ 0 0
$$148$$ 142.746 0.964500
$$149$$ 210.311i 1.41148i 0.708469 + 0.705742i $$0.249386\pi$$
−0.708469 + 0.705742i $$0.750614\pi$$
$$150$$ 0 0
$$151$$ −212.078 −1.40449 −0.702246 0.711934i $$-0.747820\pi$$
−0.702246 + 0.711934i $$0.747820\pi$$
$$152$$ 41.7079i 0.274394i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 94.7295i 0.611158i
$$156$$ 0 0
$$157$$ 119.173 0.759066 0.379533 0.925178i $$-0.376085\pi$$
0.379533 + 0.925178i $$0.376085\pi$$
$$158$$ − 51.0685i − 0.323218i
$$159$$ 0 0
$$160$$ 56.4697 0.352936
$$161$$ 91.1233i 0.565983i
$$162$$ 0 0
$$163$$ 266.268 1.63355 0.816773 0.576959i $$-0.195761\pi$$
0.816773 + 0.576959i $$0.195761\pi$$
$$164$$ − 264.602i − 1.61342i
$$165$$ 0 0
$$166$$ 15.6814 0.0944665
$$167$$ − 113.090i − 0.677184i −0.940933 0.338592i $$-0.890049\pi$$
0.940933 0.338592i $$-0.109951\pi$$
$$168$$ 0 0
$$169$$ −146.379 −0.866148
$$170$$ 35.7702i 0.210413i
$$171$$ 0 0
$$172$$ −104.171 −0.605645
$$173$$ − 117.615i − 0.679856i −0.940452 0.339928i $$-0.889597\pi$$
0.940452 0.339928i $$-0.110403\pi$$
$$174$$ 0 0
$$175$$ 52.0896 0.297655
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −23.0847 −0.129689
$$179$$ − 231.526i − 1.29344i −0.762727 0.646721i $$-0.776140\pi$$
0.762727 0.646721i $$-0.223860\pi$$
$$180$$ 0 0
$$181$$ −10.8105 −0.0597265 −0.0298633 0.999554i $$-0.509507\pi$$
−0.0298633 + 0.999554i $$0.509507\pi$$
$$182$$ 6.72622i 0.0369572i
$$183$$ 0 0
$$184$$ 64.4339 0.350184
$$185$$ 126.998i 0.686478i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 166.315i 0.884657i
$$189$$ 0 0
$$190$$ −18.2540 −0.0960739
$$191$$ − 49.5203i − 0.259269i −0.991562 0.129634i $$-0.958620\pi$$
0.991562 0.129634i $$-0.0413803\pi$$
$$192$$ 0 0
$$193$$ −8.40418 −0.0435450 −0.0217725 0.999763i $$-0.506931\pi$$
−0.0217725 + 0.999763i $$0.506931\pi$$
$$194$$ − 6.01918i − 0.0310267i
$$195$$ 0 0
$$196$$ −128.792 −0.657104
$$197$$ 125.190i 0.635481i 0.948178 + 0.317741i $$0.102924\pi$$
−0.948178 + 0.317741i $$0.897076\pi$$
$$198$$ 0 0
$$199$$ 255.728 1.28506 0.642532 0.766259i $$-0.277884\pi$$
0.642532 + 0.766259i $$0.277884\pi$$
$$200$$ − 36.8329i − 0.184164i
$$201$$ 0 0
$$202$$ 58.6512 0.290353
$$203$$ 18.3848i 0.0905654i
$$204$$ 0 0
$$205$$ 235.411 1.14835
$$206$$ − 51.8129i − 0.251519i
$$207$$ 0 0
$$208$$ −68.9259 −0.331374
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −149.332 −0.707736 −0.353868 0.935295i $$-0.615134\pi$$
−0.353868 + 0.935295i $$0.615134\pi$$
$$212$$ 302.942i 1.42897i
$$213$$ 0 0
$$214$$ −67.5060 −0.315449
$$215$$ − 92.6789i − 0.431065i
$$216$$ 0 0
$$217$$ 109.091 0.502725
$$218$$ − 20.5075i − 0.0940713i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 138.537i − 0.626866i
$$222$$ 0 0
$$223$$ −211.270 −0.947400 −0.473700 0.880686i $$-0.657082\pi$$
−0.473700 + 0.880686i $$0.657082\pi$$
$$224$$ − 65.0310i − 0.290317i
$$225$$ 0 0
$$226$$ 10.6123 0.0469571
$$227$$ 175.091i 0.771325i 0.922640 + 0.385662i $$0.126027\pi$$
−0.922640 + 0.385662i $$0.873973\pi$$
$$228$$ 0 0
$$229$$ −142.460 −0.622095 −0.311047 0.950394i $$-0.600680\pi$$
−0.311047 + 0.950394i $$0.600680\pi$$
$$230$$ 28.2004i 0.122610i
$$231$$ 0 0
$$232$$ 13.0000 0.0560345
$$233$$ − 264.240i − 1.13408i −0.823692 0.567038i $$-0.808089\pi$$
0.823692 0.567038i $$-0.191911\pi$$
$$234$$ 0 0
$$235$$ −147.968 −0.629650
$$236$$ 359.410i 1.52292i
$$237$$ 0 0
$$238$$ 41.1932 0.173081
$$239$$ 176.997i 0.740573i 0.928918 + 0.370287i $$0.120741\pi$$
−0.928918 + 0.370287i $$0.879259\pi$$
$$240$$ 0 0
$$241$$ 14.6084 0.0606156 0.0303078 0.999541i $$-0.490351\pi$$
0.0303078 + 0.999541i $$0.490351\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 157.093 0.643823
$$245$$ − 114.584i − 0.467690i
$$246$$ 0 0
$$247$$ 70.6976 0.286225
$$248$$ − 77.1392i − 0.311045i
$$249$$ 0 0
$$250$$ 46.8213 0.187285
$$251$$ − 217.159i − 0.865174i −0.901592 0.432587i $$-0.857601\pi$$
0.901592 0.432587i $$-0.142399\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ − 4.48782i − 0.0176686i
$$255$$ 0 0
$$256$$ 178.524 0.697360
$$257$$ − 33.4151i − 0.130020i −0.997885 0.0650099i $$-0.979292\pi$$
0.997885 0.0650099i $$-0.0207079\pi$$
$$258$$ 0 0
$$259$$ 146.252 0.564681
$$260$$ − 63.4719i − 0.244123i
$$261$$ 0 0
$$262$$ −7.25608 −0.0276950
$$263$$ 335.360i 1.27513i 0.770396 + 0.637566i $$0.220059\pi$$
−0.770396 + 0.637566i $$0.779941\pi$$
$$264$$ 0 0
$$265$$ −269.522 −1.01706
$$266$$ 21.0215i 0.0790282i
$$267$$ 0 0
$$268$$ −210.429 −0.785185
$$269$$ − 337.759i − 1.25561i −0.778371 0.627805i $$-0.783954\pi$$
0.778371 0.627805i $$-0.216046\pi$$
$$270$$ 0 0
$$271$$ −436.501 −1.61071 −0.805353 0.592796i $$-0.798024\pi$$
−0.805353 + 0.592796i $$0.798024\pi$$
$$272$$ 422.121i 1.55192i
$$273$$ 0 0
$$274$$ −12.8006 −0.0467176
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −199.130 −0.718881 −0.359440 0.933168i $$-0.617032\pi$$
−0.359440 + 0.933168i $$0.617032\pi$$
$$278$$ − 36.0282i − 0.129598i
$$279$$ 0 0
$$280$$ 38.3649 0.137018
$$281$$ − 250.442i − 0.891253i −0.895219 0.445627i $$-0.852981\pi$$
0.895219 0.445627i $$-0.147019\pi$$
$$282$$ 0 0
$$283$$ −43.0813 −0.152231 −0.0761153 0.997099i $$-0.524252\pi$$
−0.0761153 + 0.997099i $$0.524252\pi$$
$$284$$ − 352.453i − 1.24103i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 271.101i − 0.944604i
$$288$$ 0 0
$$289$$ −559.441 −1.93578
$$290$$ 5.68963i 0.0196194i
$$291$$ 0 0
$$292$$ −423.500 −1.45034
$$293$$ 7.51229i 0.0256392i 0.999918 + 0.0128196i $$0.00408072\pi$$
−0.999918 + 0.0128196i $$0.995919\pi$$
$$294$$ 0 0
$$295$$ −319.760 −1.08393
$$296$$ − 103.416i − 0.349378i
$$297$$ 0 0
$$298$$ 74.9536 0.251522
$$299$$ − 109.220i − 0.365283i
$$300$$ 0 0
$$301$$ −106.730 −0.354584
$$302$$ 75.5835i 0.250276i
$$303$$ 0 0
$$304$$ −215.414 −0.708600
$$305$$ 139.762i 0.458238i
$$306$$ 0 0
$$307$$ 360.275 1.17353 0.586767 0.809756i $$-0.300401\pi$$
0.586767 + 0.809756i $$0.300401\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 33.7610 0.108907
$$311$$ − 415.622i − 1.33641i −0.743979 0.668203i $$-0.767064\pi$$
0.743979 0.668203i $$-0.232936\pi$$
$$312$$ 0 0
$$313$$ −63.2863 −0.202193 −0.101096 0.994877i $$-0.532235\pi$$
−0.101096 + 0.994877i $$0.532235\pi$$
$$314$$ − 42.4727i − 0.135263i
$$315$$ 0 0
$$316$$ 554.969 1.75623
$$317$$ − 422.491i − 1.33278i −0.745604 0.666389i $$-0.767839\pi$$
0.745604 0.666389i $$-0.232161\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 179.615i 0.561297i
$$321$$ 0 0
$$322$$ 32.4758 0.100857
$$323$$ − 432.972i − 1.34047i
$$324$$ 0 0
$$325$$ −62.4341 −0.192105
$$326$$ − 94.8963i − 0.291093i
$$327$$ 0 0
$$328$$ −191.698 −0.584444
$$329$$ 170.401i 0.517936i
$$330$$ 0 0
$$331$$ −233.077 −0.704159 −0.352079 0.935970i $$-0.614525\pi$$
−0.352079 + 0.935970i $$0.614525\pi$$
$$332$$ 170.412i 0.513290i
$$333$$ 0 0
$$334$$ −40.3045 −0.120672
$$335$$ − 187.215i − 0.558851i
$$336$$ 0 0
$$337$$ 622.254 1.84645 0.923226 0.384257i $$-0.125542\pi$$
0.923226 + 0.384257i $$0.125542\pi$$
$$338$$ 52.1686i 0.154345i
$$339$$ 0 0
$$340$$ −388.720 −1.14329
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −326.394 −0.951585
$$344$$ 75.4694i 0.219388i
$$345$$ 0 0
$$346$$ −41.9173 −0.121148
$$347$$ − 648.204i − 1.86802i −0.357241 0.934012i $$-0.616283\pi$$
0.357241 0.934012i $$-0.383717\pi$$
$$348$$ 0 0
$$349$$ 304.125 0.871419 0.435709 0.900087i $$-0.356498\pi$$
0.435709 + 0.900087i $$0.356498\pi$$
$$350$$ − 18.5644i − 0.0530412i
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 250.753i − 0.710350i −0.934800 0.355175i $$-0.884421\pi$$
0.934800 0.355175i $$-0.115579\pi$$
$$354$$ 0 0
$$355$$ 313.571 0.883297
$$356$$ − 250.865i − 0.704676i
$$357$$ 0 0
$$358$$ −82.5145 −0.230487
$$359$$ − 114.425i − 0.318732i −0.987220 0.159366i $$-0.949055\pi$$
0.987220 0.159366i $$-0.0509450\pi$$
$$360$$ 0 0
$$361$$ −140.048 −0.387946
$$362$$ 3.85280i 0.0106431i
$$363$$ 0 0
$$364$$ −73.0948 −0.200810
$$365$$ − 376.780i − 1.03227i
$$366$$ 0 0
$$367$$ 39.9818 0.108942 0.0544711 0.998515i $$-0.482653\pi$$
0.0544711 + 0.998515i $$0.482653\pi$$
$$368$$ 332.791i 0.904323i
$$369$$ 0 0
$$370$$ 45.2615 0.122328
$$371$$ 310.384i 0.836614i
$$372$$ 0 0
$$373$$ 442.965 1.18757 0.593787 0.804622i $$-0.297632\pi$$
0.593787 + 0.804622i $$0.297632\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 120.492 0.320456
$$377$$ − 22.0358i − 0.0584505i
$$378$$ 0 0
$$379$$ −225.413 −0.594758 −0.297379 0.954760i $$-0.596113\pi$$
−0.297379 + 0.954760i $$0.596113\pi$$
$$380$$ − 198.369i − 0.522024i
$$381$$ 0 0
$$382$$ −17.6487 −0.0462009
$$383$$ − 182.924i − 0.477608i −0.971068 0.238804i $$-0.923245\pi$$
0.971068 0.238804i $$-0.0767554\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.99520i 0.00775958i
$$387$$ 0 0
$$388$$ 65.4113 0.168586
$$389$$ − 619.589i − 1.59277i −0.604787 0.796387i $$-0.706742\pi$$
0.604787 0.796387i $$-0.293258\pi$$
$$390$$ 0 0
$$391$$ −668.892 −1.71072
$$392$$ 93.3070i 0.238028i
$$393$$ 0 0
$$394$$ 44.6169 0.113241
$$395$$ 493.745i 1.24999i
$$396$$ 0 0
$$397$$ −45.3649 −0.114269 −0.0571347 0.998366i $$-0.518196\pi$$
−0.0571347 + 0.998366i $$0.518196\pi$$
$$398$$ − 91.1398i − 0.228995i
$$399$$ 0 0
$$400$$ 190.236 0.475590
$$401$$ − 361.330i − 0.901073i −0.892758 0.450536i $$-0.851233\pi$$
0.892758 0.450536i $$-0.148767\pi$$
$$402$$ 0 0
$$403$$ −130.756 −0.324456
$$404$$ 637.371i 1.57765i
$$405$$ 0 0
$$406$$ 6.55222 0.0161385
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −683.744 −1.67175 −0.835873 0.548922i $$-0.815038\pi$$
−0.835873 + 0.548922i $$0.815038\pi$$
$$410$$ − 83.8991i − 0.204632i
$$411$$ 0 0
$$412$$ 563.058 1.36665
$$413$$ 368.238i 0.891618i
$$414$$ 0 0
$$415$$ −151.613 −0.365332
$$416$$ 77.9456i 0.187369i
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 413.509i 0.986895i 0.869775 + 0.493448i $$0.164264\pi$$
−0.869775 + 0.493448i $$0.835736\pi$$
$$420$$ 0 0
$$421$$ −328.460 −0.780189 −0.390095 0.920775i $$-0.627558\pi$$
−0.390095 + 0.920775i $$0.627558\pi$$
$$422$$ 53.2211i 0.126116i
$$423$$ 0 0
$$424$$ 219.475 0.517629
$$425$$ 382.364i 0.899680i
$$426$$ 0 0
$$427$$ 160.952 0.376936
$$428$$ − 733.597i − 1.71401i
$$429$$ 0 0
$$430$$ −33.0302 −0.0768145
$$431$$ − 542.164i − 1.25792i −0.777438 0.628960i $$-0.783481\pi$$
0.777438 0.628960i $$-0.216519\pi$$
$$432$$ 0 0
$$433$$ −716.903 −1.65567 −0.827833 0.560975i $$-0.810426\pi$$
−0.827833 + 0.560975i $$0.810426\pi$$
$$434$$ − 38.8795i − 0.0895840i
$$435$$ 0 0
$$436$$ 222.858 0.511143
$$437$$ − 341.345i − 0.781110i
$$438$$ 0 0
$$439$$ −196.790 −0.448269 −0.224135 0.974558i $$-0.571956\pi$$
−0.224135 + 0.974558i $$0.571956\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −49.3739 −0.111706
$$443$$ − 524.354i − 1.18364i −0.806069 0.591821i $$-0.798409\pi$$
0.806069 0.591821i $$-0.201591\pi$$
$$444$$ 0 0
$$445$$ 223.190 0.501549
$$446$$ 75.2954i 0.168824i
$$447$$ 0 0
$$448$$ 206.846 0.461710
$$449$$ − 313.024i − 0.697159i −0.937279 0.348580i $$-0.886664\pi$$
0.937279 0.348580i $$-0.113336\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 115.325i 0.255145i
$$453$$ 0 0
$$454$$ 62.4013 0.137448
$$455$$ − 65.0310i − 0.142925i
$$456$$ 0 0
$$457$$ −4.06821 −0.00890200 −0.00445100 0.999990i $$-0.501417\pi$$
−0.00445100 + 0.999990i $$0.501417\pi$$
$$458$$ 50.7718i 0.110855i
$$459$$ 0 0
$$460$$ −306.458 −0.666212
$$461$$ 99.7105i 0.216292i 0.994135 + 0.108146i $$0.0344914\pi$$
−0.994135 + 0.108146i $$0.965509\pi$$
$$462$$ 0 0
$$463$$ 326.079 0.704273 0.352137 0.935949i $$-0.385455\pi$$
0.352137 + 0.935949i $$0.385455\pi$$
$$464$$ 67.1429i 0.144704i
$$465$$ 0 0
$$466$$ −94.1734 −0.202089
$$467$$ 422.171i 0.904007i 0.892016 + 0.452003i $$0.149291\pi$$
−0.892016 + 0.452003i $$0.850709\pi$$
$$468$$ 0 0
$$469$$ −215.598 −0.459698
$$470$$ 52.7348i 0.112202i
$$471$$ 0 0
$$472$$ 260.384 0.551661
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −195.126 −0.410791
$$476$$ 447.653i 0.940447i
$$477$$ 0 0
$$478$$ 63.0807 0.131968
$$479$$ 467.797i 0.976611i 0.872673 + 0.488305i $$0.162385\pi$$
−0.872673 + 0.488305i $$0.837615\pi$$
$$480$$ 0 0
$$481$$ −175.297 −0.364443
$$482$$ − 5.20633i − 0.0108015i
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 58.1952i 0.119990i
$$486$$ 0 0
$$487$$ −414.175 −0.850463 −0.425231 0.905085i $$-0.639807\pi$$
−0.425231 + 0.905085i $$0.639807\pi$$
$$488$$ − 113.810i − 0.233217i
$$489$$ 0 0
$$490$$ −40.8371 −0.0833410
$$491$$ − 176.284i − 0.359031i −0.983755 0.179516i $$-0.942547\pi$$
0.983755 0.179516i $$-0.0574530\pi$$
$$492$$ 0 0
$$493$$ −134.954 −0.273740
$$494$$ − 25.1962i − 0.0510044i
$$495$$ 0 0
$$496$$ 398.411 0.803249
$$497$$ − 361.110i − 0.726580i
$$498$$ 0 0
$$499$$ 605.317 1.21306 0.606530 0.795061i $$-0.292561\pi$$
0.606530 + 0.795061i $$0.292561\pi$$
$$500$$ 508.813i 1.01763i
$$501$$ 0 0
$$502$$ −77.3941 −0.154171
$$503$$ − 180.872i − 0.359587i −0.983704 0.179793i $$-0.942457\pi$$
0.983704 0.179793i $$-0.0575429\pi$$
$$504$$ 0 0
$$505$$ −567.057 −1.12288
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 48.7698 0.0960035
$$509$$ − 23.8193i − 0.0467962i −0.999726 0.0233981i $$-0.992551\pi$$
0.999726 0.0233981i $$-0.00744853\pi$$
$$510$$ 0 0
$$511$$ −433.903 −0.849126
$$512$$ − 400.150i − 0.781543i
$$513$$ 0 0
$$514$$ −11.9089 −0.0231691
$$515$$ 500.942i 0.972704i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ − 52.1235i − 0.100624i
$$519$$ 0 0
$$520$$ −45.9839 −0.0884305
$$521$$ 198.640i 0.381267i 0.981661 + 0.190633i $$0.0610542\pi$$
−0.981661 + 0.190633i $$0.938946\pi$$
$$522$$ 0 0
$$523$$ 1000.04 1.91212 0.956059 0.293173i $$-0.0947113\pi$$
0.956059 + 0.293173i $$0.0947113\pi$$
$$524$$ − 78.8529i − 0.150483i
$$525$$ 0 0
$$526$$ 119.520 0.227225
$$527$$ 800.786i 1.51952i
$$528$$ 0 0
$$529$$ 1.66120 0.00314027
$$530$$ 96.0561i 0.181238i
$$531$$ 0 0
$$532$$ −228.444 −0.429405
$$533$$ 324.940i 0.609643i
$$534$$ 0 0
$$535$$ 652.668 1.21994
$$536$$ 152.451i 0.284424i
$$537$$ 0 0
$$538$$ −120.375 −0.223746
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 491.351 0.908227 0.454113 0.890944i $$-0.349956\pi$$
0.454113 + 0.890944i $$0.349956\pi$$
$$542$$ 155.566i 0.287023i
$$543$$ 0 0
$$544$$ 477.361 0.877502
$$545$$ 198.273i 0.363803i
$$546$$ 0 0
$$547$$ 195.142 0.356750 0.178375 0.983963i $$-0.442916\pi$$
0.178375 + 0.983963i $$0.442916\pi$$
$$548$$ − 139.106i − 0.253843i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 68.8688i − 0.124989i
$$552$$ 0 0
$$553$$ 568.601 1.02821
$$554$$ 70.9687i 0.128102i
$$555$$ 0 0
$$556$$ 391.524 0.704179
$$557$$ 615.014i 1.10416i 0.833793 + 0.552078i $$0.186165\pi$$
−0.833793 + 0.552078i $$0.813835\pi$$
$$558$$ 0 0
$$559$$ 127.925 0.228847
$$560$$ 198.149i 0.353837i
$$561$$ 0 0
$$562$$ −89.2561 −0.158819
$$563$$ − 328.204i − 0.582955i −0.956578 0.291477i $$-0.905853\pi$$
0.956578 0.291477i $$-0.0941468\pi$$
$$564$$ 0 0
$$565$$ −102.603 −0.181598
$$566$$ 15.3539i 0.0271270i
$$567$$ 0 0
$$568$$ −255.344 −0.449549
$$569$$ − 830.739i − 1.46000i −0.683448 0.729999i $$-0.739520\pi$$
0.683448 0.729999i $$-0.260480\pi$$
$$570$$ 0 0
$$571$$ 1048.66 1.83652 0.918262 0.395973i $$-0.129593\pi$$
0.918262 + 0.395973i $$0.129593\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −96.6189 −0.168326
$$575$$ 301.447i 0.524256i
$$576$$ 0 0
$$577$$ −518.202 −0.898096 −0.449048 0.893508i $$-0.648237\pi$$
−0.449048 + 0.893508i $$0.648237\pi$$
$$578$$ 199.382i 0.344951i
$$579$$ 0 0
$$580$$ −61.8300 −0.106603
$$581$$ 174.598i 0.300513i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 306.816i 0.525370i
$$585$$ 0 0
$$586$$ 2.67734 0.00456883
$$587$$ 1016.52i 1.73173i 0.500282 + 0.865863i $$0.333230\pi$$
−0.500282 + 0.865863i $$0.666770\pi$$
$$588$$ 0 0
$$589$$ −408.652 −0.693807
$$590$$ 113.961i 0.193153i
$$591$$ 0 0
$$592$$ 534.127 0.902242
$$593$$ − 1073.37i − 1.81007i −0.425341 0.905033i $$-0.639846\pi$$
0.425341 0.905033i $$-0.360154\pi$$
$$594$$ 0 0
$$595$$ −398.268 −0.669358
$$596$$ 814.532i 1.36666i
$$597$$ 0 0
$$598$$ −38.9252 −0.0650923
$$599$$ − 767.105i − 1.28064i −0.768107 0.640322i $$-0.778801\pi$$
0.768107 0.640322i $$-0.221199\pi$$
$$600$$ 0 0
$$601$$ −752.890 −1.25273 −0.626365 0.779530i $$-0.715458\pi$$
−0.626365 + 0.779530i $$0.715458\pi$$
$$602$$ 38.0379i 0.0631858i
$$603$$ 0 0
$$604$$ −821.376 −1.35989
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 503.535 0.829548 0.414774 0.909925i $$-0.363861\pi$$
0.414774 + 0.909925i $$0.363861\pi$$
$$608$$ 243.604i 0.400664i
$$609$$ 0 0
$$610$$ 49.8105 0.0816566
$$611$$ − 204.241i − 0.334273i
$$612$$ 0 0
$$613$$ −748.338 −1.22078 −0.610390 0.792101i $$-0.708987\pi$$
−0.610390 + 0.792101i $$0.708987\pi$$
$$614$$ − 128.400i − 0.209120i
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 634.767i − 1.02880i −0.857552 0.514398i $$-0.828016\pi$$
0.857552 0.514398i $$-0.171984\pi$$
$$618$$ 0 0
$$619$$ −684.329 −1.10554 −0.552769 0.833334i $$-0.686429\pi$$
−0.552769 + 0.833334i $$0.686429\pi$$
$$620$$ 366.886i 0.591751i
$$621$$ 0 0
$$622$$ −148.125 −0.238143
$$623$$ − 257.027i − 0.412563i
$$624$$ 0 0
$$625$$ −124.506 −0.199210
$$626$$ 22.5549i 0.0360301i
$$627$$ 0 0
$$628$$ 461.556 0.734963
$$629$$ 1073.57i 1.70678i
$$630$$ 0 0
$$631$$ −266.046 −0.421627 −0.210813 0.977526i $$-0.567611\pi$$
−0.210813 + 0.977526i $$0.567611\pi$$
$$632$$ − 402.061i − 0.636173i
$$633$$ 0 0
$$634$$ −150.573 −0.237497
$$635$$ 43.3895i 0.0683300i
$$636$$ 0 0
$$637$$ 158.161 0.248291
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 289.893 0.452957
$$641$$ − 450.083i − 0.702158i −0.936346 0.351079i $$-0.885815\pi$$
0.936346 0.351079i $$-0.114185\pi$$
$$642$$ 0 0
$$643$$ 142.988 0.222376 0.111188 0.993799i $$-0.464534\pi$$
0.111188 + 0.993799i $$0.464534\pi$$
$$644$$ 352.919i 0.548011i
$$645$$ 0 0
$$646$$ −154.309 −0.238868
$$647$$ 796.539i 1.23113i 0.788088 + 0.615563i $$0.211071\pi$$
−0.788088 + 0.615563i $$0.788929\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 22.2511i 0.0342325i
$$651$$ 0 0
$$652$$ 1031.25 1.58167
$$653$$ 54.6582i 0.0837032i 0.999124 + 0.0418516i $$0.0133257\pi$$
−0.999124 + 0.0418516i $$0.986674\pi$$
$$654$$ 0 0
$$655$$ 70.1539 0.107105
$$656$$ − 990.087i − 1.50928i
$$657$$ 0 0
$$658$$ 60.7298 0.0922946
$$659$$ − 333.527i − 0.506111i −0.967452 0.253056i $$-0.918564\pi$$
0.967452 0.253056i $$-0.0814356\pi$$
$$660$$ 0 0
$$661$$ −489.397 −0.740389 −0.370195 0.928954i $$-0.620709\pi$$
−0.370195 + 0.928954i $$0.620709\pi$$
$$662$$ 83.0671i 0.125479i
$$663$$ 0 0
$$664$$ 123.460 0.185933
$$665$$ − 203.242i − 0.305627i
$$666$$ 0 0
$$667$$ −106.394 −0.159512
$$668$$ − 437.994i − 0.655680i
$$669$$ 0 0
$$670$$ −66.7223 −0.0995856
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 699.341 1.03914 0.519570 0.854428i $$-0.326092\pi$$
0.519570 + 0.854428i $$0.326092\pi$$
$$674$$ − 221.768i − 0.329032i
$$675$$ 0 0
$$676$$ −566.923 −0.838644
$$677$$ − 389.391i − 0.575171i −0.957755 0.287585i $$-0.907148\pi$$
0.957755 0.287585i $$-0.0928525\pi$$
$$678$$ 0 0
$$679$$ 67.0180 0.0987011
$$680$$ 281.618i 0.414144i
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 332.412i 0.486693i 0.969939 + 0.243347i $$0.0782453\pi$$
−0.969939 + 0.243347i $$0.921755\pi$$
$$684$$ 0 0
$$685$$ 123.760 0.180672
$$686$$ 116.325i 0.169570i
$$687$$ 0 0
$$688$$ −389.787 −0.566551
$$689$$ − 372.024i − 0.539947i
$$690$$ 0 0
$$691$$ 873.853 1.26462 0.632310 0.774715i $$-0.282107\pi$$
0.632310 + 0.774715i $$0.282107\pi$$
$$692$$ − 455.521i − 0.658267i
$$693$$ 0 0
$$694$$ −231.016 −0.332876
$$695$$ 348.331i 0.501196i
$$696$$ 0 0
$$697$$ 1990.02 2.85512
$$698$$ − 108.388i − 0.155284i
$$699$$ 0 0
$$700$$ 201.742 0.288203
$$701$$ − 630.096i − 0.898853i −0.893317 0.449426i $$-0.851628\pi$$
0.893317 0.449426i $$-0.148372\pi$$
$$702$$ 0 0
$$703$$ −547.856 −0.779312
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −89.3670 −0.126582
$$707$$ 653.027i 0.923660i
$$708$$ 0 0
$$709$$ −1007.61 −1.42117 −0.710587 0.703609i $$-0.751571\pi$$
−0.710587 + 0.703609i $$0.751571\pi$$
$$710$$ − 111.755i − 0.157401i
$$711$$ 0 0
$$712$$ −181.745 −0.255260
$$713$$ 631.321i 0.885443i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ − 896.697i − 1.25237i
$$717$$ 0 0
$$718$$ −40.7803 −0.0567971
$$719$$ 170.989i 0.237815i 0.992905 + 0.118907i $$0.0379391\pi$$
−0.992905 + 0.118907i $$0.962061\pi$$
$$720$$ 0 0
$$721$$ 576.889 0.800124
$$722$$ 49.9124i 0.0691308i
$$723$$ 0 0
$$724$$ −41.8689 −0.0578300
$$725$$ 60.8191i 0.0838884i
$$726$$ 0 0
$$727$$ 408.204 0.561490 0.280745 0.959782i $$-0.409418\pi$$
0.280745 + 0.959782i $$0.409418\pi$$
$$728$$ 52.9554i 0.0727409i
$$729$$ 0 0
$$730$$ −134.282 −0.183948
$$731$$ − 783.451i − 1.07175i
$$732$$ 0 0
$$733$$ 1178.02 1.60713 0.803563 0.595220i $$-0.202935\pi$$
0.803563 + 0.595220i $$0.202935\pi$$
$$734$$ − 14.2493i − 0.0194132i
$$735$$ 0 0
$$736$$ 376.340 0.511332
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 938.068 1.26938 0.634688 0.772769i $$-0.281129\pi$$
0.634688 + 0.772769i $$0.281129\pi$$
$$740$$ 491.863i 0.664679i
$$741$$ 0 0
$$742$$ 110.619 0.149082
$$743$$ 1050.56i 1.41394i 0.707242 + 0.706971i $$0.249939\pi$$
−0.707242 + 0.706971i $$0.750061\pi$$
$$744$$ 0 0
$$745$$ −724.673 −0.972716
$$746$$ − 157.870i − 0.211622i
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 751.617i − 1.00349i
$$750$$ 0 0
$$751$$ 1074.61 1.43091 0.715456 0.698658i $$-0.246219\pi$$
0.715456 + 0.698658i $$0.246219\pi$$
$$752$$ 622.320i 0.827553i
$$753$$ 0 0
$$754$$ −7.85344 −0.0104157
$$755$$ − 730.763i − 0.967898i
$$756$$ 0 0
$$757$$ −518.488 −0.684924 −0.342462 0.939532i $$-0.611261\pi$$
−0.342462 + 0.939532i $$0.611261\pi$$
$$758$$ 80.3359i 0.105984i
$$759$$ 0 0
$$760$$ −143.714 −0.189097
$$761$$ 1304.01i 1.71355i 0.515690 + 0.856775i $$0.327536\pi$$
−0.515690 + 0.856775i $$0.672464\pi$$
$$762$$ 0 0
$$763$$ 228.333 0.299256
$$764$$ − 191.791i − 0.251036i
$$765$$ 0 0
$$766$$ −65.1930 −0.0851083
$$767$$ − 441.367i − 0.575446i
$$768$$ 0 0
$$769$$ 141.048 0.183418 0.0917088 0.995786i $$-0.470767\pi$$
0.0917088 + 0.995786i $$0.470767\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −32.5492 −0.0421622
$$773$$ 234.469i 0.303323i 0.988432 + 0.151661i $$0.0484624\pi$$
−0.988432 + 0.151661i $$0.951538\pi$$
$$774$$ 0 0
$$775$$ 360.887 0.465661
$$776$$ − 47.3889i − 0.0610682i
$$777$$ 0 0
$$778$$ −220.818 −0.283828
$$779$$ 1015.54i 1.30364i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 238.389i 0.304845i
$$783$$ 0 0
$$784$$ −481.915 −0.614688
$$785$$ 410.638i 0.523106i
$$786$$ 0 0
$$787$$ −1454.04 −1.84757 −0.923784 0.382915i $$-0.874920\pi$$
−0.923784 + 0.382915i $$0.874920\pi$$
$$788$$ 484.858i 0.615302i
$$789$$ 0 0
$$790$$ 175.968 0.222744
$$791$$ 118.158i 0.149378i
$$792$$ 0 0
$$793$$ −192.915 −0.243273
$$794$$ 16.1678i 0.0203624i
$$795$$ 0 0
$$796$$ 990.429 1.24426
$$797$$ 325.332i 0.408195i 0.978951 + 0.204098i $$0.0654260\pi$$
−0.978951 + 0.204098i $$0.934574\pi$$
$$798$$ 0 0
$$799$$ −1250.83 −1.56549
$$800$$ − 215.130i − 0.268913i
$$801$$ 0 0
$$802$$ −128.776 −0.160568
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −313.985 −0.390044
$$806$$ 46.6006i 0.0578171i
$$807$$ 0 0
$$808$$ 461.760 0.571485
$$809$$ 894.483i 1.10567i 0.833292 + 0.552833i $$0.186453\pi$$
−0.833292 + 0.552833i $$0.813547\pi$$
$$810$$ 0 0
$$811$$ 92.4668 0.114016 0.0570079 0.998374i $$-0.481844\pi$$
0.0570079 + 0.998374i $$0.481844\pi$$
$$812$$ 71.2039i 0.0876896i
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 917.485i 1.12575i
$$816$$ 0 0
$$817$$ 399.806 0.489359
$$818$$ 243.682i 0.297900i
$$819$$ 0 0
$$820$$ 911.743 1.11188
$$821$$ 616.078i 0.750399i 0.926944 + 0.375200i $$0.122426\pi$$
−0.926944 + 0.375200i $$0.877574\pi$$
$$822$$ 0 0
$$823$$ 1584.99 1.92587 0.962937 0.269727i $$-0.0869336\pi$$
0.962937 + 0.269727i $$0.0869336\pi$$
$$824$$ − 407.922i − 0.495051i
$$825$$ 0 0
$$826$$ 131.238 0.158884
$$827$$ 776.862i 0.939374i 0.882833 + 0.469687i $$0.155633\pi$$
−0.882833 + 0.469687i $$0.844367\pi$$
$$828$$ 0 0
$$829$$ −1047.54 −1.26361 −0.631807 0.775126i $$-0.717687\pi$$
−0.631807 + 0.775126i $$0.717687\pi$$
$$830$$ 54.0338i 0.0651010i
$$831$$ 0 0
$$832$$ −247.924 −0.297986
$$833$$ − 968.624i − 1.16281i
$$834$$ 0 0
$$835$$ 389.675 0.466677
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 147.372 0.175862
$$839$$ − 197.206i − 0.235049i −0.993070 0.117524i $$-0.962504\pi$$
0.993070 0.117524i $$-0.0374958\pi$$
$$840$$ 0 0
$$841$$ 819.534 0.974476
$$842$$ 117.061i 0.139027i
$$843$$ 0 0
$$844$$ −578.362 −0.685263
$$845$$ − 504.381i − 0.596901i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 1133.55i 1.33673i
$$849$$ 0 0
$$850$$ 136.272 0.160320
$$851$$ 846.375i 0.994566i
$$852$$ 0 0
$$853$$ 1389.41 1.62885 0.814423 0.580271i $$-0.197053\pi$$
0.814423 + 0.580271i $$0.197053\pi$$
$$854$$ − 57.3622i − 0.0671688i
$$855$$ 0 0
$$856$$ −531.474 −0.620881
$$857$$ − 218.943i − 0.255476i −0.991808 0.127738i $$-0.959228\pi$$
0.991808 0.127738i $$-0.0407717\pi$$
$$858$$ 0 0
$$859$$ 679.659 0.791221 0.395611 0.918418i $$-0.370533\pi$$
0.395611 + 0.918418i $$0.370533\pi$$
$$860$$ − 358.944i − 0.417377i
$$861$$ 0 0
$$862$$ −193.224 −0.224158
$$863$$ 719.743i 0.834002i 0.908906 + 0.417001i $$0.136919\pi$$
−0.908906 + 0.417001i $$0.863081\pi$$
$$864$$ 0 0
$$865$$ 405.268 0.468518
$$866$$ 255.500i 0.295035i
$$867$$ 0 0
$$868$$ 422.509 0.486761
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 258.414 0.296687
$$872$$ − 161.456i − 0.185155i
$$873$$ 0 0
$$874$$ −121.653 −0.139191
$$875$$ 521.312i 0.595785i
$$876$$ 0 0
$$877$$ −920.439 −1.04953 −0.524766 0.851247i $$-0.675847\pi$$
−0.524766 + 0.851247i $$0.675847\pi$$
$$878$$ 70.1349i 0.0798802i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 309.389i − 0.351180i −0.984463 0.175590i $$-0.943817\pi$$
0.984463 0.175590i $$-0.0561832\pi$$
$$882$$ 0 0
$$883$$ −856.028 −0.969454 −0.484727 0.874665i $$-0.661081\pi$$
−0.484727 + 0.874665i $$0.661081\pi$$
$$884$$ − 536.553i − 0.606960i
$$885$$ 0 0
$$886$$ −186.877 −0.210922
$$887$$ − 508.096i − 0.572826i −0.958106 0.286413i $$-0.907537\pi$$
0.958106 0.286413i $$-0.0924629\pi$$
$$888$$ 0 0
$$889$$ 49.9677 0.0562067
$$890$$ − 79.5434i − 0.0893746i
$$891$$ 0 0
$$892$$ −818.246 −0.917316
$$893$$ − 638.316i − 0.714799i
$$894$$ 0 0
$$895$$ 797.774 0.891368
$$896$$ − 333.843i − 0.372592i
$$897$$ 0 0
$$898$$ −111.560 −0.124232
$$899$$ 127.373i 0.141684i
$$900$$ 0 0
$$901$$ −2278.38 −2.52872
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 83.5505 0.0924231
$$905$$ − 37.2500i − 0.0411602i
$$906$$ 0 0
$$907$$ −509.667 −0.561927 −0.280963 0.959719i $$-0.590654\pi$$
−0.280963 + 0.959719i $$0.590654\pi$$
$$908$$ 678.124i 0.746832i
$$909$$ 0 0
$$910$$ −23.1767 −0.0254689
$$911$$ 451.807i 0.495946i 0.968767 + 0.247973i $$0.0797645\pi$$
−0.968767 + 0.247973i $$0.920236\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 1.44989i 0.00158631i
$$915$$ 0 0
$$916$$ −551.744 −0.602341
$$917$$ − 80.7898i − 0.0881023i
$$918$$ 0 0
$$919$$ 1645.29 1.79030 0.895152 0.445762i $$-0.147067\pi$$
0.895152 + 0.445762i $$0.147067\pi$$
$$920$$ 222.021i 0.241327i
$$921$$ 0 0
$$922$$ 35.5362 0.0385426
$$923$$ 432.824i 0.468932i
$$924$$ 0 0
$$925$$ 483.820 0.523049
$$926$$ − 116.212i − 0.125499i
$$927$$ 0 0
$$928$$ 75.9293 0.0818204
$$929$$ 1005.87i 1.08274i 0.840783 + 0.541372i $$0.182095\pi$$
−0.840783 + 0.541372i $$0.817905\pi$$
$$930$$ 0 0
$$931$$ 494.303 0.530937
$$932$$ − 1023.40i − 1.09806i
$$933$$ 0 0
$$934$$ 150.459 0.161091
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −209.683 −0.223781 −0.111891 0.993721i $$-0.535691\pi$$
−0.111891 + 0.993721i $$0.535691\pi$$
$$938$$ 76.8380i 0.0819168i
$$939$$ 0 0
$$940$$ −573.077 −0.609656
$$941$$ − 601.375i − 0.639081i −0.947573 0.319541i $$-0.896471\pi$$
0.947573 0.319541i $$-0.103529\pi$$
$$942$$ 0 0
$$943$$ 1568.89 1.66372
$$944$$ 1344.84i 1.42462i
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 18.4760i − 0.0195101i −0.999952 0.00975504i $$-0.996895\pi$$
0.999952 0.00975504i $$-0.00310517\pi$$
$$948$$ 0 0
$$949$$ 520.073 0.548022
$$950$$ 69.5416i 0.0732017i
$$951$$ 0 0
$$952$$ 324.314 0.340666
$$953$$ 1285.82i 1.34923i 0.738169 + 0.674616i $$0.235691\pi$$
−0.738169 + 0.674616i $$0.764309\pi$$
$$954$$ 0 0
$$955$$ 170.633 0.178673
$$956$$ 685.506i 0.717057i
$$957$$ 0 0
$$958$$ 166.720 0.174029
$$959$$ − 142.523i − 0.148616i
$$960$$ 0 0
$$961$$ −205.194 −0.213521
$$962$$ 62.4747i 0.0649426i
$$963$$ 0 0
$$964$$ 56.5780 0.0586908
$$965$$ − 28.9584i − 0.0300088i
$$966$$ 0 0
$$967$$ 401.260 0.414953 0.207477 0.978240i $$-0.433475\pi$$
0.207477 + 0.978240i $$0.433475\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 20.7404 0.0213819
$$971$$ 1310.59i 1.34973i 0.737942 + 0.674864i $$0.235798\pi$$
−0.737942 + 0.674864i $$0.764202\pi$$
$$972$$ 0 0
$$973$$ 401.141 0.412272
$$974$$ 147.610i 0.151550i
$$975$$ 0 0
$$976$$ 587.810 0.602264
$$977$$ − 1220.43i − 1.24916i −0.780960 0.624581i $$-0.785270\pi$$
0.780960 0.624581i $$-0.214730\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ − 443.782i − 0.452839i
$$981$$ 0 0
$$982$$ −62.8266 −0.0639782
$$983$$ − 89.1639i − 0.0907059i −0.998971 0.0453530i $$-0.985559\pi$$
0.998971 0.0453530i $$-0.0144412\pi$$
$$984$$ 0 0
$$985$$ −431.369 −0.437938
$$986$$ 48.0967i 0.0487796i
$$987$$ 0 0
$$988$$ 273.810 0.277136
$$989$$ − 617.655i − 0.624525i
$$990$$ 0 0
$$991$$ 48.9921 0.0494370 0.0247185 0.999694i $$-0.492131\pi$$
0.0247185 + 0.999694i $$0.492131\pi$$
$$992$$ − 450.548i − 0.454181i
$$993$$ 0 0
$$994$$ −128.698 −0.129474
$$995$$ 881.166i 0.885594i
$$996$$ 0 0
$$997$$ −209.770 −0.210401 −0.105201 0.994451i $$-0.533549\pi$$
−0.105201 + 0.994451i $$0.533549\pi$$
$$998$$ − 215.731i − 0.216163i
$$999$$ 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.h.485.4 yes 8
3.2 odd 2 inner 1089.3.b.h.485.5 yes 8
11.10 odd 2 inner 1089.3.b.h.485.6 yes 8
33.32 even 2 inner 1089.3.b.h.485.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
1089.3.b.h.485.3 8 33.32 even 2 inner
1089.3.b.h.485.4 yes 8 1.1 even 1 trivial
1089.3.b.h.485.5 yes 8 3.2 odd 2 inner
1089.3.b.h.485.6 yes 8 11.10 odd 2 inner