# Properties

 Label 4332.2.a.o.1.1 Level $4332$ Weight $2$ Character 4332.1 Self dual yes Analytic conductor $34.591$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4332 = 2^{2} \cdot 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4332.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.5911941556$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 228) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.87939$$ of defining polynomial Character $$\chi$$ $$=$$ 4332.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -0.879385 q^{5} -2.18479 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -0.879385 q^{5} -2.18479 q^{7} +1.00000 q^{9} -0.162504 q^{11} +0.758770 q^{13} -0.879385 q^{15} +3.10607 q^{17} -2.18479 q^{21} +1.29086 q^{23} -4.22668 q^{25} +1.00000 q^{27} -6.51754 q^{29} +0.958111 q^{31} -0.162504 q^{33} +1.92127 q^{35} -1.16250 q^{37} +0.758770 q^{39} -10.5963 q^{41} -0.177052 q^{43} -0.879385 q^{45} -3.46791 q^{47} -2.22668 q^{49} +3.10607 q^{51} +8.33275 q^{53} +0.142903 q^{55} -2.78106 q^{59} +7.22668 q^{61} -2.18479 q^{63} -0.667252 q^{65} +1.06418 q^{67} +1.29086 q^{69} +5.08378 q^{71} +2.24897 q^{73} -4.22668 q^{75} +0.355037 q^{77} -5.24123 q^{79} +1.00000 q^{81} -17.1138 q^{83} -2.73143 q^{85} -6.51754 q^{87} +15.6236 q^{89} -1.65776 q^{91} +0.958111 q^{93} -9.41921 q^{97} -0.162504 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 9 q^{13} + 3 q^{15} - 3 q^{17} - 3 q^{21} - 12 q^{23} - 6 q^{25} + 3 q^{27} + 3 q^{29} + 6 q^{31} - 3 q^{33} - 3 q^{35} - 6 q^{37} - 9 q^{39} - 18 q^{41} - 21 q^{43} + 3 q^{45} - 15 q^{47} - 3 q^{51} + 6 q^{53} + 9 q^{59} + 15 q^{61} - 3 q^{63} - 21 q^{65} - 6 q^{67} - 12 q^{69} + 9 q^{71} - 6 q^{73} - 6 q^{75} - 24 q^{77} - 27 q^{79} + 3 q^{81} - 15 q^{83} - 18 q^{85} + 3 q^{87} + 12 q^{89} + 9 q^{91} + 6 q^{93} + 6 q^{97} - 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 - 9 * q^13 + 3 * q^15 - 3 * q^17 - 3 * q^21 - 12 * q^23 - 6 * q^25 + 3 * q^27 + 3 * q^29 + 6 * q^31 - 3 * q^33 - 3 * q^35 - 6 * q^37 - 9 * q^39 - 18 * q^41 - 21 * q^43 + 3 * q^45 - 15 * q^47 - 3 * q^51 + 6 * q^53 + 9 * q^59 + 15 * q^61 - 3 * q^63 - 21 * q^65 - 6 * q^67 - 12 * q^69 + 9 * q^71 - 6 * q^73 - 6 * q^75 - 24 * q^77 - 27 * q^79 + 3 * q^81 - 15 * q^83 - 18 * q^85 + 3 * q^87 + 12 * q^89 + 9 * q^91 + 6 * q^93 + 6 * q^97 - 3 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −0.879385 −0.393273 −0.196637 0.980476i $$-0.563002\pi$$
−0.196637 + 0.980476i $$0.563002\pi$$
$$6$$ 0 0
$$7$$ −2.18479 −0.825774 −0.412887 0.910782i $$-0.635480\pi$$
−0.412887 + 0.910782i $$0.635480\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −0.162504 −0.0489967 −0.0244984 0.999700i $$-0.507799\pi$$
−0.0244984 + 0.999700i $$0.507799\pi$$
$$12$$ 0 0
$$13$$ 0.758770 0.210445 0.105223 0.994449i $$-0.466445\pi$$
0.105223 + 0.994449i $$0.466445\pi$$
$$14$$ 0 0
$$15$$ −0.879385 −0.227056
$$16$$ 0 0
$$17$$ 3.10607 0.753332 0.376666 0.926349i $$-0.377070\pi$$
0.376666 + 0.926349i $$0.377070\pi$$
$$18$$ 0 0
$$19$$ 0 0
$$20$$ 0 0
$$21$$ −2.18479 −0.476761
$$22$$ 0 0
$$23$$ 1.29086 0.269163 0.134581 0.990903i $$-0.457031\pi$$
0.134581 + 0.990903i $$0.457031\pi$$
$$24$$ 0 0
$$25$$ −4.22668 −0.845336
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −6.51754 −1.21028 −0.605138 0.796120i $$-0.706882\pi$$
−0.605138 + 0.796120i $$0.706882\pi$$
$$30$$ 0 0
$$31$$ 0.958111 0.172082 0.0860409 0.996292i $$-0.472578\pi$$
0.0860409 + 0.996292i $$0.472578\pi$$
$$32$$ 0 0
$$33$$ −0.162504 −0.0282883
$$34$$ 0 0
$$35$$ 1.92127 0.324755
$$36$$ 0 0
$$37$$ −1.16250 −0.191114 −0.0955572 0.995424i $$-0.530463\pi$$
−0.0955572 + 0.995424i $$0.530463\pi$$
$$38$$ 0 0
$$39$$ 0.758770 0.121501
$$40$$ 0 0
$$41$$ −10.5963 −1.65486 −0.827429 0.561570i $$-0.810198\pi$$
−0.827429 + 0.561570i $$0.810198\pi$$
$$42$$ 0 0
$$43$$ −0.177052 −0.0270001 −0.0135001 0.999909i $$-0.504297\pi$$
−0.0135001 + 0.999909i $$0.504297\pi$$
$$44$$ 0 0
$$45$$ −0.879385 −0.131091
$$46$$ 0 0
$$47$$ −3.46791 −0.505847 −0.252923 0.967486i $$-0.581392\pi$$
−0.252923 + 0.967486i $$0.581392\pi$$
$$48$$ 0 0
$$49$$ −2.22668 −0.318097
$$50$$ 0 0
$$51$$ 3.10607 0.434936
$$52$$ 0 0
$$53$$ 8.33275 1.14459 0.572296 0.820047i $$-0.306053\pi$$
0.572296 + 0.820047i $$0.306053\pi$$
$$54$$ 0 0
$$55$$ 0.142903 0.0192691
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −2.78106 −0.362063 −0.181032 0.983477i $$-0.557944\pi$$
−0.181032 + 0.983477i $$0.557944\pi$$
$$60$$ 0 0
$$61$$ 7.22668 0.925282 0.462641 0.886546i $$-0.346902\pi$$
0.462641 + 0.886546i $$0.346902\pi$$
$$62$$ 0 0
$$63$$ −2.18479 −0.275258
$$64$$ 0 0
$$65$$ −0.667252 −0.0827624
$$66$$ 0 0
$$67$$ 1.06418 0.130010 0.0650050 0.997885i $$-0.479294\pi$$
0.0650050 + 0.997885i $$0.479294\pi$$
$$68$$ 0 0
$$69$$ 1.29086 0.155401
$$70$$ 0 0
$$71$$ 5.08378 0.603333 0.301667 0.953413i $$-0.402457\pi$$
0.301667 + 0.953413i $$0.402457\pi$$
$$72$$ 0 0
$$73$$ 2.24897 0.263222 0.131611 0.991301i $$-0.457985\pi$$
0.131611 + 0.991301i $$0.457985\pi$$
$$74$$ 0 0
$$75$$ −4.22668 −0.488055
$$76$$ 0 0
$$77$$ 0.355037 0.0404602
$$78$$ 0 0
$$79$$ −5.24123 −0.589684 −0.294842 0.955546i $$-0.595267\pi$$
−0.294842 + 0.955546i $$0.595267\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −17.1138 −1.87848 −0.939242 0.343255i $$-0.888470\pi$$
−0.939242 + 0.343255i $$0.888470\pi$$
$$84$$ 0 0
$$85$$ −2.73143 −0.296265
$$86$$ 0 0
$$87$$ −6.51754 −0.698754
$$88$$ 0 0
$$89$$ 15.6236 1.65610 0.828050 0.560655i $$-0.189451\pi$$
0.828050 + 0.560655i $$0.189451\pi$$
$$90$$ 0 0
$$91$$ −1.65776 −0.173780
$$92$$ 0 0
$$93$$ 0.958111 0.0993515
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −9.41921 −0.956376 −0.478188 0.878257i $$-0.658706\pi$$
−0.478188 + 0.878257i $$0.658706\pi$$
$$98$$ 0 0
$$99$$ −0.162504 −0.0163322
$$100$$ 0 0
$$101$$ −6.30541 −0.627411 −0.313706 0.949520i $$-0.601571\pi$$
−0.313706 + 0.949520i $$0.601571\pi$$
$$102$$ 0 0
$$103$$ −8.61587 −0.848947 −0.424473 0.905440i $$-0.639541\pi$$
−0.424473 + 0.905440i $$0.639541\pi$$
$$104$$ 0 0
$$105$$ 1.92127 0.187497
$$106$$ 0 0
$$107$$ −13.1480 −1.27106 −0.635530 0.772076i $$-0.719219\pi$$
−0.635530 + 0.772076i $$0.719219\pi$$
$$108$$ 0 0
$$109$$ −10.0496 −0.962580 −0.481290 0.876561i $$-0.659832\pi$$
−0.481290 + 0.876561i $$0.659832\pi$$
$$110$$ 0 0
$$111$$ −1.16250 −0.110340
$$112$$ 0 0
$$113$$ 8.04458 0.756770 0.378385 0.925648i $$-0.376480\pi$$
0.378385 + 0.925648i $$0.376480\pi$$
$$114$$ 0 0
$$115$$ −1.13516 −0.105854
$$116$$ 0 0
$$117$$ 0.758770 0.0701484
$$118$$ 0 0
$$119$$ −6.78611 −0.622082
$$120$$ 0 0
$$121$$ −10.9736 −0.997599
$$122$$ 0 0
$$123$$ −10.5963 −0.955433
$$124$$ 0 0
$$125$$ 8.11381 0.725721
$$126$$ 0 0
$$127$$ 20.0496 1.77912 0.889558 0.456821i $$-0.151012\pi$$
0.889558 + 0.456821i $$0.151012\pi$$
$$128$$ 0 0
$$129$$ −0.177052 −0.0155885
$$130$$ 0 0
$$131$$ −15.9094 −1.39001 −0.695006 0.719004i $$-0.744598\pi$$
−0.695006 + 0.719004i $$0.744598\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −0.879385 −0.0756854
$$136$$ 0 0
$$137$$ −5.04458 −0.430987 −0.215494 0.976505i $$-0.569136\pi$$
−0.215494 + 0.976505i $$0.569136\pi$$
$$138$$ 0 0
$$139$$ −10.5544 −0.895211 −0.447605 0.894231i $$-0.647723\pi$$
−0.447605 + 0.894231i $$0.647723\pi$$
$$140$$ 0 0
$$141$$ −3.46791 −0.292051
$$142$$ 0 0
$$143$$ −0.123303 −0.0103111
$$144$$ 0 0
$$145$$ 5.73143 0.475969
$$146$$ 0 0
$$147$$ −2.22668 −0.183654
$$148$$ 0 0
$$149$$ 13.7956 1.13018 0.565090 0.825029i $$-0.308841\pi$$
0.565090 + 0.825029i $$0.308841\pi$$
$$150$$ 0 0
$$151$$ −20.2841 −1.65069 −0.825346 0.564627i $$-0.809020\pi$$
−0.825346 + 0.564627i $$0.809020\pi$$
$$152$$ 0 0
$$153$$ 3.10607 0.251111
$$154$$ 0 0
$$155$$ −0.842549 −0.0676751
$$156$$ 0 0
$$157$$ −13.1753 −1.05150 −0.525752 0.850638i $$-0.676216\pi$$
−0.525752 + 0.850638i $$0.676216\pi$$
$$158$$ 0 0
$$159$$ 8.33275 0.660830
$$160$$ 0 0
$$161$$ −2.82026 −0.222268
$$162$$ 0 0
$$163$$ −17.4902 −1.36994 −0.684969 0.728572i $$-0.740184\pi$$
−0.684969 + 0.728572i $$0.740184\pi$$
$$164$$ 0 0
$$165$$ 0.142903 0.0111250
$$166$$ 0 0
$$167$$ 14.9067 1.15352 0.576759 0.816915i $$-0.304317\pi$$
0.576759 + 0.816915i $$0.304317\pi$$
$$168$$ 0 0
$$169$$ −12.4243 −0.955713
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 1.79561 0.136517 0.0682587 0.997668i $$-0.478256\pi$$
0.0682587 + 0.997668i $$0.478256\pi$$
$$174$$ 0 0
$$175$$ 9.23442 0.698057
$$176$$ 0 0
$$177$$ −2.78106 −0.209037
$$178$$ 0 0
$$179$$ 7.83481 0.585601 0.292801 0.956174i $$-0.405413\pi$$
0.292801 + 0.956174i $$0.405413\pi$$
$$180$$ 0 0
$$181$$ −0.419215 −0.0311600 −0.0155800 0.999879i $$-0.504959\pi$$
−0.0155800 + 0.999879i $$0.504959\pi$$
$$182$$ 0 0
$$183$$ 7.22668 0.534212
$$184$$ 0 0
$$185$$ 1.02229 0.0751602
$$186$$ 0 0
$$187$$ −0.504748 −0.0369108
$$188$$ 0 0
$$189$$ −2.18479 −0.158920
$$190$$ 0 0
$$191$$ −19.5398 −1.41385 −0.706926 0.707287i $$-0.749919\pi$$
−0.706926 + 0.707287i $$0.749919\pi$$
$$192$$ 0 0
$$193$$ −16.6408 −1.19783 −0.598917 0.800811i $$-0.704402\pi$$
−0.598917 + 0.800811i $$0.704402\pi$$
$$194$$ 0 0
$$195$$ −0.667252 −0.0477829
$$196$$ 0 0
$$197$$ −21.9959 −1.56714 −0.783571 0.621302i $$-0.786604\pi$$
−0.783571 + 0.621302i $$0.786604\pi$$
$$198$$ 0 0
$$199$$ −21.7442 −1.54141 −0.770704 0.637194i $$-0.780095\pi$$
−0.770704 + 0.637194i $$0.780095\pi$$
$$200$$ 0 0
$$201$$ 1.06418 0.0750613
$$202$$ 0 0
$$203$$ 14.2395 0.999415
$$204$$ 0 0
$$205$$ 9.31820 0.650811
$$206$$ 0 0
$$207$$ 1.29086 0.0897209
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.62630 −0.593859 −0.296929 0.954899i $$-0.595963\pi$$
−0.296929 + 0.954899i $$0.595963\pi$$
$$212$$ 0 0
$$213$$ 5.08378 0.348335
$$214$$ 0 0
$$215$$ 0.155697 0.0106184
$$216$$ 0 0
$$217$$ −2.09327 −0.142101
$$218$$ 0 0
$$219$$ 2.24897 0.151971
$$220$$ 0 0
$$221$$ 2.35679 0.158535
$$222$$ 0 0
$$223$$ 15.1584 1.01508 0.507540 0.861628i $$-0.330555\pi$$
0.507540 + 0.861628i $$0.330555\pi$$
$$224$$ 0 0
$$225$$ −4.22668 −0.281779
$$226$$ 0 0
$$227$$ −5.71419 −0.379264 −0.189632 0.981855i $$-0.560730\pi$$
−0.189632 + 0.981855i $$0.560730\pi$$
$$228$$ 0 0
$$229$$ 22.1634 1.46460 0.732301 0.680982i $$-0.238447\pi$$
0.732301 + 0.680982i $$0.238447\pi$$
$$230$$ 0 0
$$231$$ 0.355037 0.0233597
$$232$$ 0 0
$$233$$ −10.8280 −0.709366 −0.354683 0.934987i $$-0.615411\pi$$
−0.354683 + 0.934987i $$0.615411\pi$$
$$234$$ 0 0
$$235$$ 3.04963 0.198936
$$236$$ 0 0
$$237$$ −5.24123 −0.340454
$$238$$ 0 0
$$239$$ −9.86215 −0.637929 −0.318965 0.947767i $$-0.603335\pi$$
−0.318965 + 0.947767i $$0.603335\pi$$
$$240$$ 0 0
$$241$$ −6.90673 −0.444901 −0.222451 0.974944i $$-0.571406\pi$$
−0.222451 + 0.974944i $$0.571406\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 1.95811 0.125099
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −17.1138 −1.08454
$$250$$ 0 0
$$251$$ 24.8922 1.57118 0.785590 0.618747i $$-0.212359\pi$$
0.785590 + 0.618747i $$0.212359\pi$$
$$252$$ 0 0
$$253$$ −0.209770 −0.0131881
$$254$$ 0 0
$$255$$ −2.73143 −0.171049
$$256$$ 0 0
$$257$$ −14.2567 −0.889309 −0.444655 0.895702i $$-0.646674\pi$$
−0.444655 + 0.895702i $$0.646674\pi$$
$$258$$ 0 0
$$259$$ 2.53983 0.157817
$$260$$ 0 0
$$261$$ −6.51754 −0.403426
$$262$$ 0 0
$$263$$ 21.6168 1.33295 0.666475 0.745528i $$-0.267803\pi$$
0.666475 + 0.745528i $$0.267803\pi$$
$$264$$ 0 0
$$265$$ −7.32770 −0.450137
$$266$$ 0 0
$$267$$ 15.6236 0.956149
$$268$$ 0 0
$$269$$ −10.6263 −0.647897 −0.323948 0.946075i $$-0.605010\pi$$
−0.323948 + 0.946075i $$0.605010\pi$$
$$270$$ 0 0
$$271$$ −12.1506 −0.738099 −0.369050 0.929410i $$-0.620317\pi$$
−0.369050 + 0.929410i $$0.620317\pi$$
$$272$$ 0 0
$$273$$ −1.65776 −0.100332
$$274$$ 0 0
$$275$$ 0.686852 0.0414187
$$276$$ 0 0
$$277$$ 9.81521 0.589739 0.294869 0.955538i $$-0.404724\pi$$
0.294869 + 0.955538i $$0.404724\pi$$
$$278$$ 0 0
$$279$$ 0.958111 0.0573606
$$280$$ 0 0
$$281$$ 28.2148 1.68316 0.841578 0.540136i $$-0.181627\pi$$
0.841578 + 0.540136i $$0.181627\pi$$
$$282$$ 0 0
$$283$$ −15.4192 −0.916577 −0.458289 0.888803i $$-0.651537\pi$$
−0.458289 + 0.888803i $$0.651537\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 23.1506 1.36654
$$288$$ 0 0
$$289$$ −7.35235 −0.432491
$$290$$ 0 0
$$291$$ −9.41921 −0.552164
$$292$$ 0 0
$$293$$ 27.7769 1.62275 0.811373 0.584529i $$-0.198721\pi$$
0.811373 + 0.584529i $$0.198721\pi$$
$$294$$ 0 0
$$295$$ 2.44562 0.142390
$$296$$ 0 0
$$297$$ −0.162504 −0.00942943
$$298$$ 0 0
$$299$$ 0.979466 0.0566440
$$300$$ 0 0
$$301$$ 0.386821 0.0222960
$$302$$ 0 0
$$303$$ −6.30541 −0.362236
$$304$$ 0 0
$$305$$ −6.35504 −0.363888
$$306$$ 0 0
$$307$$ 27.1908 1.55186 0.775930 0.630819i $$-0.217281\pi$$
0.775930 + 0.630819i $$0.217281\pi$$
$$308$$ 0 0
$$309$$ −8.61587 −0.490140
$$310$$ 0 0
$$311$$ −5.94356 −0.337029 −0.168514 0.985699i $$-0.553897\pi$$
−0.168514 + 0.985699i $$0.553897\pi$$
$$312$$ 0 0
$$313$$ 7.15064 0.404178 0.202089 0.979367i $$-0.435227\pi$$
0.202089 + 0.979367i $$0.435227\pi$$
$$314$$ 0 0
$$315$$ 1.92127 0.108252
$$316$$ 0 0
$$317$$ 22.2909 1.25198 0.625990 0.779831i $$-0.284695\pi$$
0.625990 + 0.779831i $$0.284695\pi$$
$$318$$ 0 0
$$319$$ 1.05913 0.0592996
$$320$$ 0 0
$$321$$ −13.1480 −0.733847
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −3.20708 −0.177897
$$326$$ 0 0
$$327$$ −10.0496 −0.555746
$$328$$ 0 0
$$329$$ 7.57667 0.417715
$$330$$ 0 0
$$331$$ 6.30365 0.346480 0.173240 0.984880i $$-0.444576\pi$$
0.173240 + 0.984880i $$0.444576\pi$$
$$332$$ 0 0
$$333$$ −1.16250 −0.0637048
$$334$$ 0 0
$$335$$ −0.935822 −0.0511294
$$336$$ 0 0
$$337$$ −25.8726 −1.40937 −0.704685 0.709521i $$-0.748911\pi$$
−0.704685 + 0.709521i $$0.748911\pi$$
$$338$$ 0 0
$$339$$ 8.04458 0.436921
$$340$$ 0 0
$$341$$ −0.155697 −0.00843145
$$342$$ 0 0
$$343$$ 20.1584 1.08845
$$344$$ 0 0
$$345$$ −1.13516 −0.0611151
$$346$$ 0 0
$$347$$ −18.6040 −0.998715 −0.499358 0.866396i $$-0.666431\pi$$
−0.499358 + 0.866396i $$0.666431\pi$$
$$348$$ 0 0
$$349$$ −1.74329 −0.0933161 −0.0466581 0.998911i $$-0.514857\pi$$
−0.0466581 + 0.998911i $$0.514857\pi$$
$$350$$ 0 0
$$351$$ 0.758770 0.0405002
$$352$$ 0 0
$$353$$ −10.6723 −0.568029 −0.284015 0.958820i $$-0.591666\pi$$
−0.284015 + 0.958820i $$0.591666\pi$$
$$354$$ 0 0
$$355$$ −4.47060 −0.237275
$$356$$ 0 0
$$357$$ −6.78611 −0.359159
$$358$$ 0 0
$$359$$ −6.81252 −0.359551 −0.179776 0.983708i $$-0.557537\pi$$
−0.179776 + 0.983708i $$0.557537\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 0 0
$$363$$ −10.9736 −0.575964
$$364$$ 0 0
$$365$$ −1.97771 −0.103518
$$366$$ 0 0
$$367$$ −17.9905 −0.939097 −0.469548 0.882907i $$-0.655583\pi$$
−0.469548 + 0.882907i $$0.655583\pi$$
$$368$$ 0 0
$$369$$ −10.5963 −0.551620
$$370$$ 0 0
$$371$$ −18.2053 −0.945173
$$372$$ 0 0
$$373$$ 3.36009 0.173979 0.0869894 0.996209i $$-0.472275\pi$$
0.0869894 + 0.996209i $$0.472275\pi$$
$$374$$ 0 0
$$375$$ 8.11381 0.418995
$$376$$ 0 0
$$377$$ −4.94532 −0.254697
$$378$$ 0 0
$$379$$ 15.1584 0.778634 0.389317 0.921104i $$-0.372711\pi$$
0.389317 + 0.921104i $$0.372711\pi$$
$$380$$ 0 0
$$381$$ 20.0496 1.02717
$$382$$ 0 0
$$383$$ 7.13516 0.364590 0.182295 0.983244i $$-0.441647\pi$$
0.182295 + 0.983244i $$0.441647\pi$$
$$384$$ 0 0
$$385$$ −0.312214 −0.0159119
$$386$$ 0 0
$$387$$ −0.177052 −0.00900005
$$388$$ 0 0
$$389$$ 24.9358 1.26430 0.632148 0.774848i $$-0.282173\pi$$
0.632148 + 0.774848i $$0.282173\pi$$
$$390$$ 0 0
$$391$$ 4.00950 0.202769
$$392$$ 0 0
$$393$$ −15.9094 −0.802524
$$394$$ 0 0
$$395$$ 4.60906 0.231907
$$396$$ 0 0
$$397$$ 20.6536 1.03658 0.518288 0.855206i $$-0.326569\pi$$
0.518288 + 0.855206i $$0.326569\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 36.3218 1.81383 0.906913 0.421319i $$-0.138433\pi$$
0.906913 + 0.421319i $$0.138433\pi$$
$$402$$ 0 0
$$403$$ 0.726986 0.0362138
$$404$$ 0 0
$$405$$ −0.879385 −0.0436970
$$406$$ 0 0
$$407$$ 0.188911 0.00936399
$$408$$ 0 0
$$409$$ 4.95636 0.245076 0.122538 0.992464i $$-0.460897\pi$$
0.122538 + 0.992464i $$0.460897\pi$$
$$410$$ 0 0
$$411$$ −5.04458 −0.248831
$$412$$ 0 0
$$413$$ 6.07604 0.298982
$$414$$ 0 0
$$415$$ 15.0496 0.738757
$$416$$ 0 0
$$417$$ −10.5544 −0.516850
$$418$$ 0 0
$$419$$ −35.0966 −1.71458 −0.857290 0.514834i $$-0.827854\pi$$
−0.857290 + 0.514834i $$0.827854\pi$$
$$420$$ 0 0
$$421$$ −18.4020 −0.896858 −0.448429 0.893819i $$-0.648016\pi$$
−0.448429 + 0.893819i $$0.648016\pi$$
$$422$$ 0 0
$$423$$ −3.46791 −0.168616
$$424$$ 0 0
$$425$$ −13.1284 −0.636819
$$426$$ 0 0
$$427$$ −15.7888 −0.764074
$$428$$ 0 0
$$429$$ −0.123303 −0.00595313
$$430$$ 0 0
$$431$$ 8.79385 0.423585 0.211792 0.977315i $$-0.432070\pi$$
0.211792 + 0.977315i $$0.432070\pi$$
$$432$$ 0 0
$$433$$ 17.4115 0.836742 0.418371 0.908276i $$-0.362601\pi$$
0.418371 + 0.908276i $$0.362601\pi$$
$$434$$ 0 0
$$435$$ 5.73143 0.274801
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −28.4911 −1.35981 −0.679904 0.733301i $$-0.737978\pi$$
−0.679904 + 0.733301i $$0.737978\pi$$
$$440$$ 0 0
$$441$$ −2.22668 −0.106032
$$442$$ 0 0
$$443$$ 18.0428 0.857240 0.428620 0.903485i $$-0.359000\pi$$
0.428620 + 0.903485i $$0.359000\pi$$
$$444$$ 0 0
$$445$$ −13.7392 −0.651299
$$446$$ 0 0
$$447$$ 13.7956 0.652510
$$448$$ 0 0
$$449$$ 9.53478 0.449974 0.224987 0.974362i $$-0.427766\pi$$
0.224987 + 0.974362i $$0.427766\pi$$
$$450$$ 0 0
$$451$$ 1.72193 0.0810827
$$452$$ 0 0
$$453$$ −20.2841 −0.953028
$$454$$ 0 0
$$455$$ 1.45781 0.0683430
$$456$$ 0 0
$$457$$ 10.4584 0.489224 0.244612 0.969621i $$-0.421339\pi$$
0.244612 + 0.969621i $$0.421339\pi$$
$$458$$ 0 0
$$459$$ 3.10607 0.144979
$$460$$ 0 0
$$461$$ 41.6614 1.94036 0.970182 0.242378i $$-0.0779274\pi$$
0.970182 + 0.242378i $$0.0779274\pi$$
$$462$$ 0 0
$$463$$ 13.8179 0.642172 0.321086 0.947050i $$-0.395952\pi$$
0.321086 + 0.947050i $$0.395952\pi$$
$$464$$ 0 0
$$465$$ −0.842549 −0.0390723
$$466$$ 0 0
$$467$$ −7.45336 −0.344901 −0.172450 0.985018i $$-0.555168\pi$$
−0.172450 + 0.985018i $$0.555168\pi$$
$$468$$ 0 0
$$469$$ −2.32501 −0.107359
$$470$$ 0 0
$$471$$ −13.1753 −0.607086
$$472$$ 0 0
$$473$$ 0.0287716 0.00132292
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 8.33275 0.381530
$$478$$ 0 0
$$479$$ 23.7870 1.08686 0.543429 0.839455i $$-0.317126\pi$$
0.543429 + 0.839455i $$0.317126\pi$$
$$480$$ 0 0
$$481$$ −0.882074 −0.0402191
$$482$$ 0 0
$$483$$ −2.82026 −0.128326
$$484$$ 0 0
$$485$$ 8.28312 0.376117
$$486$$ 0 0
$$487$$ 4.85710 0.220096 0.110048 0.993926i $$-0.464900\pi$$
0.110048 + 0.993926i $$0.464900\pi$$
$$488$$ 0 0
$$489$$ −17.4902 −0.790934
$$490$$ 0 0
$$491$$ −2.99764 −0.135281 −0.0676407 0.997710i $$-0.521547\pi$$
−0.0676407 + 0.997710i $$0.521547\pi$$
$$492$$ 0 0
$$493$$ −20.2439 −0.911740
$$494$$ 0 0
$$495$$ 0.142903 0.00642303
$$496$$ 0 0
$$497$$ −11.1070 −0.498217
$$498$$ 0 0
$$499$$ 2.52435 0.113005 0.0565027 0.998402i $$-0.482005\pi$$
0.0565027 + 0.998402i $$0.482005\pi$$
$$500$$ 0 0
$$501$$ 14.9067 0.665983
$$502$$ 0 0
$$503$$ 18.3482 0.818107 0.409054 0.912510i $$-0.365859\pi$$
0.409054 + 0.912510i $$0.365859\pi$$
$$504$$ 0 0
$$505$$ 5.54488 0.246744
$$506$$ 0 0
$$507$$ −12.4243 −0.551781
$$508$$ 0 0
$$509$$ 16.2736 0.721316 0.360658 0.932698i $$-0.382552\pi$$
0.360658 + 0.932698i $$0.382552\pi$$
$$510$$ 0 0
$$511$$ −4.91353 −0.217362
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 7.57667 0.333868
$$516$$ 0 0
$$517$$ 0.563549 0.0247848
$$518$$ 0 0
$$519$$ 1.79561 0.0788184
$$520$$ 0 0
$$521$$ 28.2909 1.23945 0.619723 0.784821i $$-0.287245\pi$$
0.619723 + 0.784821i $$0.287245\pi$$
$$522$$ 0 0
$$523$$ 4.84793 0.211985 0.105992 0.994367i $$-0.466198\pi$$
0.105992 + 0.994367i $$0.466198\pi$$
$$524$$ 0 0
$$525$$ 9.23442 0.403023
$$526$$ 0 0
$$527$$ 2.97596 0.129635
$$528$$ 0 0
$$529$$ −21.3337 −0.927551
$$530$$ 0 0
$$531$$ −2.78106 −0.120688
$$532$$ 0 0
$$533$$ −8.04013 −0.348257
$$534$$ 0 0
$$535$$ 11.5621 0.499874
$$536$$ 0 0
$$537$$ 7.83481 0.338097
$$538$$ 0 0
$$539$$ 0.361844 0.0155857
$$540$$ 0 0
$$541$$ −24.3131 −1.04530 −0.522652 0.852546i $$-0.675057\pi$$
−0.522652 + 0.852546i $$0.675057\pi$$
$$542$$ 0 0
$$543$$ −0.419215 −0.0179902
$$544$$ 0 0
$$545$$ 8.83750 0.378557
$$546$$ 0 0
$$547$$ −22.7733 −0.973717 −0.486858 0.873481i $$-0.661857\pi$$
−0.486858 + 0.873481i $$0.661857\pi$$
$$548$$ 0 0
$$549$$ 7.22668 0.308427
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 11.4510 0.486946
$$554$$ 0 0
$$555$$ 1.02229 0.0433937
$$556$$ 0 0
$$557$$ 19.1343 0.810748 0.405374 0.914151i $$-0.367141\pi$$
0.405374 + 0.914151i $$0.367141\pi$$
$$558$$ 0 0
$$559$$ −0.134342 −0.00568205
$$560$$ 0 0
$$561$$ −0.504748 −0.0213105
$$562$$ 0 0
$$563$$ −21.3482 −0.899721 −0.449860 0.893099i $$-0.648526\pi$$
−0.449860 + 0.893099i $$0.648526\pi$$
$$564$$ 0 0
$$565$$ −7.07428 −0.297617
$$566$$ 0 0
$$567$$ −2.18479 −0.0917527
$$568$$ 0 0
$$569$$ 39.2121 1.64386 0.821929 0.569590i $$-0.192898\pi$$
0.821929 + 0.569590i $$0.192898\pi$$
$$570$$ 0 0
$$571$$ −7.27900 −0.304617 −0.152308 0.988333i $$-0.548671\pi$$
−0.152308 + 0.988333i $$0.548671\pi$$
$$572$$ 0 0
$$573$$ −19.5398 −0.816288
$$574$$ 0 0
$$575$$ −5.45605 −0.227533
$$576$$ 0 0
$$577$$ 14.9923 0.624136 0.312068 0.950060i $$-0.398978\pi$$
0.312068 + 0.950060i $$0.398978\pi$$
$$578$$ 0 0
$$579$$ −16.6408 −0.691570
$$580$$ 0 0
$$581$$ 37.3901 1.55120
$$582$$ 0 0
$$583$$ −1.35410 −0.0560812
$$584$$ 0 0
$$585$$ −0.667252 −0.0275875
$$586$$ 0 0
$$587$$ −12.0104 −0.495723 −0.247862 0.968795i $$-0.579728\pi$$
−0.247862 + 0.968795i $$0.579728\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −21.9959 −0.904790
$$592$$ 0 0
$$593$$ −36.3806 −1.49397 −0.746987 0.664839i $$-0.768500\pi$$
−0.746987 + 0.664839i $$0.768500\pi$$
$$594$$ 0 0
$$595$$ 5.96761 0.244648
$$596$$ 0 0
$$597$$ −21.7442 −0.889932
$$598$$ 0 0
$$599$$ 13.8060 0.564099 0.282050 0.959400i $$-0.408986\pi$$
0.282050 + 0.959400i $$0.408986\pi$$
$$600$$ 0 0
$$601$$ −0.967606 −0.0394695 −0.0197347 0.999805i $$-0.506282\pi$$
−0.0197347 + 0.999805i $$0.506282\pi$$
$$602$$ 0 0
$$603$$ 1.06418 0.0433367
$$604$$ 0 0
$$605$$ 9.65002 0.392329
$$606$$ 0 0
$$607$$ 1.53890 0.0624619 0.0312309 0.999512i $$-0.490057\pi$$
0.0312309 + 0.999512i $$0.490057\pi$$
$$608$$ 0 0
$$609$$ 14.2395 0.577013
$$610$$ 0 0
$$611$$ −2.63135 −0.106453
$$612$$ 0 0
$$613$$ 41.1753 1.66305 0.831527 0.555484i $$-0.187467\pi$$
0.831527 + 0.555484i $$0.187467\pi$$
$$614$$ 0 0
$$615$$ 9.31820 0.375746
$$616$$ 0 0
$$617$$ 30.6860 1.23537 0.617687 0.786424i $$-0.288070\pi$$
0.617687 + 0.786424i $$0.288070\pi$$
$$618$$ 0 0
$$619$$ −11.9581 −0.480637 −0.240319 0.970694i $$-0.577252\pi$$
−0.240319 + 0.970694i $$0.577252\pi$$
$$620$$ 0 0
$$621$$ 1.29086 0.0518004
$$622$$ 0 0
$$623$$ −34.1343 −1.36756
$$624$$ 0 0
$$625$$ 13.9982 0.559930
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −3.61081 −0.143973
$$630$$ 0 0
$$631$$ −48.1762 −1.91787 −0.958933 0.283634i $$-0.908460\pi$$
−0.958933 + 0.283634i $$0.908460\pi$$
$$632$$ 0 0
$$633$$ −8.62630 −0.342864
$$634$$ 0 0
$$635$$ −17.6313 −0.699679
$$636$$ 0 0
$$637$$ −1.68954 −0.0669420
$$638$$ 0 0
$$639$$ 5.08378 0.201111
$$640$$ 0 0
$$641$$ 6.72967 0.265806 0.132903 0.991129i $$-0.457570\pi$$
0.132903 + 0.991129i $$0.457570\pi$$
$$642$$ 0 0
$$643$$ −25.6928 −1.01323 −0.506613 0.862173i $$-0.669103\pi$$
−0.506613 + 0.862173i $$0.669103\pi$$
$$644$$ 0 0
$$645$$ 0.155697 0.00613055
$$646$$ 0 0
$$647$$ −50.4380 −1.98292 −0.991461 0.130403i $$-0.958373\pi$$
−0.991461 + 0.130403i $$0.958373\pi$$
$$648$$ 0 0
$$649$$ 0.451933 0.0177399
$$650$$ 0 0
$$651$$ −2.09327 −0.0820419
$$652$$ 0 0
$$653$$ −35.8607 −1.40334 −0.701669 0.712503i $$-0.747562\pi$$
−0.701669 + 0.712503i $$0.747562\pi$$
$$654$$ 0 0
$$655$$ 13.9905 0.546654
$$656$$ 0 0
$$657$$ 2.24897 0.0877407
$$658$$ 0 0
$$659$$ −12.6108 −0.491248 −0.245624 0.969365i $$-0.578993\pi$$
−0.245624 + 0.969365i $$0.578993\pi$$
$$660$$ 0 0
$$661$$ −44.5699 −1.73357 −0.866783 0.498685i $$-0.833816\pi$$
−0.866783 + 0.498685i $$0.833816\pi$$
$$662$$ 0 0
$$663$$ 2.35679 0.0915302
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −8.41323 −0.325762
$$668$$ 0 0
$$669$$ 15.1584 0.586057
$$670$$ 0 0
$$671$$ −1.17436 −0.0453358
$$672$$ 0 0
$$673$$ −12.8203 −0.494185 −0.247092 0.968992i $$-0.579475\pi$$
−0.247092 + 0.968992i $$0.579475\pi$$
$$674$$ 0 0
$$675$$ −4.22668 −0.162685
$$676$$ 0 0
$$677$$ −29.3688 −1.12873 −0.564367 0.825524i $$-0.690880\pi$$
−0.564367 + 0.825524i $$0.690880\pi$$
$$678$$ 0 0
$$679$$ 20.5790 0.789751
$$680$$ 0 0
$$681$$ −5.71419 −0.218968
$$682$$ 0 0
$$683$$ 38.0033 1.45416 0.727078 0.686555i $$-0.240878\pi$$
0.727078 + 0.686555i $$0.240878\pi$$
$$684$$ 0 0
$$685$$ 4.43613 0.169496
$$686$$ 0 0
$$687$$ 22.1634 0.845588
$$688$$ 0 0
$$689$$ 6.32264 0.240874
$$690$$ 0 0
$$691$$ 32.2918 1.22844 0.614219 0.789136i $$-0.289471\pi$$
0.614219 + 0.789136i $$0.289471\pi$$
$$692$$ 0 0
$$693$$ 0.355037 0.0134867
$$694$$ 0 0
$$695$$ 9.28136 0.352062
$$696$$ 0 0
$$697$$ −32.9127 −1.24666
$$698$$ 0 0
$$699$$ −10.8280 −0.409553
$$700$$ 0 0
$$701$$ 23.1138 0.872996 0.436498 0.899705i $$-0.356219\pi$$
0.436498 + 0.899705i $$0.356219\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 3.04963 0.114856
$$706$$ 0 0
$$707$$ 13.7760 0.518100
$$708$$ 0 0
$$709$$ 36.7588 1.38050 0.690252 0.723569i $$-0.257500\pi$$
0.690252 + 0.723569i $$0.257500\pi$$
$$710$$ 0 0
$$711$$ −5.24123 −0.196561
$$712$$ 0 0
$$713$$ 1.23679 0.0463180
$$714$$ 0 0
$$715$$ 0.108431 0.00405509
$$716$$ 0 0
$$717$$ −9.86215 −0.368309
$$718$$ 0 0
$$719$$ 32.9504 1.22884 0.614421 0.788979i $$-0.289390\pi$$
0.614421 + 0.788979i $$0.289390\pi$$
$$720$$ 0 0
$$721$$ 18.8239 0.701038
$$722$$ 0 0
$$723$$ −6.90673 −0.256864
$$724$$ 0 0
$$725$$ 27.5476 1.02309
$$726$$ 0 0
$$727$$ −7.92303 −0.293849 −0.146924 0.989148i $$-0.546937\pi$$
−0.146924 + 0.989148i $$0.546937\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −0.549935 −0.0203401
$$732$$ 0 0
$$733$$ −7.43470 −0.274607 −0.137303 0.990529i $$-0.543844\pi$$
−0.137303 + 0.990529i $$0.543844\pi$$
$$734$$ 0 0
$$735$$ 1.95811 0.0722260
$$736$$ 0 0
$$737$$ −0.172933 −0.00637007
$$738$$ 0 0
$$739$$ 19.3500 0.711801 0.355900 0.934524i $$-0.384174\pi$$
0.355900 + 0.934524i $$0.384174\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 31.3833 1.15134 0.575671 0.817681i $$-0.304741\pi$$
0.575671 + 0.817681i $$0.304741\pi$$
$$744$$ 0 0
$$745$$ −12.1317 −0.444469
$$746$$ 0 0
$$747$$ −17.1138 −0.626161
$$748$$ 0 0
$$749$$ 28.7256 1.04961
$$750$$ 0 0
$$751$$ 5.22256 0.190574 0.0952870 0.995450i $$-0.469623\pi$$
0.0952870 + 0.995450i $$0.469623\pi$$
$$752$$ 0 0
$$753$$ 24.8922 0.907121
$$754$$ 0 0
$$755$$ 17.8375 0.649173
$$756$$ 0 0
$$757$$ −29.7716 −1.08207 −0.541033 0.841001i $$-0.681967\pi$$
−0.541033 + 0.841001i $$0.681967\pi$$
$$758$$ 0 0
$$759$$ −0.209770 −0.00761415
$$760$$ 0 0
$$761$$ −9.35267 −0.339034 −0.169517 0.985527i $$-0.554221\pi$$
−0.169517 + 0.985527i $$0.554221\pi$$
$$762$$ 0 0
$$763$$ 21.9564 0.794873
$$764$$ 0 0
$$765$$ −2.73143 −0.0987550
$$766$$ 0 0
$$767$$ −2.11019 −0.0761944
$$768$$ 0 0
$$769$$ 38.8334 1.40037 0.700184 0.713963i $$-0.253101\pi$$
0.700184 + 0.713963i $$0.253101\pi$$
$$770$$ 0 0
$$771$$ −14.2567 −0.513443
$$772$$ 0 0
$$773$$ −23.7674 −0.854856 −0.427428 0.904049i $$-0.640580\pi$$
−0.427428 + 0.904049i $$0.640580\pi$$
$$774$$ 0 0
$$775$$ −4.04963 −0.145467
$$776$$ 0 0
$$777$$ 2.53983 0.0911159
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −0.826133 −0.0295614
$$782$$ 0 0
$$783$$ −6.51754 −0.232918
$$784$$ 0 0
$$785$$ 11.5862 0.413528
$$786$$ 0 0
$$787$$ −13.7561 −0.490351 −0.245176 0.969479i $$-0.578846\pi$$
−0.245176 + 0.969479i $$0.578846\pi$$
$$788$$ 0 0
$$789$$ 21.6168 0.769578
$$790$$ 0 0
$$791$$ −17.5757 −0.624921
$$792$$ 0 0
$$793$$ 5.48339 0.194721
$$794$$ 0 0
$$795$$ −7.32770 −0.259887
$$796$$ 0 0
$$797$$ −7.77238 −0.275312 −0.137656 0.990480i $$-0.543957\pi$$
−0.137656 + 0.990480i $$0.543957\pi$$
$$798$$ 0 0
$$799$$ −10.7716 −0.381071
$$800$$ 0 0
$$801$$ 15.6236 0.552033
$$802$$ 0 0
$$803$$ −0.365466 −0.0128970
$$804$$ 0 0
$$805$$ 2.48009 0.0874119
$$806$$ 0 0
$$807$$ −10.6263 −0.374063
$$808$$ 0 0
$$809$$ 13.6919 0.481382 0.240691 0.970602i $$-0.422626\pi$$
0.240691 + 0.970602i $$0.422626\pi$$
$$810$$ 0 0
$$811$$ −22.1084 −0.776332 −0.388166 0.921589i $$-0.626891\pi$$
−0.388166 + 0.921589i $$0.626891\pi$$
$$812$$ 0 0
$$813$$ −12.1506 −0.426142
$$814$$ 0 0
$$815$$ 15.3806 0.538760
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −1.65776 −0.0579267
$$820$$ 0 0
$$821$$ −21.5776 −0.753063 −0.376532 0.926404i $$-0.622883\pi$$
−0.376532 + 0.926404i $$0.622883\pi$$
$$822$$ 0 0
$$823$$ −9.35235 −0.326002 −0.163001 0.986626i $$-0.552117\pi$$
−0.163001 + 0.986626i $$0.552117\pi$$
$$824$$ 0 0
$$825$$ 0.686852 0.0239131
$$826$$ 0 0
$$827$$ −33.5800 −1.16769 −0.583845 0.811865i $$-0.698452\pi$$
−0.583845 + 0.811865i $$0.698452\pi$$
$$828$$ 0 0
$$829$$ 0.650340 0.0225872 0.0112936 0.999936i $$-0.496405\pi$$
0.0112936 + 0.999936i $$0.496405\pi$$
$$830$$ 0 0
$$831$$ 9.81521 0.340486
$$832$$ 0 0
$$833$$ −6.91622 −0.239633
$$834$$ 0 0
$$835$$ −13.1088 −0.453647
$$836$$ 0 0
$$837$$ 0.958111 0.0331172
$$838$$ 0 0
$$839$$ −22.9077 −0.790860 −0.395430 0.918496i $$-0.629404\pi$$
−0.395430 + 0.918496i $$0.629404\pi$$
$$840$$ 0 0
$$841$$ 13.4783 0.464770
$$842$$ 0 0
$$843$$ 28.2148 0.971770
$$844$$ 0 0
$$845$$ 10.9257 0.375856
$$846$$ 0 0
$$847$$ 23.9750 0.823792
$$848$$ 0 0
$$849$$ −15.4192 −0.529186
$$850$$ 0 0
$$851$$ −1.50063 −0.0514409
$$852$$ 0 0
$$853$$ 30.5158 1.04484 0.522420 0.852688i $$-0.325029\pi$$
0.522420 + 0.852688i $$0.325029\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18.0473 0.616483 0.308241 0.951308i $$-0.400260\pi$$
0.308241 + 0.951308i $$0.400260\pi$$
$$858$$ 0 0
$$859$$ 36.8857 1.25852 0.629262 0.777193i $$-0.283357\pi$$
0.629262 + 0.777193i $$0.283357\pi$$
$$860$$ 0 0
$$861$$ 23.1506 0.788972
$$862$$ 0 0
$$863$$ −28.1712 −0.958958 −0.479479 0.877553i $$-0.659174\pi$$
−0.479479 + 0.877553i $$0.659174\pi$$
$$864$$ 0 0
$$865$$ −1.57903 −0.0536886
$$866$$ 0 0
$$867$$ −7.35235 −0.249699
$$868$$ 0 0
$$869$$ 0.851720 0.0288926
$$870$$ 0 0
$$871$$ 0.807467 0.0273600
$$872$$ 0 0
$$873$$ −9.41921 −0.318792
$$874$$ 0 0
$$875$$ −17.7270 −0.599282
$$876$$ 0 0
$$877$$ 12.7115 0.429237 0.214619 0.976698i $$-0.431149\pi$$
0.214619 + 0.976698i $$0.431149\pi$$
$$878$$ 0 0
$$879$$ 27.7769 0.936893
$$880$$ 0 0
$$881$$ 36.8530 1.24161 0.620804 0.783966i $$-0.286806\pi$$
0.620804 + 0.783966i $$0.286806\pi$$
$$882$$ 0 0
$$883$$ −39.6245 −1.33347 −0.666736 0.745294i $$-0.732309\pi$$
−0.666736 + 0.745294i $$0.732309\pi$$
$$884$$ 0 0
$$885$$ 2.44562 0.0822087
$$886$$ 0 0
$$887$$ −47.5057 −1.59508 −0.797542 0.603263i $$-0.793867\pi$$
−0.797542 + 0.603263i $$0.793867\pi$$
$$888$$ 0 0
$$889$$ −43.8043 −1.46915
$$890$$ 0 0
$$891$$ −0.162504 −0.00544408
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −6.88981 −0.230301
$$896$$ 0 0
$$897$$ 0.979466 0.0327034
$$898$$ 0 0
$$899$$ −6.24453 −0.208267
$$900$$ 0 0
$$901$$ 25.8821 0.862257
$$902$$ 0 0
$$903$$ 0.386821 0.0128726
$$904$$ 0 0
$$905$$ 0.368651 0.0122544
$$906$$ 0 0
$$907$$ 23.0060 0.763901 0.381951 0.924183i $$-0.375252\pi$$
0.381951 + 0.924183i $$0.375252\pi$$
$$908$$ 0 0
$$909$$ −6.30541 −0.209137
$$910$$ 0 0
$$911$$ 13.7888 0.456843 0.228422 0.973562i $$-0.426644\pi$$
0.228422 + 0.973562i $$0.426644\pi$$
$$912$$ 0 0
$$913$$ 2.78106 0.0920396
$$914$$ 0 0
$$915$$ −6.35504 −0.210091
$$916$$ 0 0
$$917$$ 34.7588 1.14784
$$918$$ 0 0
$$919$$ 54.6623 1.80314 0.901572 0.432630i $$-0.142414\pi$$
0.901572 + 0.432630i $$0.142414\pi$$
$$920$$ 0 0
$$921$$ 27.1908 0.895967
$$922$$ 0 0
$$923$$ 3.85742 0.126969
$$924$$ 0 0
$$925$$ 4.91353 0.161556
$$926$$ 0 0
$$927$$ −8.61587 −0.282982
$$928$$ 0 0
$$929$$ 46.8084 1.53573 0.767867 0.640609i $$-0.221318\pi$$
0.767867 + 0.640609i $$0.221318\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −5.94356 −0.194584
$$934$$ 0 0
$$935$$ 0.443868 0.0145160
$$936$$ 0 0
$$937$$ −38.3209 −1.25189 −0.625944 0.779868i $$-0.715286\pi$$
−0.625944 + 0.779868i $$0.715286\pi$$
$$938$$ 0 0
$$939$$ 7.15064 0.233352
$$940$$ 0 0
$$941$$ −29.4584 −0.960317 −0.480158 0.877182i $$-0.659421\pi$$
−0.480158 + 0.877182i $$0.659421\pi$$
$$942$$ 0 0
$$943$$ −13.6783 −0.445426
$$944$$ 0 0
$$945$$ 1.92127 0.0624991
$$946$$ 0 0
$$947$$ −37.9273 −1.23247 −0.616235 0.787562i $$-0.711343\pi$$
−0.616235 + 0.787562i $$0.711343\pi$$
$$948$$ 0 0
$$949$$ 1.70645 0.0553938
$$950$$ 0 0
$$951$$ 22.2909 0.722831
$$952$$ 0 0
$$953$$ −8.23267 −0.266682 −0.133341 0.991070i $$-0.542571\pi$$
−0.133341 + 0.991070i $$0.542571\pi$$
$$954$$ 0 0
$$955$$ 17.1830 0.556030
$$956$$ 0 0
$$957$$ 1.05913 0.0342367
$$958$$ 0 0
$$959$$ 11.0214 0.355898
$$960$$ 0 0
$$961$$ −30.0820 −0.970388
$$962$$ 0 0
$$963$$ −13.1480 −0.423687
$$964$$ 0 0
$$965$$ 14.6337 0.471076
$$966$$ 0 0
$$967$$ −16.2294 −0.521901 −0.260951 0.965352i $$-0.584036\pi$$
−0.260951 + 0.965352i $$0.584036\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 33.1962 1.06532 0.532658 0.846331i $$-0.321193\pi$$
0.532658 + 0.846331i $$0.321193\pi$$
$$972$$ 0 0
$$973$$ 23.0591 0.739242
$$974$$ 0 0
$$975$$ −3.20708 −0.102709
$$976$$ 0 0
$$977$$ 23.9932 0.767610 0.383805 0.923414i $$-0.374613\pi$$
0.383805 + 0.923414i $$0.374613\pi$$
$$978$$ 0 0
$$979$$ −2.53890 −0.0811435
$$980$$ 0 0
$$981$$ −10.0496 −0.320860
$$982$$ 0 0
$$983$$ 46.4962 1.48300 0.741499 0.670954i $$-0.234115\pi$$
0.741499 + 0.670954i $$0.234115\pi$$
$$984$$ 0 0
$$985$$ 19.3429 0.616315
$$986$$ 0 0
$$987$$ 7.57667 0.241168
$$988$$ 0 0
$$989$$ −0.228549 −0.00726743
$$990$$ 0 0
$$991$$ 35.2513 1.11980 0.559898 0.828562i $$-0.310840\pi$$
0.559898 + 0.828562i $$0.310840\pi$$
$$992$$ 0 0
$$993$$ 6.30365 0.200040
$$994$$ 0 0
$$995$$ 19.1215 0.606194
$$996$$ 0 0
$$997$$ 55.5877 1.76048 0.880240 0.474529i $$-0.157381\pi$$
0.880240 + 0.474529i $$0.157381\pi$$
$$998$$ 0 0
$$999$$ −1.16250 −0.0367800
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 4332.2.a.o.1.1 3
19.6 even 9 228.2.q.a.169.1 yes 6
19.16 even 9 228.2.q.a.85.1 6
19.18 odd 2 4332.2.a.n.1.1 3
57.35 odd 18 684.2.bo.a.541.1 6
57.44 odd 18 684.2.bo.a.397.1 6
76.35 odd 18 912.2.bo.e.769.1 6
76.63 odd 18 912.2.bo.e.625.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
228.2.q.a.85.1 6 19.16 even 9
228.2.q.a.169.1 yes 6 19.6 even 9
684.2.bo.a.397.1 6 57.44 odd 18
684.2.bo.a.541.1 6 57.35 odd 18
912.2.bo.e.625.1 6 76.63 odd 18
912.2.bo.e.769.1 6 76.35 odd 18
4332.2.a.n.1.1 3 19.18 odd 2
4332.2.a.o.1.1 3 1.1 even 1 trivial