Properties

 Label 9200.2.a.cd.1.1 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-2.14510$$ of defining polynomial Character $$\chi$$ $$=$$ 9200.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-2.14510 q^{3} -1.14510 q^{7} +1.60147 q^{9} +O(q^{10})$$ $$q-2.14510 q^{3} -1.14510 q^{7} +1.60147 q^{9} -5.89167 q^{11} +4.89167 q^{13} +5.89167 q^{17} +2.34803 q^{19} +2.45636 q^{21} +1.00000 q^{23} +3.00000 q^{27} +3.74657 q^{29} -5.68874 q^{31} +12.6382 q^{33} -4.00000 q^{37} -10.4931 q^{39} -1.05783 q^{41} +11.4931 q^{43} +7.74657 q^{47} -5.68874 q^{49} -12.6382 q^{51} -12.9863 q^{53} -5.03677 q^{57} -0.797069 q^{59} +13.8917 q^{61} -1.83384 q^{63} -15.5667 q^{67} -2.14510 q^{69} -2.94217 q^{71} -6.32698 q^{73} +6.74657 q^{77} -11.2397 q^{81} -0.912726 q^{83} -8.03677 q^{87} -15.4931 q^{89} -5.60147 q^{91} +12.2029 q^{93} -13.3848 q^{97} -9.43531 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^7 + 3 * q^9 $$3 q + 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{17} - 3 q^{19} + 12 q^{21} + 3 q^{23} + 9 q^{27} + 3 q^{29} - 6 q^{31} + 15 q^{33} - 12 q^{37} - 15 q^{39} - 6 q^{41} + 18 q^{43} + 15 q^{47} - 6 q^{49} - 15 q^{51} - 6 q^{53} + 6 q^{57} - 6 q^{59} + 27 q^{61} + 12 q^{63} + 12 q^{67} - 6 q^{71} + 15 q^{73} + 12 q^{77} - 9 q^{81} - 12 q^{83} - 3 q^{87} - 30 q^{89} - 15 q^{91} + 33 q^{93} - 9 q^{97} - 9 q^{99}+O(q^{100})$$ 3 * q + 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^17 - 3 * q^19 + 12 * q^21 + 3 * q^23 + 9 * q^27 + 3 * q^29 - 6 * q^31 + 15 * q^33 - 12 * q^37 - 15 * q^39 - 6 * q^41 + 18 * q^43 + 15 * q^47 - 6 * q^49 - 15 * q^51 - 6 * q^53 + 6 * q^57 - 6 * q^59 + 27 * q^61 + 12 * q^63 + 12 * q^67 - 6 * q^71 + 15 * q^73 + 12 * q^77 - 9 * q^81 - 12 * q^83 - 3 * q^87 - 30 * q^89 - 15 * q^91 + 33 * q^93 - 9 * q^97 - 9 * 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
$$3$$ −2.14510 −1.23848 −0.619238 0.785204i $$-0.712558\pi$$
−0.619238 + 0.785204i $$0.712558\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.14510 −0.432808 −0.216404 0.976304i $$-0.569433\pi$$
−0.216404 + 0.976304i $$0.569433\pi$$
$$8$$ 0 0
$$9$$ 1.60147 0.533822
$$10$$ 0 0
$$11$$ −5.89167 −1.77641 −0.888203 0.459452i $$-0.848046\pi$$
−0.888203 + 0.459452i $$0.848046\pi$$
$$12$$ 0 0
$$13$$ 4.89167 1.35671 0.678353 0.734736i $$-0.262694\pi$$
0.678353 + 0.734736i $$0.262694\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.89167 1.42894 0.714470 0.699666i $$-0.246668\pi$$
0.714470 + 0.699666i $$0.246668\pi$$
$$18$$ 0 0
$$19$$ 2.34803 0.538676 0.269338 0.963046i $$-0.413195\pi$$
0.269338 + 0.963046i $$0.413195\pi$$
$$20$$ 0 0
$$21$$ 2.45636 0.536022
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 3.00000 0.577350
$$28$$ 0 0
$$29$$ 3.74657 0.695720 0.347860 0.937546i $$-0.386908\pi$$
0.347860 + 0.937546i $$0.386908\pi$$
$$30$$ 0 0
$$31$$ −5.68874 −1.02173 −0.510864 0.859662i $$-0.670674\pi$$
−0.510864 + 0.859662i $$0.670674\pi$$
$$32$$ 0 0
$$33$$ 12.6382 2.20004
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ −10.4931 −1.68025
$$40$$ 0 0
$$41$$ −1.05783 −0.165205 −0.0826025 0.996583i $$-0.526323\pi$$
−0.0826025 + 0.996583i $$0.526323\pi$$
$$42$$ 0 0
$$43$$ 11.4931 1.75269 0.876343 0.481687i $$-0.159976\pi$$
0.876343 + 0.481687i $$0.159976\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 7.74657 1.12995 0.564977 0.825107i $$-0.308885\pi$$
0.564977 + 0.825107i $$0.308885\pi$$
$$48$$ 0 0
$$49$$ −5.68874 −0.812677
$$50$$ 0 0
$$51$$ −12.6382 −1.76971
$$52$$ 0 0
$$53$$ −12.9863 −1.78380 −0.891901 0.452231i $$-0.850628\pi$$
−0.891901 + 0.452231i $$0.850628\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −5.03677 −0.667137
$$58$$ 0 0
$$59$$ −0.797069 −0.103770 −0.0518848 0.998653i $$-0.516523\pi$$
−0.0518848 + 0.998653i $$0.516523\pi$$
$$60$$ 0 0
$$61$$ 13.8917 1.77865 0.889323 0.457279i $$-0.151176\pi$$
0.889323 + 0.457279i $$0.151176\pi$$
$$62$$ 0 0
$$63$$ −1.83384 −0.231042
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −15.5667 −1.90177 −0.950887 0.309540i $$-0.899825\pi$$
−0.950887 + 0.309540i $$0.899825\pi$$
$$68$$ 0 0
$$69$$ −2.14510 −0.258240
$$70$$ 0 0
$$71$$ −2.94217 −0.349172 −0.174586 0.984642i $$-0.555859\pi$$
−0.174586 + 0.984642i $$0.555859\pi$$
$$72$$ 0 0
$$73$$ −6.32698 −0.740517 −0.370258 0.928929i $$-0.620731\pi$$
−0.370258 + 0.928929i $$0.620731\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.74657 0.768843
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ −11.2397 −1.24886
$$82$$ 0 0
$$83$$ −0.912726 −0.100185 −0.0500923 0.998745i $$-0.515952\pi$$
−0.0500923 + 0.998745i $$0.515952\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −8.03677 −0.861633
$$88$$ 0 0
$$89$$ −15.4931 −1.64227 −0.821135 0.570735i $$-0.806659\pi$$
−0.821135 + 0.570735i $$0.806659\pi$$
$$90$$ 0 0
$$91$$ −5.60147 −0.587193
$$92$$ 0 0
$$93$$ 12.2029 1.26539
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −13.3848 −1.35902 −0.679511 0.733666i $$-0.737808\pi$$
−0.679511 + 0.733666i $$0.737808\pi$$
$$98$$ 0 0
$$99$$ −9.43531 −0.948284
$$100$$ 0 0
$$101$$ 2.79707 0.278319 0.139159 0.990270i $$-0.455560\pi$$
0.139159 + 0.990270i $$0.455560\pi$$
$$102$$ 0 0
$$103$$ −9.89167 −0.974655 −0.487328 0.873219i $$-0.662028\pi$$
−0.487328 + 0.873219i $$0.662028\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.6961 1.22738 0.613688 0.789549i $$-0.289685\pi$$
0.613688 + 0.789549i $$0.289685\pi$$
$$108$$ 0 0
$$109$$ 3.65197 0.349795 0.174897 0.984587i $$-0.444041\pi$$
0.174897 + 0.984587i $$0.444041\pi$$
$$110$$ 0 0
$$111$$ 8.58041 0.814417
$$112$$ 0 0
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 7.83384 0.724239
$$118$$ 0 0
$$119$$ −6.74657 −0.618457
$$120$$ 0 0
$$121$$ 23.7118 2.15562
$$122$$ 0 0
$$123$$ 2.26915 0.204602
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 12.1524 1.07835 0.539177 0.842193i $$-0.318735\pi$$
0.539177 + 0.842193i $$0.318735\pi$$
$$128$$ 0 0
$$129$$ −24.6540 −2.17066
$$130$$ 0 0
$$131$$ 6.94950 0.607181 0.303590 0.952803i $$-0.401815\pi$$
0.303590 + 0.952803i $$0.401815\pi$$
$$132$$ 0 0
$$133$$ −2.68874 −0.233143
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.44904 0.209235 0.104618 0.994513i $$-0.466638\pi$$
0.104618 + 0.994513i $$0.466638\pi$$
$$138$$ 0 0
$$139$$ −9.23970 −0.783702 −0.391851 0.920029i $$-0.628165\pi$$
−0.391851 + 0.920029i $$0.628165\pi$$
$$140$$ 0 0
$$141$$ −16.6172 −1.39942
$$142$$ 0 0
$$143$$ −28.8201 −2.41006
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 12.2029 1.00648
$$148$$ 0 0
$$149$$ −2.05783 −0.168584 −0.0842919 0.996441i $$-0.526863\pi$$
−0.0842919 + 0.996441i $$0.526863\pi$$
$$150$$ 0 0
$$151$$ −9.34803 −0.760732 −0.380366 0.924836i $$-0.624202\pi$$
−0.380366 + 0.924836i $$0.624202\pi$$
$$152$$ 0 0
$$153$$ 9.43531 0.762799
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3.60879 0.288013 0.144007 0.989577i $$-0.454001\pi$$
0.144007 + 0.989577i $$0.454001\pi$$
$$158$$ 0 0
$$159$$ 27.8569 2.20920
$$160$$ 0 0
$$161$$ −1.14510 −0.0902467
$$162$$ 0 0
$$163$$ 21.2554 1.66485 0.832427 0.554135i $$-0.186951\pi$$
0.832427 + 0.554135i $$0.186951\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −10.9863 −0.850143 −0.425072 0.905160i $$-0.639751\pi$$
−0.425072 + 0.905160i $$0.639751\pi$$
$$168$$ 0 0
$$169$$ 10.9284 0.840650
$$170$$ 0 0
$$171$$ 3.76030 0.287557
$$172$$ 0 0
$$173$$ 6.28288 0.477678 0.238839 0.971059i $$-0.423233\pi$$
0.238839 + 0.971059i $$0.423233\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1.70979 0.128516
$$178$$ 0 0
$$179$$ −10.1524 −0.758828 −0.379414 0.925227i $$-0.623874\pi$$
−0.379414 + 0.925227i $$0.623874\pi$$
$$180$$ 0 0
$$181$$ 11.1451 0.828409 0.414204 0.910184i $$-0.364060\pi$$
0.414204 + 0.910184i $$0.364060\pi$$
$$182$$ 0 0
$$183$$ −29.7991 −2.20281
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −34.7118 −2.53838
$$188$$ 0 0
$$189$$ −3.43531 −0.249882
$$190$$ 0 0
$$191$$ 14.6961 1.06337 0.531685 0.846942i $$-0.321559\pi$$
0.531685 + 0.846942i $$0.321559\pi$$
$$192$$ 0 0
$$193$$ 26.4005 1.90035 0.950176 0.311715i $$-0.100903\pi$$
0.950176 + 0.311715i $$0.100903\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4.26076 0.303567 0.151783 0.988414i $$-0.451498\pi$$
0.151783 + 0.988414i $$0.451498\pi$$
$$198$$ 0 0
$$199$$ −18.0735 −1.28120 −0.640600 0.767875i $$-0.721314\pi$$
−0.640600 + 0.767875i $$0.721314\pi$$
$$200$$ 0 0
$$201$$ 33.3921 2.35530
$$202$$ 0 0
$$203$$ −4.29021 −0.301113
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.60147 0.111310
$$208$$ 0 0
$$209$$ −13.8338 −0.956907
$$210$$ 0 0
$$211$$ 11.8990 0.819161 0.409580 0.912274i $$-0.365675\pi$$
0.409580 + 0.912274i $$0.365675\pi$$
$$212$$ 0 0
$$213$$ 6.31126 0.432440
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 6.51419 0.442212
$$218$$ 0 0
$$219$$ 13.5720 0.917112
$$220$$ 0 0
$$221$$ 28.8201 1.93865
$$222$$ 0 0
$$223$$ 4.58041 0.306727 0.153363 0.988170i $$-0.450989\pi$$
0.153363 + 0.988170i $$0.450989\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2.21666 0.147125 0.0735624 0.997291i $$-0.476563\pi$$
0.0735624 + 0.997291i $$0.476563\pi$$
$$228$$ 0 0
$$229$$ −19.1608 −1.26618 −0.633091 0.774077i $$-0.718214\pi$$
−0.633091 + 0.774077i $$0.718214\pi$$
$$230$$ 0 0
$$231$$ −14.4721 −0.952193
$$232$$ 0 0
$$233$$ 21.7466 1.42467 0.712333 0.701842i $$-0.247639\pi$$
0.712333 + 0.701842i $$0.247639\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −20.9495 −1.35511 −0.677555 0.735472i $$-0.736961\pi$$
−0.677555 + 0.735472i $$0.736961\pi$$
$$240$$ 0 0
$$241$$ 24.0735 1.55071 0.775357 0.631523i $$-0.217570\pi$$
0.775357 + 0.631523i $$0.217570\pi$$
$$242$$ 0 0
$$243$$ 15.1103 0.969328
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 11.4858 0.730825
$$248$$ 0 0
$$249$$ 1.95789 0.124076
$$250$$ 0 0
$$251$$ 7.82651 0.494005 0.247003 0.969015i $$-0.420554\pi$$
0.247003 + 0.969015i $$0.420554\pi$$
$$252$$ 0 0
$$253$$ −5.89167 −0.370406
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −9.82012 −0.612562 −0.306281 0.951941i $$-0.599085\pi$$
−0.306281 + 0.951941i $$0.599085\pi$$
$$258$$ 0 0
$$259$$ 4.58041 0.284613
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ −14.6887 −0.905746 −0.452873 0.891575i $$-0.649601\pi$$
−0.452873 + 0.891575i $$0.649601\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 33.2344 2.03391
$$268$$ 0 0
$$269$$ −0.253432 −0.0154520 −0.00772600 0.999970i $$-0.502459\pi$$
−0.00772600 + 0.999970i $$0.502459\pi$$
$$270$$ 0 0
$$271$$ 19.6099 1.19121 0.595607 0.803276i $$-0.296912\pi$$
0.595607 + 0.803276i $$0.296912\pi$$
$$272$$ 0 0
$$273$$ 12.0157 0.727224
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −29.1976 −1.75431 −0.877157 0.480204i $$-0.840563\pi$$
−0.877157 + 0.480204i $$0.840563\pi$$
$$278$$ 0 0
$$279$$ −9.11032 −0.545421
$$280$$ 0 0
$$281$$ −27.1755 −1.62115 −0.810577 0.585633i $$-0.800846\pi$$
−0.810577 + 0.585633i $$0.800846\pi$$
$$282$$ 0 0
$$283$$ 18.1471 1.07873 0.539366 0.842071i $$-0.318664\pi$$
0.539366 + 0.842071i $$0.318664\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1.21132 0.0715021
$$288$$ 0 0
$$289$$ 17.7118 1.04187
$$290$$ 0 0
$$291$$ 28.7118 1.68311
$$292$$ 0 0
$$293$$ −0.681412 −0.0398085 −0.0199043 0.999802i $$-0.506336\pi$$
−0.0199043 + 0.999802i $$0.506336\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −17.6750 −1.02561
$$298$$ 0 0
$$299$$ 4.89167 0.282893
$$300$$ 0 0
$$301$$ −13.1608 −0.758577
$$302$$ 0 0
$$303$$ −6.00000 −0.344691
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 26.5877 1.51744 0.758721 0.651415i $$-0.225824\pi$$
0.758721 + 0.651415i $$0.225824\pi$$
$$308$$ 0 0
$$309$$ 21.2186 1.20709
$$310$$ 0 0
$$311$$ −13.5299 −0.767211 −0.383605 0.923497i $$-0.625318\pi$$
−0.383605 + 0.923497i $$0.625318\pi$$
$$312$$ 0 0
$$313$$ −19.8412 −1.12149 −0.560745 0.827989i $$-0.689485\pi$$
−0.560745 + 0.827989i $$0.689485\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −3.55096 −0.199442 −0.0997210 0.995015i $$-0.531795\pi$$
−0.0997210 + 0.995015i $$0.531795\pi$$
$$318$$ 0 0
$$319$$ −22.0735 −1.23588
$$320$$ 0 0
$$321$$ −27.2344 −1.52007
$$322$$ 0 0
$$323$$ 13.8338 0.769736
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −7.83384 −0.433212
$$328$$ 0 0
$$329$$ −8.87062 −0.489053
$$330$$ 0 0
$$331$$ 13.3133 0.731763 0.365881 0.930662i $$-0.380768\pi$$
0.365881 + 0.930662i $$0.380768\pi$$
$$332$$ 0 0
$$333$$ −6.40586 −0.351039
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 20.4951 1.11644 0.558220 0.829693i $$-0.311484\pi$$
0.558220 + 0.829693i $$0.311484\pi$$
$$338$$ 0 0
$$339$$ −21.4510 −1.16506
$$340$$ 0 0
$$341$$ 33.5162 1.81500
$$342$$ 0 0
$$343$$ 14.5299 0.784541
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −7.60147 −0.408068 −0.204034 0.978964i $$-0.565405\pi$$
−0.204034 + 0.978964i $$0.565405\pi$$
$$348$$ 0 0
$$349$$ 5.55736 0.297479 0.148739 0.988876i $$-0.452478\pi$$
0.148739 + 0.988876i $$0.452478\pi$$
$$350$$ 0 0
$$351$$ 14.6750 0.783294
$$352$$ 0 0
$$353$$ −1.96323 −0.104492 −0.0522460 0.998634i $$-0.516638\pi$$
−0.0522460 + 0.998634i $$0.516638\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 14.4721 0.765944
$$358$$ 0 0
$$359$$ −10.5069 −0.554531 −0.277266 0.960793i $$-0.589428\pi$$
−0.277266 + 0.960793i $$0.589428\pi$$
$$360$$ 0 0
$$361$$ −13.4867 −0.709828
$$362$$ 0 0
$$363$$ −50.8642 −2.66968
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 33.7412 1.76128 0.880639 0.473788i $$-0.157114\pi$$
0.880639 + 0.473788i $$0.157114\pi$$
$$368$$ 0 0
$$369$$ −1.69408 −0.0881901
$$370$$ 0 0
$$371$$ 14.8706 0.772044
$$372$$ 0 0
$$373$$ 17.4510 0.903580 0.451790 0.892124i $$-0.350786\pi$$
0.451790 + 0.892124i $$0.350786\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 18.3270 0.943887
$$378$$ 0 0
$$379$$ 33.4584 1.71864 0.859320 0.511438i $$-0.170887\pi$$
0.859320 + 0.511438i $$0.170887\pi$$
$$380$$ 0 0
$$381$$ −26.0682 −1.33551
$$382$$ 0 0
$$383$$ 13.0873 0.668728 0.334364 0.942444i $$-0.391478\pi$$
0.334364 + 0.942444i $$0.391478\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 18.4059 0.935623
$$388$$ 0 0
$$389$$ 14.1083 0.715321 0.357660 0.933852i $$-0.383575\pi$$
0.357660 + 0.933852i $$0.383575\pi$$
$$390$$ 0 0
$$391$$ 5.89167 0.297955
$$392$$ 0 0
$$393$$ −14.9074 −0.751978
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −16.6245 −0.834360 −0.417180 0.908824i $$-0.636982\pi$$
−0.417180 + 0.908824i $$0.636982\pi$$
$$398$$ 0 0
$$399$$ 5.76762 0.288742
$$400$$ 0 0
$$401$$ 22.2628 1.11175 0.555874 0.831266i $$-0.312384\pi$$
0.555874 + 0.831266i $$0.312384\pi$$
$$402$$ 0 0
$$403$$ −27.8274 −1.38618
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 23.5667 1.16816
$$408$$ 0 0
$$409$$ 15.6887 0.775758 0.387879 0.921710i $$-0.373208\pi$$
0.387879 + 0.921710i $$0.373208\pi$$
$$410$$ 0 0
$$411$$ −5.25343 −0.259133
$$412$$ 0 0
$$413$$ 0.912726 0.0449123
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 19.8201 0.970595
$$418$$ 0 0
$$419$$ 5.66769 0.276885 0.138442 0.990371i $$-0.455790\pi$$
0.138442 + 0.990371i $$0.455790\pi$$
$$420$$ 0 0
$$421$$ 15.0716 0.734543 0.367271 0.930114i $$-0.380292\pi$$
0.367271 + 0.930114i $$0.380292\pi$$
$$422$$ 0 0
$$423$$ 12.4059 0.603194
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −15.9074 −0.769813
$$428$$ 0 0
$$429$$ 61.8221 2.98480
$$430$$ 0 0
$$431$$ −29.3500 −1.41374 −0.706870 0.707343i $$-0.749894\pi$$
−0.706870 + 0.707343i $$0.749894\pi$$
$$432$$ 0 0
$$433$$ −29.6245 −1.42366 −0.711832 0.702350i $$-0.752134\pi$$
−0.711832 + 0.702350i $$0.752134\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.34803 0.112322
$$438$$ 0 0
$$439$$ 20.7907 0.992285 0.496142 0.868241i $$-0.334749\pi$$
0.496142 + 0.868241i $$0.334749\pi$$
$$440$$ 0 0
$$441$$ −9.11032 −0.433825
$$442$$ 0 0
$$443$$ 21.2975 1.01188 0.505938 0.862570i $$-0.331146\pi$$
0.505938 + 0.862570i $$0.331146\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 4.41425 0.208787
$$448$$ 0 0
$$449$$ −19.3197 −0.911751 −0.455875 0.890044i $$-0.650674\pi$$
−0.455875 + 0.890044i $$0.650674\pi$$
$$450$$ 0 0
$$451$$ 6.23238 0.293471
$$452$$ 0 0
$$453$$ 20.0525 0.942148
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −0.912726 −0.0426955 −0.0213478 0.999772i $$-0.506796\pi$$
−0.0213478 + 0.999772i $$0.506796\pi$$
$$458$$ 0 0
$$459$$ 17.6750 0.824999
$$460$$ 0 0
$$461$$ 2.83384 0.131985 0.0659926 0.997820i $$-0.478979\pi$$
0.0659926 + 0.997820i $$0.478979\pi$$
$$462$$ 0 0
$$463$$ −17.2618 −0.802225 −0.401112 0.916029i $$-0.631376\pi$$
−0.401112 + 0.916029i $$0.631376\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −21.4510 −0.992635 −0.496318 0.868141i $$-0.665315\pi$$
−0.496318 + 0.868141i $$0.665315\pi$$
$$468$$ 0 0
$$469$$ 17.8255 0.823103
$$470$$ 0 0
$$471$$ −7.74123 −0.356697
$$472$$ 0 0
$$473$$ −67.7138 −3.11348
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −20.7971 −0.952232
$$478$$ 0 0
$$479$$ −9.49314 −0.433752 −0.216876 0.976199i $$-0.569587\pi$$
−0.216876 + 0.976199i $$0.569587\pi$$
$$480$$ 0 0
$$481$$ −19.5667 −0.892164
$$482$$ 0 0
$$483$$ 2.45636 0.111768
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 38.3270 1.73676 0.868381 0.495898i $$-0.165161\pi$$
0.868381 + 0.495898i $$0.165161\pi$$
$$488$$ 0 0
$$489$$ −45.5951 −2.06188
$$490$$ 0 0
$$491$$ 13.9947 0.631570 0.315785 0.948831i $$-0.397732\pi$$
0.315785 + 0.948831i $$0.397732\pi$$
$$492$$ 0 0
$$493$$ 22.0735 0.994143
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3.36909 0.151124
$$498$$ 0 0
$$499$$ 0.0642277 0.00287522 0.00143761 0.999999i $$-0.499542\pi$$
0.00143761 + 0.999999i $$0.499542\pi$$
$$500$$ 0 0
$$501$$ 23.5667 1.05288
$$502$$ 0 0
$$503$$ −24.2785 −1.08252 −0.541262 0.840854i $$-0.682053\pi$$
−0.541262 + 0.840854i $$0.682053\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −23.4426 −1.04112
$$508$$ 0 0
$$509$$ −2.65929 −0.117871 −0.0589356 0.998262i $$-0.518771\pi$$
−0.0589356 + 0.998262i $$0.518771\pi$$
$$510$$ 0 0
$$511$$ 7.24504 0.320502
$$512$$ 0 0
$$513$$ 7.04410 0.311005
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −45.6402 −2.00726
$$518$$ 0 0
$$519$$ −13.4774 −0.591593
$$520$$ 0 0
$$521$$ −17.0598 −0.747404 −0.373702 0.927549i $$-0.621912\pi$$
−0.373702 + 0.927549i $$0.621912\pi$$
$$522$$ 0 0
$$523$$ −17.5667 −0.768137 −0.384069 0.923305i $$-0.625477\pi$$
−0.384069 + 0.923305i $$0.625477\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −33.5162 −1.45999
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −1.27648 −0.0553944
$$532$$ 0 0
$$533$$ −5.17455 −0.224135
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 21.7780 0.939790
$$538$$ 0 0
$$539$$ 33.5162 1.44364
$$540$$ 0 0
$$541$$ −8.36909 −0.359815 −0.179908 0.983684i $$-0.557580\pi$$
−0.179908 + 0.983684i $$0.557580\pi$$
$$542$$ 0 0
$$543$$ −23.9074 −1.02596
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −16.7486 −0.716117 −0.358058 0.933699i $$-0.616561\pi$$
−0.358058 + 0.933699i $$0.616561\pi$$
$$548$$ 0 0
$$549$$ 22.2470 0.949480
$$550$$ 0 0
$$551$$ 8.79707 0.374768
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 7.26182 0.307693 0.153847 0.988095i $$-0.450834\pi$$
0.153847 + 0.988095i $$0.450834\pi$$
$$558$$ 0 0
$$559$$ 56.2206 2.37788
$$560$$ 0 0
$$561$$ 74.4603 3.14372
$$562$$ 0 0
$$563$$ 34.5530 1.45623 0.728117 0.685453i $$-0.240396\pi$$
0.728117 + 0.685453i $$0.240396\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 12.8706 0.540515
$$568$$ 0 0
$$569$$ 34.7275 1.45585 0.727926 0.685655i $$-0.240484\pi$$
0.727926 + 0.685655i $$0.240484\pi$$
$$570$$ 0 0
$$571$$ −14.7034 −0.615318 −0.307659 0.951497i $$-0.599546\pi$$
−0.307659 + 0.951497i $$0.599546\pi$$
$$572$$ 0 0
$$573$$ −31.5246 −1.31696
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6.13777 0.255519 0.127759 0.991805i $$-0.459221\pi$$
0.127759 + 0.991805i $$0.459221\pi$$
$$578$$ 0 0
$$579$$ −56.6318 −2.35354
$$580$$ 0 0
$$581$$ 1.04516 0.0433607
$$582$$ 0 0
$$583$$ 76.5108 3.16876
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −1.33338 −0.0550344 −0.0275172 0.999621i $$-0.508760\pi$$
−0.0275172 + 0.999621i $$0.508760\pi$$
$$588$$ 0 0
$$589$$ −13.3574 −0.550380
$$590$$ 0 0
$$591$$ −9.13977 −0.375960
$$592$$ 0 0
$$593$$ −1.88434 −0.0773807 −0.0386903 0.999251i $$-0.512319\pi$$
−0.0386903 + 0.999251i $$0.512319\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 38.7696 1.58673
$$598$$ 0 0
$$599$$ 43.7632 1.78812 0.894058 0.447951i $$-0.147846\pi$$
0.894058 + 0.447951i $$0.147846\pi$$
$$600$$ 0 0
$$601$$ 4.50046 0.183578 0.0917889 0.995778i $$-0.470742\pi$$
0.0917889 + 0.995778i $$0.470742\pi$$
$$602$$ 0 0
$$603$$ −24.9295 −1.01521
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 6.17455 0.250617 0.125309 0.992118i $$-0.460008\pi$$
0.125309 + 0.992118i $$0.460008\pi$$
$$608$$ 0 0
$$609$$ 9.20293 0.372922
$$610$$ 0 0
$$611$$ 37.8937 1.53301
$$612$$ 0 0
$$613$$ 9.72445 0.392767 0.196383 0.980527i $$-0.437080\pi$$
0.196383 + 0.980527i $$0.437080\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0.730849 0.0294229 0.0147114 0.999892i $$-0.495317\pi$$
0.0147114 + 0.999892i $$0.495317\pi$$
$$618$$ 0 0
$$619$$ −11.1598 −0.448549 −0.224274 0.974526i $$-0.572001\pi$$
−0.224274 + 0.974526i $$0.572001\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 0 0
$$623$$ 17.7412 0.710787
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 29.6750 1.18511
$$628$$ 0 0
$$629$$ −23.5667 −0.939665
$$630$$ 0 0
$$631$$ 4.87062 0.193896 0.0969481 0.995289i $$-0.469092\pi$$
0.0969481 + 0.995289i $$0.469092\pi$$
$$632$$ 0 0
$$633$$ −25.5246 −1.01451
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −27.8274 −1.10256
$$638$$ 0 0
$$639$$ −4.71179 −0.186395
$$640$$ 0 0
$$641$$ 32.6540 1.28975 0.644877 0.764286i $$-0.276909\pi$$
0.644877 + 0.764286i $$0.276909\pi$$
$$642$$ 0 0
$$643$$ 47.9579 1.89127 0.945637 0.325223i $$-0.105439\pi$$
0.945637 + 0.325223i $$0.105439\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.1103 0.476106 0.238053 0.971252i $$-0.423491\pi$$
0.238053 + 0.971252i $$0.423491\pi$$
$$648$$ 0 0
$$649$$ 4.69607 0.184337
$$650$$ 0 0
$$651$$ −13.9736 −0.547669
$$652$$ 0 0
$$653$$ 43.2133 1.69107 0.845534 0.533922i $$-0.179282\pi$$
0.845534 + 0.533922i $$0.179282\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −10.1324 −0.395304
$$658$$ 0 0
$$659$$ −20.4333 −0.795969 −0.397984 0.917392i $$-0.630290\pi$$
−0.397984 + 0.917392i $$0.630290\pi$$
$$660$$ 0 0
$$661$$ 48.1398 1.87242 0.936210 0.351441i $$-0.114308\pi$$
0.936210 + 0.351441i $$0.114308\pi$$
$$662$$ 0 0
$$663$$ −61.8221 −2.40097
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.74657 0.145068
$$668$$ 0 0
$$669$$ −9.82545 −0.379874
$$670$$ 0 0
$$671$$ −81.8452 −3.15960
$$672$$ 0 0
$$673$$ −16.5436 −0.637710 −0.318855 0.947803i $$-0.603298\pi$$
−0.318855 + 0.947803i $$0.603298\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 48.9863 1.88270 0.941348 0.337438i $$-0.109560\pi$$
0.941348 + 0.337438i $$0.109560\pi$$
$$678$$ 0 0
$$679$$ 15.3270 0.588195
$$680$$ 0 0
$$681$$ −4.75496 −0.182210
$$682$$ 0 0
$$683$$ −21.8779 −0.837136 −0.418568 0.908185i $$-0.637468\pi$$
−0.418568 + 0.908185i $$0.637468\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 41.1019 1.56814
$$688$$ 0 0
$$689$$ −63.5246 −2.42009
$$690$$ 0 0
$$691$$ 3.30393 0.125688 0.0628438 0.998023i $$-0.479983\pi$$
0.0628438 + 0.998023i $$0.479983\pi$$
$$692$$ 0 0
$$693$$ 10.8044 0.410425
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −6.23238 −0.236068
$$698$$ 0 0
$$699$$ −46.6486 −1.76441
$$700$$ 0 0
$$701$$ −4.46369 −0.168591 −0.0842956 0.996441i $$-0.526864\pi$$
−0.0842956 + 0.996441i $$0.526864\pi$$
$$702$$ 0 0
$$703$$ −9.39214 −0.354231
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −3.20293 −0.120459
$$708$$ 0 0
$$709$$ −2.39853 −0.0900789 −0.0450394 0.998985i $$-0.514341\pi$$
−0.0450394 + 0.998985i $$0.514341\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −5.68874 −0.213045
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 44.9388 1.67827
$$718$$ 0 0
$$719$$ −12.0725 −0.450228 −0.225114 0.974332i $$-0.572275\pi$$
−0.225114 + 0.974332i $$0.572275\pi$$
$$720$$ 0 0
$$721$$ 11.3270 0.421839
$$722$$ 0 0
$$723$$ −51.6402 −1.92052
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 11.4269 0.423801 0.211900 0.977291i $$-0.432035\pi$$
0.211900 + 0.977291i $$0.432035\pi$$
$$728$$ 0 0
$$729$$ 1.30592 0.0483676
$$730$$ 0 0
$$731$$ 67.7138 2.50448
$$732$$ 0 0
$$733$$ 3.08727 0.114031 0.0570155 0.998373i $$-0.481842\pi$$
0.0570155 + 0.998373i $$0.481842\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 91.7138 3.37832
$$738$$ 0 0
$$739$$ 8.36909 0.307862 0.153931 0.988082i $$-0.450807\pi$$
0.153931 + 0.988082i $$0.450807\pi$$
$$740$$ 0 0
$$741$$ −24.6382 −0.905108
$$742$$ 0 0
$$743$$ 15.1598 0.556158 0.278079 0.960558i $$-0.410302\pi$$
0.278079 + 0.960558i $$0.410302\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −1.46170 −0.0534808
$$748$$ 0 0
$$749$$ −14.5383 −0.531218
$$750$$ 0 0
$$751$$ 33.6823 1.22909 0.614543 0.788883i $$-0.289340\pi$$
0.614543 + 0.788883i $$0.289340\pi$$
$$752$$ 0 0
$$753$$ −16.7887 −0.611813
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 29.1608 1.05987 0.529934 0.848039i $$-0.322217\pi$$
0.529934 + 0.848039i $$0.322217\pi$$
$$758$$ 0 0
$$759$$ 12.6382 0.458739
$$760$$ 0 0
$$761$$ −6.91912 −0.250818 −0.125409 0.992105i $$-0.540024\pi$$
−0.125409 + 0.992105i $$0.540024\pi$$
$$762$$ 0 0
$$763$$ −4.18188 −0.151394
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −3.89900 −0.140785
$$768$$ 0 0
$$769$$ −14.3491 −0.517442 −0.258721 0.965952i $$-0.583301\pi$$
−0.258721 + 0.965952i $$0.583301\pi$$
$$770$$ 0 0
$$771$$ 21.0652 0.758643
$$772$$ 0 0
$$773$$ −50.6265 −1.82091 −0.910454 0.413609i $$-0.864268\pi$$
−0.910454 + 0.413609i $$0.864268\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −9.82545 −0.352486
$$778$$ 0 0
$$779$$ −2.48382 −0.0889920
$$780$$ 0 0
$$781$$ 17.3343 0.620270
$$782$$ 0 0
$$783$$ 11.2397 0.401674
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 20.9442 0.746579 0.373289 0.927715i $$-0.378230\pi$$
0.373289 + 0.927715i $$0.378230\pi$$
$$788$$ 0 0
$$789$$ 31.5089 1.12174
$$790$$ 0 0
$$791$$ −11.4510 −0.407152
$$792$$ 0 0
$$793$$ 67.9535 2.41310
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −19.0009 −0.673047 −0.336524 0.941675i $$-0.609251\pi$$
−0.336524 + 0.941675i $$0.609251\pi$$
$$798$$ 0 0
$$799$$ 45.6402 1.61464
$$800$$ 0 0
$$801$$ −24.8117 −0.876679
$$802$$ 0 0
$$803$$ 37.2765 1.31546
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0.543637 0.0191369
$$808$$ 0 0
$$809$$ 18.4300 0.647963 0.323982 0.946063i $$-0.394978\pi$$
0.323982 + 0.946063i $$0.394978\pi$$
$$810$$ 0 0
$$811$$ 21.1387 0.742280 0.371140 0.928577i $$-0.378967\pi$$
0.371140 + 0.928577i $$0.378967\pi$$
$$812$$ 0 0
$$813$$ −42.0652 −1.47529
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 26.9863 0.944130
$$818$$ 0 0
$$819$$ −8.97055 −0.313457
$$820$$ 0 0
$$821$$ −16.7550 −0.584752 −0.292376 0.956303i $$-0.594446\pi$$
−0.292376 + 0.956303i $$0.594446\pi$$
$$822$$ 0 0
$$823$$ 24.9220 0.868728 0.434364 0.900737i $$-0.356973\pi$$
0.434364 + 0.900737i $$0.356973\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6.62252 −0.230288 −0.115144 0.993349i $$-0.536733\pi$$
−0.115144 + 0.993349i $$0.536733\pi$$
$$828$$ 0 0
$$829$$ 52.7001 1.83035 0.915174 0.403058i $$-0.132053\pi$$
0.915174 + 0.403058i $$0.132053\pi$$
$$830$$ 0 0
$$831$$ 62.6318 2.17267
$$832$$ 0 0
$$833$$ −33.5162 −1.16127
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −17.0662 −0.589895
$$838$$ 0 0
$$839$$ −42.5951 −1.47054 −0.735272 0.677772i $$-0.762946\pi$$
−0.735272 + 0.677772i $$0.762946\pi$$
$$840$$ 0 0
$$841$$ −14.9632 −0.515973
$$842$$ 0 0
$$843$$ 58.2942 2.00776
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −27.1524 −0.932969
$$848$$ 0 0
$$849$$ −38.9274 −1.33598
$$850$$ 0 0
$$851$$ −4.00000 −0.137118
$$852$$ 0 0
$$853$$ 57.7255 1.97648 0.988242 0.152898i $$-0.0488606\pi$$
0.988242 + 0.152898i $$0.0488606\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 55.3721 1.89148 0.945738 0.324930i $$-0.105341\pi$$
0.945738 + 0.324930i $$0.105341\pi$$
$$858$$ 0 0
$$859$$ 14.3544 0.489767 0.244883 0.969553i $$-0.421250\pi$$
0.244883 + 0.969553i $$0.421250\pi$$
$$860$$ 0 0
$$861$$ −2.59841 −0.0885536
$$862$$ 0 0
$$863$$ 15.9211 0.541961 0.270981 0.962585i $$-0.412652\pi$$
0.270981 + 0.962585i $$0.412652\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −37.9936 −1.29033
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −76.1471 −2.58015
$$872$$ 0 0
$$873$$ −21.4353 −0.725475
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30.2975 1.02308 0.511538 0.859261i $$-0.329076\pi$$
0.511538 + 0.859261i $$0.329076\pi$$
$$878$$ 0 0
$$879$$ 1.46170 0.0493019
$$880$$ 0 0
$$881$$ −11.1883 −0.376943 −0.188471 0.982079i $$-0.560353\pi$$
−0.188471 + 0.982079i $$0.560353\pi$$
$$882$$ 0 0
$$883$$ 25.2882 0.851016 0.425508 0.904955i $$-0.360095\pi$$
0.425508 + 0.904955i $$0.360095\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −32.2849 −1.08402 −0.542010 0.840372i $$-0.682336\pi$$
−0.542010 + 0.840372i $$0.682336\pi$$
$$888$$ 0 0
$$889$$ −13.9158 −0.466720
$$890$$ 0 0
$$891$$ 66.2206 2.21847
$$892$$ 0 0
$$893$$ 18.1892 0.608679
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −10.4931 −0.350356
$$898$$ 0 0
$$899$$ −21.3133 −0.710837
$$900$$ 0 0
$$901$$ −76.5108 −2.54895
$$902$$ 0 0
$$903$$ 28.2313 0.939479
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −1.47848 −0.0490922 −0.0245461 0.999699i $$-0.507814\pi$$
−0.0245461 + 0.999699i $$0.507814\pi$$
$$908$$ 0 0
$$909$$ 4.47941 0.148573
$$910$$ 0 0
$$911$$ −5.70979 −0.189174 −0.0945870 0.995517i $$-0.530153\pi$$
−0.0945870 + 0.995517i $$0.530153\pi$$
$$912$$ 0 0
$$913$$ 5.37748 0.177969
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −7.95789 −0.262793
$$918$$ 0 0
$$919$$ −22.6265 −0.746379 −0.373190 0.927755i $$-0.621736\pi$$
−0.373190 + 0.927755i $$0.621736\pi$$
$$920$$ 0 0
$$921$$ −57.0334 −1.87932
$$922$$ 0 0
$$923$$ −14.3921 −0.473723
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −15.8412 −0.520292
$$928$$ 0 0
$$929$$ −50.0829 −1.64317 −0.821583 0.570089i $$-0.806909\pi$$
−0.821583 + 0.570089i $$0.806909\pi$$
$$930$$ 0 0
$$931$$ −13.3574 −0.437770
$$932$$ 0 0
$$933$$ 29.0230 0.950172
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −9.98428 −0.326172 −0.163086 0.986612i $$-0.552145\pi$$
−0.163086 + 0.986612i $$0.552145\pi$$
$$938$$ 0 0
$$939$$ 42.5613 1.38894
$$940$$ 0 0
$$941$$ −36.0809 −1.17620 −0.588101 0.808787i $$-0.700124\pi$$
−0.588101 + 0.808787i $$0.700124\pi$$
$$942$$ 0 0
$$943$$ −1.05783 −0.0344476
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −32.4730 −1.05523 −0.527616 0.849483i $$-0.676914\pi$$
−0.527616 + 0.849483i $$0.676914\pi$$
$$948$$ 0 0
$$949$$ −30.9495 −1.00466
$$950$$ 0 0
$$951$$ 7.61718 0.247004
$$952$$ 0 0
$$953$$ 39.3427 1.27443 0.637217 0.770684i $$-0.280085\pi$$
0.637217 + 0.770684i $$0.280085\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 47.3500 1.53061
$$958$$ 0 0
$$959$$ −2.80440 −0.0905587
$$960$$ 0 0
$$961$$ 1.36176 0.0439278
$$962$$ 0 0
$$963$$ 20.3323 0.655200
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 23.5720 0.758025 0.379013 0.925392i $$-0.376264\pi$$
0.379013 + 0.925392i $$0.376264\pi$$
$$968$$ 0 0
$$969$$ −29.6750 −0.953299
$$970$$ 0 0
$$971$$ −12.9368 −0.415163 −0.207581 0.978218i $$-0.566559\pi$$
−0.207581 + 0.978218i $$0.566559\pi$$
$$972$$ 0 0
$$973$$ 10.5804 0.339192
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 22.7118 0.726614 0.363307 0.931669i $$-0.381648\pi$$
0.363307 + 0.931669i $$0.381648\pi$$
$$978$$ 0 0
$$979$$ 91.2805 2.91734
$$980$$ 0 0
$$981$$ 5.84850 0.186728
$$982$$ 0 0
$$983$$ 26.1819 0.835072 0.417536 0.908660i $$-0.362894\pi$$
0.417536 + 0.908660i $$0.362894\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 19.0284 0.605680
$$988$$ 0 0
$$989$$ 11.4931 0.365460
$$990$$ 0 0
$$991$$ 56.3143 1.78888 0.894442 0.447185i $$-0.147573\pi$$
0.894442 + 0.447185i $$0.147573\pi$$
$$992$$ 0 0
$$993$$ −28.5583 −0.906270
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 3.74123 0.118486 0.0592430 0.998244i $$-0.481131\pi$$
0.0592430 + 0.998244i $$0.481131\pi$$
$$998$$ 0 0
$$999$$ −12.0000 −0.379663
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 9200.2.a.cd.1.1 3
4.3 odd 2 4600.2.a.y.1.3 3
5.4 even 2 1840.2.a.t.1.3 3
20.3 even 4 4600.2.e.r.4049.5 6
20.7 even 4 4600.2.e.r.4049.2 6
20.19 odd 2 920.2.a.g.1.1 3
40.19 odd 2 7360.2.a.cb.1.3 3
40.29 even 2 7360.2.a.ca.1.1 3
60.59 even 2 8280.2.a.bo.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.1 3 20.19 odd 2
1840.2.a.t.1.3 3 5.4 even 2
4600.2.a.y.1.3 3 4.3 odd 2
4600.2.e.r.4049.2 6 20.7 even 4
4600.2.e.r.4049.5 6 20.3 even 4
7360.2.a.ca.1.1 3 40.29 even 2
7360.2.a.cb.1.3 3 40.19 odd 2
8280.2.a.bo.1.1 3 60.59 even 2
9200.2.a.cd.1.1 3 1.1 even 1 trivial