# Properties

 Label 6900.2.a.x.1.2 Level $6900$ Weight $2$ Character 6900.1 Self dual yes Analytic conductor $55.097$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$55.0967773947$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1380) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.526440$$ of defining polynomial Character $$\chi$$ $$=$$ 6900.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.52644 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.52644 q^{7} +1.00000 q^{9} -3.59821 q^{11} -5.59821 q^{13} -4.07177 q^{17} -1.59821 q^{19} +1.52644 q^{21} -1.00000 q^{23} -1.00000 q^{27} -4.07177 q^{29} +2.47356 q^{31} +3.59821 q^{33} -9.66998 q^{37} +5.59821 q^{39} +5.01889 q^{41} -7.05288 q^{43} +4.54533 q^{47} -4.66998 q^{49} +4.07177 q^{51} -5.01889 q^{53} +1.59821 q^{57} +4.07177 q^{59} +6.54533 q^{61} -1.52644 q^{63} -2.47356 q^{67} +1.00000 q^{69} -5.01889 q^{71} +8.79463 q^{73} +5.49245 q^{77} +2.94712 q^{79} +1.00000 q^{81} -9.12465 q^{83} +4.07177 q^{87} +12.2493 q^{89} +8.54533 q^{91} -2.47356 q^{93} +13.0907 q^{97} -3.59821 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 2 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 2 q^{7} + 3 q^{9} + 4 q^{11} - 2 q^{13} + 10 q^{19} + 2 q^{21} - 3 q^{23} - 3 q^{27} + 10 q^{31} - 4 q^{33} - 2 q^{37} + 2 q^{39} + 8 q^{41} - 16 q^{43} + 4 q^{47} + 13 q^{49} - 8 q^{53} - 10 q^{57} + 10 q^{61} - 2 q^{63} - 10 q^{67} + 3 q^{69} - 8 q^{71} - 18 q^{73} + 12 q^{77} + 14 q^{79} + 3 q^{81} - 10 q^{83} + 2 q^{89} + 16 q^{91} - 10 q^{93} + 20 q^{97} + 4 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 2 * q^7 + 3 * q^9 + 4 * q^11 - 2 * q^13 + 10 * q^19 + 2 * q^21 - 3 * q^23 - 3 * q^27 + 10 * q^31 - 4 * q^33 - 2 * q^37 + 2 * q^39 + 8 * q^41 - 16 * q^43 + 4 * q^47 + 13 * q^49 - 8 * q^53 - 10 * q^57 + 10 * q^61 - 2 * q^63 - 10 * q^67 + 3 * q^69 - 8 * q^71 - 18 * q^73 + 12 * q^77 + 14 * q^79 + 3 * q^81 - 10 * q^83 + 2 * q^89 + 16 * q^91 - 10 * q^93 + 20 * q^97 + 4 * 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 0
$$6$$ 0 0
$$7$$ −1.52644 −0.576940 −0.288470 0.957489i $$-0.593147\pi$$
−0.288470 + 0.957489i $$0.593147\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.59821 −1.08490 −0.542451 0.840088i $$-0.682503\pi$$
−0.542451 + 0.840088i $$0.682503\pi$$
$$12$$ 0 0
$$13$$ −5.59821 −1.55266 −0.776332 0.630324i $$-0.782922\pi$$
−0.776332 + 0.630324i $$0.782922\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.07177 −0.987550 −0.493775 0.869590i $$-0.664383\pi$$
−0.493775 + 0.869590i $$0.664383\pi$$
$$18$$ 0 0
$$19$$ −1.59821 −0.366655 −0.183327 0.983052i $$-0.558687\pi$$
−0.183327 + 0.983052i $$0.558687\pi$$
$$20$$ 0 0
$$21$$ 1.52644 0.333096
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −4.07177 −0.756109 −0.378054 0.925783i $$-0.623407\pi$$
−0.378054 + 0.925783i $$0.623407\pi$$
$$30$$ 0 0
$$31$$ 2.47356 0.444265 0.222132 0.975017i $$-0.428698\pi$$
0.222132 + 0.975017i $$0.428698\pi$$
$$32$$ 0 0
$$33$$ 3.59821 0.626368
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.66998 −1.58974 −0.794868 0.606783i $$-0.792460\pi$$
−0.794868 + 0.606783i $$0.792460\pi$$
$$38$$ 0 0
$$39$$ 5.59821 0.896431
$$40$$ 0 0
$$41$$ 5.01889 0.783819 0.391910 0.920004i $$-0.371815\pi$$
0.391910 + 0.920004i $$0.371815\pi$$
$$42$$ 0 0
$$43$$ −7.05288 −1.07555 −0.537777 0.843087i $$-0.680736\pi$$
−0.537777 + 0.843087i $$0.680736\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.54533 0.663005 0.331502 0.943454i $$-0.392445\pi$$
0.331502 + 0.943454i $$0.392445\pi$$
$$48$$ 0 0
$$49$$ −4.66998 −0.667140
$$50$$ 0 0
$$51$$ 4.07177 0.570162
$$52$$ 0 0
$$53$$ −5.01889 −0.689398 −0.344699 0.938713i $$-0.612019\pi$$
−0.344699 + 0.938713i $$0.612019\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.59821 0.211688
$$58$$ 0 0
$$59$$ 4.07177 0.530099 0.265050 0.964235i $$-0.414612\pi$$
0.265050 + 0.964235i $$0.414612\pi$$
$$60$$ 0 0
$$61$$ 6.54533 0.838044 0.419022 0.907976i $$-0.362373\pi$$
0.419022 + 0.907976i $$0.362373\pi$$
$$62$$ 0 0
$$63$$ −1.52644 −0.192313
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.47356 −0.302193 −0.151097 0.988519i $$-0.548280\pi$$
−0.151097 + 0.988519i $$0.548280\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −5.01889 −0.595633 −0.297816 0.954623i $$-0.596258\pi$$
−0.297816 + 0.954623i $$0.596258\pi$$
$$72$$ 0 0
$$73$$ 8.79463 1.02933 0.514667 0.857390i $$-0.327916\pi$$
0.514667 + 0.857390i $$0.327916\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 5.49245 0.625923
$$78$$ 0 0
$$79$$ 2.94712 0.331577 0.165788 0.986161i $$-0.446983\pi$$
0.165788 + 0.986161i $$0.446983\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −9.12465 −1.00156 −0.500780 0.865574i $$-0.666954\pi$$
−0.500780 + 0.865574i $$0.666954\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 4.07177 0.436540
$$88$$ 0 0
$$89$$ 12.2493 1.29842 0.649212 0.760608i $$-0.275099\pi$$
0.649212 + 0.760608i $$0.275099\pi$$
$$90$$ 0 0
$$91$$ 8.54533 0.895794
$$92$$ 0 0
$$93$$ −2.47356 −0.256496
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 13.0907 1.32916 0.664578 0.747219i $$-0.268611\pi$$
0.664578 + 0.747219i $$0.268611\pi$$
$$98$$ 0 0
$$99$$ −3.59821 −0.361634
$$100$$ 0 0
$$101$$ −12.2153 −1.21547 −0.607735 0.794140i $$-0.707922\pi$$
−0.607735 + 0.794140i $$0.707922\pi$$
$$102$$ 0 0
$$103$$ 0.143542 0.0141436 0.00707181 0.999975i $$-0.497749\pi$$
0.00707181 + 0.999975i $$0.497749\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −5.26819 −0.509295 −0.254648 0.967034i $$-0.581960\pi$$
−0.254648 + 0.967034i $$0.581960\pi$$
$$108$$ 0 0
$$109$$ 12.7946 1.22550 0.612752 0.790275i $$-0.290063\pi$$
0.612752 + 0.790275i $$0.290063\pi$$
$$110$$ 0 0
$$111$$ 9.66998 0.917834
$$112$$ 0 0
$$113$$ −5.01889 −0.472138 −0.236069 0.971736i $$-0.575859\pi$$
−0.236069 + 0.971736i $$0.575859\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −5.59821 −0.517555
$$118$$ 0 0
$$119$$ 6.21531 0.569757
$$120$$ 0 0
$$121$$ 1.94712 0.177011
$$122$$ 0 0
$$123$$ −5.01889 −0.452538
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 17.7040 1.57097 0.785487 0.618879i $$-0.212413\pi$$
0.785487 + 0.618879i $$0.212413\pi$$
$$128$$ 0 0
$$129$$ 7.05288 0.620971
$$130$$ 0 0
$$131$$ −6.94712 −0.606973 −0.303486 0.952836i $$-0.598151\pi$$
−0.303486 + 0.952836i $$0.598151\pi$$
$$132$$ 0 0
$$133$$ 2.43957 0.211538
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.2493 1.04653 0.523264 0.852171i $$-0.324714\pi$$
0.523264 + 0.852171i $$0.324714\pi$$
$$138$$ 0 0
$$139$$ −6.61710 −0.561255 −0.280628 0.959817i $$-0.590543\pi$$
−0.280628 + 0.959817i $$0.590543\pi$$
$$140$$ 0 0
$$141$$ −4.54533 −0.382786
$$142$$ 0 0
$$143$$ 20.1435 1.68449
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.66998 0.385174
$$148$$ 0 0
$$149$$ 9.59821 0.786316 0.393158 0.919471i $$-0.371383\pi$$
0.393158 + 0.919471i $$0.371383\pi$$
$$150$$ 0 0
$$151$$ −4.94712 −0.402591 −0.201295 0.979531i $$-0.564515\pi$$
−0.201295 + 0.979531i $$0.564515\pi$$
$$152$$ 0 0
$$153$$ −4.07177 −0.329183
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 7.63220 0.609116 0.304558 0.952494i $$-0.401491\pi$$
0.304558 + 0.952494i $$0.401491\pi$$
$$158$$ 0 0
$$159$$ 5.01889 0.398024
$$160$$ 0 0
$$161$$ 1.52644 0.120300
$$162$$ 0 0
$$163$$ −23.3400 −1.82813 −0.914064 0.405571i $$-0.867073\pi$$
−0.914064 + 0.405571i $$0.867073\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 7.45467 0.576860 0.288430 0.957501i $$-0.406867\pi$$
0.288430 + 0.957501i $$0.406867\pi$$
$$168$$ 0 0
$$169$$ 18.3400 1.41077
$$170$$ 0 0
$$171$$ −1.59821 −0.122218
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −4.07177 −0.306053
$$178$$ 0 0
$$179$$ −22.2871 −1.66581 −0.832907 0.553412i $$-0.813325\pi$$
−0.832907 + 0.553412i $$0.813325\pi$$
$$180$$ 0 0
$$181$$ 20.2493 1.50512 0.752559 0.658524i $$-0.228819\pi$$
0.752559 + 0.658524i $$0.228819\pi$$
$$182$$ 0 0
$$183$$ −6.54533 −0.483845
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 14.6511 1.07139
$$188$$ 0 0
$$189$$ 1.52644 0.111032
$$190$$ 0 0
$$191$$ −21.8475 −1.58083 −0.790415 0.612571i $$-0.790135\pi$$
−0.790415 + 0.612571i $$0.790135\pi$$
$$192$$ 0 0
$$193$$ −12.0378 −0.866499 −0.433249 0.901274i $$-0.642633\pi$$
−0.433249 + 0.901274i $$0.642633\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.2493 1.30021 0.650104 0.759845i $$-0.274725\pi$$
0.650104 + 0.759845i $$0.274725\pi$$
$$198$$ 0 0
$$199$$ −17.1964 −1.21902 −0.609511 0.792778i $$-0.708634\pi$$
−0.609511 + 0.792778i $$0.708634\pi$$
$$200$$ 0 0
$$201$$ 2.47356 0.174471
$$202$$ 0 0
$$203$$ 6.21531 0.436229
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ 5.75070 0.397784
$$210$$ 0 0
$$211$$ −5.66998 −0.390338 −0.195169 0.980770i $$-0.562525\pi$$
−0.195169 + 0.980770i $$0.562525\pi$$
$$212$$ 0 0
$$213$$ 5.01889 0.343889
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.77574 −0.256314
$$218$$ 0 0
$$219$$ −8.79463 −0.594286
$$220$$ 0 0
$$221$$ 22.7946 1.53333
$$222$$ 0 0
$$223$$ −11.3400 −0.759380 −0.379690 0.925114i $$-0.623969\pi$$
−0.379690 + 0.925114i $$0.623969\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 26.3928 1.75175 0.875877 0.482534i $$-0.160284\pi$$
0.875877 + 0.482534i $$0.160284\pi$$
$$228$$ 0 0
$$229$$ −4.24930 −0.280802 −0.140401 0.990095i $$-0.544839\pi$$
−0.140401 + 0.990095i $$0.544839\pi$$
$$230$$ 0 0
$$231$$ −5.49245 −0.361377
$$232$$ 0 0
$$233$$ 1.89424 0.124096 0.0620479 0.998073i $$-0.480237\pi$$
0.0620479 + 0.998073i $$0.480237\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −2.94712 −0.191436
$$238$$ 0 0
$$239$$ −6.98111 −0.451570 −0.225785 0.974177i $$-0.572495\pi$$
−0.225785 + 0.974177i $$0.572495\pi$$
$$240$$ 0 0
$$241$$ 3.20537 0.206476 0.103238 0.994657i $$-0.467080\pi$$
0.103238 + 0.994657i $$0.467080\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.94712 0.569292
$$248$$ 0 0
$$249$$ 9.12465 0.578251
$$250$$ 0 0
$$251$$ 13.4457 0.848686 0.424343 0.905501i $$-0.360505\pi$$
0.424343 + 0.905501i $$0.360505\pi$$
$$252$$ 0 0
$$253$$ 3.59821 0.226218
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −18.9382 −1.18133 −0.590665 0.806917i $$-0.701135\pi$$
−0.590665 + 0.806917i $$0.701135\pi$$
$$258$$ 0 0
$$259$$ 14.7606 0.917182
$$260$$ 0 0
$$261$$ −4.07177 −0.252036
$$262$$ 0 0
$$263$$ 0.981108 0.0604977 0.0302489 0.999542i $$-0.490370\pi$$
0.0302489 + 0.999542i $$0.490370\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −12.2493 −0.749645
$$268$$ 0 0
$$269$$ −3.12465 −0.190513 −0.0952567 0.995453i $$-0.530367\pi$$
−0.0952567 + 0.995453i $$0.530367\pi$$
$$270$$ 0 0
$$271$$ −9.52644 −0.578690 −0.289345 0.957225i $$-0.593437\pi$$
−0.289345 + 0.957225i $$0.593437\pi$$
$$272$$ 0 0
$$273$$ −8.54533 −0.517187
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −4.39284 −0.263940 −0.131970 0.991254i $$-0.542130\pi$$
−0.131970 + 0.991254i $$0.542130\pi$$
$$278$$ 0 0
$$279$$ 2.47356 0.148088
$$280$$ 0 0
$$281$$ 12.9382 0.771827 0.385913 0.922535i $$-0.373886\pi$$
0.385913 + 0.922535i $$0.373886\pi$$
$$282$$ 0 0
$$283$$ −19.7757 −1.17555 −0.587773 0.809026i $$-0.699995\pi$$
−0.587773 + 0.809026i $$0.699995\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −7.66104 −0.452217
$$288$$ 0 0
$$289$$ −0.420681 −0.0247459
$$290$$ 0 0
$$291$$ −13.0907 −0.767388
$$292$$ 0 0
$$293$$ 11.2682 0.658295 0.329147 0.944279i $$-0.393239\pi$$
0.329147 + 0.944279i $$0.393239\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.59821 0.208789
$$298$$ 0 0
$$299$$ 5.59821 0.323753
$$300$$ 0 0
$$301$$ 10.7658 0.620530
$$302$$ 0 0
$$303$$ 12.2153 0.701751
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 25.8475 1.47520 0.737598 0.675240i $$-0.235960\pi$$
0.737598 + 0.675240i $$0.235960\pi$$
$$308$$ 0 0
$$309$$ −0.143542 −0.00816582
$$310$$ 0 0
$$311$$ −29.0529 −1.64744 −0.823719 0.566998i $$-0.808105\pi$$
−0.823719 + 0.566998i $$0.808105\pi$$
$$312$$ 0 0
$$313$$ 21.9571 1.24109 0.620543 0.784172i $$-0.286912\pi$$
0.620543 + 0.784172i $$0.286912\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.65109 −0.148900 −0.0744500 0.997225i $$-0.523720\pi$$
−0.0744500 + 0.997225i $$0.523720\pi$$
$$318$$ 0 0
$$319$$ 14.6511 0.820304
$$320$$ 0 0
$$321$$ 5.26819 0.294042
$$322$$ 0 0
$$323$$ 6.50755 0.362090
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −12.7946 −0.707545
$$328$$ 0 0
$$329$$ −6.93817 −0.382514
$$330$$ 0 0
$$331$$ 7.77574 0.427393 0.213697 0.976900i $$-0.431450\pi$$
0.213697 + 0.976900i $$0.431450\pi$$
$$332$$ 0 0
$$333$$ −9.66998 −0.529912
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 16.9471 0.923168 0.461584 0.887096i $$-0.347281\pi$$
0.461584 + 0.887096i $$0.347281\pi$$
$$338$$ 0 0
$$339$$ 5.01889 0.272589
$$340$$ 0 0
$$341$$ −8.90039 −0.481983
$$342$$ 0 0
$$343$$ 17.8135 0.961840
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −2.90934 −0.156181 −0.0780907 0.996946i $$-0.524882\pi$$
−0.0780907 + 0.996946i $$0.524882\pi$$
$$348$$ 0 0
$$349$$ −18.8664 −1.00990 −0.504948 0.863150i $$-0.668488\pi$$
−0.504948 + 0.863150i $$0.668488\pi$$
$$350$$ 0 0
$$351$$ 5.59821 0.298810
$$352$$ 0 0
$$353$$ −18.9382 −1.00798 −0.503989 0.863710i $$-0.668135\pi$$
−0.503989 + 0.863710i $$0.668135\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −6.21531 −0.328949
$$358$$ 0 0
$$359$$ −4.54533 −0.239893 −0.119947 0.992780i $$-0.538272\pi$$
−0.119947 + 0.992780i $$0.538272\pi$$
$$360$$ 0 0
$$361$$ −16.4457 −0.865564
$$362$$ 0 0
$$363$$ −1.94712 −0.102197
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 33.9571 1.77254 0.886272 0.463165i $$-0.153286\pi$$
0.886272 + 0.463165i $$0.153286\pi$$
$$368$$ 0 0
$$369$$ 5.01889 0.261273
$$370$$ 0 0
$$371$$ 7.66104 0.397741
$$372$$ 0 0
$$373$$ 12.2115 0.632288 0.316144 0.948711i $$-0.397612\pi$$
0.316144 + 0.948711i $$0.397612\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 22.7946 1.17398
$$378$$ 0 0
$$379$$ 13.0529 0.670481 0.335241 0.942133i $$-0.391182\pi$$
0.335241 + 0.942133i $$0.391182\pi$$
$$380$$ 0 0
$$381$$ −17.7040 −0.907002
$$382$$ 0 0
$$383$$ 26.4268 1.35035 0.675174 0.737659i $$-0.264069\pi$$
0.675174 + 0.737659i $$0.264069\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −7.05288 −0.358518
$$388$$ 0 0
$$389$$ −15.5893 −0.790407 −0.395204 0.918594i $$-0.629326\pi$$
−0.395204 + 0.918594i $$0.629326\pi$$
$$390$$ 0 0
$$391$$ 4.07177 0.205918
$$392$$ 0 0
$$393$$ 6.94712 0.350436
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 35.4457 1.77897 0.889485 0.456965i $$-0.151063\pi$$
0.889485 + 0.456965i $$0.151063\pi$$
$$398$$ 0 0
$$399$$ −2.43957 −0.122131
$$400$$ 0 0
$$401$$ 27.5893 1.37774 0.688871 0.724884i $$-0.258107\pi$$
0.688871 + 0.724884i $$0.258107\pi$$
$$402$$ 0 0
$$403$$ −13.8475 −0.689794
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 34.7946 1.72471
$$408$$ 0 0
$$409$$ 12.7606 0.630973 0.315487 0.948930i $$-0.397832\pi$$
0.315487 + 0.948930i $$0.397832\pi$$
$$410$$ 0 0
$$411$$ −12.2493 −0.604213
$$412$$ 0 0
$$413$$ −6.21531 −0.305836
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 6.61710 0.324041
$$418$$ 0 0
$$419$$ −22.7946 −1.11359 −0.556795 0.830650i $$-0.687969\pi$$
−0.556795 + 0.830650i $$0.687969\pi$$
$$420$$ 0 0
$$421$$ 26.6889 1.30074 0.650368 0.759619i $$-0.274615\pi$$
0.650368 + 0.759619i $$0.274615\pi$$
$$422$$ 0 0
$$423$$ 4.54533 0.221002
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −9.99105 −0.483501
$$428$$ 0 0
$$429$$ −20.1435 −0.972539
$$430$$ 0 0
$$431$$ −27.4079 −1.32019 −0.660097 0.751180i $$-0.729485\pi$$
−0.660097 + 0.751180i $$0.729485\pi$$
$$432$$ 0 0
$$433$$ 8.57932 0.412296 0.206148 0.978521i $$-0.433907\pi$$
0.206148 + 0.978521i $$0.433907\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.59821 0.0764528
$$438$$ 0 0
$$439$$ 9.96222 0.475471 0.237735 0.971330i $$-0.423595\pi$$
0.237735 + 0.971330i $$0.423595\pi$$
$$440$$ 0 0
$$441$$ −4.66998 −0.222380
$$442$$ 0 0
$$443$$ 1.63599 0.0777284 0.0388642 0.999245i $$-0.487626\pi$$
0.0388642 + 0.999245i $$0.487626\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −9.59821 −0.453980
$$448$$ 0 0
$$449$$ 38.6082 1.82203 0.911016 0.412372i $$-0.135299\pi$$
0.911016 + 0.412372i $$0.135299\pi$$
$$450$$ 0 0
$$451$$ −18.0590 −0.850366
$$452$$ 0 0
$$453$$ 4.94712 0.232436
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.6700 0.826567 0.413283 0.910602i $$-0.364382\pi$$
0.413283 + 0.910602i $$0.364382\pi$$
$$458$$ 0 0
$$459$$ 4.07177 0.190054
$$460$$ 0 0
$$461$$ −15.0907 −0.702842 −0.351421 0.936217i $$-0.614301\pi$$
−0.351421 + 0.936217i $$0.614301\pi$$
$$462$$ 0 0
$$463$$ −22.1346 −1.02868 −0.514341 0.857586i $$-0.671963\pi$$
−0.514341 + 0.857586i $$0.671963\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −29.2682 −1.35437 −0.677185 0.735813i $$-0.736800\pi$$
−0.677185 + 0.735813i $$0.736800\pi$$
$$468$$ 0 0
$$469$$ 3.77574 0.174348
$$470$$ 0 0
$$471$$ −7.63220 −0.351673
$$472$$ 0 0
$$473$$ 25.3777 1.16687
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −5.01889 −0.229799
$$478$$ 0 0
$$479$$ 18.9382 0.865307 0.432654 0.901560i $$-0.357577\pi$$
0.432654 + 0.901560i $$0.357577\pi$$
$$480$$ 0 0
$$481$$ 54.1346 2.46833
$$482$$ 0 0
$$483$$ −1.52644 −0.0694554
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 5.20537 0.235878 0.117939 0.993021i $$-0.462371\pi$$
0.117939 + 0.993021i $$0.462371\pi$$
$$488$$ 0 0
$$489$$ 23.3400 1.05547
$$490$$ 0 0
$$491$$ −12.2153 −0.551269 −0.275635 0.961262i $$-0.588888\pi$$
−0.275635 + 0.961262i $$0.588888\pi$$
$$492$$ 0 0
$$493$$ 16.5793 0.746695
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 7.66104 0.343644
$$498$$ 0 0
$$499$$ 35.1157 1.57199 0.785997 0.618230i $$-0.212150\pi$$
0.785997 + 0.618230i $$0.212150\pi$$
$$500$$ 0 0
$$501$$ −7.45467 −0.333050
$$502$$ 0 0
$$503$$ 6.21531 0.277127 0.138564 0.990354i $$-0.455751\pi$$
0.138564 + 0.990354i $$0.455751\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −18.3400 −0.814506
$$508$$ 0 0
$$509$$ 9.85646 0.436880 0.218440 0.975850i $$-0.429903\pi$$
0.218440 + 0.975850i $$0.429903\pi$$
$$510$$ 0 0
$$511$$ −13.4245 −0.593864
$$512$$ 0 0
$$513$$ 1.59821 0.0705627
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −16.3551 −0.719295
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −41.2253 −1.80611 −0.903056 0.429524i $$-0.858682\pi$$
−0.903056 + 0.429524i $$0.858682\pi$$
$$522$$ 0 0
$$523$$ 22.6799 0.991724 0.495862 0.868401i $$-0.334852\pi$$
0.495862 + 0.868401i $$0.334852\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −10.0718 −0.438733
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 4.07177 0.176700
$$532$$ 0 0
$$533$$ −28.0968 −1.21701
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 22.2871 0.961759
$$538$$ 0 0
$$539$$ 16.8036 0.723781
$$540$$ 0 0
$$541$$ 22.1435 0.952025 0.476013 0.879438i $$-0.342082\pi$$
0.476013 + 0.879438i $$0.342082\pi$$
$$542$$ 0 0
$$543$$ −20.2493 −0.868981
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ 6.54533 0.279348
$$550$$ 0 0
$$551$$ 6.50755 0.277231
$$552$$ 0 0
$$553$$ −4.49860 −0.191300
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 9.37395 0.397187 0.198594 0.980082i $$-0.436363\pi$$
0.198594 + 0.980082i $$0.436363\pi$$
$$558$$ 0 0
$$559$$ 39.4835 1.66997
$$560$$ 0 0
$$561$$ −14.6511 −0.618570
$$562$$ 0 0
$$563$$ −27.3060 −1.15081 −0.575405 0.817869i $$-0.695155\pi$$
−0.575405 + 0.817869i $$0.695155\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −1.52644 −0.0641044
$$568$$ 0 0
$$569$$ 23.3022 0.976878 0.488439 0.872598i $$-0.337566\pi$$
0.488439 + 0.872598i $$0.337566\pi$$
$$570$$ 0 0
$$571$$ 32.9382 1.37842 0.689210 0.724562i $$-0.257958\pi$$
0.689210 + 0.724562i $$0.257958\pi$$
$$572$$ 0 0
$$573$$ 21.8475 0.912693
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −41.3400 −1.72101 −0.860503 0.509446i $$-0.829850\pi$$
−0.860503 + 0.509446i $$0.829850\pi$$
$$578$$ 0 0
$$579$$ 12.0378 0.500273
$$580$$ 0 0
$$581$$ 13.9282 0.577840
$$582$$ 0 0
$$583$$ 18.0590 0.747929
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −23.0529 −0.951494 −0.475747 0.879582i $$-0.657822\pi$$
−0.475747 + 0.879582i $$0.657822\pi$$
$$588$$ 0 0
$$589$$ −3.95327 −0.162892
$$590$$ 0 0
$$591$$ −18.2493 −0.750676
$$592$$ 0 0
$$593$$ −31.9533 −1.31216 −0.656082 0.754690i $$-0.727787\pi$$
−0.656082 + 0.754690i $$0.727787\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 17.1964 0.703803
$$598$$ 0 0
$$599$$ 26.6421 1.08857 0.544284 0.838901i $$-0.316801\pi$$
0.544284 + 0.838901i $$0.316801\pi$$
$$600$$ 0 0
$$601$$ 7.45846 0.304237 0.152119 0.988362i $$-0.451390\pi$$
0.152119 + 0.988362i $$0.451390\pi$$
$$602$$ 0 0
$$603$$ −2.47356 −0.100731
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −13.4925 −0.547642 −0.273821 0.961781i $$-0.588288\pi$$
−0.273821 + 0.961781i $$0.588288\pi$$
$$608$$ 0 0
$$609$$ −6.21531 −0.251857
$$610$$ 0 0
$$611$$ −25.4457 −1.02942
$$612$$ 0 0
$$613$$ 41.4457 1.67398 0.836988 0.547221i $$-0.184314\pi$$
0.836988 + 0.547221i $$0.184314\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −25.2304 −1.01574 −0.507869 0.861434i $$-0.669567\pi$$
−0.507869 + 0.861434i $$0.669567\pi$$
$$618$$ 0 0
$$619$$ −5.12845 −0.206130 −0.103065 0.994675i $$-0.532865\pi$$
−0.103065 + 0.994675i $$0.532865\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ −18.6978 −0.749112
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −5.75070 −0.229661
$$628$$ 0 0
$$629$$ 39.3740 1.56994
$$630$$ 0 0
$$631$$ −11.6360 −0.463222 −0.231611 0.972809i $$-0.574400\pi$$
−0.231611 + 0.972809i $$0.574400\pi$$
$$632$$ 0 0
$$633$$ 5.66998 0.225362
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 26.1435 1.03584
$$638$$ 0 0
$$639$$ −5.01889 −0.198544
$$640$$ 0 0
$$641$$ 17.7418 0.700757 0.350379 0.936608i $$-0.386053\pi$$
0.350379 + 0.936608i $$0.386053\pi$$
$$642$$ 0 0
$$643$$ −31.7757 −1.25311 −0.626556 0.779376i $$-0.715536\pi$$
−0.626556 + 0.779376i $$0.715536\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 46.7946 1.83969 0.919843 0.392286i $$-0.128316\pi$$
0.919843 + 0.392286i $$0.128316\pi$$
$$648$$ 0 0
$$649$$ −14.6511 −0.575106
$$650$$ 0 0
$$651$$ 3.77574 0.147983
$$652$$ 0 0
$$653$$ −3.59821 −0.140809 −0.0704044 0.997519i $$-0.522429\pi$$
−0.0704044 + 0.997519i $$0.522429\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8.79463 0.343111
$$658$$ 0 0
$$659$$ −5.49245 −0.213956 −0.106978 0.994261i $$-0.534117\pi$$
−0.106978 + 0.994261i $$0.534117\pi$$
$$660$$ 0 0
$$661$$ 4.90934 0.190951 0.0954755 0.995432i $$-0.469563\pi$$
0.0954755 + 0.995432i $$0.469563\pi$$
$$662$$ 0 0
$$663$$ −22.7946 −0.885270
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.07177 0.157660
$$668$$ 0 0
$$669$$ 11.3400 0.438428
$$670$$ 0 0
$$671$$ −23.5515 −0.909195
$$672$$ 0 0
$$673$$ −25.2253 −0.972362 −0.486181 0.873858i $$-0.661610\pi$$
−0.486181 + 0.873858i $$0.661610\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −49.6610 −1.90863 −0.954314 0.298805i $$-0.903412\pi$$
−0.954314 + 0.298805i $$0.903412\pi$$
$$678$$ 0 0
$$679$$ −19.9821 −0.766843
$$680$$ 0 0
$$681$$ −26.3928 −1.01138
$$682$$ 0 0
$$683$$ −9.84751 −0.376805 −0.188402 0.982092i $$-0.560331\pi$$
−0.188402 + 0.982092i $$0.560331\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 4.24930 0.162121
$$688$$ 0 0
$$689$$ 28.0968 1.07040
$$690$$ 0 0
$$691$$ −17.4457 −0.663667 −0.331833 0.943338i $$-0.607667\pi$$
−0.331833 + 0.943338i $$0.607667\pi$$
$$692$$ 0 0
$$693$$ 5.49245 0.208641
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −20.4358 −0.774060
$$698$$ 0 0
$$699$$ −1.89424 −0.0716468
$$700$$ 0 0
$$701$$ 24.5075 0.925637 0.462819 0.886453i $$-0.346838\pi$$
0.462819 + 0.886453i $$0.346838\pi$$
$$702$$ 0 0
$$703$$ 15.4547 0.582884
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18.6459 0.701253
$$708$$ 0 0
$$709$$ −41.4547 −1.55686 −0.778431 0.627730i $$-0.783984\pi$$
−0.778431 + 0.627730i $$0.783984\pi$$
$$710$$ 0 0
$$711$$ 2.94712 0.110526
$$712$$ 0 0
$$713$$ −2.47356 −0.0926356
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.98111 0.260714
$$718$$ 0 0
$$719$$ 30.9811 1.15540 0.577700 0.816249i $$-0.303950\pi$$
0.577700 + 0.816249i $$0.303950\pi$$
$$720$$ 0 0
$$721$$ −0.219108 −0.00816002
$$722$$ 0 0
$$723$$ −3.20537 −0.119209
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −15.8513 −0.587892 −0.293946 0.955822i $$-0.594969\pi$$
−0.293946 + 0.955822i $$0.594969\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 28.7177 1.06216
$$732$$ 0 0
$$733$$ −0.0807173 −0.00298136 −0.00149068 0.999999i $$-0.500474\pi$$
−0.00149068 + 0.999999i $$0.500474\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8.90039 0.327850
$$738$$ 0 0
$$739$$ −41.2215 −1.51636 −0.758178 0.652048i $$-0.773910\pi$$
−0.758178 + 0.652048i $$0.773910\pi$$
$$740$$ 0 0
$$741$$ −8.94712 −0.328681
$$742$$ 0 0
$$743$$ −7.71292 −0.282959 −0.141480 0.989941i $$-0.545186\pi$$
−0.141480 + 0.989941i $$0.545186\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −9.12465 −0.333854
$$748$$ 0 0
$$749$$ 8.04158 0.293833
$$750$$ 0 0
$$751$$ 24.7946 0.904769 0.452384 0.891823i $$-0.350573\pi$$
0.452384 + 0.891823i $$0.350573\pi$$
$$752$$ 0 0
$$753$$ −13.4457 −0.489989
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 30.6171 1.11280 0.556399 0.830915i $$-0.312183\pi$$
0.556399 + 0.830915i $$0.312183\pi$$
$$758$$ 0 0
$$759$$ −3.59821 −0.130607
$$760$$ 0 0
$$761$$ 37.1624 1.34714 0.673569 0.739125i $$-0.264761\pi$$
0.673569 + 0.739125i $$0.264761\pi$$
$$762$$ 0 0
$$763$$ −19.5302 −0.707042
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −22.7946 −0.823066
$$768$$ 0 0
$$769$$ −40.4217 −1.45764 −0.728822 0.684704i $$-0.759932\pi$$
−0.728822 + 0.684704i $$0.759932\pi$$
$$770$$ 0 0
$$771$$ 18.9382 0.682042
$$772$$ 0 0
$$773$$ −25.1964 −0.906252 −0.453126 0.891446i $$-0.649691\pi$$
−0.453126 + 0.891446i $$0.649691\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −14.7606 −0.529535
$$778$$ 0 0
$$779$$ −8.02125 −0.287391
$$780$$ 0 0
$$781$$ 18.0590 0.646203
$$782$$ 0 0
$$783$$ 4.07177 0.145513
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −9.66998 −0.344698 −0.172349 0.985036i $$-0.555136\pi$$
−0.172349 + 0.985036i $$0.555136\pi$$
$$788$$ 0 0
$$789$$ −0.981108 −0.0349284
$$790$$ 0 0
$$791$$ 7.66104 0.272395
$$792$$ 0 0
$$793$$ −36.6421 −1.30120
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −37.6610 −1.33402 −0.667011 0.745048i $$-0.732427\pi$$
−0.667011 + 0.745048i $$0.732427\pi$$
$$798$$ 0 0
$$799$$ −18.5075 −0.654750
$$800$$ 0 0
$$801$$ 12.2493 0.432808
$$802$$ 0 0
$$803$$ −31.6449 −1.11673
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 3.12465 0.109993
$$808$$ 0 0
$$809$$ −11.2002 −0.393779 −0.196889 0.980426i $$-0.563084\pi$$
−0.196889 + 0.980426i $$0.563084\pi$$
$$810$$ 0 0
$$811$$ 37.0099 1.29959 0.649797 0.760107i $$-0.274854\pi$$
0.649797 + 0.760107i $$0.274854\pi$$
$$812$$ 0 0
$$813$$ 9.52644 0.334107
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 11.2720 0.394357
$$818$$ 0 0
$$819$$ 8.54533 0.298598
$$820$$ 0 0
$$821$$ −12.2493 −0.427504 −0.213752 0.976888i $$-0.568568\pi$$
−0.213752 + 0.976888i $$0.568568\pi$$
$$822$$ 0 0
$$823$$ 48.5742 1.69319 0.846595 0.532238i $$-0.178649\pi$$
0.846595 + 0.532238i $$0.178649\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 10.5704 0.367568 0.183784 0.982967i $$-0.441165\pi$$
0.183784 + 0.982967i $$0.441165\pi$$
$$828$$ 0 0
$$829$$ −12.1865 −0.423254 −0.211627 0.977351i $$-0.567876\pi$$
−0.211627 + 0.977351i $$0.567876\pi$$
$$830$$ 0 0
$$831$$ 4.39284 0.152386
$$832$$ 0 0
$$833$$ 19.0151 0.658834
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −2.47356 −0.0854988
$$838$$ 0 0
$$839$$ −25.4457 −0.878484 −0.439242 0.898369i $$-0.644753\pi$$
−0.439242 + 0.898369i $$0.644753\pi$$
$$840$$ 0 0
$$841$$ −12.4207 −0.428299
$$842$$ 0 0
$$843$$ −12.9382 −0.445614
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.97216 −0.102125
$$848$$ 0 0
$$849$$ 19.7757 0.678702
$$850$$ 0 0
$$851$$ 9.66998 0.331483
$$852$$ 0 0
$$853$$ −17.2720 −0.591382 −0.295691 0.955284i $$-0.595550\pi$$
−0.295691 + 0.955284i $$0.595550\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −41.7328 −1.42557 −0.712783 0.701385i $$-0.752565\pi$$
−0.712783 + 0.701385i $$0.752565\pi$$
$$858$$ 0 0
$$859$$ 34.6171 1.18112 0.590560 0.806994i $$-0.298907\pi$$
0.590560 + 0.806994i $$0.298907\pi$$
$$860$$ 0 0
$$861$$ 7.66104 0.261087
$$862$$ 0 0
$$863$$ −38.4608 −1.30922 −0.654611 0.755966i $$-0.727167\pi$$
−0.654611 + 0.755966i $$0.727167\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0.420681 0.0142871
$$868$$ 0 0
$$869$$ −10.6044 −0.359728
$$870$$ 0 0
$$871$$ 13.8475 0.469205
$$872$$ 0 0
$$873$$ 13.0907 0.443052
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 27.3022 0.921929 0.460965 0.887419i $$-0.347504\pi$$
0.460965 + 0.887419i $$0.347504\pi$$
$$878$$ 0 0
$$879$$ −11.2682 −0.380067
$$880$$ 0 0
$$881$$ −4.79463 −0.161535 −0.0807676 0.996733i $$-0.525737\pi$$
−0.0807676 + 0.996733i $$0.525737\pi$$
$$882$$ 0 0
$$883$$ 52.6710 1.77252 0.886260 0.463188i $$-0.153295\pi$$
0.886260 + 0.463188i $$0.153295\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −48.9292 −1.64288 −0.821441 0.570293i $$-0.806830\pi$$
−0.821441 + 0.570293i $$0.806830\pi$$
$$888$$ 0 0
$$889$$ −27.0240 −0.906357
$$890$$ 0 0
$$891$$ −3.59821 −0.120545
$$892$$ 0 0
$$893$$ −7.26440 −0.243094
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −5.59821 −0.186919
$$898$$ 0 0
$$899$$ −10.0718 −0.335912
$$900$$ 0 0
$$901$$ 20.4358 0.680814
$$902$$ 0 0
$$903$$ −10.7658 −0.358263
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 5.73796 0.190526 0.0952629 0.995452i $$-0.469631\pi$$
0.0952629 + 0.995452i $$0.469631\pi$$
$$908$$ 0 0
$$909$$ −12.2153 −0.405156
$$910$$ 0 0
$$911$$ −42.6799 −1.41405 −0.707025 0.707189i $$-0.749963\pi$$
−0.707025 + 0.707189i $$0.749963\pi$$
$$912$$ 0 0
$$913$$ 32.8324 1.08659
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 10.6044 0.350187
$$918$$ 0 0
$$919$$ 25.5515 0.842866 0.421433 0.906860i $$-0.361527\pi$$
0.421433 + 0.906860i $$0.361527\pi$$
$$920$$ 0 0
$$921$$ −25.8475 −0.851704
$$922$$ 0 0
$$923$$ 28.0968 0.924818
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0.143542 0.00471454
$$928$$ 0 0
$$929$$ −53.9481 −1.76998 −0.884990 0.465610i $$-0.845835\pi$$
−0.884990 + 0.465610i $$0.845835\pi$$
$$930$$ 0 0
$$931$$ 7.46362 0.244610
$$932$$ 0 0
$$933$$ 29.0529 0.951149
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −7.48351 −0.244475 −0.122238 0.992501i $$-0.539007\pi$$
−0.122238 + 0.992501i $$0.539007\pi$$
$$938$$ 0 0
$$939$$ −21.9571 −0.716542
$$940$$ 0 0
$$941$$ 22.6133 0.737173 0.368586 0.929594i $$-0.379842\pi$$
0.368586 + 0.929594i $$0.379842\pi$$
$$942$$ 0 0
$$943$$ −5.01889 −0.163438
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 7.19642 0.233852 0.116926 0.993141i $$-0.462696\pi$$
0.116926 + 0.993141i $$0.462696\pi$$
$$948$$ 0 0
$$949$$ −49.2342 −1.59821
$$950$$ 0 0
$$951$$ 2.65109 0.0859675
$$952$$ 0 0
$$953$$ 60.6799 1.96562 0.982808 0.184631i $$-0.0591092\pi$$
0.982808 + 0.184631i $$0.0591092\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −14.6511 −0.473602
$$958$$ 0 0
$$959$$ −18.6978 −0.603784
$$960$$ 0 0
$$961$$ −24.8815 −0.802629
$$962$$ 0 0
$$963$$ −5.26819 −0.169765
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −4.40179 −0.141552 −0.0707760 0.997492i $$-0.522548\pi$$
−0.0707760 + 0.997492i $$0.522548\pi$$
$$968$$ 0 0
$$969$$ −6.50755 −0.209053
$$970$$ 0 0
$$971$$ 1.01510 0.0325760 0.0162880 0.999867i $$-0.494815\pi$$
0.0162880 + 0.999867i $$0.494815\pi$$
$$972$$ 0 0
$$973$$ 10.1006 0.323811
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 18.4646 0.590735 0.295368 0.955384i $$-0.404558\pi$$
0.295368 + 0.955384i $$0.404558\pi$$
$$978$$ 0 0
$$979$$ −44.0756 −1.40866
$$980$$ 0 0
$$981$$ 12.7946 0.408501
$$982$$ 0 0
$$983$$ −44.0917 −1.40631 −0.703153 0.711039i $$-0.748225\pi$$
−0.703153 + 0.711039i $$0.748225\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 6.93817 0.220845
$$988$$ 0 0
$$989$$ 7.05288 0.224269
$$990$$ 0 0
$$991$$ 8.22426 0.261252 0.130626 0.991432i $$-0.458301\pi$$
0.130626 + 0.991432i $$0.458301\pi$$
$$992$$ 0 0
$$993$$ −7.77574 −0.246756
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −14.4306 −0.457023 −0.228511 0.973541i $$-0.573386\pi$$
−0.228511 + 0.973541i $$0.573386\pi$$
$$998$$ 0 0
$$999$$ 9.66998 0.305945
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 6900.2.a.x.1.2 3
5.2 odd 4 6900.2.f.r.6349.5 6
5.3 odd 4 6900.2.f.r.6349.2 6
5.4 even 2 1380.2.a.j.1.2 3
15.14 odd 2 4140.2.a.s.1.2 3
20.19 odd 2 5520.2.a.bv.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.j.1.2 3 5.4 even 2
4140.2.a.s.1.2 3 15.14 odd 2
5520.2.a.bv.1.2 3 20.19 odd 2
6900.2.a.x.1.2 3 1.1 even 1 trivial
6900.2.f.r.6349.2 6 5.3 odd 4
6900.2.f.r.6349.5 6 5.2 odd 4