Properties

 Label 2646.2.d.c.2645.1 Level $2646$ Weight $2$ Character 2646.2645 Analytic conductor $21.128$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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Newspace parameters

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$21.1284163748$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{31}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 2645.1 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2646.2645 Dual form 2646.2.d.c.2645.4

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000i q^{11} -3.46410i q^{13} +1.00000 q^{16} +3.00000 q^{22} -5.00000 q^{25} -3.46410 q^{26} -9.00000i q^{29} +1.73205i q^{31} -1.00000i q^{32} -8.00000 q^{37} +10.3923 q^{41} +4.00000 q^{43} -3.00000i q^{44} +10.3923 q^{47} +5.00000i q^{50} +3.46410i q^{52} -6.00000i q^{53} -9.00000 q^{58} +5.19615 q^{59} -13.8564i q^{61} +1.73205 q^{62} -1.00000 q^{64} +2.00000 q^{67} -12.0000i q^{71} -5.19615i q^{73} +8.00000i q^{74} -13.0000 q^{79} -10.3923i q^{82} -5.19615 q^{83} -4.00000i q^{86} -3.00000 q^{88} -10.3923 q^{89} -10.3923i q^{94} +8.66025i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + O(q^{10})$$ $$4 q - 4 q^{4} + 4 q^{16} + 12 q^{22} - 20 q^{25} - 32 q^{37} + 16 q^{43} - 36 q^{58} - 4 q^{64} + 8 q^{67} - 52 q^{79} - 12 q^{88} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 1.00000i 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000i 0.904534i 0.891883 + 0.452267i $$0.149385\pi$$
−0.891883 + 0.452267i $$0.850615\pi$$
$$12$$ 0 0
$$13$$ − 3.46410i − 0.960769i −0.877058 0.480384i $$-0.840497\pi$$
0.877058 0.480384i $$-0.159503\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.00000 0.639602
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ −3.46410 −0.679366
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 9.00000i − 1.67126i −0.549294 0.835629i $$-0.685103\pi$$
0.549294 0.835629i $$-0.314897\pi$$
$$30$$ 0 0
$$31$$ 1.73205i 0.311086i 0.987829 + 0.155543i $$0.0497126\pi$$
−0.987829 + 0.155543i $$0.950287\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 10.3923 1.62301 0.811503 0.584349i $$-0.198650\pi$$
0.811503 + 0.584349i $$0.198650\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ − 3.00000i − 0.452267i
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.3923 1.51587 0.757937 0.652328i $$-0.226208\pi$$
0.757937 + 0.652328i $$0.226208\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 5.00000i 0.707107i
$$51$$ 0 0
$$52$$ 3.46410i 0.480384i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −9.00000 −1.18176
$$59$$ 5.19615 0.676481 0.338241 0.941060i $$-0.390168\pi$$
0.338241 + 0.941060i $$0.390168\pi$$
$$60$$ 0 0
$$61$$ − 13.8564i − 1.77413i −0.461644 0.887066i $$-0.652740\pi$$
0.461644 0.887066i $$-0.347260\pi$$
$$62$$ 1.73205 0.219971
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 12.0000i − 1.42414i −0.702109 0.712069i $$-0.747758\pi$$
0.702109 0.712069i $$-0.252242\pi$$
$$72$$ 0 0
$$73$$ − 5.19615i − 0.608164i −0.952646 0.304082i $$-0.901650\pi$$
0.952646 0.304082i $$-0.0983496\pi$$
$$74$$ 8.00000i 0.929981i
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −13.0000 −1.46261 −0.731307 0.682048i $$-0.761089\pi$$
−0.731307 + 0.682048i $$0.761089\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 10.3923i − 1.14764i
$$83$$ −5.19615 −0.570352 −0.285176 0.958475i $$-0.592052\pi$$
−0.285176 + 0.958475i $$0.592052\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ − 4.00000i − 0.431331i
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ −10.3923 −1.10158 −0.550791 0.834643i $$-0.685674\pi$$
−0.550791 + 0.834643i $$0.685674\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ − 10.3923i − 1.07188i
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 8.66025i 0.879316i 0.898165 + 0.439658i $$0.144900\pi$$
−0.898165 + 0.439658i $$0.855100\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 5.00000 0.500000
$$101$$ 5.19615 0.517036 0.258518 0.966006i $$-0.416766\pi$$
0.258518 + 0.966006i $$0.416766\pi$$
$$102$$ 0 0
$$103$$ − 17.3205i − 1.70664i −0.521387 0.853320i $$-0.674585\pi$$
0.521387 0.853320i $$-0.325415\pi$$
$$104$$ 3.46410 0.339683
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ −8.00000 −0.766261 −0.383131 0.923694i $$-0.625154\pi$$
−0.383131 + 0.923694i $$0.625154\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.0000i 1.12887i 0.825479 + 0.564433i $$0.190905\pi$$
−0.825479 + 0.564433i $$0.809095\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 9.00000i 0.835629i
$$117$$ 0 0
$$118$$ − 5.19615i − 0.478345i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 2.00000 0.181818
$$122$$ −13.8564 −1.25450
$$123$$ 0 0
$$124$$ − 1.73205i − 0.155543i
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 15.5885 1.36197 0.680985 0.732297i $$-0.261552\pi$$
0.680985 + 0.732297i $$0.261552\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ − 2.00000i − 0.172774i
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ − 6.92820i − 0.587643i −0.955860 0.293821i $$-0.905073\pi$$
0.955860 0.293821i $$-0.0949270\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ 10.3923 0.869048
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −5.19615 −0.430037
$$147$$ 0 0
$$148$$ 8.00000 0.657596
$$149$$ 15.0000i 1.22885i 0.788976 + 0.614424i $$0.210612\pi$$
−0.788976 + 0.614424i $$0.789388\pi$$
$$150$$ 0 0
$$151$$ −23.0000 −1.87171 −0.935857 0.352381i $$-0.885372\pi$$
−0.935857 + 0.352381i $$0.885372\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 10.3923i − 0.829396i −0.909959 0.414698i $$-0.863887\pi$$
0.909959 0.414698i $$-0.136113\pi$$
$$158$$ 13.0000i 1.03422i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ −10.3923 −0.811503
$$165$$ 0 0
$$166$$ 5.19615i 0.403300i
$$167$$ 10.3923 0.804181 0.402090 0.915600i $$-0.368284\pi$$
0.402090 + 0.915600i $$0.368284\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −25.9808 −1.97528 −0.987640 0.156737i $$-0.949903\pi$$
−0.987640 + 0.156737i $$0.949903\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3.00000i 0.226134i
$$177$$ 0 0
$$178$$ 10.3923i 0.778936i
$$179$$ − 9.00000i − 0.672692i −0.941739 0.336346i $$-0.890809\pi$$
0.941739 0.336346i $$-0.109191\pi$$
$$180$$ 0 0
$$181$$ − 20.7846i − 1.54491i −0.635071 0.772454i $$-0.719029\pi$$
0.635071 0.772454i $$-0.280971\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −10.3923 −0.757937
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.00000i 0.434145i 0.976156 + 0.217072i $$0.0696508\pi$$
−0.976156 + 0.217072i $$0.930349\pi$$
$$192$$ 0 0
$$193$$ −11.0000 −0.791797 −0.395899 0.918294i $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ 8.66025 0.621770
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 15.0000i − 1.06871i −0.845262 0.534353i $$-0.820555\pi$$
0.845262 0.534353i $$-0.179445\pi$$
$$198$$ 0 0
$$199$$ − 1.73205i − 0.122782i −0.998114 0.0613909i $$-0.980446\pi$$
0.998114 0.0613909i $$-0.0195536\pi$$
$$200$$ − 5.00000i − 0.353553i
$$201$$ 0 0
$$202$$ − 5.19615i − 0.365600i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −17.3205 −1.20678
$$207$$ 0 0
$$208$$ − 3.46410i − 0.240192i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 8.00000i 0.541828i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 5.19615i − 0.347960i −0.984749 0.173980i $$-0.944337\pi$$
0.984749 0.173980i $$-0.0556628\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ 25.9808 1.72440 0.862202 0.506565i $$-0.169085\pi$$
0.862202 + 0.506565i $$0.169085\pi$$
$$228$$ 0 0
$$229$$ 6.92820i 0.457829i 0.973447 + 0.228914i $$0.0735176\pi$$
−0.973447 + 0.228914i $$0.926482\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 9.00000 0.590879
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −5.19615 −0.338241
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.0000i 0.776215i 0.921614 + 0.388108i $$0.126871\pi$$
−0.921614 + 0.388108i $$0.873129\pi$$
$$240$$ 0 0
$$241$$ 12.1244i 0.780998i 0.920603 + 0.390499i $$0.127698\pi$$
−0.920603 + 0.390499i $$0.872302\pi$$
$$242$$ − 2.00000i − 0.128565i
$$243$$ 0 0
$$244$$ 13.8564i 0.887066i
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −1.73205 −0.109985
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −25.9808 −1.63989 −0.819946 0.572441i $$-0.805996\pi$$
−0.819946 + 0.572441i $$0.805996\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ − 8.00000i − 0.501965i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 20.7846 1.29651 0.648254 0.761424i $$-0.275499\pi$$
0.648254 + 0.761424i $$0.275499\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 15.5885i − 0.963058i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −2.00000 −0.122169
$$269$$ 15.5885 0.950445 0.475223 0.879866i $$-0.342368\pi$$
0.475223 + 0.879866i $$0.342368\pi$$
$$270$$ 0 0
$$271$$ 3.46410i 0.210429i 0.994450 + 0.105215i $$0.0335529\pi$$
−0.994450 + 0.105215i $$0.966447\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ − 15.0000i − 0.904534i
$$276$$ 0 0
$$277$$ 14.0000 0.841178 0.420589 0.907251i $$-0.361823\pi$$
0.420589 + 0.907251i $$0.361823\pi$$
$$278$$ −6.92820 −0.415526
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000i 0.357930i 0.983855 + 0.178965i $$0.0572749\pi$$
−0.983855 + 0.178965i $$0.942725\pi$$
$$282$$ 0 0
$$283$$ − 6.92820i − 0.411839i −0.978569 0.205919i $$-0.933982\pi$$
0.978569 0.205919i $$-0.0660185\pi$$
$$284$$ 12.0000i 0.712069i
$$285$$ 0 0
$$286$$ − 10.3923i − 0.614510i
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 5.19615i 0.304082i
$$293$$ 5.19615 0.303562 0.151781 0.988414i $$-0.451499\pi$$
0.151781 + 0.988414i $$0.451499\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ − 8.00000i − 0.464991i
$$297$$ 0 0
$$298$$ 15.0000 0.868927
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 23.0000i 1.32350i
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 17.3205i 0.988534i 0.869310 + 0.494267i $$0.164563\pi$$
−0.869310 + 0.494267i $$0.835437\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −31.1769 −1.76788 −0.883940 0.467600i $$-0.845119\pi$$
−0.883940 + 0.467600i $$0.845119\pi$$
$$312$$ 0 0
$$313$$ − 20.7846i − 1.17482i −0.809291 0.587408i $$-0.800148\pi$$
0.809291 0.587408i $$-0.199852\pi$$
$$314$$ −10.3923 −0.586472
$$315$$ 0 0
$$316$$ 13.0000 0.731307
$$317$$ 9.00000i 0.505490i 0.967533 + 0.252745i $$0.0813334\pi$$
−0.967533 + 0.252745i $$0.918667\pi$$
$$318$$ 0 0
$$319$$ 27.0000 1.51171
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 17.3205i 0.960769i
$$326$$ 10.0000i 0.553849i
$$327$$ 0 0
$$328$$ 10.3923i 0.573819i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 5.19615 0.285176
$$333$$ 0 0
$$334$$ − 10.3923i − 0.568642i
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 13.0000 0.708155 0.354078 0.935216i $$-0.384795\pi$$
0.354078 + 0.935216i $$0.384795\pi$$
$$338$$ − 1.00000i − 0.0543928i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −5.19615 −0.281387
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 4.00000i 0.215666i
$$345$$ 0 0
$$346$$ 25.9808i 1.39673i
$$347$$ − 3.00000i − 0.161048i −0.996753 0.0805242i $$-0.974341\pi$$
0.996753 0.0805242i $$-0.0256594\pi$$
$$348$$ 0 0
$$349$$ 27.7128i 1.48343i 0.670714 + 0.741716i $$0.265988\pi$$
−0.670714 + 0.741716i $$0.734012\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 3.00000 0.159901
$$353$$ −10.3923 −0.553127 −0.276563 0.960996i $$-0.589196\pi$$
−0.276563 + 0.960996i $$0.589196\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.3923 0.550791
$$357$$ 0 0
$$358$$ −9.00000 −0.475665
$$359$$ 12.0000i 0.633336i 0.948536 + 0.316668i $$0.102564\pi$$
−0.948536 + 0.316668i $$0.897436\pi$$
$$360$$ 0 0
$$361$$ 19.0000 1.00000
$$362$$ −20.7846 −1.09241
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 10.3923i − 0.542474i −0.962513 0.271237i $$-0.912567\pi$$
0.962513 0.271237i $$-0.0874327\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 10.3923i 0.535942i
$$377$$ −31.1769 −1.60569
$$378$$ 0 0
$$379$$ −26.0000 −1.33553 −0.667765 0.744372i $$-0.732749\pi$$
−0.667765 + 0.744372i $$0.732749\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 6.00000 0.306987
$$383$$ 20.7846 1.06204 0.531022 0.847358i $$-0.321808\pi$$
0.531022 + 0.847358i $$0.321808\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 11.0000i 0.559885i
$$387$$ 0 0
$$388$$ − 8.66025i − 0.439658i
$$389$$ 9.00000i 0.456318i 0.973624 + 0.228159i $$0.0732706\pi$$
−0.973624 + 0.228159i $$0.926729\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −15.0000 −0.755689
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 24.2487i 1.21701i 0.793551 + 0.608504i $$0.208230\pi$$
−0.793551 + 0.608504i $$0.791770\pi$$
$$398$$ −1.73205 −0.0868199
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ 30.0000i 1.49813i 0.662497 + 0.749064i $$0.269497\pi$$
−0.662497 + 0.749064i $$0.730503\pi$$
$$402$$ 0 0
$$403$$ 6.00000 0.298881
$$404$$ −5.19615 −0.258518
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 24.0000i − 1.18964i
$$408$$ 0 0
$$409$$ − 6.92820i − 0.342578i −0.985221 0.171289i $$-0.945207\pi$$
0.985221 0.171289i $$-0.0547931\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 17.3205i 0.853320i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −3.46410 −0.169842
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 31.1769 1.52309 0.761546 0.648111i $$-0.224441\pi$$
0.761546 + 0.648111i $$0.224441\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 2.00000i 0.0973585i
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 18.0000i − 0.867029i −0.901146 0.433515i $$-0.857273\pi$$
0.901146 0.433515i $$-0.142727\pi$$
$$432$$ 0 0
$$433$$ − 12.1244i − 0.582659i −0.956623 0.291330i $$-0.905902\pi$$
0.956623 0.291330i $$-0.0940977\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 8.00000 0.383131
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 15.5885i 0.743996i 0.928233 + 0.371998i $$0.121327\pi$$
−0.928233 + 0.371998i $$0.878673\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 27.0000i − 1.28281i −0.767203 0.641404i $$-0.778352\pi$$
0.767203 0.641404i $$-0.221648\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −5.19615 −0.246045
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 30.0000i − 1.41579i −0.706319 0.707894i $$-0.749646\pi$$
0.706319 0.707894i $$-0.250354\pi$$
$$450$$ 0 0
$$451$$ 31.1769i 1.46806i
$$452$$ − 12.0000i − 0.564433i
$$453$$ 0 0
$$454$$ − 25.9808i − 1.21934i
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 6.92820 0.323734
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −15.5885 −0.726027 −0.363013 0.931784i $$-0.618252\pi$$
−0.363013 + 0.931784i $$0.618252\pi$$
$$462$$ 0 0
$$463$$ 5.00000 0.232370 0.116185 0.993228i $$-0.462933\pi$$
0.116185 + 0.993228i $$0.462933\pi$$
$$464$$ − 9.00000i − 0.417815i
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ 5.19615 0.240449 0.120225 0.992747i $$-0.461639\pi$$
0.120225 + 0.992747i $$0.461639\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 5.19615i 0.239172i
$$473$$ 12.0000i 0.551761i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 12.0000 0.548867
$$479$$ −10.3923 −0.474837 −0.237418 0.971408i $$-0.576301\pi$$
−0.237418 + 0.971408i $$0.576301\pi$$
$$480$$ 0 0
$$481$$ 27.7128i 1.26360i
$$482$$ 12.1244 0.552249
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −11.0000 −0.498458 −0.249229 0.968445i $$-0.580177\pi$$
−0.249229 + 0.968445i $$0.580177\pi$$
$$488$$ 13.8564 0.627250
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 36.0000i 1.62466i 0.583200 + 0.812329i $$0.301800\pi$$
−0.583200 + 0.812329i $$0.698200\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 1.73205i 0.0777714i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 16.0000 0.716258 0.358129 0.933672i $$-0.383415\pi$$
0.358129 + 0.933672i $$0.383415\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 25.9808i 1.15958i
$$503$$ 31.1769 1.39011 0.695055 0.718957i $$-0.255380\pi$$
0.695055 + 0.718957i $$0.255380\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ −5.19615 −0.230315 −0.115158 0.993347i $$-0.536737\pi$$
−0.115158 + 0.993347i $$0.536737\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 0 0
$$514$$ − 20.7846i − 0.916770i
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 31.1769i 1.37116i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 20.7846 0.910590 0.455295 0.890341i $$-0.349534\pi$$
0.455295 + 0.890341i $$0.349534\pi$$
$$522$$ 0 0
$$523$$ 38.1051i 1.66622i 0.553107 + 0.833110i $$0.313442\pi$$
−0.553107 + 0.833110i $$0.686558\pi$$
$$524$$ −15.5885 −0.680985
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 36.0000i − 1.55933i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 2.00000i 0.0863868i
$$537$$ 0 0
$$538$$ − 15.5885i − 0.672066i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 3.46410 0.148796
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 14.0000 0.598597 0.299298 0.954160i $$-0.403247\pi$$
0.299298 + 0.954160i $$0.403247\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ 0 0
$$550$$ −15.0000 −0.639602
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ − 14.0000i − 0.594803i
$$555$$ 0 0
$$556$$ 6.92820i 0.293821i
$$557$$ 21.0000i 0.889799i 0.895581 + 0.444899i $$0.146761\pi$$
−0.895581 + 0.444899i $$0.853239\pi$$
$$558$$ 0 0
$$559$$ − 13.8564i − 0.586064i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 31.1769 1.31395 0.656975 0.753912i $$-0.271836\pi$$
0.656975 + 0.753912i $$0.271836\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −6.92820 −0.291214
$$567$$ 0 0
$$568$$ 12.0000 0.503509
$$569$$ 12.0000i 0.503066i 0.967849 + 0.251533i $$0.0809347\pi$$
−0.967849 + 0.251533i $$0.919065\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −10.3923 −0.434524
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8.66025i 0.360531i 0.983618 + 0.180266i $$0.0576957\pi$$
−0.983618 + 0.180266i $$0.942304\pi$$
$$578$$ 17.0000i 0.707107i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 18.0000 0.745484
$$584$$ 5.19615 0.215018
$$585$$ 0 0
$$586$$ − 5.19615i − 0.214651i
$$587$$ −10.3923 −0.428936 −0.214468 0.976731i $$-0.568802\pi$$
−0.214468 + 0.976731i $$0.568802\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −8.00000 −0.328798
$$593$$ −20.7846 −0.853522 −0.426761 0.904365i $$-0.640345\pi$$
−0.426761 + 0.904365i $$0.640345\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ − 15.0000i − 0.614424i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 30.0000i − 1.22577i −0.790173 0.612883i $$-0.790010\pi$$
0.790173 0.612883i $$-0.209990\pi$$
$$600$$ 0 0
$$601$$ − 34.6410i − 1.41304i −0.707695 0.706518i $$-0.750265\pi$$
0.707695 0.706518i $$-0.249735\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 23.0000 0.935857
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 5.19615i − 0.210905i −0.994424 0.105453i $$-0.966371\pi$$
0.994424 0.105453i $$-0.0336291\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 36.0000i − 1.45640i
$$612$$ 0 0
$$613$$ −10.0000 −0.403896 −0.201948 0.979396i $$-0.564727\pi$$
−0.201948 + 0.979396i $$0.564727\pi$$
$$614$$ 17.3205 0.698999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ − 34.6410i − 1.39234i −0.717877 0.696170i $$-0.754886\pi$$
0.717877 0.696170i $$-0.245114\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 31.1769i 1.25008i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ −20.7846 −0.830720
$$627$$ 0 0
$$628$$ 10.3923i 0.414698i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −37.0000 −1.47295 −0.736473 0.676467i $$-0.763510\pi$$
−0.736473 + 0.676467i $$0.763510\pi$$
$$632$$ − 13.0000i − 0.517112i
$$633$$ 0 0
$$634$$ 9.00000 0.357436
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ − 27.0000i − 1.06894i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 42.0000i 1.65890i 0.558581 + 0.829450i $$0.311346\pi$$
−0.558581 + 0.829450i $$0.688654\pi$$
$$642$$ 0 0
$$643$$ 3.46410i 0.136611i 0.997664 + 0.0683054i $$0.0217592\pi$$
−0.997664 + 0.0683054i $$0.978241\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 10.3923 0.408564 0.204282 0.978912i $$-0.434514\pi$$
0.204282 + 0.978912i $$0.434514\pi$$
$$648$$ 0 0
$$649$$ 15.5885i 0.611900i
$$650$$ 17.3205 0.679366
$$651$$ 0 0
$$652$$ 10.0000 0.391630
$$653$$ − 30.0000i − 1.17399i −0.809590 0.586995i $$-0.800311\pi$$
0.809590 0.586995i $$-0.199689\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 10.3923 0.405751
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 15.0000i 0.584317i 0.956370 + 0.292159i $$0.0943735\pi$$
−0.956370 + 0.292159i $$0.905627\pi$$
$$660$$ 0 0
$$661$$ − 10.3923i − 0.404214i −0.979363 0.202107i $$-0.935221\pi$$
0.979363 0.202107i $$-0.0647788\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ 0 0
$$664$$ − 5.19615i − 0.201650i
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −10.3923 −0.402090
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 41.5692 1.60476
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ − 13.0000i − 0.500741i
$$675$$ 0 0
$$676$$ −1.00000 −0.0384615
$$677$$ 15.5885 0.599113 0.299557 0.954079i $$-0.403161\pi$$
0.299557 + 0.954079i $$0.403161\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 5.19615i 0.198971i
$$683$$ − 21.0000i − 0.803543i −0.915740 0.401771i $$-0.868395\pi$$
0.915740 0.401771i $$-0.131605\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ −20.7846 −0.791831
$$690$$ 0 0
$$691$$ 13.8564i 0.527123i 0.964643 + 0.263561i $$0.0848971\pi$$
−0.964643 + 0.263561i $$0.915103\pi$$
$$692$$ 25.9808 0.987640
$$693$$ 0 0
$$694$$ −3.00000 −0.113878
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 27.7128 1.04895
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.00000i 0.226617i 0.993560 + 0.113308i $$0.0361448\pi$$
−0.993560 + 0.113308i $$0.963855\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ − 3.00000i − 0.113067i
$$705$$ 0 0
$$706$$ 10.3923i 0.391120i
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 10.3923i − 0.389468i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 9.00000i 0.336346i
$$717$$ 0 0
$$718$$ 12.0000 0.447836
$$719$$ −10.3923 −0.387568 −0.193784 0.981044i $$-0.562076\pi$$
−0.193784 + 0.981044i $$0.562076\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 19.0000i − 0.707107i
$$723$$ 0 0
$$724$$ 20.7846i 0.772454i
$$725$$ 45.0000i 1.67126i
$$726$$ 0 0
$$727$$ 24.2487i 0.899335i 0.893196 + 0.449667i $$0.148458\pi$$
−0.893196 + 0.449667i $$0.851542\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 13.8564i 0.511798i 0.966704 + 0.255899i $$0.0823715\pi$$
−0.966704 + 0.255899i $$0.917629\pi$$
$$734$$ −10.3923 −0.383587
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.00000i 0.221013i
$$738$$ 0 0
$$739$$ 26.0000 0.956425 0.478213 0.878244i $$-0.341285\pi$$
0.478213 + 0.878244i $$0.341285\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 6.00000i 0.220119i 0.993925 + 0.110059i $$0.0351041\pi$$
−0.993925 + 0.110059i $$0.964896\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.00000i 0.146450i
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 10.3923 0.378968
$$753$$ 0 0
$$754$$ 31.1769i 1.13540i
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 26.0000i 0.944363i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −41.5692 −1.50688 −0.753442 0.657515i $$-0.771608\pi$$
−0.753442 + 0.657515i $$0.771608\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ − 6.00000i − 0.217072i
$$765$$ 0 0
$$766$$ − 20.7846i − 0.750978i
$$767$$ − 18.0000i − 0.649942i
$$768$$ 0 0
$$769$$ 15.5885i 0.562134i 0.959688 + 0.281067i $$0.0906883\pi$$
−0.959688 + 0.281067i $$0.909312\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 11.0000 0.395899
$$773$$ −41.5692 −1.49514 −0.747570 0.664183i $$-0.768780\pi$$
−0.747570 + 0.664183i $$0.768780\pi$$
$$774$$ 0 0
$$775$$ − 8.66025i − 0.311086i
$$776$$ −8.66025 −0.310885
$$777$$ 0 0
$$778$$ 9.00000 0.322666
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 24.2487i 0.864373i 0.901784 + 0.432187i $$0.142258\pi$$
−0.901784 + 0.432187i $$0.857742\pi$$
$$788$$ 15.0000i 0.534353i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −48.0000 −1.70453
$$794$$ 24.2487 0.860555
$$795$$ 0 0
$$796$$ 1.73205i 0.0613909i
$$797$$ 15.5885 0.552171 0.276086 0.961133i $$-0.410963\pi$$
0.276086 + 0.961133i $$0.410963\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 5.00000i 0.176777i
$$801$$ 0 0
$$802$$ 30.0000 1.05934
$$803$$ 15.5885 0.550105
$$804$$ 0 0
$$805$$ 0 0
$$806$$ − 6.00000i − 0.211341i
$$807$$ 0 0
$$808$$ 5.19615i 0.182800i
$$809$$ 6.00000i 0.210949i 0.994422 + 0.105474i $$0.0336361\pi$$
−0.994422 + 0.105474i $$0.966364\pi$$
$$810$$ 0 0
$$811$$ 10.3923i 0.364923i 0.983213 + 0.182462i $$0.0584065\pi$$
−0.983213 + 0.182462i $$0.941593\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −6.92820 −0.242239
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 33.0000i − 1.15171i −0.817553 0.575854i $$-0.804670\pi$$
0.817553 0.575854i $$-0.195330\pi$$
$$822$$ 0 0
$$823$$ 11.0000 0.383436 0.191718 0.981450i $$-0.438594\pi$$
0.191718 + 0.981450i $$0.438594\pi$$
$$824$$ 17.3205 0.603388
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 3.00000i 0.104320i 0.998639 + 0.0521601i $$0.0166106\pi$$
−0.998639 + 0.0521601i $$0.983389\pi$$
$$828$$ 0 0
$$829$$ 31.1769i 1.08282i 0.840759 + 0.541409i $$0.182109\pi$$
−0.840759 + 0.541409i $$0.817891\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 3.46410i 0.120096i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ − 31.1769i − 1.07699i
$$839$$ 20.7846 0.717564 0.358782 0.933421i $$-0.383192\pi$$
0.358782 + 0.933421i $$0.383192\pi$$
$$840$$ 0 0
$$841$$ −52.0000 −1.79310
$$842$$ 10.0000i 0.344623i
$$843$$ 0 0
$$844$$ 2.00000 0.0688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 6.00000i − 0.206041i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 10.3923i − 0.355826i −0.984046 0.177913i $$-0.943065\pi$$
0.984046 0.177913i $$-0.0569345\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 10.3923 0.354994 0.177497 0.984121i $$-0.443200\pi$$
0.177497 + 0.984121i $$0.443200\pi$$
$$858$$ 0 0
$$859$$ − 48.4974i − 1.65471i −0.561679 0.827355i $$-0.689844\pi$$
0.561679 0.827355i $$-0.310156\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −18.0000 −0.613082
$$863$$ − 6.00000i − 0.204242i −0.994772 0.102121i $$-0.967437\pi$$
0.994772 0.102121i $$-0.0325630\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −12.1244 −0.412002
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 39.0000i − 1.32298i
$$870$$ 0 0
$$871$$ − 6.92820i − 0.234753i
$$872$$ − 8.00000i − 0.270914i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 15.5885 0.526085
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −31.1769 −1.05038 −0.525188 0.850986i $$-0.676005\pi$$
−0.525188 + 0.850986i $$0.676005\pi$$
$$882$$ 0 0
$$883$$ −2.00000 −0.0673054 −0.0336527 0.999434i $$-0.510714\pi$$
−0.0336527 + 0.999434i $$0.510714\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −27.0000 −0.907083
$$887$$ 31.1769 1.04682 0.523409 0.852081i $$-0.324660\pi$$
0.523409 + 0.852081i $$0.324660\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 5.19615i 0.173980i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ 15.5885 0.519904
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 31.1769 1.03808
$$903$$ 0 0
$$904$$ −12.0000 −0.399114
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 22.0000 0.730498 0.365249 0.930910i $$-0.380984\pi$$
0.365249 + 0.930910i $$0.380984\pi$$
$$908$$ −25.9808 −0.862202
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 36.0000i 1.19273i 0.802712 + 0.596367i $$0.203390\pi$$
−0.802712 + 0.596367i $$0.796610\pi$$
$$912$$ 0 0
$$913$$ − 15.5885i − 0.515903i
$$914$$ − 22.0000i − 0.727695i
$$915$$ 0 0
$$916$$ − 6.92820i − 0.228914i
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −35.0000 −1.15454 −0.577272 0.816552i $$-0.695883\pi$$
−0.577272 + 0.816552i $$0.695883\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 15.5885i 0.513378i
$$923$$ −41.5692 −1.36827
$$924$$ 0 0
$$925$$ 40.0000 1.31519
$$926$$ − 5.00000i − 0.164310i
$$927$$ 0 0
$$928$$ −9.00000 −0.295439
$$929$$ −41.5692 −1.36384 −0.681921 0.731426i $$-0.738855\pi$$
−0.681921 + 0.731426i $$0.738855\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 18.0000i − 0.589610i
$$933$$ 0 0
$$934$$ − 5.19615i − 0.170023i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 34.6410i 1.13167i 0.824518 + 0.565836i $$0.191447\pi$$
−0.824518 + 0.565836i $$0.808553\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 5.19615 0.169390 0.0846949 0.996407i $$-0.473008\pi$$
0.0846949 + 0.996407i $$0.473008\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 5.19615 0.169120
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ − 9.00000i − 0.292461i −0.989251 0.146230i $$-0.953286\pi$$
0.989251 0.146230i $$-0.0467141\pi$$
$$948$$ 0 0
$$949$$ −18.0000 −0.584305
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 18.0000i 0.583077i 0.956559 + 0.291539i $$0.0941672\pi$$
−0.956559 + 0.291539i $$0.905833\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ − 12.0000i − 0.388108i
$$957$$ 0 0
$$958$$ 10.3923i 0.335760i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 28.0000 0.903226
$$962$$ 27.7128 0.893497
$$963$$ 0 0
$$964$$ − 12.1244i − 0.390499i
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 56.0000 1.80084 0.900419 0.435023i $$-0.143260\pi$$
0.900419 + 0.435023i $$0.143260\pi$$
$$968$$ 2.00000i 0.0642824i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 31.1769 1.00051 0.500257 0.865877i $$-0.333239\pi$$
0.500257 + 0.865877i $$0.333239\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 11.0000i 0.352463i
$$975$$ 0 0
$$976$$ − 13.8564i − 0.443533i
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ 0 0
$$979$$ − 31.1769i − 0.996419i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 36.0000 1.14881
$$983$$ −20.7846 −0.662926 −0.331463 0.943468i $$-0.607542\pi$$
−0.331463 + 0.943468i $$0.607542\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 1.73205 0.0549927
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 38.1051i − 1.20680i −0.797438 0.603401i $$-0.793812\pi$$
0.797438 0.603401i $$-0.206188\pi$$
$$998$$ − 16.0000i − 0.506471i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2646.2.d.c.2645.1 4
3.2 odd 2 inner 2646.2.d.c.2645.3 4
7.2 even 3 378.2.k.a.269.1 yes 4
7.3 odd 6 378.2.k.a.215.2 yes 4
7.6 odd 2 inner 2646.2.d.c.2645.2 4
21.2 odd 6 378.2.k.a.269.2 yes 4
21.17 even 6 378.2.k.a.215.1 4
21.20 even 2 inner 2646.2.d.c.2645.4 4
63.2 odd 6 1134.2.t.a.1025.1 4
63.16 even 3 1134.2.t.a.1025.2 4
63.23 odd 6 1134.2.l.d.269.2 4
63.31 odd 6 1134.2.t.a.593.1 4
63.38 even 6 1134.2.l.d.215.2 4
63.52 odd 6 1134.2.l.d.215.1 4
63.58 even 3 1134.2.l.d.269.1 4
63.59 even 6 1134.2.t.a.593.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.a.215.1 4 21.17 even 6
378.2.k.a.215.2 yes 4 7.3 odd 6
378.2.k.a.269.1 yes 4 7.2 even 3
378.2.k.a.269.2 yes 4 21.2 odd 6
1134.2.l.d.215.1 4 63.52 odd 6
1134.2.l.d.215.2 4 63.38 even 6
1134.2.l.d.269.1 4 63.58 even 3
1134.2.l.d.269.2 4 63.23 odd 6
1134.2.t.a.593.1 4 63.31 odd 6
1134.2.t.a.593.2 4 63.59 even 6
1134.2.t.a.1025.1 4 63.2 odd 6
1134.2.t.a.1025.2 4 63.16 even 3
2646.2.d.c.2645.1 4 1.1 even 1 trivial
2646.2.d.c.2645.2 4 7.6 odd 2 inner
2646.2.d.c.2645.3 4 3.2 odd 2 inner
2646.2.d.c.2645.4 4 21.20 even 2 inner