# Properties

 Label 384.3.h.b.65.2 Level $384$ Weight $3$ Character 384.65 Analytic conductor $10.463$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 384.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 65.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.65 Dual form 384.3.h.b.65.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.00000 + 2.82843i) q^{3} +(-7.00000 - 5.65685i) q^{9} +O(q^{10})$$ $$q+(-1.00000 + 2.82843i) q^{3} +(-7.00000 - 5.65685i) q^{9} -14.0000 q^{11} -33.9411i q^{17} +16.9706i q^{19} -25.0000 q^{25} +(23.0000 - 14.1421i) q^{27} +(14.0000 - 39.5980i) q^{33} -67.8823i q^{41} -84.8528i q^{43} -49.0000 q^{49} +(96.0000 + 33.9411i) q^{51} +(-48.0000 - 16.9706i) q^{57} -82.0000 q^{59} +118.794i q^{67} +142.000 q^{73} +(25.0000 - 70.7107i) q^{75} +(17.0000 + 79.1960i) q^{81} -158.000 q^{83} +101.823i q^{89} -94.0000 q^{97} +(98.0000 + 79.1960i) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 14q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 14q^{9} - 28q^{11} - 50q^{25} + 46q^{27} + 28q^{33} - 98q^{49} + 192q^{51} - 96q^{57} - 164q^{59} + 284q^{73} + 50q^{75} + 34q^{81} - 316q^{83} - 188q^{97} + 196q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 + 2.82843i −0.333333 + 0.942809i
$$4$$ 0 0
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ −7.00000 5.65685i −0.777778 0.628539i
$$10$$ 0 0
$$11$$ −14.0000 −1.27273 −0.636364 0.771389i $$-0.719562\pi$$
−0.636364 + 0.771389i $$0.719562\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 33.9411i 1.99654i −0.0588235 0.998268i $$-0.518735\pi$$
0.0588235 0.998268i $$-0.481265\pi$$
$$18$$ 0 0
$$19$$ 16.9706i 0.893188i 0.894737 + 0.446594i $$0.147363\pi$$
−0.894737 + 0.446594i $$0.852637\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ −25.0000 −1.00000
$$26$$ 0 0
$$27$$ 23.0000 14.1421i 0.851852 0.523783i
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 14.0000 39.5980i 0.424242 1.19994i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 67.8823i 1.65566i −0.560976 0.827832i $$-0.689574\pi$$
0.560976 0.827832i $$-0.310426\pi$$
$$42$$ 0 0
$$43$$ 84.8528i 1.97332i −0.162791 0.986661i $$-0.552050\pi$$
0.162791 0.986661i $$-0.447950\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −49.0000 −1.00000
$$50$$ 0 0
$$51$$ 96.0000 + 33.9411i 1.88235 + 0.665512i
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −48.0000 16.9706i −0.842105 0.297729i
$$58$$ 0 0
$$59$$ −82.0000 −1.38983 −0.694915 0.719092i $$-0.744558\pi$$
−0.694915 + 0.719092i $$0.744558\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 118.794i 1.77304i 0.462687 + 0.886522i $$0.346886\pi$$
−0.462687 + 0.886522i $$0.653114\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 142.000 1.94521 0.972603 0.232473i $$-0.0746819\pi$$
0.972603 + 0.232473i $$0.0746819\pi$$
$$74$$ 0 0
$$75$$ 25.0000 70.7107i 0.333333 0.942809i
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 17.0000 + 79.1960i 0.209877 + 0.977728i
$$82$$ 0 0
$$83$$ −158.000 −1.90361 −0.951807 0.306697i $$-0.900776\pi$$
−0.951807 + 0.306697i $$0.900776\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 101.823i 1.14408i 0.820225 + 0.572041i $$0.193848\pi$$
−0.820225 + 0.572041i $$0.806152\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −94.0000 −0.969072 −0.484536 0.874771i $$-0.661012\pi$$
−0.484536 + 0.874771i $$0.661012\pi$$
$$98$$ 0 0
$$99$$ 98.0000 + 79.1960i 0.989899 + 0.799959i
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −178.000 −1.66355 −0.831776 0.555112i $$-0.812675\pi$$
−0.831776 + 0.555112i $$0.812675\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 203.647i 1.80218i 0.433628 + 0.901092i $$0.357233\pi$$
−0.433628 + 0.901092i $$0.642767\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 75.0000 0.619835
$$122$$ 0 0
$$123$$ 192.000 + 67.8823i 1.56098 + 0.551888i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 240.000 + 84.8528i 1.86047 + 0.657774i
$$130$$ 0 0
$$131$$ 62.0000 0.473282 0.236641 0.971597i $$-0.423953\pi$$
0.236641 + 0.971597i $$0.423953\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 135.765i 0.990982i 0.868613 + 0.495491i $$0.165012\pi$$
−0.868613 + 0.495491i $$0.834988\pi$$
$$138$$ 0 0
$$139$$ 186.676i 1.34299i −0.741007 0.671497i $$-0.765652\pi$$
0.741007 0.671497i $$-0.234348\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 49.0000 138.593i 0.333333 0.942809i
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ −192.000 + 237.588i −1.25490 + 1.55286i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 50.9117i 0.312342i −0.987730 0.156171i $$-0.950085\pi$$
0.987730 0.156171i $$-0.0499150\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 169.000 1.00000
$$170$$ 0 0
$$171$$ 96.0000 118.794i 0.561404 0.694701i
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 82.0000 231.931i 0.463277 1.31034i
$$178$$ 0 0
$$179$$ −34.0000 −0.189944 −0.0949721 0.995480i $$-0.530276\pi$$
−0.0949721 + 0.995480i $$0.530276\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 475.176i 2.54105i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ −98.0000 −0.507772 −0.253886 0.967234i $$-0.581709\pi$$
−0.253886 + 0.967234i $$0.581709\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ −336.000 118.794i −1.67164 0.591015i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 237.588i 1.13678i
$$210$$ 0 0
$$211$$ 356.382i 1.68901i −0.535545 0.844507i $$-0.679894\pi$$
0.535545 0.844507i $$-0.320106\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −142.000 + 401.637i −0.648402 + 1.83396i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 175.000 + 141.421i 0.777778 + 0.628539i
$$226$$ 0 0
$$227$$ −446.000 −1.96476 −0.982379 0.186900i $$-0.940156\pi$$
−0.982379 + 0.186900i $$0.940156\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 169.706i 0.728350i −0.931330 0.364175i $$-0.881351\pi$$
0.931330 0.364175i $$-0.118649\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −194.000 −0.804979 −0.402490 0.915425i $$-0.631855\pi$$
−0.402490 + 0.915425i $$0.631855\pi$$
$$242$$ 0 0
$$243$$ −241.000 31.1127i −0.991770 0.128036i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 158.000 446.891i 0.634538 1.79474i
$$250$$ 0 0
$$251$$ 466.000 1.85657 0.928287 0.371865i $$-0.121282\pi$$
0.928287 + 0.371865i $$0.121282\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 339.411i 1.32067i −0.750973 0.660333i $$-0.770415\pi$$
0.750973 0.660333i $$-0.229585\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −288.000 101.823i −1.07865 0.381361i
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 350.000 1.27273
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 509.117i 1.81180i −0.423488 0.905902i $$-0.639194\pi$$
0.423488 0.905902i $$-0.360806\pi$$
$$282$$ 0 0
$$283$$ 560.029i 1.97890i 0.144876 + 0.989450i $$0.453722\pi$$
−0.144876 + 0.989450i $$0.546278\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −863.000 −2.98616
$$290$$ 0 0
$$291$$ 94.0000 265.872i 0.323024 0.913650i
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −322.000 + 197.990i −1.08418 + 0.666633i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 288.500i 0.939738i −0.882736 0.469869i $$-0.844301\pi$$
0.882736 0.469869i $$-0.155699\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 526.000 1.68051 0.840256 0.542191i $$-0.182405\pi$$
0.840256 + 0.542191i $$0.182405\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 178.000 503.460i 0.554517 1.56841i
$$322$$ 0 0
$$323$$ 576.000 1.78328
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 661.852i 1.99955i −0.0211480 0.999776i $$-0.506732\pi$$
0.0211480 0.999776i $$-0.493268\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 478.000 1.41840 0.709199 0.705009i $$-0.249057\pi$$
0.709199 + 0.705009i $$0.249057\pi$$
$$338$$ 0 0
$$339$$ −576.000 203.647i −1.69912 0.600728i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 658.000 1.89625 0.948127 0.317892i $$-0.102975\pi$$
0.948127 + 0.317892i $$0.102975\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 678.823i 1.92301i 0.274788 + 0.961505i $$0.411392\pi$$
−0.274788 + 0.961505i $$0.588608\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 73.0000 0.202216
$$362$$ 0 0
$$363$$ −75.0000 + 212.132i −0.206612 + 0.584386i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ −384.000 + 475.176i −1.04065 + 1.28774i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 322.441i 0.850767i 0.905013 + 0.425383i $$0.139861\pi$$
−0.905013 + 0.425383i $$0.860139\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −480.000 + 593.970i −1.24031 + 1.53481i
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −62.0000 + 175.362i −0.157761 + 0.446215i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 237.588i 0.592488i −0.955112 0.296244i $$-0.904266\pi$$
0.955112 0.296244i $$-0.0957342\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −334.000 −0.816626 −0.408313 0.912842i $$-0.633883\pi$$
−0.408313 + 0.912842i $$0.633883\pi$$
$$410$$ 0 0
$$411$$ −384.000 135.765i −0.934307 0.330327i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 528.000 + 186.676i 1.26619 + 0.447665i
$$418$$ 0 0
$$419$$ 514.000 1.22673 0.613365 0.789799i $$-0.289815\pi$$
0.613365 + 0.789799i $$0.289815\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 848.528i 1.99654i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 578.000 1.33487 0.667436 0.744667i $$-0.267392\pi$$
0.667436 + 0.744667i $$0.267392\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 343.000 + 277.186i 0.777778 + 0.628539i
$$442$$ 0 0
$$443$$ −878.000 −1.98194 −0.990971 0.134079i $$-0.957192\pi$$
−0.990971 + 0.134079i $$0.957192\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 237.588i 0.529149i 0.964365 + 0.264574i $$0.0852315\pi$$
−0.964365 + 0.264574i $$0.914769\pi$$
$$450$$ 0 0
$$451$$ 950.352i 2.10721i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −238.000 −0.520788 −0.260394 0.965502i $$-0.583852\pi$$
−0.260394 + 0.965502i $$0.583852\pi$$
$$458$$ 0 0
$$459$$ −480.000 780.646i −1.04575 1.70075i
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 34.0000 0.0728051 0.0364026 0.999337i $$-0.488410\pi$$
0.0364026 + 0.999337i $$0.488410\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1187.94i 2.51150i
$$474$$ 0 0
$$475$$ 424.264i 0.893188i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 0 0
$$489$$ 144.000 + 50.9117i 0.294479 + 0.104114i
$$490$$ 0 0
$$491$$ 782.000 1.59267 0.796334 0.604857i $$-0.206770\pi$$
0.796334 + 0.604857i $$0.206770\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 593.970i 1.19032i 0.803607 + 0.595160i $$0.202911\pi$$
−0.803607 + 0.595160i $$0.797089\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −169.000 + 478.004i −0.333333 + 0.942809i
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 240.000 + 390.323i 0.467836 + 0.760863i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 271.529i 0.521169i −0.965451 0.260584i $$-0.916085\pi$$
0.965451 0.260584i $$-0.0839152\pi$$
$$522$$ 0 0
$$523$$ 967.322i 1.84956i −0.380497 0.924782i $$-0.624247\pi$$
0.380497 0.924782i $$-0.375753\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 529.000 1.00000
$$530$$ 0 0
$$531$$ 574.000 + 463.862i 1.08098 + 0.873563i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 34.0000 96.1665i 0.0633147 0.179081i
$$538$$ 0 0
$$539$$ 686.000 1.27273
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 390.323i 0.713570i −0.934186 0.356785i $$-0.883873\pi$$
0.934186 0.356785i $$-0.116127\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −1344.00 475.176i −2.39572 0.847016i
$$562$$ 0 0
$$563$$ 226.000 0.401421 0.200710 0.979651i $$-0.435675\pi$$
0.200710 + 0.979651i $$0.435675\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 950.352i 1.67021i −0.550088 0.835107i $$-0.685406\pi$$
0.550088 0.835107i $$-0.314594\pi$$
$$570$$ 0 0
$$571$$ 933.381i 1.63464i −0.576182 0.817321i $$-0.695458\pi$$
0.576182 0.817321i $$-0.304542\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −2.00000 −0.00346620 −0.00173310 0.999998i $$-0.500552\pi$$
−0.00173310 + 0.999998i $$0.500552\pi$$
$$578$$ 0 0
$$579$$ 98.0000 277.186i 0.169257 0.478732i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −1138.00 −1.93867 −0.969336 0.245741i $$-0.920969\pi$$
−0.969336 + 0.245741i $$0.920969\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 814.587i 1.37367i 0.726813 + 0.686836i $$0.241001\pi$$
−0.726813 + 0.686836i $$0.758999\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ −914.000 −1.52080 −0.760399 0.649456i $$-0.774997\pi$$
−0.760399 + 0.649456i $$0.774997\pi$$
$$602$$ 0 0
$$603$$ 672.000 831.558i 1.11443 1.37903i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1187.94i 1.92535i −0.270665 0.962674i $$-0.587243\pi$$
0.270665 0.962674i $$-0.412757\pi$$
$$618$$ 0 0
$$619$$ 1103.09i 1.78205i 0.453958 + 0.891023i $$0.350012\pi$$
−0.453958 + 0.891023i $$0.649988\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 1.00000
$$626$$ 0 0
$$627$$ 672.000 + 237.588i 1.07177 + 0.378928i
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 1008.00 + 356.382i 1.59242 + 0.563004i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1187.94i 1.85326i −0.375975 0.926630i $$-0.622692\pi$$
0.375975 0.926630i $$-0.377308\pi$$
$$642$$ 0 0
$$643$$ 424.264i 0.659820i 0.944012 + 0.329910i $$0.107018\pi$$
−0.944012 + 0.329910i $$0.892982\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 1148.00 1.76888
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −994.000 803.273i −1.51294 1.22264i
$$658$$ 0 0
$$659$$ −994.000 −1.50835 −0.754173 0.656676i $$-0.771962\pi$$
−0.754173 + 0.656676i $$0.771962\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −1246.00 −1.85141 −0.925706 0.378244i $$-0.876528\pi$$
−0.925706 + 0.378244i $$0.876528\pi$$
$$674$$ 0 0
$$675$$ −575.000 + 353.553i −0.851852 + 0.523783i
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 446.000 1261.48i 0.654919 1.85239i
$$682$$ 0 0
$$683$$ −398.000 −0.582723 −0.291362 0.956613i $$-0.594108\pi$$
−0.291362 + 0.956613i $$0.594108\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1170.97i 1.69460i 0.531114 + 0.847300i $$0.321773\pi$$
−0.531114 + 0.847300i $$0.678227\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −2304.00 −3.30560
$$698$$ 0 0
$$699$$ 480.000 + 169.706i 0.686695 + 0.242783i
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 194.000 548.715i 0.268326 0.758942i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ 329.000 650.538i 0.451303 0.892371i
$$730$$ 0 0
$$731$$ −2880.00 −3.93981
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1663.12i 2.25660i
$$738$$ 0 0
$$739$$ 1442.50i 1.95196i −0.217862 0.975980i $$-0.569908\pi$$
0.217862 0.975980i $$-0.430092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 1106.00 + 893.783i 1.48059 + 1.19650i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 0 0
$$753$$ −466.000 + 1318.05i −0.618858 + 1.75039i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 610.940i 0.802812i −0.915900 0.401406i $$-0.868522\pi$$
0.915900 0.401406i $$-0.131478\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 1054.00 1.37061 0.685306 0.728256i $$-0.259669\pi$$
0.685306 + 0.728256i $$0.259669\pi$$
$$770$$ 0 0
$$771$$ 960.000 + 339.411i 1.24514 + 0.440222i
$$772$$ 0 0
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1152.00 1.47882
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1272.79i 1.61727i −0.588310 0.808635i $$-0.700207\pi$$
0.588310 0.808635i $$-0.299793\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 576.000 712.764i 0.719101 0.889842i
$$802$$ 0 0
$$803$$ −1988.00 −2.47572
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 339.411i 0.419544i 0.977750 + 0.209772i $$0.0672722\pi$$
−0.977750 + 0.209772i $$0.932728\pi$$
$$810$$ 0 0
$$811$$ 1612.20i