# Properties

 Label 39.4.f.a.8.1 Level $39$ Weight $4$ Character 39.8 Analytic conductor $2.301$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.30107449022$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## Embedding invariants

 Embedding label 8.1 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 39.8 Dual form 39.4.f.a.5.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.19615 q^{3} +8.00000i q^{4} +(-25.5885 + 25.5885i) q^{7} +27.0000 q^{9} +O(q^{10})$$ $$q-5.19615 q^{3} +8.00000i q^{4} +(-25.5885 + 25.5885i) q^{7} +27.0000 q^{9} -41.5692i q^{12} +(31.1769 - 35.0000i) q^{13} -64.0000 q^{16} +(49.9423 + 49.9423i) q^{19} +(132.962 - 132.962i) q^{21} +125.000i q^{25} -140.296 q^{27} +(-204.708 - 204.708i) q^{28} +(231.942 + 231.942i) q^{31} +216.000i q^{36} +(-163.238 + 163.238i) q^{37} +(-162.000 + 181.865i) q^{39} -218.238i q^{43} +332.554 q^{48} -966.538i q^{49} +(280.000 + 249.415i) q^{52} +(-259.508 - 259.508i) q^{57} +935.307 q^{61} +(-690.888 + 690.888i) q^{63} -512.000i q^{64} +(112.642 + 112.642i) q^{67} +(-407.939 + 407.939i) q^{73} -649.519i q^{75} +(-399.538 + 399.538i) q^{76} -1091.19 q^{79} +729.000 q^{81} +(1063.69 + 1063.69i) q^{84} +(97.8269 + 1693.37i) q^{91} +(-1205.21 - 1205.21i) q^{93} +(20.8921 + 20.8921i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 40 q^{7} + 108 q^{9}+O(q^{10})$$ 4 * q - 40 * q^7 + 108 * q^9 $$4 q - 40 q^{7} + 108 q^{9} - 256 q^{16} - 112 q^{19} + 324 q^{21} - 320 q^{28} + 616 q^{31} + 220 q^{37} - 648 q^{39} + 1120 q^{52} - 1620 q^{57} - 1080 q^{63} + 1760 q^{67} - 2380 q^{73} + 896 q^{76} + 2916 q^{81} + 2592 q^{84} - 544 q^{91} - 1620 q^{93} - 2660 q^{97}+O(q^{100})$$ 4 * q - 40 * q^7 + 108 * q^9 - 256 * q^16 - 112 * q^19 + 324 * q^21 - 320 * q^28 + 616 * q^31 + 220 * q^37 - 648 * q^39 + 1120 * q^52 - 1620 * q^57 - 1080 * q^63 + 1760 * q^67 - 2380 * q^73 + 896 * q^76 + 2916 * q^81 + 2592 * q^84 - 544 * q^91 - 1620 * q^93 - 2660 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$$.

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$-1$$ $$e\left(\frac{1}{4}\right)$$

## 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 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ −5.19615 −1.00000
$$4$$ 8.00000i 1.00000i
$$5$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$6$$ 0 0
$$7$$ −25.5885 + 25.5885i −1.38165 + 1.38165i −0.539949 + 0.841698i $$0.681557\pi$$
−0.841698 + 0.539949i $$0.818443\pi$$
$$8$$ 0 0
$$9$$ 27.0000 1.00000
$$10$$ 0 0
$$11$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$12$$ 41.5692i 1.00000i
$$13$$ 31.1769 35.0000i 0.665148 0.746712i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −64.0000 −1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 49.9423 + 49.9423i 0.603029 + 0.603029i 0.941115 0.338086i $$-0.109780\pi$$
−0.338086 + 0.941115i $$0.609780\pi$$
$$20$$ 0 0
$$21$$ 132.962 132.962i 1.38165 1.38165i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 125.000i 1.00000i
$$26$$ 0 0
$$27$$ −140.296 −1.00000
$$28$$ −204.708 204.708i −1.38165 1.38165i
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 231.942 + 231.942i 1.34381 + 1.34381i 0.892233 + 0.451576i $$0.149138\pi$$
0.451576 + 0.892233i $$0.350862\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 216.000i 1.00000i
$$37$$ −163.238 + 163.238i −0.725303 + 0.725303i −0.969680 0.244377i $$-0.921417\pi$$
0.244377 + 0.969680i $$0.421417\pi$$
$$38$$ 0 0
$$39$$ −162.000 + 181.865i −0.665148 + 0.746712i
$$40$$ 0 0
$$41$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$42$$ 0 0
$$43$$ 218.238i 0.773978i −0.922084 0.386989i $$-0.873515\pi$$
0.922084 0.386989i $$-0.126485\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$48$$ 332.554 1.00000
$$49$$ 966.538i 2.81790i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 280.000 + 249.415i 0.746712 + 0.665148i
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −259.508 259.508i −0.603029 0.603029i
$$58$$ 0 0
$$59$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$60$$ 0 0
$$61$$ 935.307 1.96318 0.981589 0.191006i $$-0.0611749\pi$$
0.981589 + 0.191006i $$0.0611749\pi$$
$$62$$ 0 0
$$63$$ −690.888 + 690.888i −1.38165 + 1.38165i
$$64$$ 512.000i 1.00000i
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 112.642 + 112.642i 0.205395 + 0.205395i 0.802307 0.596912i $$-0.203606\pi$$
−0.596912 + 0.802307i $$0.703606\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$72$$ 0 0
$$73$$ −407.939 + 407.939i −0.654049 + 0.654049i −0.953966 0.299916i $$-0.903041\pi$$
0.299916 + 0.953966i $$0.403041\pi$$
$$74$$ 0 0
$$75$$ 649.519i 1.00000i
$$76$$ −399.538 + 399.538i −0.603029 + 0.603029i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1091.19 −1.55403 −0.777017 0.629480i $$-0.783268\pi$$
−0.777017 + 0.629480i $$0.783268\pi$$
$$80$$ 0 0
$$81$$ 729.000 1.00000
$$82$$ 0 0
$$83$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$84$$ 1063.69 + 1063.69i 1.38165 + 1.38165i
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$90$$ 0 0
$$91$$ 97.8269 + 1693.37i 0.112693 + 1.95069i
$$92$$ 0 0
$$93$$ −1205.21 1205.21i −1.34381 1.34381i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 20.8921 + 20.8921i 0.0218688 + 0.0218688i 0.717957 0.696088i $$-0.245078\pi$$
−0.696088 + 0.717957i $$0.745078\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −1000.00 −1.00000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 1028.84i 0.984218i −0.870534 0.492109i $$-0.836226\pi$$
0.870534 0.492109i $$-0.163774\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 1122.37i 1.00000i
$$109$$ 1414.19 + 1414.19i 1.24271 + 1.24271i 0.958874 + 0.283833i $$0.0916061\pi$$
0.283833 + 0.958874i $$0.408394\pi$$
$$110$$ 0 0
$$111$$ 848.212 848.212i 0.725303 0.725303i
$$112$$ 1637.66 1637.66i 1.38165 1.38165i
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 841.777 945.000i 0.665148 0.746712i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1331.00i 1.00000i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −1855.54 + 1855.54i −1.34381 + 1.34381i
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 380.000i 0.265508i 0.991149 + 0.132754i $$0.0423821\pi$$
−0.991149 + 0.132754i $$0.957618\pi$$
$$128$$ 0 0
$$129$$ 1134.00i 0.773978i
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ −2555.89 −1.66635
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$138$$ 0 0
$$139$$ 2576.00 1.57190 0.785948 0.618293i $$-0.212175\pi$$
0.785948 + 0.618293i $$0.212175\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1728.00 −1.00000
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 5022.28i 2.81790i
$$148$$ −1305.91 1305.91i −0.725303 0.725303i
$$149$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$150$$ 0 0
$$151$$ −762.788 + 762.788i −0.411091 + 0.411091i −0.882119 0.471027i $$-0.843883\pi$$
0.471027 + 0.882119i $$0.343883\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −1454.92 1296.00i −0.746712 0.665148i
$$157$$ 3850.00 1.95709 0.978546 0.206028i $$-0.0660539\pi$$
0.978546 + 0.206028i $$0.0660539\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −499.689 + 499.689i −0.240114 + 0.240114i −0.816897 0.576783i $$-0.804308\pi$$
0.576783 + 0.816897i $$0.304308\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$168$$ 0 0
$$169$$ −253.000 2182.38i −0.115157 0.993347i
$$170$$ 0 0
$$171$$ 1348.44 + 1348.44i 0.603029 + 0.603029i
$$172$$ 1745.91 0.773978
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −3198.56 3198.56i −1.38165 1.38165i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 3429.46i 1.40834i 0.710031 + 0.704171i $$0.248681\pi$$
−0.710031 + 0.704171i $$0.751319\pi$$
$$182$$ 0 0
$$183$$ −4860.00 −1.96318
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 3589.96 3589.96i 1.38165 1.38165i
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 2660.43i 1.00000i
$$193$$ 3193.86 3193.86i 1.19119 1.19119i 0.214453 0.976734i $$-0.431203\pi$$
0.976734 0.214453i $$-0.0687968\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 7732.31 2.81790
$$197$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$198$$ 0 0
$$199$$ 5236.00i 1.86518i −0.360942 0.932588i $$-0.617545\pi$$
0.360942 0.932588i $$-0.382455\pi$$
$$200$$ 0 0
$$201$$ −585.307 585.307i −0.205395 0.205395i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −1995.32 + 2240.00i −0.665148 + 0.746712i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −1091.19 −0.356023 −0.178011 0.984028i $$-0.556966\pi$$
−0.178011 + 0.984028i $$0.556966\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −11870.1 −3.71334
$$218$$ 0 0
$$219$$ 2119.71 2119.71i 0.654049 0.654049i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 4525.04 + 4525.04i 1.35883 + 1.35883i 0.875362 + 0.483469i $$0.160623\pi$$
0.483469 + 0.875362i $$0.339377\pi$$
$$224$$ 0 0
$$225$$ 3375.00i 1.00000i
$$226$$ 0 0
$$227$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$228$$ 2076.06 2076.06i 0.603029 0.603029i
$$229$$ −4883.04 + 4883.04i −1.40908 + 1.40908i −0.644370 + 0.764714i $$0.722880\pi$$
−0.764714 + 0.644370i $$0.777120\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 5670.00 1.55403
$$238$$ 0 0
$$239$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$240$$ 0 0
$$241$$ −4312.54 + 4312.54i −1.15268 + 1.15268i −0.166662 + 0.986014i $$0.553299\pi$$
−0.986014 + 0.166662i $$0.946701\pi$$
$$242$$ 0 0
$$243$$ −3788.00 −1.00000
$$244$$ 7482.46i 1.96318i
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3305.03 190.934i 0.851392 0.0491855i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ −5527.11 5527.11i −1.38165 1.38165i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 4096.00 1.00000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 8354.04i 2.00423i
$$260$$ 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
$$267$$ 0 0
$$268$$ −901.139 + 901.139i −0.205395 + 0.205395i
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 4848.71 4848.71i 1.08686 1.08686i 0.0910064 0.995850i $$-0.470992\pi$$
0.995850 0.0910064i $$-0.0290084\pi$$
$$272$$ 0 0
$$273$$ −508.323 8798.98i −0.112693 1.95069i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8293.06i 1.79885i 0.437074 + 0.899425i $$0.356015\pi$$
−0.437074 + 0.899425i $$0.643985\pi$$
$$278$$ 0 0
$$279$$ 6262.44 + 6262.44i 1.34381 + 1.34381i
$$280$$ 0 0
$$281$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$282$$ 0 0
$$283$$ 5600.00i 1.17627i −0.808761 0.588137i $$-0.799862\pi$$
0.808761 0.588137i $$-0.200138\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4913.00 −1.00000
$$290$$ 0 0
$$291$$ −108.559 108.559i −0.0218688 0.0218688i
$$292$$ −3263.51 3263.51i −0.654049 0.654049i
$$293$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 5196.15 1.00000
$$301$$ 5584.38 + 5584.38i 1.06936 + 1.06936i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −3196.31 3196.31i −0.603029 0.603029i
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 6115.01 6115.01i 1.13681 1.13681i 0.147797 0.989018i $$-0.452782\pi$$
0.989018 0.147797i $$-0.0472182\pi$$
$$308$$ 0 0
$$309$$ 5346.00i 0.984218i
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −4738.89 −0.855776 −0.427888 0.903832i $$-0.640742\pi$$
−0.427888 + 0.903832i $$0.640742\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 8729.54i 1.55403i
$$317$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 5832.00i 1.00000i
$$325$$ 4375.00 + 3897.11i 0.746712 + 0.665148i
$$326$$ 0 0
$$327$$ −7348.36 7348.36i −1.24271 1.24271i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −6497.56 6497.56i −1.07897 1.07897i −0.996602 0.0823644i $$-0.973753\pi$$
−0.0823644 0.996602i $$-0.526247\pi$$
$$332$$ 0 0
$$333$$ −4407.44 + 4407.44i −0.725303 + 0.725303i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −8509.54 + 8509.54i −1.38165 + 1.38165i
$$337$$ 4930.00i 0.796897i 0.917191 + 0.398448i $$0.130451\pi$$
−0.917191 + 0.398448i $$0.869549\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 15955.4 + 15955.4i 2.51169 + 2.51169i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 0 0
$$349$$ 3306.96 3306.96i 0.507214 0.507214i −0.406456 0.913670i $$-0.633236\pi$$
0.913670 + 0.406456i $$0.133236\pi$$
$$350$$ 0 0
$$351$$ −4374.00 + 4910.36i −0.665148 + 0.746712i
$$352$$ 0 0
$$353$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$360$$ 0 0
$$361$$ 1870.54i 0.272713i
$$362$$ 0 0
$$363$$ 6916.08i 1.00000i
$$364$$ −13546.9 + 782.615i −1.95069 + 0.112693i
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −4340.00 −0.617292 −0.308646 0.951177i $$-0.599876\pi$$
−0.308646 + 0.951177i $$0.599876\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 9641.66 9641.66i 1.34381 1.34381i
$$373$$ 7420.11 1.03002 0.515011 0.857183i $$-0.327788\pi$$
0.515011 + 0.857183i $$0.327788\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −10293.6 10293.6i −1.39510 1.39510i −0.813402 0.581702i $$-0.802387\pi$$
−0.581702 0.813402i $$-0.697613\pi$$
$$380$$ 0 0
$$381$$ 1974.54i 0.265508i
$$382$$ 0 0
$$383$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 5892.44i 0.773978i
$$388$$ −167.137 + 167.137i −0.0218688 + 0.0218688i
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 7292.76 7292.76i 0.921947 0.921947i −0.0752196 0.997167i $$-0.523966\pi$$
0.997167 + 0.0752196i $$0.0239658\pi$$
$$398$$ 0 0
$$399$$ 13280.8 1.66635
$$400$$ 8000.00i 1.00000i
$$401$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$402$$ 0 0
$$403$$ 15349.2 886.735i 1.89727 0.109607i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 3047.69 + 3047.69i 0.368456 + 0.368456i 0.866914 0.498458i $$-0.166100\pi$$
−0.498458 + 0.866914i $$0.666100\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8230.71 0.984218
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −13385.3 −1.57190
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ −7477.81 7477.81i −0.865668 0.865668i 0.126322 0.991989i $$-0.459683\pi$$
−0.991989 + 0.126322i $$0.959683\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −23933.1 + 23933.1i −2.71242 + 2.71242i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$432$$ 8978.95 1.00000
$$433$$ 2590.00i 0.287454i 0.989617 + 0.143727i $$0.0459087\pi$$
−0.989617 + 0.143727i $$0.954091\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −11313.5 + 11313.5i −1.24271 + 1.24271i
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 10756.0i 1.16938i −0.811257 0.584690i $$-0.801216\pi$$
0.811257 0.584690i $$-0.198784\pi$$
$$440$$ 0 0
$$441$$ 26096.5i 2.81790i
$$442$$ 0 0
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 6785.69 + 6785.69i 0.725303 + 0.725303i
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 13101.3 + 13101.3i 1.38165 + 1.38165i
$$449$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 3963.56 3963.56i 0.411091 0.411091i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1065.11 1065.11i −0.109023 0.109023i 0.650491 0.759514i $$-0.274563\pi$$
−0.759514 + 0.650491i $$0.774563\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$462$$ 0 0
$$463$$ −8689.69 + 8689.69i −0.872233 + 0.872233i −0.992716 0.120482i $$-0.961556\pi$$
0.120482 + 0.992716i $$0.461556\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 7560.00 + 6734.21i 0.746712 + 0.665148i
$$469$$ −5764.69 −0.567566
$$470$$ 0 0
$$471$$ −20005.2 −1.95709
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −6242.79 + 6242.79i −0.603029 + 0.603029i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$480$$ 0 0
$$481$$ 624.074 + 10802.6i 0.0591587 + 1.02403i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 10648.0 1.00000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12959.7 + 12959.7i 1.20588 + 1.20588i 0.972351 + 0.233526i $$0.0750265\pi$$
0.233526 + 0.972351i $$0.424974\pi$$
$$488$$ 0 0
$$489$$ 2596.46 2596.46i 0.240114 0.240114i
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −14844.3 14844.3i −1.34381 1.34381i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 615.940 + 615.940i 0.0552570 + 0.0552570i 0.734195 0.678938i $$-0.237560\pi$$
−0.678938 + 0.734195i $$0.737560\pi$$
$$500$$ 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
$$507$$ 1314.63 + 11340.0i 0.115157 + 0.993347i
$$508$$ −3040.00 −0.265508
$$509$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$510$$ 0 0
$$511$$ 20877.0i 1.80733i
$$512$$ 0 0
$$513$$ −7006.71 7006.71i −0.603029 0.603029i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ −9072.00 −0.773978
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 12040.0 1.00664 0.503320 0.864100i $$-0.332112\pi$$
0.503320 + 0.864100i $$0.332112\pi$$
$$524$$ 0 0
$$525$$ 16620.2 + 16620.2i 1.38165 + 1.38165i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −12167.0 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 20447.1i 1.66635i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 16795.0 16795.0i 1.33470 1.33470i 0.433586 0.901112i $$-0.357248\pi$$
0.901112 0.433586i $$-0.142752\pi$$
$$542$$ 0 0
$$543$$ 17820.0i 1.40834i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1640.00 0.128193 0.0640963 0.997944i $$-0.479584\pi$$
0.0640963 + 0.997944i $$0.479584\pi$$
$$548$$ 0 0
$$549$$ 25253.3 1.96318
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 27921.9 27921.9i 2.14713 2.14713i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 20608.0i 1.57190i
$$557$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$558$$ 0 0
$$559$$ −7638.34 6804.00i −0.577938 0.514810i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −18654.0 + 18654.0i −1.38165 + 1.38165i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 23312.0i 1.70854i 0.519829 + 0.854270i $$0.325996\pi$$
−0.519829 + 0.854270i $$0.674004\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 13824.0i 1.00000i
$$577$$ −19517.5 19517.5i −1.40819 1.40819i −0.769300 0.638888i $$-0.779395\pi$$
−0.638888 0.769300i $$-0.720605\pi$$
$$578$$ 0 0
$$579$$ −16595.8 + 16595.8i −1.19119 + 1.19119i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$588$$ −40178.2 −2.81790
$$589$$ 23167.5i 1.62071i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 10447.3 10447.3i 0.725303 0.725303i
$$593$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 27207.1i 1.86518i
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ −3117.69 −0.211603 −0.105801 0.994387i $$-0.533741\pi$$
−0.105801 + 0.994387i $$0.533741\pi$$
$$602$$ 0 0
$$603$$ 3041.34 + 3041.34i 0.205395 + 0.205395i
$$604$$ −6102.30 6102.30i −0.411091 0.411091i
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 28420.0 1.90038 0.950191 0.311667i $$-0.100887\pi$$
0.950191 + 0.311667i $$0.100887\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 21134.6 + 21134.6i 1.39253 + 1.39253i 0.819625 + 0.572900i $$0.194182\pi$$
0.572900 + 0.819625i $$0.305818\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$618$$ 0 0
$$619$$ 5611.71 5611.71i 0.364384 0.364384i −0.501040 0.865424i $$-0.667049\pi$$
0.865424 + 0.501040i $$0.167049\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 10368.0 11639.4i 0.665148 0.746712i
$$625$$ −15625.0 −1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 30800.0i 1.95709i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −16768.3 + 16768.3i −1.05790 + 1.05790i −0.0596825 + 0.998217i $$0.519009\pi$$
−0.998217 + 0.0596825i $$0.980991\pi$$
$$632$$ 0 0
$$633$$ 5670.00 0.356023
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −33828.8 30133.7i −2.10416 1.87432i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ −21498.2 21498.2i −1.31851 1.31851i −0.914953 0.403561i $$-0.867772\pi$$
−0.403561 0.914953i $$-0.632228\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 61678.8 3.71334
$$652$$ −3997.51 3997.51i −0.240114 0.240114i
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −11014.3 + 11014.3i −0.654049 + 0.654049i
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 3320.96 3320.96i 0.195416 0.195416i −0.602615 0.798032i $$-0.705875\pi$$
0.798032 + 0.602615i $$0.205875\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −23512.8 23512.8i −1.35883 1.35883i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 25315.7i 1.45000i 0.688751 + 0.724998i $$0.258159\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$674$$ 0 0
$$675$$ 17537.0i 1.00000i
$$676$$ 17459.1 2024.00i 0.993347 0.115157i
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ −1069.19 −0.0604299
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$684$$ −10787.5 + 10787.5i −0.603029 + 0.603029i
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 25373.0 25373.0i 1.40908 1.40908i
$$688$$ 13967.3i 0.773978i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 8253.94 + 8253.94i 0.454406 + 0.454406i 0.896814 0.442408i $$-0.145876\pi$$
−0.442408 + 0.896814i $$0.645876\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 25588.5 25588.5i 1.38165 1.38165i
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ −16305.0 −0.874758
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 23529.0 23529.0i 1.24633 1.24633i 0.289003 0.957328i $$-0.406676\pi$$
0.957328 0.289003i $$-0.0933236\pi$$
$$710$$ 0 0
$$711$$ −29462.2 −1.55403
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 26326.4 + 26326.4i 1.35984 + 1.35984i
$$722$$ 0 0
$$723$$ 22408.6 22408.6i 1.15268 1.15268i
$$724$$ −27435.7 −1.40834
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 10780.0i 0.549942i 0.961452 + 0.274971i $$0.0886683\pi$$
−0.961452 + 0.274971i $$0.911332\pi$$
$$728$$ 0 0
$$729$$ 19683.0 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 38880.0i 1.96318i
$$733$$ −10838.2 10838.2i −0.546137 0.546137i 0.379184 0.925321i $$-0.376205\pi$$
−0.925321 + 0.379184i $$0.876205\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −3139.29 + 3139.29i −0.156266 + 0.156266i −0.780910 0.624644i $$-0.785244\pi$$
0.624644 + 0.780910i $$0.285244\pi$$
$$740$$ 0 0
$$741$$ −17173.4 + 992.120i −0.851392 + 0.0491855i
$$742$$ 0 0
$$743$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 33827.0i 1.64363i 0.569757 + 0.821813i $$0.307037\pi$$
−0.569757 + 0.821813i $$0.692963\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 28719.7 + 28719.7i 1.38165 + 1.38165i
$$757$$ −3928.29 −0.188608 −0.0943039 0.995543i $$-0.530063\pi$$
−0.0943039 + 0.995543i $$0.530063\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$762$$ 0 0
$$763$$ −72374.0 −3.43396
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −21283.4 −1.00000
$$769$$ −23503.3 23503.3i −1.10215 1.10215i −0.994151 0.107995i $$-0.965557\pi$$
−0.107995 0.994151i $$-0.534443\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 25550.9 + 25550.9i 1.19119 + 1.19119i
$$773$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$774$$ 0 0
$$775$$ −28992.8 + 28992.8i −1.34381 + 1.34381i
$$776$$ 0 0
$$777$$ 43408.9i 2.00423i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 61858.5i 2.81790i
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 17631.4 17631.4i 0.798592 0.798592i −0.184281 0.982874i $$-0.558996\pi$$
0.982874 + 0.184281i $$0.0589958\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 29160.0 32735.8i 1.30580 1.46593i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 41888.0 1.86518
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$