Properties

 Label 105.4.g.a.104.4 Level $105$ Weight $4$ Character 105.104 Analytic conductor $6.195$ Analytic rank $0$ Dimension $4$ CM discriminant -35 Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.g (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.19520055060$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-5}, \sqrt{7})$$ Defining polynomial: $$x^{4} - x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

 Embedding label 104.4 Root $$-1.32288 + 1.11803i$$ of defining polynomial Character $$\chi$$ $$=$$ 105.104 Dual form 105.4.g.a.104.3

$q$-expansion

 $$f(q)$$ $$=$$ $$q+(2.64575 + 4.47214i) q^{3} -8.00000 q^{4} -11.1803i q^{5} -18.5203 q^{7} +(-13.0000 + 23.6643i) q^{9} +O(q^{10})$$ $$q+(2.64575 + 4.47214i) q^{3} -8.00000 q^{4} -11.1803i q^{5} -18.5203 q^{7} +(-13.0000 + 23.6643i) q^{9} -11.8322i q^{11} +(-21.1660 - 35.7771i) q^{12} -84.6640 q^{13} +(50.0000 - 29.5804i) q^{15} +64.0000 q^{16} -102.859i q^{17} +89.4427i q^{20} +(-49.0000 - 82.8251i) q^{21} -125.000 q^{25} +(-140.225 + 4.47214i) q^{27} +148.162 q^{28} +307.636i q^{29} +(52.9150 - 31.3050i) q^{33} +207.063i q^{35} +(104.000 - 189.315i) q^{36} +(-224.000 - 378.629i) q^{39} +94.6573i q^{44} +(264.575 + 145.344i) q^{45} +178.885i q^{47} +(169.328 + 286.217i) q^{48} +343.000 q^{49} +(460.000 - 272.140i) q^{51} +677.312 q^{52} -132.288 q^{55} +(-400.000 + 236.643i) q^{60} +(240.763 - 438.269i) q^{63} -512.000 q^{64} +946.573i q^{65} +822.873i q^{68} -863.748i q^{71} -1132.38 q^{73} +(-330.719 - 559.017i) q^{75} +219.135i q^{77} +236.000 q^{79} -715.542i q^{80} +(-391.000 - 615.272i) q^{81} -1511.58i q^{83} +(392.000 + 662.601i) q^{84} -1150.00 q^{85} +(-1375.79 + 813.929i) q^{87} +1568.00 q^{91} +963.053 q^{97} +(280.000 + 153.818i) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 32q^{4} - 52q^{9} + O(q^{10})$$ $$4q - 32q^{4} - 52q^{9} + 200q^{15} + 256q^{16} - 196q^{21} - 500q^{25} + 416q^{36} - 896q^{39} + 1372q^{49} + 1840q^{51} - 1600q^{60} - 2048q^{64} + 944q^{79} - 1564q^{81} + 1568q^{84} - 4600q^{85} + 6272q^{91} + 1120q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/105\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 2.64575 + 4.47214i 0.509175 + 0.860663i
$$4$$ −8.00000 −1.00000
$$5$$ 11.1803i 1.00000i
$$6$$ 0 0
$$7$$ −18.5203 −1.00000
$$8$$ 0 0
$$9$$ −13.0000 + 23.6643i −0.481481 + 0.876456i
$$10$$ 0 0
$$11$$ 11.8322i 0.324321i −0.986764 0.162160i $$-0.948154\pi$$
0.986764 0.162160i $$-0.0518462\pi$$
$$12$$ −21.1660 35.7771i −0.509175 0.860663i
$$13$$ −84.6640 −1.80628 −0.903138 0.429351i $$-0.858742\pi$$
−0.903138 + 0.429351i $$0.858742\pi$$
$$14$$ 0 0
$$15$$ 50.0000 29.5804i 0.860663 0.509175i
$$16$$ 64.0000 1.00000
$$17$$ 102.859i 1.46747i −0.679435 0.733735i $$-0.737775\pi$$
0.679435 0.733735i $$-0.262225\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 89.4427i 1.00000i
$$21$$ −49.0000 82.8251i −0.509175 0.860663i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −125.000 −1.00000
$$26$$ 0 0
$$27$$ −140.225 + 4.47214i −0.999492 + 0.0318764i
$$28$$ 148.162 1.00000
$$29$$ 307.636i 1.96988i 0.172889 + 0.984941i $$0.444690\pi$$
−0.172889 + 0.984941i $$0.555310\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 52.9150 31.3050i 0.279131 0.165136i
$$34$$ 0 0
$$35$$ 207.063i 1.00000i
$$36$$ 104.000 189.315i 0.481481 0.876456i
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ −224.000 378.629i −0.919710 1.55459i
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 94.6573i 0.324321i
$$45$$ 264.575 + 145.344i 0.876456 + 0.481481i
$$46$$ 0 0
$$47$$ 178.885i 0.555173i 0.960701 + 0.277586i $$0.0895345\pi$$
−0.960701 + 0.277586i $$0.910466\pi$$
$$48$$ 169.328 + 286.217i 0.509175 + 0.860663i
$$49$$ 343.000 1.00000
$$50$$ 0 0
$$51$$ 460.000 272.140i 1.26300 0.747200i
$$52$$ 677.312 1.80628
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −132.288 −0.324321
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ −400.000 + 236.643i −0.860663 + 0.509175i
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 240.763 438.269i 0.481481 0.876456i
$$64$$ −512.000 −1.00000
$$65$$ 946.573i 1.80628i
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 822.873i 1.46747i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 863.748i 1.44377i −0.692011 0.721887i $$-0.743275\pi$$
0.692011 0.721887i $$-0.256725\pi$$
$$72$$ 0 0
$$73$$ −1132.38 −1.81555 −0.907776 0.419456i $$-0.862221\pi$$
−0.907776 + 0.419456i $$0.862221\pi$$
$$74$$ 0 0
$$75$$ −330.719 559.017i −0.509175 0.860663i
$$76$$ 0 0
$$77$$ 219.135i 0.324321i
$$78$$ 0 0
$$79$$ 236.000 0.336102 0.168051 0.985778i $$-0.446253\pi$$
0.168051 + 0.985778i $$0.446253\pi$$
$$80$$ 715.542i 1.00000i
$$81$$ −391.000 615.272i −0.536351 0.843995i
$$82$$ 0 0
$$83$$ 1511.58i 1.99901i −0.0314901 0.999504i $$-0.510025\pi$$
0.0314901 0.999504i $$-0.489975\pi$$
$$84$$ 392.000 + 662.601i 0.509175 + 0.860663i
$$85$$ −1150.00 −1.46747
$$86$$ 0 0
$$87$$ −1375.79 + 813.929i −1.69541 + 1.00302i
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 1568.00 1.80628
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 963.053 1.00807 0.504037 0.863682i $$-0.331847\pi$$
0.504037 + 0.863682i $$0.331847\pi$$
$$98$$ 0 0
$$99$$ 280.000 + 153.818i 0.284253 + 0.156155i
$$100$$ 1000.00 1.00000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ −799.017 −0.764364 −0.382182 0.924087i $$-0.624827\pi$$
−0.382182 + 0.924087i $$0.624827\pi$$
$$104$$ 0 0
$$105$$ −926.013 + 547.837i −0.860663 + 0.509175i
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 1121.80 35.7771i 0.999492 0.0318764i
$$109$$ −2266.00 −1.99122 −0.995612 0.0935765i $$-0.970170\pi$$
−0.995612 + 0.0935765i $$0.970170\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1185.30 −1.00000
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2461.09i 1.96988i
$$117$$ 1100.63 2003.52i 0.869688 1.58312i
$$118$$ 0 0
$$119$$ 1904.98i 1.46747i
$$120$$ 0 0
$$121$$ 1191.00 0.894816
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1397.54i 1.00000i
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ −423.320 + 250.440i −0.279131 + 0.165136i
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 50.0000 + 1567.76i 0.0318764 + 0.999492i
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 1656.50i 1.00000i
$$141$$ −800.000 + 473.286i −0.477817 + 0.282680i
$$142$$ 0 0
$$143$$ 1001.76i 0.585813i
$$144$$ −832.000 + 1514.52i −0.481481 + 0.876456i
$$145$$ 3439.48 1.96988
$$146$$ 0 0
$$147$$ 907.493 + 1533.94i 0.509175 + 0.860663i
$$148$$ 0 0
$$149$$ 2674.07i 1.47026i −0.677928 0.735128i $$-0.737122\pi$$
0.677928 0.735128i $$-0.262878\pi$$
$$150$$ 0 0
$$151$$ −2788.00 −1.50254 −0.751272 0.659992i $$-0.770559\pi$$
−0.751272 + 0.659992i $$0.770559\pi$$
$$152$$ 0 0
$$153$$ 2434.09 + 1337.17i 1.28617 + 0.706560i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 1792.00 + 3029.03i 0.919710 + 1.55459i
$$157$$ 486.818 0.247467 0.123734 0.992315i $$-0.460513\pi$$
0.123734 + 0.992315i $$0.460513\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ −350.000 591.608i −0.165136 0.279131i
$$166$$ 0 0
$$167$$ 2558.06i 1.18532i −0.805452 0.592661i $$-0.798077\pi$$
0.805452 0.592661i $$-0.201923\pi$$
$$168$$ 0 0
$$169$$ 4971.00 2.26263
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 4405.05i 1.93590i 0.251150 + 0.967948i $$0.419191\pi$$
−0.251150 + 0.967948i $$0.580809\pi$$
$$174$$ 0 0
$$175$$ 2315.03 1.00000
$$176$$ 757.258i 0.324321i
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4697.37i 1.96144i −0.195419 0.980720i $$-0.562607\pi$$
0.195419 0.980720i $$-0.437393\pi$$
$$180$$ −2116.60 1162.76i −0.876456 0.481481i
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1217.05 −0.475931
$$188$$ 1431.08i 0.555173i
$$189$$ 2597.00 82.8251i 0.999492 0.0318764i
$$190$$ 0 0
$$191$$ 3182.85i 1.20577i 0.797826 + 0.602887i $$0.205983\pi$$
−0.797826 + 0.602887i $$0.794017\pi$$
$$192$$ −1354.62 2289.73i −0.509175 0.860663i
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ −4233.20 + 2504.40i −1.55459 + 0.919710i
$$196$$ −2744.00 −1.00000
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 5697.50i 1.96988i
$$204$$ −3680.00 + 2177.12i −1.26300 + 0.747200i
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −5418.50 −1.80628
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −2392.00 −0.780436 −0.390218 0.920722i $$-0.627600\pi$$
−0.390218 + 0.920722i $$0.627600\pi$$
$$212$$ 0 0
$$213$$ 3862.80 2285.26i 1.24260 0.735134i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −2996.00 5064.16i −0.924433 1.56258i
$$220$$ 1058.30 0.324321
$$221$$ 8708.47i 2.65066i
$$222$$ 0 0
$$223$$ −5180.38 −1.55562 −0.777812 0.628498i $$-0.783670\pi$$
−0.777812 + 0.628498i $$0.783670\pi$$
$$224$$ 0 0
$$225$$ 1625.00 2958.04i 0.481481 0.876456i
$$226$$ 0 0
$$227$$ 4767.30i 1.39391i 0.717117 + 0.696953i $$0.245461\pi$$
−0.717117 + 0.696953i $$0.754539\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ −980.000 + 579.776i −0.279131 + 0.165136i
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 2000.00 0.555173
$$236$$ 0 0
$$237$$ 624.397 + 1055.42i 0.171135 + 0.289271i
$$238$$ 0 0
$$239$$ 4673.70i 1.26492i 0.774592 + 0.632462i $$0.217955\pi$$
−0.774592 + 0.632462i $$0.782045\pi$$
$$240$$ 3200.00 1893.15i 0.860663 0.509175i
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 1717.09 3376.46i 0.453299 0.891359i
$$244$$ 0 0
$$245$$ 3834.86i 1.00000i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 6760.00 3999.27i 1.72047 1.01785i
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ −1926.11 + 3506.15i −0.481481 + 0.876456i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −3042.61 5142.96i −0.747200 1.26300i
$$256$$ 4096.00 1.00000
$$257$$ 3358.57i 0.815183i 0.913164 + 0.407592i $$0.133631\pi$$
−0.913164 + 0.407592i $$0.866369\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 7572.58i 1.80628i
$$261$$ −7280.00 3999.27i −1.72652 0.948462i
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 6582.98i 1.46747i
$$273$$ 4148.54 + 7012.31i 0.919710 + 1.55459i
$$274$$ 0 0
$$275$$ 1479.02i 0.324321i
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4780.19i 1.01481i 0.861707 + 0.507406i $$0.169396\pi$$
−0.861707 + 0.507406i $$0.830604\pi$$
$$282$$ 0 0
$$283$$ −2227.72 −0.467931 −0.233965 0.972245i $$-0.575170\pi$$
−0.233965 + 0.972245i $$0.575170\pi$$
$$284$$ 6909.98i 1.44377i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −5667.00 −1.15347
$$290$$ 0 0
$$291$$ 2548.00 + 4306.91i 0.513287 + 0.867613i
$$292$$ 9059.05 1.81555
$$293$$ 6739.51i 1.34378i 0.740653 + 0.671888i $$0.234516\pi$$
−0.740653 + 0.671888i $$0.765484\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 52.9150 + 1659.16i 0.0103382 + 0.324156i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 2645.75 + 4472.14i 0.509175 + 0.860663i
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 9011.43 1.67527 0.837637 0.546227i $$-0.183936\pi$$
0.837637 + 0.546227i $$0.183936\pi$$
$$308$$ 1753.08i 0.324321i
$$309$$ −2114.00 3573.31i −0.389195 0.657860i
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 2677.50 0.483518 0.241759 0.970336i $$-0.422276\pi$$
0.241759 + 0.970336i $$0.422276\pi$$
$$314$$ 0 0
$$315$$ −4900.00 2691.82i −0.876456 0.481481i
$$316$$ −1888.00 −0.336102
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 3640.00 0.638874
$$320$$ 5724.33i 1.00000i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 3128.00 + 4922.18i 0.536351 + 0.843995i
$$325$$ 10583.0 1.80628
$$326$$ 0 0
$$327$$ −5995.27 10133.9i −1.01388 1.71377i
$$328$$ 0 0
$$329$$ 3313.00i 0.555173i
$$330$$ 0 0
$$331$$ −7432.00 −1.23414 −0.617069 0.786909i $$-0.711680\pi$$
−0.617069 + 0.786909i $$0.711680\pi$$
$$332$$ 12092.7i 1.99901i
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −3136.00 5300.81i −0.509175 0.860663i
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 9200.00 1.46747
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −6352.45 −1.00000
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 11006.3 6511.43i 1.69541 1.00302i
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 11872.0 378.629i 1.80536 0.0575776i
$$352$$ 0 0
$$353$$ 6220.74i 0.937951i −0.883211 0.468975i $$-0.844623\pi$$
0.883211 0.468975i $$-0.155377\pi$$
$$354$$ 0 0
$$355$$ −9656.99 −1.44377
$$356$$ 0 0
$$357$$ −8519.32 + 5040.10i −1.26300 + 0.747200i
$$358$$ 0 0
$$359$$ 11086.7i 1.62990i −0.579529 0.814952i $$-0.696763\pi$$
0.579529 0.814952i $$-0.303237\pi$$
$$360$$ 0 0
$$361$$ 6859.00 1.00000
$$362$$ 0 0
$$363$$ 3151.09 + 5326.31i 0.455618 + 0.770135i
$$364$$ −12544.0 −1.80628
$$365$$ 12660.4i 1.81555i
$$366$$ 0 0
$$367$$ −14038.4 −1.99672 −0.998360 0.0572477i $$-0.981768\pi$$
−0.998360 + 0.0572477i $$0.981768\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ −6250.00 + 3697.55i −0.860663 + 0.509175i
$$376$$ 0 0
$$377$$ 26045.7i 3.55815i
$$378$$ 0 0
$$379$$ 14096.0 1.91046 0.955228 0.295870i $$-0.0956097\pi$$
0.955228 + 0.295870i $$0.0956097\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 4686.80i 0.625285i 0.949871 + 0.312643i $$0.101214\pi$$
−0.949871 + 0.312643i $$0.898786\pi$$
$$384$$ 0 0
$$385$$ 2450.00 0.324321
$$386$$ 0 0
$$387$$ 0 0
$$388$$ −7704.43 −1.00807
$$389$$ 6697.00i 0.872883i 0.899733 + 0.436442i $$0.143761\pi$$
−0.899733 + 0.436442i $$0.856239\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 2638.56i 0.336102i
$$396$$ −2240.00 1230.54i −0.284253 0.156155i
$$397$$ −275.158 −0.0347854 −0.0173927 0.999849i $$-0.505537\pi$$
−0.0173927 + 0.999849i $$0.505537\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −8000.00 −1.00000
$$401$$ 9702.37i 1.20826i −0.796885 0.604131i $$-0.793520\pi$$
0.796885 0.604131i $$-0.206480\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −6878.95 + 4371.51i −0.843995 + 0.536351i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 6392.14 0.764364
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −16900.0 −1.99901
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 7408.10 4382.69i 0.860663 0.509175i
$$421$$ −3922.00 −0.454030 −0.227015 0.973891i $$-0.572897\pi$$
−0.227015 + 0.973891i $$0.572897\pi$$
$$422$$ 0 0
$$423$$ −4233.20 2325.51i −0.486585 0.267305i
$$424$$ 0 0
$$425$$ 12857.4i 1.46747i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −4480.00 + 2650.40i −0.504188 + 0.298281i
$$430$$ 0 0
$$431$$ 17476.1i 1.95312i −0.215249 0.976559i $$-0.569056\pi$$
0.215249 0.976559i $$-0.430944\pi$$
$$432$$ −8974.39 + 286.217i −0.999492 + 0.0318764i
$$433$$ −12562.0 −1.39421 −0.697105 0.716970i $$-0.745529\pi$$
−0.697105 + 0.716970i $$0.745529\pi$$
$$434$$ 0 0
$$435$$ 9100.00 + 15381.8i 1.00302 + 1.69541i
$$436$$ 18128.0 1.99122
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ −4459.00 + 8116.86i −0.481481 + 0.876456i
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 11958.8 7074.92i 1.26540 0.748618i
$$448$$ 9482.37 1.00000
$$449$$ 15239.8i 1.60181i −0.598793 0.800904i $$-0.704353\pi$$
0.598793 0.800904i $$-0.295647\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −7376.35 12468.3i −0.765058 1.29318i
$$454$$ 0 0
$$455$$ 17530.8i 1.80628i
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 460.000 + 14423.4i 0.0467777 + 1.46673i
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 19688.7i 1.96988i
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 17164.1i 1.70077i 0.526164 + 0.850383i $$0.323630\pi$$
−0.526164 + 0.850383i $$0.676370\pi$$
$$468$$ −8805.06 + 16028.1i −0.869688 + 1.58312i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 1288.00 + 2177.12i 0.126004 + 0.212986i
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 15239.8i 1.46747i
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −9528.00 −0.894816
$$485$$ 10767.3i 1.00807i
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 17689.1i 1.62586i −0.582362 0.812930i $$-0.697871\pi$$
0.582362 0.812930i $$-0.302129\pi$$
$$492$$ 0 0
$$493$$ 31643.2 2.89075
$$494$$ 0 0
$$495$$ 1719.74 3130.50i 0.156155 0.284253i
$$496$$ 0 0
$$497$$ 15996.8i 1.44377i
$$498$$ 0 0
$$499$$ 7544.00 0.676785 0.338393 0.941005i $$-0.390117\pi$$
0.338393 + 0.941005i $$0.390117\pi$$
$$500$$ 11180.3i 1.00000i
$$501$$ 11440.0 6768.00i 1.02016 0.603536i
$$502$$ 0 0
$$503$$ 2432.84i 0.215656i 0.994170 + 0.107828i $$0.0343896\pi$$
−0.994170 + 0.107828i $$0.965610\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 13152.0 + 22231.0i 1.15208 + 1.94736i
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 20972.0 1.81555
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 8933.28i 0.764364i
$$516$$ 0 0
$$517$$ 2116.60 0.180054
$$518$$ 0 0
$$519$$ −19700.0 + 11654.7i −1.66615 + 0.985710i
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 7296.98 0.610086 0.305043 0.952339i $$-0.401329\pi$$
0.305043 + 0.952339i $$0.401329\pi$$
$$524$$ 0 0
$$525$$ 6125.00 + 10353.1i 0.509175 + 0.860663i
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 3386.56 2003.52i 0.279131 0.165136i
$$529$$ −12167.0 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 21007.3 12428.1i 1.68814 0.998716i
$$538$$ 0 0
$$539$$ 4058.43i 0.324321i
$$540$$ −400.000 12542.1i −0.0318764 0.999492i
$$541$$ −9718.00 −0.772291 −0.386146 0.922438i $$-0.626194\pi$$
−0.386146 + 0.922438i $$0.626194\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 25334.7i 1.99122i
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −4370.78 −0.336102
$$554$$ 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
$$560$$ 13252.0i 1.00000i
$$561$$ −3220.00 5442.79i −0.242332 0.409617i
$$562$$ 0 0
$$563$$ 13908.3i 1.04115i −0.853816 0.520574i $$-0.825718\pi$$
0.853816 0.520574i $$-0.174282\pi$$
$$564$$ 6400.00 3786.29i 0.477817 0.282680i
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 7241.42 + 11395.0i 0.536351 + 0.843995i
$$568$$ 0 0
$$569$$ 25226.2i 1.85859i 0.369343 + 0.929293i $$0.379583\pi$$
−0.369343 + 0.929293i $$0.620417\pi$$
$$570$$ 0 0
$$571$$ −22048.0 −1.61590 −0.807951 0.589250i $$-0.799423\pi$$
−0.807951 + 0.589250i $$0.799423\pi$$
$$572$$ 8014.07i 0.585813i
$$573$$ −14234.1 + 8421.03i −1.03777 + 0.613951i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 6656.00 12116.1i 0.481481 0.876456i
$$577$$ −18848.3 −1.35991 −0.679954 0.733255i $$-0.738000\pi$$
−0.679954 + 0.733255i $$0.738000\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ −27515.8 −1.96988
$$581$$ 27994.9i 1.99901i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −22400.0 12305.4i −1.58312 0.869688i
$$586$$ 0 0
$$587$$ 15035.3i 1.05720i −0.848872 0.528598i $$-0.822718\pi$$
0.848872 0.528598i $$-0.177282\pi$$
$$588$$ −7259.94 12271.5i −0.509175 0.860663i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 1104.62i 0.0764944i 0.999268 + 0.0382472i $$0.0121774\pi$$
−0.999268 + 0.0382472i $$0.987823\pi$$
$$594$$ 0 0
$$595$$ 21298.3 1.46747
$$596$$ 21392.5i 1.47026i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 27273.1i 1.86035i −0.367116 0.930175i $$-0.619655\pi$$
0.367116 0.930175i $$-0.380345\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 22304.0 1.50254
$$605$$ 13315.8i 0.894816i
$$606$$ 0 0
$$607$$ −9085.51 −0.607528 −0.303764 0.952747i $$-0.598243\pi$$
−0.303764 + 0.952747i $$0.598243\pi$$
$$608$$ 0 0
$$609$$ 25480.0 15074.2i 1.69541 1.00302i
$$610$$ 0 0
$$611$$ 15145.2i 1.00280i
$$612$$ −19472.7 10697.3i −1.28617 0.706560i
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ −14336.0 24232.3i −0.919710 1.55459i
$$625$$ 15625.0 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −3894.55 −0.247467
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 14852.0 0.937003 0.468501 0.883463i $$-0.344794\pi$$
0.468501 + 0.883463i $$0.344794\pi$$
$$632$$ 0 0
$$633$$ −6328.64 10697.3i −0.397379 0.671693i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −29039.8 −1.80628
$$638$$ 0 0
$$639$$ 20440.0 + 11228.7i 1.26541 + 0.695151i
$$640$$ 0 0
$$641$$ 15003.2i 0.924477i 0.886756 + 0.462239i $$0.152954\pi$$
−0.886756 + 0.462239i $$0.847046\pi$$
$$642$$ 0 0
$$643$$ 30061.0 1.84369 0.921844 0.387562i $$-0.126683\pi$$
0.921844 + 0.387562i $$0.126683\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28997.3i 1.76198i 0.473133 + 0.880991i $$0.343123\pi$$
−0.473133 + 0.880991i $$0.656877\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 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
$$657$$ 14721.0 26797.0i 0.874154 1.59125i
$$658$$ 0 0
$$659$$ 11489.0i 0.679133i 0.940582 + 0.339567i $$0.110280\pi$$
−0.940582 + 0.339567i $$0.889720\pi$$
$$660$$ 2800.00 + 4732.86i 0.165136 + 0.279131i
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ −38945.5 + 23040.4i −2.28132 + 1.34965i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 20464.5i 1.18532i
$$669$$ −13706.0 23167.4i −0.792085 1.33887i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 17528.1 559.017i 0.999492 0.0318764i
$$676$$ −39768.0 −2.26263
$$677$$ 31577.8i 1.79266i −0.443386 0.896331i $$-0.646223\pi$$
0.443386 0.896331i $$-0.353777\pi$$
$$678$$ 0 0
$$679$$ −17836.0 −1.00807
$$680$$ 0 0
$$681$$ −21320.0 + 12613.1i −1.19968 + 0.709742i
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 35240.4i 1.93590i
$$693$$ −5185.67 2848.75i −0.284253 0.156155i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ −18520.3 −1.00000
$$701$$ 16068.1i 0.865739i 0.901457 + 0.432869i $$0.142499\pi$$
−0.901457 + 0.432869i $$0.857501\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 6058.07i 0.324321i
$$705$$ 5291.50 + 8944.27i 0.282680 + 0.477817i
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2126.00 0.112614 0.0563072 0.998413i $$-0.482067\pi$$
0.0563072 + 0.998413i $$0.482067\pi$$
$$710$$ 0 0
$$711$$ −3068.00 + 5584.78i −0.161827 + 0.294579i
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 11200.0 0.585813
$$716$$ 37578.9i 1.96144i
$$717$$ −20901.4 + 12365.5i −1.08867 + 0.644068i
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 16932.8 + 9302.04i 0.876456 + 0.481481i
$$721$$ 14798.0 0.764364
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 38454.5i 1.96988i
$$726$$ 0 0
$$727$$ 11392.6 0.581194 0.290597 0.956845i $$-0.406146\pi$$
0.290597 + 0.956845i $$0.406146\pi$$
$$728$$ 0 0
$$729$$ 19643.0 1254.21i 0.997968 0.0637204i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 16488.3 0.830846 0.415423 0.909628i $$-0.363634\pi$$
0.415423 + 0.909628i $$0.363634\pi$$
$$734$$ 0 0
$$735$$ 17150.0 10146.1i 0.860663 0.509175i
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −23056.0 −1.14767 −0.573835 0.818971i $$-0.694545\pi$$
−0.573835 + 0.818971i $$0.694545\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −29897.0 −1.47026
$$746$$ 0 0
$$747$$ 35770.6 + 19650.6i 1.75204 + 0.962485i
$$748$$ 9736.36 0.475931
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 27092.0 1.31638 0.658190 0.752852i $$-0.271322\pi$$
0.658190 + 0.752852i $$0.271322\pi$$
$$752$$ 11448.7i 0.555173i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 31170.8i 1.50254i
$$756$$ −20776.0 + 662.601i −0.999492 + 0.0318764i
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 41966.9 1.99122
$$764$$ 25462.8i 1.20577i
$$765$$ 14950.0 27214.0i 0.706560 1.28617i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 10837.0 + 18317.9i 0.509175 + 0.860663i
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ −15020.0 + 8885.95i −0.701598 + 0.415071i
$$772$$ 0 0
$$773$$ 5174.26i 0.240757i −0.992728 0.120379i $$-0.961589\pi$$
0.992728 0.120379i $$-0.0384108\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 33865.6 20035.2i 1.55459 0.919710i
$$781$$ −10220.0 −0.468246
$$782$$ 0 0
$$783$$ −1375.79 43138.2i −0.0627928 1.96888i
$$784$$ 21952.0 1.00000
$$785$$ 5442.79i 0.247467i
$$786$$ 0 0
$$787$$ 15202.5 0.688577 0.344289 0.938864i $$-0.388120\pi$$
0.344289 + 0.938864i $$0.388120\pi$$
$$788$$ 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
$$797$$ 41031.8i 1.82362i 0.410616 + 0.911808i $$0.365314\pi$$
−0.410616 + 0.911808i $$0.634686\pi$$
$$798$$ 0 0
$$799$$ 18400.0 0.814700
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0