# Properties

 Label 5070.2.b.g.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.g.1351.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +1.00000 q^{22} +3.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} -1.00000 q^{29} -1.00000 q^{30} -3.00000i q^{31} +1.00000i q^{32} +1.00000i q^{33} -2.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +5.00000i q^{37} -6.00000 q^{38} -1.00000 q^{40} +10.0000i q^{41} -2.00000 q^{42} -5.00000 q^{43} +1.00000i q^{44} -1.00000i q^{45} +3.00000i q^{46} -3.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +2.00000 q^{51} +14.0000 q^{53} -1.00000i q^{54} -1.00000 q^{55} -2.00000 q^{56} -6.00000i q^{57} -1.00000i q^{58} +5.00000i q^{59} -1.00000i q^{60} -10.0000 q^{61} +3.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -1.00000 q^{66} +2.00000 q^{68} -3.00000 q^{69} -2.00000i q^{70} +4.00000i q^{71} -1.00000i q^{72} +2.00000i q^{73} -5.00000 q^{74} +1.00000 q^{75} -6.00000i q^{76} -2.00000 q^{77} +5.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} -6.00000i q^{83} -2.00000i q^{84} +2.00000i q^{85} -5.00000i q^{86} +1.00000 q^{87} -1.00000 q^{88} -10.0000i q^{89} +1.00000 q^{90} -3.00000 q^{92} +3.00000i q^{93} +3.00000 q^{94} +6.00000 q^{95} -1.00000i q^{96} -10.0000i q^{97} +3.00000i q^{98} -1.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} + 2q^{12} + 4q^{14} + 2q^{16} - 4q^{17} + 2q^{22} + 6q^{23} - 2q^{25} - 2q^{27} - 2q^{29} - 2q^{30} - 4q^{35} - 2q^{36} - 12q^{38} - 2q^{40} - 4q^{42} - 10q^{43} - 2q^{48} + 6q^{49} + 4q^{51} + 28q^{53} - 2q^{55} - 4q^{56} - 20q^{61} + 6q^{62} - 2q^{64} - 2q^{66} + 4q^{68} - 6q^{69} - 10q^{74} + 2q^{75} - 4q^{77} + 10q^{79} + 2q^{81} - 20q^{82} + 2q^{87} - 2q^{88} + 2q^{90} - 6q^{92} + 6q^{94} + 12q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 1.00000i − 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ − 1.00000i − 0.301511i −0.988571 0.150756i $$-0.951829\pi$$
0.988571 0.150756i $$-0.0481707\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 6.00000i 1.37649i 0.725476 + 0.688247i $$0.241620\pi$$
−0.725476 + 0.688247i $$0.758380\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ 2.00000i 0.436436i
$$22$$ 1.00000 0.213201
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000i 0.377964i
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ − 3.00000i − 0.538816i −0.963026 0.269408i $$-0.913172\pi$$
0.963026 0.269408i $$-0.0868280\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 1.00000i 0.174078i
$$34$$ − 2.00000i − 0.342997i
$$35$$ −2.00000 −0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 5.00000i 0.821995i 0.911636 + 0.410997i $$0.134819\pi$$
−0.911636 + 0.410997i $$0.865181\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 10.0000i 1.56174i 0.624695 + 0.780869i $$0.285223\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −5.00000 −0.762493 −0.381246 0.924473i $$-0.624505\pi$$
−0.381246 + 0.924473i $$0.624505\pi$$
$$44$$ 1.00000i 0.150756i
$$45$$ − 1.00000i − 0.149071i
$$46$$ 3.00000i 0.442326i
$$47$$ − 3.00000i − 0.437595i −0.975770 0.218797i $$-0.929787\pi$$
0.975770 0.218797i $$-0.0702134\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 3.00000 0.428571
$$50$$ − 1.00000i − 0.141421i
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 14.0000 1.92305 0.961524 0.274721i $$-0.0885855\pi$$
0.961524 + 0.274721i $$0.0885855\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ −1.00000 −0.134840
$$56$$ −2.00000 −0.267261
$$57$$ − 6.00000i − 0.794719i
$$58$$ − 1.00000i − 0.131306i
$$59$$ 5.00000i 0.650945i 0.945552 + 0.325472i $$0.105523\pi$$
−0.945552 + 0.325472i $$0.894477\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 3.00000 0.381000
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 2.00000 0.242536
$$69$$ −3.00000 −0.361158
$$70$$ − 2.00000i − 0.239046i
$$71$$ 4.00000i 0.474713i 0.971423 + 0.237356i $$0.0762809\pi$$
−0.971423 + 0.237356i $$0.923719\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ −5.00000 −0.581238
$$75$$ 1.00000 0.115470
$$76$$ − 6.00000i − 0.688247i
$$77$$ −2.00000 −0.227921
$$78$$ 0 0
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ −10.0000 −1.10432
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ − 2.00000i − 0.218218i
$$85$$ 2.00000i 0.216930i
$$86$$ − 5.00000i − 0.539164i
$$87$$ 1.00000 0.107211
$$88$$ −1.00000 −0.106600
$$89$$ − 10.0000i − 1.06000i −0.847998 0.529999i $$-0.822192\pi$$
0.847998 0.529999i $$-0.177808\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −3.00000 −0.312772
$$93$$ 3.00000i 0.311086i
$$94$$ 3.00000 0.309426
$$95$$ 6.00000 0.615587
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ − 1.00000i − 0.100504i
$$100$$ 1.00000 0.100000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ 6.00000 0.591198 0.295599 0.955312i $$-0.404481\pi$$
0.295599 + 0.955312i $$0.404481\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 14.0000i 1.35980i
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ − 6.00000i − 0.574696i −0.957826 0.287348i $$-0.907226\pi$$
0.957826 0.287348i $$-0.0927736\pi$$
$$110$$ − 1.00000i − 0.0953463i
$$111$$ − 5.00000i − 0.474579i
$$112$$ − 2.00000i − 0.188982i
$$113$$ 17.0000 1.59923 0.799613 0.600516i $$-0.205038\pi$$
0.799613 + 0.600516i $$0.205038\pi$$
$$114$$ 6.00000 0.561951
$$115$$ − 3.00000i − 0.279751i
$$116$$ 1.00000 0.0928477
$$117$$ 0 0
$$118$$ −5.00000 −0.460287
$$119$$ 4.00000i 0.366679i
$$120$$ 1.00000 0.0912871
$$121$$ 10.0000 0.909091
$$122$$ − 10.0000i − 0.905357i
$$123$$ − 10.0000i − 0.901670i
$$124$$ 3.00000i 0.269408i
$$125$$ 1.00000i 0.0894427i
$$126$$ 2.00000 0.178174
$$127$$ 14.0000 1.24230 0.621150 0.783692i $$-0.286666\pi$$
0.621150 + 0.783692i $$0.286666\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 5.00000 0.440225
$$130$$ 0 0
$$131$$ 13.0000 1.13582 0.567908 0.823092i $$-0.307753\pi$$
0.567908 + 0.823092i $$0.307753\pi$$
$$132$$ − 1.00000i − 0.0870388i
$$133$$ 12.0000 1.04053
$$134$$ 0 0
$$135$$ 1.00000i 0.0860663i
$$136$$ 2.00000i 0.171499i
$$137$$ 9.00000i 0.768922i 0.923141 + 0.384461i $$0.125613\pi$$
−0.923141 + 0.384461i $$0.874387\pi$$
$$138$$ − 3.00000i − 0.255377i
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 3.00000i 0.252646i
$$142$$ −4.00000 −0.335673
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 1.00000i 0.0830455i
$$146$$ −2.00000 −0.165521
$$147$$ −3.00000 −0.247436
$$148$$ − 5.00000i − 0.410997i
$$149$$ 11.0000i 0.901155i 0.892737 + 0.450578i $$0.148782\pi$$
−0.892737 + 0.450578i $$0.851218\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ − 24.0000i − 1.95309i −0.215308 0.976546i $$-0.569076\pi$$
0.215308 0.976546i $$-0.430924\pi$$
$$152$$ 6.00000 0.486664
$$153$$ −2.00000 −0.161690
$$154$$ − 2.00000i − 0.161165i
$$155$$ −3.00000 −0.240966
$$156$$ 0 0
$$157$$ 25.0000 1.99522 0.997609 0.0691164i $$-0.0220180\pi$$
0.997609 + 0.0691164i $$0.0220180\pi$$
$$158$$ 5.00000i 0.397779i
$$159$$ −14.0000 −1.11027
$$160$$ 1.00000 0.0790569
$$161$$ − 6.00000i − 0.472866i
$$162$$ 1.00000i 0.0785674i
$$163$$ − 17.0000i − 1.33154i −0.746156 0.665771i $$-0.768103\pi$$
0.746156 0.665771i $$-0.231897\pi$$
$$164$$ − 10.0000i − 0.780869i
$$165$$ 1.00000 0.0778499
$$166$$ 6.00000 0.465690
$$167$$ 7.00000i 0.541676i 0.962625 + 0.270838i $$0.0873008\pi$$
−0.962625 + 0.270838i $$0.912699\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 0 0
$$170$$ −2.00000 −0.153393
$$171$$ 6.00000i 0.458831i
$$172$$ 5.00000 0.381246
$$173$$ −4.00000 −0.304114 −0.152057 0.988372i $$-0.548590\pi$$
−0.152057 + 0.988372i $$0.548590\pi$$
$$174$$ 1.00000i 0.0758098i
$$175$$ 2.00000i 0.151186i
$$176$$ − 1.00000i − 0.0753778i
$$177$$ − 5.00000i − 0.375823i
$$178$$ 10.0000 0.749532
$$179$$ 7.00000 0.523205 0.261602 0.965176i $$-0.415749\pi$$
0.261602 + 0.965176i $$0.415749\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ − 3.00000i − 0.221163i
$$185$$ 5.00000 0.367607
$$186$$ −3.00000 −0.219971
$$187$$ 2.00000i 0.146254i
$$188$$ 3.00000i 0.218797i
$$189$$ 2.00000i 0.145479i
$$190$$ 6.00000i 0.435286i
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 4.00000i 0.287926i 0.989583 + 0.143963i $$0.0459847\pi$$
−0.989583 + 0.143963i $$0.954015\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ 0 0
$$202$$ − 14.0000i − 0.985037i
$$203$$ 2.00000i 0.140372i
$$204$$ −2.00000 −0.140028
$$205$$ 10.0000 0.698430
$$206$$ 6.00000i 0.418040i
$$207$$ 3.00000 0.208514
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 2.00000i 0.138013i
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ −14.0000 −0.961524
$$213$$ − 4.00000i − 0.274075i
$$214$$ − 6.00000i − 0.410152i
$$215$$ 5.00000i 0.340997i
$$216$$ 1.00000i 0.0680414i
$$217$$ −6.00000 −0.407307
$$218$$ 6.00000 0.406371
$$219$$ − 2.00000i − 0.135147i
$$220$$ 1.00000 0.0674200
$$221$$ 0 0
$$222$$ 5.00000 0.335578
$$223$$ 10.0000i 0.669650i 0.942280 + 0.334825i $$0.108677\pi$$
−0.942280 + 0.334825i $$0.891323\pi$$
$$224$$ 2.00000 0.133631
$$225$$ −1.00000 −0.0666667
$$226$$ 17.0000i 1.13082i
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 6.00000i 0.397360i
$$229$$ − 22.0000i − 1.45380i −0.686743 0.726900i $$-0.740960\pi$$
0.686743 0.726900i $$-0.259040\pi$$
$$230$$ 3.00000 0.197814
$$231$$ 2.00000 0.131590
$$232$$ 1.00000i 0.0656532i
$$233$$ −3.00000 −0.196537 −0.0982683 0.995160i $$-0.531330\pi$$
−0.0982683 + 0.995160i $$0.531330\pi$$
$$234$$ 0 0
$$235$$ −3.00000 −0.195698
$$236$$ − 5.00000i − 0.325472i
$$237$$ −5.00000 −0.324785
$$238$$ −4.00000 −0.259281
$$239$$ − 8.00000i − 0.517477i −0.965947 0.258738i $$-0.916693\pi$$
0.965947 0.258738i $$-0.0833068\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ 7.00000i 0.450910i 0.974254 + 0.225455i $$0.0723868\pi$$
−0.974254 + 0.225455i $$0.927613\pi$$
$$242$$ 10.0000i 0.642824i
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ − 3.00000i − 0.191663i
$$246$$ 10.0000 0.637577
$$247$$ 0 0
$$248$$ −3.00000 −0.190500
$$249$$ 6.00000i 0.380235i
$$250$$ −1.00000 −0.0632456
$$251$$ −3.00000 −0.189358 −0.0946792 0.995508i $$-0.530183\pi$$
−0.0946792 + 0.995508i $$0.530183\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ − 3.00000i − 0.188608i
$$254$$ 14.0000i 0.878438i
$$255$$ − 2.00000i − 0.125245i
$$256$$ 1.00000 0.0625000
$$257$$ −21.0000 −1.30994 −0.654972 0.755653i $$-0.727320\pi$$
−0.654972 + 0.755653i $$0.727320\pi$$
$$258$$ 5.00000i 0.311286i
$$259$$ 10.0000 0.621370
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ 13.0000i 0.803143i
$$263$$ 23.0000 1.41824 0.709120 0.705087i $$-0.249092\pi$$
0.709120 + 0.705087i $$0.249092\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ − 14.0000i − 0.860013i
$$266$$ 12.0000i 0.735767i
$$267$$ 10.0000i 0.611990i
$$268$$ 0 0
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 29.0000i 1.76162i 0.473466 + 0.880812i $$0.343003\pi$$
−0.473466 + 0.880812i $$0.656997\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ −9.00000 −0.543710
$$275$$ 1.00000i 0.0603023i
$$276$$ 3.00000 0.180579
$$277$$ 19.0000 1.14160 0.570800 0.821089i $$-0.306633\pi$$
0.570800 + 0.821089i $$0.306633\pi$$
$$278$$ 10.0000i 0.599760i
$$279$$ − 3.00000i − 0.179605i
$$280$$ 2.00000i 0.119523i
$$281$$ 30.0000i 1.78965i 0.446417 + 0.894825i $$0.352700\pi$$
−0.446417 + 0.894825i $$0.647300\pi$$
$$282$$ −3.00000 −0.178647
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ − 4.00000i − 0.237356i
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ 20.0000 1.18056
$$288$$ 1.00000i 0.0589256i
$$289$$ −13.0000 −0.764706
$$290$$ −1.00000 −0.0587220
$$291$$ 10.0000i 0.586210i
$$292$$ − 2.00000i − 0.117041i
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ − 3.00000i − 0.174964i
$$295$$ 5.00000 0.291111
$$296$$ 5.00000 0.290619
$$297$$ 1.00000i 0.0580259i
$$298$$ −11.0000 −0.637213
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 10.0000i 0.576390i
$$302$$ 24.0000 1.38104
$$303$$ 14.0000 0.804279
$$304$$ 6.00000i 0.344124i
$$305$$ 10.0000i 0.572598i
$$306$$ − 2.00000i − 0.114332i
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 2.00000 0.113961
$$309$$ −6.00000 −0.341328
$$310$$ − 3.00000i − 0.170389i
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −12.0000 −0.678280 −0.339140 0.940736i $$-0.610136\pi$$
−0.339140 + 0.940736i $$0.610136\pi$$
$$314$$ 25.0000i 1.41083i
$$315$$ −2.00000 −0.112687
$$316$$ −5.00000 −0.281272
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ − 14.0000i − 0.785081i
$$319$$ 1.00000i 0.0559893i
$$320$$ 1.00000i 0.0559017i
$$321$$ 6.00000 0.334887
$$322$$ 6.00000 0.334367
$$323$$ − 12.0000i − 0.667698i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 17.0000 0.941543
$$327$$ 6.00000i 0.331801i
$$328$$ 10.0000 0.552158
$$329$$ −6.00000 −0.330791
$$330$$ 1.00000i 0.0550482i
$$331$$ 4.00000i 0.219860i 0.993939 + 0.109930i $$0.0350627\pi$$
−0.993939 + 0.109930i $$0.964937\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ 5.00000i 0.273998i
$$334$$ −7.00000 −0.383023
$$335$$ 0 0
$$336$$ 2.00000i 0.109109i
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 0 0
$$339$$ −17.0000 −0.923313
$$340$$ − 2.00000i − 0.108465i
$$341$$ −3.00000 −0.162459
$$342$$ −6.00000 −0.324443
$$343$$ − 20.0000i − 1.07990i
$$344$$ 5.00000i 0.269582i
$$345$$ 3.00000i 0.161515i
$$346$$ − 4.00000i − 0.215041i
$$347$$ −18.0000 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$348$$ −1.00000 −0.0536056
$$349$$ − 8.00000i − 0.428230i −0.976808 0.214115i $$-0.931313\pi$$
0.976808 0.214115i $$-0.0686868\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ − 14.0000i − 0.745145i −0.928003 0.372572i $$-0.878476\pi$$
0.928003 0.372572i $$-0.121524\pi$$
$$354$$ 5.00000 0.265747
$$355$$ 4.00000 0.212298
$$356$$ 10.0000i 0.529999i
$$357$$ − 4.00000i − 0.211702i
$$358$$ 7.00000i 0.369961i
$$359$$ − 12.0000i − 0.633336i −0.948536 0.316668i $$-0.897436\pi$$
0.948536 0.316668i $$-0.102564\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −17.0000 −0.894737
$$362$$ − 16.0000i − 0.840941i
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 10.0000i 0.522708i
$$367$$ 36.0000 1.87918 0.939592 0.342296i $$-0.111204\pi$$
0.939592 + 0.342296i $$0.111204\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 10.0000i 0.520579i
$$370$$ 5.00000i 0.259938i
$$371$$ − 28.0000i − 1.45369i
$$372$$ − 3.00000i − 0.155543i
$$373$$ 37.0000 1.91579 0.957894 0.287123i $$-0.0926989\pi$$
0.957894 + 0.287123i $$0.0926989\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ − 1.00000i − 0.0516398i
$$376$$ −3.00000 −0.154713
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ − 30.0000i − 1.54100i −0.637442 0.770498i $$-0.720007\pi$$
0.637442 0.770498i $$-0.279993\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ −14.0000 −0.717242
$$382$$ 24.0000i 1.22795i
$$383$$ − 27.0000i − 1.37964i −0.723983 0.689818i $$-0.757691\pi$$
0.723983 0.689818i $$-0.242309\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 2.00000i 0.101929i
$$386$$ −4.00000 −0.203595
$$387$$ −5.00000 −0.254164
$$388$$ 10.0000i 0.507673i
$$389$$ 1.00000 0.0507020 0.0253510 0.999679i $$-0.491930\pi$$
0.0253510 + 0.999679i $$0.491930\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ − 3.00000i − 0.151523i
$$393$$ −13.0000 −0.655763
$$394$$ 12.0000 0.604551
$$395$$ − 5.00000i − 0.251577i
$$396$$ 1.00000i 0.0502519i
$$397$$ 13.0000i 0.652451i 0.945292 + 0.326226i $$0.105777\pi$$
−0.945292 + 0.326226i $$0.894223\pi$$
$$398$$ 0 0
$$399$$ −12.0000 −0.600751
$$400$$ −1.00000 −0.0500000
$$401$$ 12.0000i 0.599251i 0.954057 + 0.299626i $$0.0968618\pi$$
−0.954057 + 0.299626i $$0.903138\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ − 1.00000i − 0.0496904i
$$406$$ −2.00000 −0.0992583
$$407$$ 5.00000 0.247841
$$408$$ − 2.00000i − 0.0990148i
$$409$$ − 26.0000i − 1.28562i −0.766027 0.642809i $$-0.777769\pi$$
0.766027 0.642809i $$-0.222231\pi$$
$$410$$ 10.0000i 0.493865i
$$411$$ − 9.00000i − 0.443937i
$$412$$ −6.00000 −0.295599
$$413$$ 10.0000 0.492068
$$414$$ 3.00000i 0.147442i
$$415$$ −6.00000 −0.294528
$$416$$ 0 0
$$417$$ −10.0000 −0.489702
$$418$$ 6.00000i 0.293470i
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 28.0000i 1.36464i 0.731055 + 0.682318i $$0.239028\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ − 8.00000i − 0.389434i
$$423$$ − 3.00000i − 0.145865i
$$424$$ − 14.0000i − 0.679900i
$$425$$ 2.00000 0.0970143
$$426$$ 4.00000 0.193801
$$427$$ 20.0000i 0.967868i
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ −5.00000 −0.241121
$$431$$ 12.0000i 0.578020i 0.957326 + 0.289010i $$0.0933260\pi$$
−0.957326 + 0.289010i $$0.906674\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ − 6.00000i − 0.288009i
$$435$$ − 1.00000i − 0.0479463i
$$436$$ 6.00000i 0.287348i
$$437$$ 18.0000i 0.861057i
$$438$$ 2.00000 0.0955637
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 1.00000i 0.0476731i
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ 16.0000 0.760183 0.380091 0.924949i $$-0.375893\pi$$
0.380091 + 0.924949i $$0.375893\pi$$
$$444$$ 5.00000i 0.237289i
$$445$$ −10.0000 −0.474045
$$446$$ −10.0000 −0.473514
$$447$$ − 11.0000i − 0.520282i
$$448$$ 2.00000i 0.0944911i
$$449$$ 36.0000i 1.69895i 0.527633 + 0.849473i $$0.323080\pi$$
−0.527633 + 0.849473i $$0.676920\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ 10.0000 0.470882
$$452$$ −17.0000 −0.799613
$$453$$ 24.0000i 1.12762i
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ −6.00000 −0.280976
$$457$$ 14.0000i 0.654892i 0.944870 + 0.327446i $$0.106188\pi$$
−0.944870 + 0.327446i $$0.893812\pi$$
$$458$$ 22.0000 1.02799
$$459$$ 2.00000 0.0933520
$$460$$ 3.00000i 0.139876i
$$461$$ 27.0000i 1.25752i 0.777601 + 0.628758i $$0.216436\pi$$
−0.777601 + 0.628758i $$0.783564\pi$$
$$462$$ 2.00000i 0.0930484i
$$463$$ − 10.0000i − 0.464739i −0.972628 0.232370i $$-0.925352\pi$$
0.972628 0.232370i $$-0.0746479\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 3.00000 0.139122
$$466$$ − 3.00000i − 0.138972i
$$467$$ 2.00000 0.0925490 0.0462745 0.998929i $$-0.485265\pi$$
0.0462745 + 0.998929i $$0.485265\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ − 3.00000i − 0.138380i
$$471$$ −25.0000 −1.15194
$$472$$ 5.00000 0.230144
$$473$$ 5.00000i 0.229900i
$$474$$ − 5.00000i − 0.229658i
$$475$$ − 6.00000i − 0.275299i
$$476$$ − 4.00000i − 0.183340i
$$477$$ 14.0000 0.641016
$$478$$ 8.00000 0.365911
$$479$$ 4.00000i 0.182765i 0.995816 + 0.0913823i $$0.0291285\pi$$
−0.995816 + 0.0913823i $$0.970871\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ −7.00000 −0.318841
$$483$$ 6.00000i 0.273009i
$$484$$ −10.0000 −0.454545
$$485$$ −10.0000 −0.454077
$$486$$ − 1.00000i − 0.0453609i
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ 17.0000i 0.768767i
$$490$$ 3.00000 0.135526
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ 10.0000i 0.450835i
$$493$$ 2.00000 0.0900755
$$494$$ 0 0
$$495$$ −1.00000 −0.0449467
$$496$$ − 3.00000i − 0.134704i
$$497$$ 8.00000 0.358849
$$498$$ −6.00000 −0.268866
$$499$$ − 40.0000i − 1.79065i −0.445418 0.895323i $$-0.646945\pi$$
0.445418 0.895323i $$-0.353055\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ − 7.00000i − 0.312737i
$$502$$ − 3.00000i − 0.133897i
$$503$$ 40.0000 1.78351 0.891756 0.452517i $$-0.149474\pi$$
0.891756 + 0.452517i $$0.149474\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 14.0000i 0.622992i
$$506$$ 3.00000 0.133366
$$507$$ 0 0
$$508$$ −14.0000 −0.621150
$$509$$ − 9.00000i − 0.398918i −0.979906 0.199459i $$-0.936082\pi$$
0.979906 0.199459i $$-0.0639185\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ 4.00000 0.176950
$$512$$ 1.00000i 0.0441942i
$$513$$ − 6.00000i − 0.264906i
$$514$$ − 21.0000i − 0.926270i
$$515$$ − 6.00000i − 0.264392i
$$516$$ −5.00000 −0.220113
$$517$$ −3.00000 −0.131940
$$518$$ 10.0000i 0.439375i
$$519$$ 4.00000 0.175581
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ − 1.00000i − 0.0437688i
$$523$$ 33.0000 1.44299 0.721495 0.692420i $$-0.243455\pi$$
0.721495 + 0.692420i $$0.243455\pi$$
$$524$$ −13.0000 −0.567908
$$525$$ − 2.00000i − 0.0872872i
$$526$$ 23.0000i 1.00285i
$$527$$ 6.00000i 0.261364i
$$528$$ 1.00000i 0.0435194i
$$529$$ −14.0000 −0.608696
$$530$$ 14.0000 0.608121
$$531$$ 5.00000i 0.216982i
$$532$$ −12.0000 −0.520266
$$533$$ 0 0
$$534$$ −10.0000 −0.432742
$$535$$ 6.00000i 0.259403i
$$536$$ 0 0
$$537$$ −7.00000 −0.302072
$$538$$ 18.0000i 0.776035i
$$539$$ − 3.00000i − 0.129219i
$$540$$ − 1.00000i − 0.0430331i
$$541$$ 8.00000i 0.343947i 0.985102 + 0.171973i $$0.0550143\pi$$
−0.985102 + 0.171973i $$0.944986\pi$$
$$542$$ −29.0000 −1.24566
$$543$$ 16.0000 0.686626
$$544$$ − 2.00000i − 0.0857493i
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ − 9.00000i − 0.384461i
$$549$$ −10.0000 −0.426790
$$550$$ −1.00000 −0.0426401
$$551$$ − 6.00000i − 0.255609i
$$552$$ 3.00000i 0.127688i
$$553$$ − 10.0000i − 0.425243i
$$554$$ 19.0000i 0.807233i
$$555$$ −5.00000 −0.212238
$$556$$ −10.0000 −0.424094
$$557$$ 34.0000i 1.44063i 0.693649 + 0.720313i $$0.256002\pi$$
−0.693649 + 0.720313i $$0.743998\pi$$
$$558$$ 3.00000 0.127000
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ − 2.00000i − 0.0844401i
$$562$$ −30.0000 −1.26547
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ − 3.00000i − 0.126323i
$$565$$ − 17.0000i − 0.715195i
$$566$$ 13.0000i 0.546431i
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 4.00000 0.167836
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ − 6.00000i − 0.251312i
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 20.0000i 0.834784i
$$575$$ −3.00000 −0.125109
$$576$$ −1.00000 −0.0416667
$$577$$ − 18.0000i − 0.749350i −0.927156 0.374675i $$-0.877754\pi$$
0.927156 0.374675i $$-0.122246\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ − 4.00000i − 0.166234i
$$580$$ − 1.00000i − 0.0415227i
$$581$$ −12.0000 −0.497844
$$582$$ −10.0000 −0.414513
$$583$$ − 14.0000i − 0.579821i
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ − 18.0000i − 0.742940i −0.928445 0.371470i $$-0.878854\pi$$
0.928445 0.371470i $$-0.121146\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 18.0000 0.741677
$$590$$ 5.00000i 0.205847i
$$591$$ 12.0000i 0.493614i
$$592$$ 5.00000i 0.205499i
$$593$$ 35.0000i 1.43728i 0.695383 + 0.718639i $$0.255235\pi$$
−0.695383 + 0.718639i $$0.744765\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 4.00000 0.163984
$$596$$ − 11.0000i − 0.450578i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ −11.0000 −0.448699 −0.224350 0.974509i $$-0.572026\pi$$
−0.224350 + 0.974509i $$0.572026\pi$$
$$602$$ −10.0000 −0.407570
$$603$$ 0 0
$$604$$ 24.0000i 0.976546i
$$605$$ − 10.0000i − 0.406558i
$$606$$ 14.0000i 0.568711i
$$607$$ −10.0000 −0.405887 −0.202944 0.979190i $$-0.565051\pi$$
−0.202944 + 0.979190i $$0.565051\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ − 2.00000i − 0.0810441i
$$610$$ −10.0000 −0.404888
$$611$$ 0 0
$$612$$ 2.00000 0.0808452
$$613$$ − 37.0000i − 1.49442i −0.664590 0.747208i $$-0.731394\pi$$
0.664590 0.747208i $$-0.268606\pi$$
$$614$$ 0 0
$$615$$ −10.0000 −0.403239
$$616$$ 2.00000i 0.0805823i
$$617$$ 3.00000i 0.120775i 0.998175 + 0.0603877i $$0.0192337\pi$$
−0.998175 + 0.0603877i $$0.980766\pi$$
$$618$$ − 6.00000i − 0.241355i
$$619$$ 10.0000i 0.401934i 0.979598 + 0.200967i $$0.0644084\pi$$
−0.979598 + 0.200967i $$0.935592\pi$$
$$620$$ 3.00000 0.120483
$$621$$ −3.00000 −0.120386
$$622$$ 0 0
$$623$$ −20.0000 −0.801283
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 12.0000i − 0.479616i
$$627$$ −6.00000 −0.239617
$$628$$ −25.0000 −0.997609
$$629$$ − 10.0000i − 0.398726i
$$630$$ − 2.00000i − 0.0796819i
$$631$$ − 48.0000i − 1.91085i −0.295234 0.955425i $$-0.595398\pi$$
0.295234 0.955425i $$-0.404602\pi$$
$$632$$ − 5.00000i − 0.198889i
$$633$$ 8.00000 0.317971
$$634$$ −12.0000 −0.476581
$$635$$ − 14.0000i − 0.555573i
$$636$$ 14.0000 0.555136
$$637$$ 0 0
$$638$$ −1.00000 −0.0395904
$$639$$ 4.00000i 0.158238i
$$640$$ −1.00000 −0.0395285
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 6.00000i 0.236801i
$$643$$ − 40.0000i − 1.57745i −0.614749 0.788723i $$-0.710743\pi$$
0.614749 0.788723i $$-0.289257\pi$$
$$644$$ 6.00000i 0.236433i
$$645$$ − 5.00000i − 0.196875i
$$646$$ 12.0000 0.472134
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 5.00000 0.196267
$$650$$ 0 0
$$651$$ 6.00000 0.235159
$$652$$ 17.0000i 0.665771i
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ − 13.0000i − 0.507952i
$$656$$ 10.0000i 0.390434i
$$657$$ 2.00000i 0.0780274i
$$658$$ − 6.00000i − 0.233904i
$$659$$ −13.0000 −0.506408 −0.253204 0.967413i $$-0.581484\pi$$
−0.253204 + 0.967413i $$0.581484\pi$$
$$660$$ −1.00000 −0.0389249
$$661$$ 12.0000i 0.466746i 0.972387 + 0.233373i $$0.0749763\pi$$
−0.972387 + 0.233373i $$0.925024\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ − 12.0000i − 0.465340i
$$666$$ −5.00000 −0.193746
$$667$$ −3.00000 −0.116160
$$668$$ − 7.00000i − 0.270838i
$$669$$ − 10.0000i − 0.386622i
$$670$$ 0 0
$$671$$ 10.0000i 0.386046i
$$672$$ −2.00000 −0.0771517
$$673$$ 16.0000 0.616755 0.308377 0.951264i $$-0.400214\pi$$
0.308377 + 0.951264i $$0.400214\pi$$
$$674$$ 22.0000i 0.847408i
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ − 17.0000i − 0.652881i
$$679$$ −20.0000 −0.767530
$$680$$ 2.00000 0.0766965
$$681$$ − 20.0000i − 0.766402i
$$682$$ − 3.00000i − 0.114876i
$$683$$ 44.0000i 1.68361i 0.539779 + 0.841807i $$0.318508\pi$$
−0.539779 + 0.841807i $$0.681492\pi$$
$$684$$ − 6.00000i − 0.229416i
$$685$$ 9.00000 0.343872
$$686$$ 20.0000 0.763604
$$687$$ 22.0000i 0.839352i
$$688$$ −5.00000 −0.190623
$$689$$ 0 0
$$690$$ −3.00000 −0.114208
$$691$$ − 14.0000i − 0.532585i −0.963892 0.266293i $$-0.914201\pi$$
0.963892 0.266293i $$-0.0857987\pi$$
$$692$$ 4.00000 0.152057
$$693$$ −2.00000 −0.0759737
$$694$$ − 18.0000i − 0.683271i
$$695$$ − 10.0000i − 0.379322i
$$696$$ − 1.00000i − 0.0379049i
$$697$$ − 20.0000i − 0.757554i
$$698$$ 8.00000 0.302804
$$699$$ 3.00000 0.113470
$$700$$ − 2.00000i − 0.0755929i
$$701$$ −23.0000 −0.868698 −0.434349 0.900745i $$-0.643022\pi$$
−0.434349 + 0.900745i $$0.643022\pi$$
$$702$$ 0 0
$$703$$ −30.0000 −1.13147
$$704$$ 1.00000i 0.0376889i
$$705$$ 3.00000 0.112987
$$706$$ 14.0000 0.526897
$$707$$ 28.0000i 1.05305i
$$708$$ 5.00000i 0.187912i
$$709$$ 16.0000i 0.600893i 0.953799 + 0.300446i $$0.0971356\pi$$
−0.953799 + 0.300446i $$0.902864\pi$$
$$710$$ 4.00000i 0.150117i
$$711$$ 5.00000 0.187515
$$712$$ −10.0000 −0.374766
$$713$$ − 9.00000i − 0.337053i
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ −7.00000 −0.261602
$$717$$ 8.00000i 0.298765i
$$718$$ 12.0000 0.447836
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ − 12.0000i − 0.446903i
$$722$$ − 17.0000i − 0.632674i
$$723$$ − 7.00000i − 0.260333i
$$724$$ 16.0000 0.594635
$$725$$ 1.00000 0.0371391
$$726$$ − 10.0000i − 0.371135i
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000i 0.0740233i
$$731$$ 10.0000 0.369863
$$732$$ −10.0000 −0.369611
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 36.0000i 1.32878i
$$735$$ 3.00000i 0.110657i
$$736$$ 3.00000i 0.110581i
$$737$$ 0 0
$$738$$ −10.0000 −0.368105
$$739$$ − 24.0000i − 0.882854i −0.897297 0.441427i $$-0.854472\pi$$
0.897297 0.441427i $$-0.145528\pi$$
$$740$$ −5.00000 −0.183804
$$741$$ 0 0
$$742$$ 28.0000 1.02791
$$743$$ − 15.0000i − 0.550297i −0.961402 0.275148i $$-0.911273\pi$$
0.961402 0.275148i $$-0.0887270\pi$$
$$744$$ 3.00000 0.109985
$$745$$ 11.0000 0.403009
$$746$$ 37.0000i 1.35467i
$$747$$ − 6.00000i − 0.219529i
$$748$$ − 2.00000i − 0.0731272i
$$749$$ 12.0000i 0.438470i
$$750$$ 1.00000 0.0365148
$$751$$ −41.0000 −1.49611 −0.748056 0.663636i $$-0.769012\pi$$
−0.748056 + 0.663636i $$0.769012\pi$$
$$752$$ − 3.00000i − 0.109399i
$$753$$ 3.00000 0.109326
$$754$$ 0 0
$$755$$ −24.0000 −0.873449
$$756$$ − 2.00000i − 0.0727393i
$$757$$ −54.0000 −1.96266 −0.981332 0.192323i $$-0.938398\pi$$
−0.981332 + 0.192323i $$0.938398\pi$$
$$758$$ 30.0000 1.08965
$$759$$ 3.00000i 0.108893i
$$760$$ − 6.00000i − 0.217643i
$$761$$ 20.0000i 0.724999i 0.931984 + 0.362500i $$0.118077\pi$$
−0.931984 + 0.362500i $$0.881923\pi$$
$$762$$ − 14.0000i − 0.507166i
$$763$$ −12.0000 −0.434429
$$764$$ −24.0000 −0.868290
$$765$$ 2.00000i 0.0723102i
$$766$$ 27.0000 0.975550
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ − 29.0000i − 1.04577i −0.852404 0.522883i $$-0.824856\pi$$
0.852404 0.522883i $$-0.175144\pi$$
$$770$$ −2.00000 −0.0720750
$$771$$ 21.0000 0.756297
$$772$$ − 4.00000i − 0.143963i
$$773$$ − 16.0000i − 0.575480i −0.957709 0.287740i $$-0.907096\pi$$
0.957709 0.287740i $$-0.0929039\pi$$
$$774$$ − 5.00000i − 0.179721i
$$775$$ 3.00000i 0.107763i
$$776$$ −10.0000 −0.358979
$$777$$ −10.0000 −0.358748
$$778$$ 1.00000i 0.0358517i
$$779$$ −60.0000 −2.14972
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ − 6.00000i − 0.214560i
$$783$$ 1.00000 0.0357371
$$784$$ 3.00000 0.107143
$$785$$ − 25.0000i − 0.892288i
$$786$$ − 13.0000i − 0.463695i
$$787$$ − 25.0000i − 0.891154i −0.895244 0.445577i $$-0.852999\pi$$
0.895244 0.445577i $$-0.147001\pi$$
$$788$$ 12.0000i 0.427482i
$$789$$ −23.0000 −0.818822
$$790$$ 5.00000 0.177892
$$791$$ − 34.0000i − 1.20890i
$$792$$ −1.00000 −0.0355335
$$793$$ 0 0
$$794$$ −13.0000 −0.461353
$$795$$ 14.0000i 0.496529i
$$796$$ 0 0
$$797$$ 52.0000 1.84193 0.920967 0.389640i $$-0.127401\pi$$
0.920967 + 0.389640i $$0.127401\pi$$
$$798$$ − 12.0000i − 0.424795i
$$799$$ 6.00000i 0.212265i
$$800$$ − 1.00000i − 0.0353553i
$$801$$ − 10.0000i − 0.353333i
$$802$$ −12.0000 −0.423735
$$803$$ 2.00000 0.0705785
$$804$$ 0 0
$$805$$ −6.00000 −0.211472
$$806$$ 0 0
$$807$$ −18.0000 −0.633630
$$808$$ 14.0000i 0.492518i
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 52.0000i 1.82597i 0.407997 + 0.912983i $$0.366228\pi$$
−0.407997 + 0.912983i $$0.633772\pi$$
$$812$$ − 2.00000i − 0.0701862i
$$813$$ − 29.0000i − 1.01707i
$$814$$ 5.00000i 0.175250i
$$815$$ −17.0000 −0.595484
$$816$$ 2.00000 0.0700140
$$817$$ − 30.0000i − 1.04957i
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ −10.0000 −0.349215
$$821$$ 23.0000i 0.802706i 0.915924 + 0.401353i $$0.131460\pi$$
−0.915924 + 0.401353i $$0.868540\pi$$
$$822$$ 9.00000 0.313911
$$823$$ 22.0000 0.766872 0.383436 0.923567i $$-0.374741\pi$$
0.383436 + 0.923567i $$0.374741\pi$$
$$824$$ − 6.00000i − 0.209020i
$$825$$ − 1.00000i − 0.0348155i
$$826$$ 10.0000i 0.347945i
$$827$$ 18.0000i 0.625921i 0.949766 + 0.312961i $$0.101321\pi$$
−0.949766 + 0.312961i $$0.898679\pi$$
$$828$$ −3.00000 −0.104257
$$829$$ −26.0000 −0.903017 −0.451509 0.892267i $$-0.649114\pi$$
−0.451509 + 0.892267i $$0.649114\pi$$
$$830$$ − 6.00000i − 0.208263i
$$831$$ −19.0000 −0.659103
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ − 10.0000i − 0.346272i
$$835$$ 7.00000 0.242245
$$836$$ −6.00000 −0.207514
$$837$$ 3.00000i 0.103695i
$$838$$ 4.00000i 0.138178i
$$839$$ − 38.0000i − 1.31191i −0.754802 0.655953i $$-0.772267\pi$$
0.754802 0.655953i $$-0.227733\pi$$
$$840$$ − 2.00000i − 0.0690066i
$$841$$ −28.0000 −0.965517
$$842$$ −28.0000 −0.964944
$$843$$ − 30.0000i − 1.03325i
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ − 20.0000i − 0.687208i
$$848$$ 14.0000 0.480762
$$849$$ −13.0000 −0.446159
$$850$$ 2.00000i 0.0685994i
$$851$$ 15.0000i 0.514193i
$$852$$ 4.00000i 0.137038i
$$853$$ − 7.00000i − 0.239675i −0.992793 0.119838i $$-0.961763\pi$$
0.992793 0.119838i $$-0.0382374\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ 6.00000 0.205196
$$856$$ 6.00000i 0.205076i
$$857$$ 35.0000 1.19558 0.597789 0.801654i $$-0.296046\pi$$
0.597789 + 0.801654i $$0.296046\pi$$
$$858$$ 0 0
$$859$$ 18.0000 0.614152 0.307076 0.951685i $$-0.400649\pi$$
0.307076 + 0.951685i $$0.400649\pi$$
$$860$$ − 5.00000i − 0.170499i
$$861$$ −20.0000 −0.681598
$$862$$ −12.0000 −0.408722
$$863$$ − 51.0000i − 1.73606i −0.496512 0.868030i $$-0.665386\pi$$
0.496512 0.868030i $$-0.334614\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 4.00000i 0.136004i
$$866$$ − 16.0000i − 0.543702i
$$867$$ 13.0000 0.441503
$$868$$ 6.00000 0.203653
$$869$$ − 5.00000i − 0.169613i
$$870$$ 1.00000 0.0339032
$$871$$ 0 0
$$872$$ −6.00000 −0.203186
$$873$$ − 10.0000i − 0.338449i
$$874$$ −18.0000 −0.608859
$$875$$ 2.00000 0.0676123
$$876$$ 2.00000i 0.0675737i
$$877$$ 13.0000i 0.438979i 0.975615 + 0.219489i $$0.0704391\pi$$
−0.975615 + 0.219489i $$0.929561\pi$$
$$878$$ − 28.0000i − 0.944954i
$$879$$ − 14.0000i − 0.472208i
$$880$$ −1.00000 −0.0337100
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ 1.00000 0.0336527 0.0168263 0.999858i $$-0.494644\pi$$
0.0168263 + 0.999858i $$0.494644\pi$$
$$884$$ 0 0
$$885$$ −5.00000 −0.168073
$$886$$ 16.0000i 0.537531i
$$887$$ 1.00000 0.0335767 0.0167884 0.999859i $$-0.494656\pi$$
0.0167884 + 0.999859i $$0.494656\pi$$
$$888$$ −5.00000 −0.167789
$$889$$ − 28.0000i − 0.939090i
$$890$$ − 10.0000i − 0.335201i
$$891$$ − 1.00000i − 0.0335013i
$$892$$ − 10.0000i − 0.334825i
$$893$$ 18.0000 0.602347
$$894$$ 11.0000 0.367895
$$895$$ − 7.00000i − 0.233984i
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ −36.0000 −1.20134
$$899$$ 3.00000i 0.100056i
$$900$$ 1.00000 0.0333333
$$901$$ −28.0000 −0.932815
$$902$$ 10.0000i 0.332964i
$$903$$ − 10.0000i − 0.332779i
$$904$$ − 17.0000i − 0.565412i
$$905$$ 16.0000i 0.531858i
$$906$$ −24.0000 −0.797347
$$907$$ 53.0000 1.75984 0.879918 0.475125i $$-0.157597\pi$$
0.879918 + 0.475125i $$0.157597\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ 34.0000 1.12647 0.563235 0.826297i $$-0.309557\pi$$
0.563235 + 0.826297i $$0.309557\pi$$
$$912$$ − 6.00000i − 0.198680i
$$913$$ −6.00000 −0.198571
$$914$$ −14.0000 −0.463079
$$915$$ − 10.0000i − 0.330590i
$$916$$ 22.0000i 0.726900i
$$917$$ − 26.0000i − 0.858596i
$$918$$ 2.00000i 0.0660098i
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ −3.00000 −0.0989071
$$921$$ 0 0
$$922$$ −27.0000 −0.889198
$$923$$ 0 0
$$924$$ −2.00000 −0.0657952
$$925$$ − 5.00000i − 0.164399i
$$926$$ 10.0000 0.328620
$$927$$ 6.00000 0.197066
$$928$$ − 1.00000i − 0.0328266i
$$929$$ 24.0000i 0.787414i 0.919236 + 0.393707i $$0.128808\pi$$
−0.919236 + 0.393707i $$0.871192\pi$$
$$930$$ 3.00000i 0.0983739i
$$931$$ 18.0000i 0.589926i
$$932$$ 3.00000 0.0982683
$$933$$ 0 0
$$934$$ 2.00000i 0.0654420i
$$935$$ 2.00000 0.0654070
$$936$$ 0 0
$$937$$ −18.0000 −0.588034 −0.294017 0.955800i $$-0.594992\pi$$
−0.294017 + 0.955800i $$0.594992\pi$$
$$938$$ 0 0
$$939$$ 12.0000 0.391605
$$940$$ 3.00000 0.0978492
$$941$$ 26.0000i 0.847576i 0.905761 + 0.423788i $$0.139300\pi$$
−0.905761 + 0.423788i $$0.860700\pi$$
$$942$$ − 25.0000i − 0.814544i
$$943$$ 30.0000i 0.976934i
$$944$$ 5.00000i 0.162736i
$$945$$ 2.00000 0.0650600
$$946$$ −5.00000 −0.162564
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 5.00000 0.162392
$$949$$ 0 0
$$950$$ 6.00000 0.194666
$$951$$ − 12.0000i − 0.389127i
$$952$$ 4.00000 0.129641
$$953$$ −15.0000 −0.485898 −0.242949 0.970039i $$-0.578115\pi$$
−0.242949 + 0.970039i $$0.578115\pi$$
$$954$$ 14.0000i 0.453267i
$$955$$ − 24.0000i − 0.776622i
$$956$$ 8.00000i 0.258738i
$$957$$ − 1.00000i − 0.0323254i
$$958$$ −4.00000 −0.129234
$$959$$ 18.0000 0.581250
$$960$$ − 1.00000i − 0.0322749i
$$961$$ 22.0000 0.709677
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ − 7.00000i − 0.225455i
$$965$$ 4.00000 0.128765
$$966$$ −6.00000 −0.193047
$$967$$ 16.0000i 0.514525i 0.966342 + 0.257263i $$0.0828206\pi$$
−0.966342 + 0.257263i $$0.917179\pi$$
$$968$$ − 10.0000i − 0.321412i
$$969$$ 12.0000i 0.385496i
$$970$$ − 10.0000i − 0.321081i
$$971$$ −40.0000 −1.28366 −0.641831 0.766846i $$-0.721825\pi$$
−0.641831 + 0.766846i $$0.721825\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ − 20.0000i − 0.641171i
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ − 39.0000i − 1.24772i −0.781536 0.623860i $$-0.785563\pi$$
0.781536 0.623860i $$-0.214437\pi$$
$$978$$ −17.0000 −0.543600
$$979$$ −10.0000 −0.319601
$$980$$ 3.00000i 0.0958315i
$$981$$ − 6.00000i − 0.191565i
$$982$$ − 24.0000i − 0.765871i
$$983$$ 25.0000i 0.797376i 0.917087 + 0.398688i $$0.130534\pi$$
−0.917087 + 0.398688i $$0.869466\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ −12.0000 −0.382352
$$986$$ 2.00000i 0.0636930i
$$987$$ 6.00000 0.190982
$$988$$ 0 0
$$989$$ −15.0000 −0.476972
$$990$$ − 1.00000i − 0.0317821i
$$991$$ 39.0000 1.23888 0.619438 0.785046i $$-0.287361\pi$$
0.619438 + 0.785046i $$0.287361\pi$$
$$992$$ 3.00000 0.0952501
$$993$$ − 4.00000i − 0.126936i
$$994$$ 8.00000i 0.253745i
$$995$$ 0 0
$$996$$ − 6.00000i − 0.190117i
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ 40.0000 1.26618
$$999$$ − 5.00000i − 0.158193i
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 5070.2.b.g.1351.2 2
13.2 odd 12 390.2.i.a.61.1 2
13.5 odd 4 5070.2.a.o.1.1 1
13.6 odd 12 390.2.i.a.211.1 yes 2
13.8 odd 4 5070.2.a.f.1.1 1
13.12 even 2 inner 5070.2.b.g.1351.1 2
39.2 even 12 1170.2.i.k.451.1 2
39.32 even 12 1170.2.i.k.991.1 2
65.2 even 12 1950.2.z.h.1699.1 4
65.19 odd 12 1950.2.i.s.601.1 2
65.28 even 12 1950.2.z.h.1699.2 4
65.32 even 12 1950.2.z.h.1849.2 4
65.54 odd 12 1950.2.i.s.451.1 2
65.58 even 12 1950.2.z.h.1849.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.a.61.1 2 13.2 odd 12
390.2.i.a.211.1 yes 2 13.6 odd 12
1170.2.i.k.451.1 2 39.2 even 12
1170.2.i.k.991.1 2 39.32 even 12
1950.2.i.s.451.1 2 65.54 odd 12
1950.2.i.s.601.1 2 65.19 odd 12
1950.2.z.h.1699.1 4 65.2 even 12
1950.2.z.h.1699.2 4 65.28 even 12
1950.2.z.h.1849.1 4 65.58 even 12
1950.2.z.h.1849.2 4 65.32 even 12
5070.2.a.f.1.1 1 13.8 odd 4
5070.2.a.o.1.1 1 13.5 odd 4
5070.2.b.g.1351.1 2 13.12 even 2 inner
5070.2.b.g.1351.2 2 1.1 even 1 trivial