# Properties

 Label 5070.2.b.g.1351.1 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

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

## $q$-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 $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 1.00000i 0.301511i 0.988571 + 0.150756i $$0.0481707\pi$$
−0.988571 + 0.150756i $$0.951829\pi$$
$$12$$ 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 $$-0.758380\pi$$
0.725476 0.688247i $$-0.241620\pi$$
$$20$$ − 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 $$0.0868280\pi$$
−0.963026 + 0.269408i $$0.913172\pi$$
$$32$$ − 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 $$-0.865181\pi$$
0.911636 0.410997i $$-0.134819\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ − 10.0000i − 1.56174i −0.624695 0.780869i $$-0.714777\pi$$
0.624695 0.780869i $$-0.285223\pi$$
$$42$$ −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 $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ −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 $$-0.894477\pi$$
0.945552 0.325472i $$-0.105523\pi$$
$$60$$ 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 $$-0.923719\pi$$
0.971423 0.237356i $$-0.0762809\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ −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 $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 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 $$0.177808\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 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 $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ − 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 $$0.0927736\pi$$
−0.957826 + 0.287348i $$0.907226\pi$$
$$110$$ 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 $$-0.874387\pi$$
0.923141 0.384461i $$-0.125613\pi$$
$$138$$ 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 $$-0.851218\pi$$
0.892737 0.450578i $$-0.148782\pi$$
$$150$$ − 1.00000i − 0.0816497i
$$151$$ 24.0000i 1.95309i 0.215308 + 0.976546i $$0.430924\pi$$
−0.215308 + 0.976546i $$0.569076\pi$$
$$152$$ 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 $$0.231897\pi$$
−0.746156 + 0.665771i $$0.768103\pi$$
$$164$$ 10.0000i 0.780869i
$$165$$ 1.00000 0.0778499
$$166$$ 6.00000 0.465690
$$167$$ − 7.00000i − 0.541676i −0.962625 0.270838i $$-0.912699\pi$$
0.962625 0.270838i $$-0.0873008\pi$$
$$168$$ 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 $$-0.954015\pi$$
0.989583 0.143963i $$-0.0459847\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 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 $$-0.891323\pi$$
0.942280 0.334825i $$-0.108677\pi$$
$$224$$ 2.00000 0.133631
$$225$$ −1.00000 −0.0666667
$$226$$ − 17.0000i − 1.13082i
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ − 6.00000i − 0.397360i
$$229$$ 22.0000i 1.45380i 0.686743 + 0.726900i $$0.259040\pi$$
−0.686743 + 0.726900i $$0.740960\pi$$
$$230$$ 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 $$0.0833068\pi$$
−0.965947 + 0.258738i $$0.916693\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ − 7.00000i − 0.450910i −0.974254 0.225455i $$-0.927613\pi$$
0.974254 0.225455i $$-0.0723868\pi$$
$$242$$ − 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 $$-0.656997\pi$$
0.473466 0.880812i $$-0.343003\pi$$
$$272$$ −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 $$-0.647300\pi$$
0.446417 0.894825i $$-0.352700\pi$$
$$282$$ −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 $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ 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 $$-0.890590\pi$$
0.941507 0.336994i $$-0.109410\pi$$
$$318$$ 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 $$-0.964937\pi$$
0.993939 0.109930i $$-0.0350627\pi$$
$$332$$ − 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 $$0.0686868\pi$$
−0.976808 + 0.214115i $$0.931313\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ 14.0000i 0.745145i 0.928003 + 0.372572i $$0.121524\pi$$
−0.928003 + 0.372572i $$0.878476\pi$$
$$354$$ 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 $$0.102564\pi$$
−0.948536 + 0.316668i $$0.897436\pi$$
$$360$$ −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 $$0.279993\pi$$
−0.637442 + 0.770498i $$0.720007\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ −14.0000 −0.717242
$$382$$ − 24.0000i − 1.22795i
$$383$$ 27.0000i 1.37964i 0.723983 + 0.689818i $$0.242309\pi$$
−0.723983 + 0.689818i $$0.757691\pi$$
$$384$$ − 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 $$-0.894223\pi$$
0.945292 0.326226i $$-0.105777\pi$$
$$398$$ 0 0
$$399$$ −12.0000 −0.600751
$$400$$ −1.00000 −0.0500000
$$401$$ − 12.0000i − 0.599251i −0.954057 0.299626i $$-0.903138\pi$$
0.954057 0.299626i $$-0.0968618\pi$$
$$402$$ 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 $$0.222231\pi$$
−0.766027 + 0.642809i $$0.777769\pi$$
$$410$$ − 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 $$-0.760972\pi$$
0.731055 0.682318i $$-0.239028\pi$$
$$422$$ 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 $$-0.906674\pi$$
0.957326 0.289010i $$-0.0933260\pi$$
$$432$$ −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 $$-0.676920\pi$$
0.527633 0.849473i $$-0.323080\pi$$
$$450$$ 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 $$-0.893812\pi$$
0.944870 0.327446i $$-0.106188\pi$$
$$458$$ 22.0000 1.02799
$$459$$ 2.00000 0.0933520
$$460$$ − 3.00000i − 0.139876i
$$461$$ − 27.0000i − 1.25752i −0.777601 0.628758i $$-0.783564\pi$$
0.777601 0.628758i $$-0.216436\pi$$
$$462$$ − 2.00000i − 0.0930484i
$$463$$ 10.0000i 0.464739i 0.972628 + 0.232370i $$0.0746479\pi$$
−0.972628 + 0.232370i $$0.925352\pi$$
$$464$$ −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 $$-0.970871\pi$$
0.995816 0.0913823i $$-0.0291285\pi$$
$$480$$ −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 $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ − 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 $$0.353055\pi$$
−0.445418 + 0.895323i $$0.646945\pi$$
$$500$$ 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 $$0.0639185\pi$$
−0.979906 + 0.199459i $$0.936082\pi$$
$$510$$ 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 $$-0.944986\pi$$
0.985102 0.171973i $$-0.0550143\pi$$
$$542$$ −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 $$-0.743998\pi$$
0.693649 0.720313i $$-0.256002\pi$$
$$558$$ 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 $$0.122246\pi$$
−0.927156 + 0.374675i $$0.877754\pi$$
$$578$$ 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 $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ 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 $$-0.744765\pi$$
0.695383 0.718639i $$-0.255235\pi$$
$$594$$ −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 $$0.268606\pi$$
−0.664590 + 0.747208i $$0.731394\pi$$
$$614$$ 0 0
$$615$$ −10.0000 −0.403239
$$616$$ − 2.00000i − 0.0805823i
$$617$$ − 3.00000i − 0.120775i −0.998175 0.0603877i $$-0.980766\pi$$
0.998175 0.0603877i $$-0.0192337\pi$$
$$618$$ 6.00000i 0.241355i
$$619$$ − 10.0000i − 0.401934i −0.979598 0.200967i $$-0.935592\pi$$
0.979598 0.200967i $$-0.0644084\pi$$
$$620$$ 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 $$0.404602\pi$$
−0.295234 + 0.955425i $$0.595398\pi$$
$$632$$ 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 $$0.289257\pi$$
−0.614749 + 0.788723i $$0.710743\pi$$
$$644$$ − 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 $$-0.925024\pi$$
0.972387 0.233373i $$-0.0749763\pi$$
$$662$$ −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 $$-0.681492\pi$$
0.539779 0.841807i $$-0.318508\pi$$
$$684$$ 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 $$0.0857987\pi$$
−0.963892 + 0.266293i $$0.914201\pi$$
$$692$$ 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 $$-0.902864\pi$$
0.953799 0.300446i $$-0.0971356\pi$$
$$710$$ − 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 $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ − 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 $$0.145528\pi$$
−0.897297 + 0.441427i $$0.854472\pi$$
$$740$$ −5.00000 −0.183804
$$741$$ 0 0
$$742$$ 28.0000 1.02791
$$743$$ 15.0000i 0.550297i 0.961402 + 0.275148i $$0.0887270\pi$$
−0.961402 + 0.275148i $$0.911273\pi$$
$$744$$ 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 $$-0.881923\pi$$
0.931984 0.362500i $$-0.118077\pi$$
$$762$$ 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 $$0.175144\pi$$
−0.852404 + 0.522883i $$0.824856\pi$$
$$770$$ −2.00000 −0.0720750
$$771$$ 21.0000 0.756297
$$772$$ 4.00000i 0.143963i
$$773$$ 16.0000i 0.575480i 0.957709 + 0.287740i $$0.0929039\pi$$
−0.957709 + 0.287740i $$0.907096\pi$$
$$774$$ 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 $$0.147001\pi$$
−0.895244 + 0.445577i $$0.852999\pi$$
$$788$$ − 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 $$-0.633772\pi$$
0.407997 0.912983i $$-0.366228\pi$$
$$812$$ 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 $$-0.868540\pi$$
0.915924 0.401353i $$-0.131460\pi$$
$$822$$ 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 $$-0.898679\pi$$
0.949766 0.312961i $$-0.101321\pi$$
$$828$$ −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 $$0.227733\pi$$
−0.754802 + 0.655953i $$0.772267\pi$$
$$840$$ 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 $$0.0382374\pi$$
−0.992793 + 0.119838i $$0.961763\pi$$
$$854$$ −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 $$0.334614\pi$$
−0.496512 + 0.868030i $$0.665386\pi$$
$$864$$ 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 $$-0.929561\pi$$
0.975615 0.219489i $$-0.0704391\pi$$
$$878$$ 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 $$-0.871192\pi$$
0.919236 0.393707i $$-0.128808\pi$$
$$930$$ − 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 $$-0.860700\pi$$
0.905761 0.423788i $$-0.139300\pi$$
$$942$$ 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 $$0.320332\pi$$
−0.534946 + 0.844886i $$0.679668\pi$$
$$948$$ 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 $$-0.917179\pi$$
0.966342 0.257263i $$-0.0828206\pi$$
$$968$$ 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 $$0.214437\pi$$
−0.781536 + 0.623860i $$0.785563\pi$$
$$978$$ −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 $$-0.869466\pi$$
0.917087 0.398688i $$-0.130534\pi$$
$$984$$ −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.1351.1 2
13.5 odd 4 5070.2.a.f.1.1 1
13.7 odd 12 390.2.i.a.211.1 yes 2
13.8 odd 4 5070.2.a.o.1.1 1
13.11 odd 12 390.2.i.a.61.1 2
13.12 even 2 inner 5070.2.b.g.1351.2 2
39.11 even 12 1170.2.i.k.451.1 2
39.20 even 12 1170.2.i.k.991.1 2
65.7 even 12 1950.2.z.h.1849.2 4
65.24 odd 12 1950.2.i.s.451.1 2
65.33 even 12 1950.2.z.h.1849.1 4
65.37 even 12 1950.2.z.h.1699.1 4
65.59 odd 12 1950.2.i.s.601.1 2
65.63 even 12 1950.2.z.h.1699.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.a.61.1 2 13.11 odd 12
390.2.i.a.211.1 yes 2 13.7 odd 12
1170.2.i.k.451.1 2 39.11 even 12
1170.2.i.k.991.1 2 39.20 even 12
1950.2.i.s.451.1 2 65.24 odd 12
1950.2.i.s.601.1 2 65.59 odd 12
1950.2.z.h.1699.1 4 65.37 even 12
1950.2.z.h.1699.2 4 65.63 even 12
1950.2.z.h.1849.1 4 65.33 even 12
1950.2.z.h.1849.2 4 65.7 even 12
5070.2.a.f.1.1 1 13.5 odd 4
5070.2.a.o.1.1 1 13.8 odd 4
5070.2.b.g.1351.1 2 1.1 even 1 trivial
5070.2.b.g.1351.2 2 13.12 even 2 inner