# Properties

 Label 2850.2.d.e.799.1 Level $2850$ Weight $2$ Character 2850.799 Analytic conductor $22.757$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.7573645761$$ Analytic rank: $$1$$ 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 570) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2850.799 Dual form 2850.2.d.e.799.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +6.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} -4.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -2.00000 q^{29} -2.00000 q^{31} -1.00000i q^{32} -8.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} +6.00000 q^{39} -12.0000 q^{41} -2.00000i q^{42} +4.00000i q^{43} -4.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} -6.00000i q^{52} +10.0000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} +2.00000i q^{58} -6.00000 q^{59} -14.0000 q^{61} +2.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +12.0000i q^{67} +8.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -10.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} -6.00000i q^{78} -14.0000 q^{79} +1.00000 q^{81} +12.0000i q^{82} +2.00000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +2.00000i q^{87} -12.0000 q^{91} +4.00000i q^{92} +2.00000i q^{93} -12.0000 q^{94} -1.00000 q^{96} -2.00000i q^{97} -3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} + 2 q^{24} + 12 q^{26} - 4 q^{29} - 4 q^{31} - 16 q^{34} + 2 q^{36} + 12 q^{39} - 24 q^{41} - 8 q^{46} + 6 q^{49} - 16 q^{51} + 2 q^{54} - 4 q^{56} - 12 q^{59} - 28 q^{61} - 2 q^{64} - 8 q^{69} - 16 q^{71} + 4 q^{74} + 2 q^{76} - 28 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} - 24 q^{91} - 24 q^{94} - 2 q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1027$$ $$1351$$ $$1901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 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
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 8.00000i − 1.94029i −0.242536 0.970143i $$-0.577979\pi$$
0.242536 0.970143i $$-0.422021\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ 1.00000i 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ −8.00000 −1.37199
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ − 6.00000i − 0.832050i
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 1.00000i 0.132453i
$$58$$ 2.00000i 0.262613i
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 2.00000i 0.254000i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 8.00000i 0.970143i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ − 6.00000i − 0.679366i
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 12.0000i 1.32518i
$$83$$ 2.00000i 0.219529i 0.993958 + 0.109764i $$0.0350096\pi$$
−0.993958 + 0.109764i $$0.964990\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 2.00000i 0.214423i
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −12.0000 −1.25794
$$92$$ 4.00000i 0.417029i
$$93$$ 2.00000i 0.207390i
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 8.00000i 0.792118i
$$103$$ − 16.0000i − 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 16.0000i 1.54678i 0.633932 + 0.773389i $$0.281440\pi$$
−0.633932 + 0.773389i $$0.718560\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 12.0000 1.14939 0.574696 0.818367i $$-0.305120\pi$$
0.574696 + 0.818367i $$0.305120\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 2.00000i 0.188982i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ − 6.00000i − 0.554700i
$$118$$ 6.00000i 0.552345i
$$119$$ 16.0000 1.46672
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 14.0000i 1.26750i
$$123$$ 12.0000i 1.08200i
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ − 2.00000i − 0.173422i
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 8.00000 0.685994
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 4.00000i 0.340503i
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ 8.00000i 0.671345i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ − 3.00000i − 0.247436i
$$148$$ − 2.00000i − 0.164399i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 18.0000 1.46482 0.732410 0.680864i $$-0.238396\pi$$
0.732410 + 0.680864i $$0.238396\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ 8.00000i 0.646762i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ 4.00000i 0.319235i 0.987179 + 0.159617i $$0.0510260\pi$$
−0.987179 + 0.159617i $$0.948974\pi$$
$$158$$ 14.0000i 1.11378i
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 8.00000i 0.626608i 0.949653 + 0.313304i $$0.101436\pi$$
−0.949653 + 0.313304i $$0.898564\pi$$
$$164$$ 12.0000 0.937043
$$165$$ 0 0
$$166$$ 2.00000 0.155230
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 4.00000i − 0.304997i
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.00000i 0.450988i
$$178$$ 0 0
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 12.0000i 0.889499i
$$183$$ 14.0000i 1.03491i
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 2.00000 0.146647
$$187$$ 0 0
$$188$$ 12.0000i 0.875190i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 6.00000i 0.422159i
$$203$$ − 4.00000i − 0.280745i
$$204$$ 8.00000 0.560112
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ 4.00000i 0.278019i
$$208$$ 6.00000i 0.416025i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ − 10.0000i − 0.686803i
$$213$$ 8.00000i 0.548151i
$$214$$ 16.0000 1.09374
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ − 4.00000i − 0.271538i
$$218$$ − 12.0000i − 0.812743i
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ 48.0000 3.22883
$$222$$ − 2.00000i − 0.134231i
$$223$$ − 4.00000i − 0.267860i −0.990991 0.133930i $$-0.957240\pi$$
0.990991 0.133930i $$-0.0427597\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 2.00000i − 0.131306i
$$233$$ 24.0000i 1.57229i 0.618041 + 0.786146i $$0.287927\pi$$
−0.618041 + 0.786146i $$0.712073\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ 14.0000i 0.909398i
$$238$$ − 16.0000i − 1.03713i
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 12.0000 0.765092
$$247$$ − 6.00000i − 0.381771i
$$248$$ − 2.00000i − 0.127000i
$$249$$ 2.00000 0.126745
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 30.0000i 1.87135i 0.352865 + 0.935674i $$0.385208\pi$$
−0.352865 + 0.935674i $$0.614792\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ 0 0
$$268$$ − 12.0000i − 0.733017i
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ − 8.00000i − 0.485071i
$$273$$ 12.0000i 0.726273i
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 4.00000i − 0.240337i −0.992754 0.120168i $$-0.961657\pi$$
0.992754 0.120168i $$-0.0383434\pi$$
$$278$$ − 12.0000i − 0.719712i
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ −20.0000 −1.19310 −0.596550 0.802576i $$-0.703462\pi$$
−0.596550 + 0.802576i $$0.703462\pi$$
$$282$$ 12.0000i 0.714590i
$$283$$ 20.0000i 1.18888i 0.804141 + 0.594438i $$0.202626\pi$$
−0.804141 + 0.594438i $$0.797374\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 24.0000i − 1.41668i
$$288$$ 1.00000i 0.0589256i
$$289$$ −47.0000 −2.76471
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 10.0000i 0.585206i
$$293$$ − 18.0000i − 1.05157i −0.850617 0.525786i $$-0.823771\pi$$
0.850617 0.525786i $$-0.176229\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 10.0000i 0.579284i
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ − 18.0000i − 1.03578i
$$303$$ 6.00000i 0.344691i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 8.00000 0.457330
$$307$$ 28.0000i 1.59804i 0.601302 + 0.799022i $$0.294649\pi$$
−0.601302 + 0.799022i $$0.705351\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 6.00000i 0.339683i
$$313$$ 6.00000i 0.339140i 0.985518 + 0.169570i $$0.0542379\pi$$
−0.985518 + 0.169570i $$0.945762\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ − 10.0000i − 0.560772i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 16.0000 0.893033
$$322$$ − 8.00000i − 0.445823i
$$323$$ 8.00000i 0.445132i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ − 12.0000i − 0.663602i
$$328$$ − 12.0000i − 0.662589i
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ − 2.00000i − 0.109764i
$$333$$ − 2.00000i − 0.109599i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 1.00000i − 0.0540738i
$$343$$ 20.0000i 1.07990i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ 10.0000i 0.536828i 0.963304 + 0.268414i $$0.0864995\pi$$
−0.963304 + 0.268414i $$0.913500\pi$$
$$348$$ − 2.00000i − 0.107211i
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 0 0
$$353$$ 20.0000i 1.06449i 0.846590 + 0.532246i $$0.178652\pi$$
−0.846590 + 0.532246i $$0.821348\pi$$
$$354$$ 6.00000 0.318896
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 16.0000i − 0.846810i
$$358$$ 10.0000i 0.528516i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 12.0000i 0.630706i
$$363$$ 11.0000i 0.577350i
$$364$$ 12.0000 0.628971
$$365$$ 0 0
$$366$$ 14.0000 0.731792
$$367$$ − 34.0000i − 1.77479i −0.461014 0.887393i $$-0.652514\pi$$
0.461014 0.887393i $$-0.347486\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ −20.0000 −1.03835
$$372$$ − 2.00000i − 0.103695i
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ − 12.0000i − 0.618031i
$$378$$ 2.00000i 0.102869i
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ − 4.00000i − 0.203331i
$$388$$ 2.00000i 0.101535i
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ −32.0000 −1.61831
$$392$$ 3.00000i 0.151523i
$$393$$ 0 0
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 8.00000i − 0.401508i −0.979642 0.200754i $$-0.935661\pi$$
0.979642 0.200754i $$-0.0643393\pi$$
$$398$$ 4.00000i 0.200502i
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ − 12.0000i − 0.598506i
$$403$$ − 12.0000i − 0.597763i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ −4.00000 −0.198517
$$407$$ 0 0
$$408$$ − 8.00000i − 0.396059i
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 16.0000i 0.788263i
$$413$$ − 12.0000i − 0.590481i
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ − 12.0000i − 0.587643i
$$418$$ 0 0
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ 4.00000 0.194948 0.0974740 0.995238i $$-0.468924\pi$$
0.0974740 + 0.995238i $$0.468924\pi$$
$$422$$ 16.0000i 0.778868i
$$423$$ 12.0000i 0.583460i
$$424$$ −10.0000 −0.485643
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ − 28.0000i − 1.35501i
$$428$$ − 16.0000i − 0.773389i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 34.0000i − 1.63394i −0.576683 0.816968i $$-0.695653\pi$$
0.576683 0.816968i $$-0.304347\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ −12.0000 −0.574696
$$437$$ 4.00000i 0.191346i
$$438$$ 10.0000i 0.477818i
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 48.0000i − 2.28313i
$$443$$ − 34.0000i − 1.61539i −0.589601 0.807694i $$-0.700715\pi$$
0.589601 0.807694i $$-0.299285\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ −4.00000 −0.189405
$$447$$ 10.0000i 0.472984i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ −36.0000 −1.69895 −0.849473 0.527633i $$-0.823080\pi$$
−0.849473 + 0.527633i $$0.823080\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 6.00000i − 0.282216i
$$453$$ − 18.0000i − 0.845714i
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ − 10.0000i − 0.467780i −0.972263 0.233890i $$-0.924854\pi$$
0.972263 0.233890i $$-0.0751456\pi$$
$$458$$ 6.00000i 0.280362i
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ 22.0000i 1.02243i 0.859454 + 0.511213i $$0.170804\pi$$
−0.859454 + 0.511213i $$0.829196\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ 22.0000i 1.01804i 0.860755 + 0.509019i $$0.169992\pi$$
−0.860755 + 0.509019i $$0.830008\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ 4.00000 0.184310
$$472$$ − 6.00000i − 0.276172i
$$473$$ 0 0
$$474$$ 14.0000 0.643041
$$475$$ 0 0
$$476$$ −16.0000 −0.733359
$$477$$ − 10.0000i − 0.457869i
$$478$$ − 8.00000i − 0.365911i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 14.0000i 0.637683i
$$483$$ − 8.00000i − 0.364013i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 40.0000i − 1.81257i −0.422664 0.906287i $$-0.638905\pi$$
0.422664 0.906287i $$-0.361095\pi$$
$$488$$ − 14.0000i − 0.633750i
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ −16.0000 −0.722070 −0.361035 0.932552i $$-0.617576\pi$$
−0.361035 + 0.932552i $$0.617576\pi$$
$$492$$ − 12.0000i − 0.541002i
$$493$$ 16.0000i 0.720604i
$$494$$ −6.00000 −0.269953
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ − 16.0000i − 0.717698i
$$498$$ − 2.00000i − 0.0896221i
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ 12.0000i 0.535586i
$$503$$ − 16.0000i − 0.713405i −0.934218 0.356702i $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 23.0000i 1.02147i
$$508$$ − 8.00000i − 0.354943i
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ 20.0000 0.884748
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ 30.0000 1.32324
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 4.00000i 0.175750i
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ 16.0000i 0.696971i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 2.00000i 0.0867110i
$$533$$ − 72.0000i − 3.11867i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 10.0000i 0.431532i
$$538$$ − 30.0000i − 1.29339i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 4.00000i 0.171815i
$$543$$ 12.0000i 0.514969i
$$544$$ −8.00000 −0.342997
$$545$$ 0 0
$$546$$ 12.0000 0.513553
$$547$$ 36.0000i 1.53925i 0.638497 + 0.769624i $$0.279557\pi$$
−0.638497 + 0.769624i $$0.720443\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ − 4.00000i − 0.170251i
$$553$$ − 28.0000i − 1.19068i
$$554$$ −4.00000 −0.169944
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ − 2.00000i − 0.0846668i
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 20.0000i 0.843649i
$$563$$ 24.0000i 1.01148i 0.862686 + 0.505740i $$0.168780\pi$$
−0.862686 + 0.505740i $$0.831220\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 2.00000i 0.0839921i
$$568$$ − 8.00000i − 0.335673i
$$569$$ −4.00000 −0.167689 −0.0838444 0.996479i $$-0.526720\pi$$
−0.0838444 + 0.996479i $$0.526720\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000i 1.24892i 0.781058 + 0.624458i $$0.214680\pi$$
−0.781058 + 0.624458i $$0.785320\pi$$
$$578$$ 47.0000i 1.95494i
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 2.00000i 0.0829027i
$$583$$ 0 0
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 26.0000i 1.07313i 0.843857 + 0.536567i $$0.180279\pi$$
−0.843857 + 0.536567i $$0.819721\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ 2.00000 0.0824086
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ 2.00000i 0.0821995i
$$593$$ 36.0000i 1.47834i 0.673517 + 0.739171i $$0.264783\pi$$
−0.673517 + 0.739171i $$0.735217\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 4.00000i 0.163709i
$$598$$ − 24.0000i − 0.981433i
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −14.0000 −0.571072 −0.285536 0.958368i $$-0.592172\pi$$
−0.285536 + 0.958368i $$0.592172\pi$$
$$602$$ 8.00000i 0.326056i
$$603$$ − 12.0000i − 0.488678i
$$604$$ −18.0000 −0.732410
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ − 24.0000i − 0.974130i −0.873366 0.487065i $$-0.838067\pi$$
0.873366 0.487065i $$-0.161933\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ −4.00000 −0.162088
$$610$$ 0 0
$$611$$ 72.0000 2.91281
$$612$$ − 8.00000i − 0.323381i
$$613$$ − 16.0000i − 0.646234i −0.946359 0.323117i $$-0.895269\pi$$
0.946359 0.323117i $$-0.104731\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 20.0000i − 0.805170i −0.915383 0.402585i $$-0.868112\pi$$
0.915383 0.402585i $$-0.131888\pi$$
$$618$$ 16.0000i 0.643614i
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 24.0000i − 0.962312i
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ 6.00000 0.239808
$$627$$ 0 0
$$628$$ − 4.00000i − 0.159617i
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ − 14.0000i − 0.556890i
$$633$$ 16.0000i 0.635943i
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ −10.0000 −0.396526
$$637$$ 18.0000i 0.713186i
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −20.0000 −0.789953 −0.394976 0.918691i $$-0.629247\pi$$
−0.394976 + 0.918691i $$0.629247\pi$$
$$642$$ − 16.0000i − 0.631470i
$$643$$ − 28.0000i − 1.10421i −0.833774 0.552106i $$-0.813824\pi$$
0.833774 0.552106i $$-0.186176\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ 16.0000i 0.629025i 0.949253 + 0.314512i $$0.101841\pi$$
−0.949253 + 0.314512i $$0.898159\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −4.00000 −0.156772
$$652$$ − 8.00000i − 0.313304i
$$653$$ − 42.0000i − 1.64359i −0.569785 0.821794i $$-0.692974\pi$$
0.569785 0.821794i $$-0.307026\pi$$
$$654$$ −12.0000 −0.469237
$$655$$ 0 0
$$656$$ −12.0000 −0.468521
$$657$$ 10.0000i 0.390137i
$$658$$ − 24.0000i − 0.935617i
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ − 48.0000i − 1.86417i
$$664$$ −2.00000 −0.0776151
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 8.00000i 0.309761i
$$668$$ 8.00000i 0.309529i
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 2.00000i − 0.0771517i
$$673$$ 2.00000i 0.0770943i 0.999257 + 0.0385472i $$0.0122730\pi$$
−0.999257 + 0.0385472i $$0.987727\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ − 6.00000i − 0.230429i
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 0 0
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 6.00000i 0.228914i
$$688$$ 4.00000i 0.152499i
$$689$$ −60.0000 −2.28582
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ − 2.00000i − 0.0760286i
$$693$$ 0 0
$$694$$ 10.0000 0.379595
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ 96.0000i 3.63626i
$$698$$ 14.0000i 0.529908i
$$699$$ 24.0000 0.907763
$$700$$ 0 0
$$701$$ 14.0000 0.528773 0.264386 0.964417i $$-0.414831\pi$$
0.264386 + 0.964417i $$0.414831\pi$$
$$702$$ 6.00000i 0.226455i
$$703$$ − 2.00000i − 0.0754314i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 20.0000 0.752710
$$707$$ − 12.0000i − 0.451306i
$$708$$ − 6.00000i − 0.225494i
$$709$$ 18.0000 0.676004 0.338002 0.941145i $$-0.390249\pi$$
0.338002 + 0.941145i $$0.390249\pi$$
$$710$$ 0 0
$$711$$ 14.0000 0.525041
$$712$$ 0 0
$$713$$ 8.00000i 0.299602i
$$714$$ −16.0000 −0.598785
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ − 8.00000i − 0.298765i
$$718$$ 0 0
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ − 1.00000i − 0.0372161i
$$723$$ 14.0000i 0.520666i
$$724$$ 12.0000 0.445976
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ − 42.0000i − 1.55769i −0.627214 0.778847i $$-0.715805\pi$$
0.627214 0.778847i $$-0.284195\pi$$
$$728$$ − 12.0000i − 0.444750i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 32.0000 1.18356
$$732$$ − 14.0000i − 0.517455i
$$733$$ 8.00000i 0.295487i 0.989026 + 0.147743i $$0.0472010\pi$$
−0.989026 + 0.147743i $$0.952799\pi$$
$$734$$ −34.0000 −1.25496
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 0 0
$$738$$ − 12.0000i − 0.441726i
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ −6.00000 −0.220416
$$742$$ 20.0000i 0.734223i
$$743$$ 40.0000i 1.46746i 0.679442 + 0.733729i $$0.262222\pi$$
−0.679442 + 0.733729i $$0.737778\pi$$
$$744$$ −2.00000 −0.0733236
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ − 2.00000i − 0.0731762i
$$748$$ 0 0
$$749$$ −32.0000 −1.16925
$$750$$ 0 0
$$751$$ −2.00000 −0.0729810 −0.0364905 0.999334i $$-0.511618\pi$$
−0.0364905 + 0.999334i $$0.511618\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ 12.0000i 0.437304i
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ − 40.0000i − 1.45382i −0.686730 0.726912i $$-0.740955\pi$$
0.686730 0.726912i $$-0.259045\pi$$
$$758$$ − 16.0000i − 0.581146i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ − 8.00000i − 0.289809i
$$763$$ 24.0000i 0.868858i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 36.0000i − 1.29988i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 0 0
$$771$$ 30.0000 1.08042
$$772$$ 2.00000i 0.0719816i
$$773$$ 10.0000i 0.359675i 0.983696 + 0.179838i $$0.0575572\pi$$
−0.983696 + 0.179838i $$0.942443\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 4.00000i 0.143499i
$$778$$ 18.0000i 0.645331i
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 32.0000i 1.14432i
$$783$$ − 2.00000i − 0.0714742i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 20.0000i 0.712923i 0.934310 + 0.356462i $$0.116017\pi$$
−0.934310 + 0.356462i $$0.883983\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ − 84.0000i − 2.98293i
$$794$$ −8.00000 −0.283909
$$795$$ 0 0
$$796$$ 4.00000 0.141776
$$797$$ 18.0000i 0.637593i 0.947823 + 0.318796i $$0.103279\pi$$
−0.947823 + 0.318796i $$0.896721\pi$$
$$798$$ 2.00000i 0.0707992i
$$799$$ −96.0000 −3.39624
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ −12.0000 −0.422682
$$807$$ − 30.0000i − 1.05605i
$$808$$ − 6.00000i − 0.211079i
$$809$$ −38.0000 −1.33601 −0.668004 0.744157i $$-0.732851\pi$$
−0.668004 + 0.744157i $$0.732851\pi$$
$$810$$ 0 0
$$811$$ 16.0000 0.561836 0.280918 0.959732i $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ 4.00000i 0.140372i
$$813$$ 4.00000i 0.140286i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −8.00000 −0.280056
$$817$$ − 4.00000i − 0.139942i
$$818$$ − 6.00000i − 0.209785i
$$819$$ 12.0000 0.419314
$$820$$ 0 0
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ 14.0000i 0.488009i 0.969774 + 0.244005i $$0.0784612\pi$$
−0.969774 + 0.244005i $$0.921539\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ − 32.0000i − 1.11275i −0.830932 0.556375i $$-0.812192\pi$$
0.830932 0.556375i $$-0.187808\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ −28.0000 −0.972480 −0.486240 0.873825i $$-0.661632\pi$$
−0.486240 + 0.873825i $$0.661632\pi$$
$$830$$ 0 0
$$831$$ −4.00000 −0.138758
$$832$$ − 6.00000i − 0.208013i
$$833$$ − 24.0000i − 0.831551i
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 2.00000i − 0.0691301i
$$838$$ − 28.0000i − 0.967244i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 4.00000i − 0.137849i
$$843$$ 20.0000i 0.688837i
$$844$$ 16.0000 0.550743
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ − 22.0000i − 0.755929i
$$848$$ 10.0000i 0.343401i
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 28.0000i − 0.958702i −0.877623 0.479351i $$-0.840872\pi$$
0.877623 0.479351i $$-0.159128\pi$$
$$854$$ −28.0000 −0.958140
$$855$$ 0 0
$$856$$ −16.0000 −0.546869
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ 0 0
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ − 24.0000i − 0.817443i
$$863$$ 8.00000i 0.272323i 0.990687 + 0.136162i $$0.0434766\pi$$
−0.990687 + 0.136162i $$0.956523\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ 47.0000i 1.59620i
$$868$$ 4.00000i 0.135769i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −72.0000 −2.43963
$$872$$ 12.0000i 0.406371i
$$873$$ 2.00000i 0.0676897i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ − 18.0000i − 0.607817i −0.952701 0.303908i $$-0.901708\pi$$
0.952701 0.303908i $$-0.0982917\pi$$
$$878$$ − 10.0000i − 0.337484i
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ − 52.0000i − 1.74994i −0.484178 0.874970i $$-0.660881\pi$$
0.484178 0.874970i $$-0.339119\pi$$
$$884$$ −48.0000 −1.61441
$$885$$ 0 0
$$886$$ −34.0000 −1.14225
$$887$$ − 32.0000i − 1.07445i −0.843437 0.537227i $$-0.819472\pi$$
0.843437 0.537227i $$-0.180528\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 4.00000i 0.133930i
$$893$$ 12.0000i 0.401565i
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ − 24.0000i − 0.801337i
$$898$$ 36.0000i 1.20134i
$$899$$ 4.00000 0.133407
$$900$$ 0 0
$$901$$ 80.0000 2.66519
$$902$$ 0 0
$$903$$ 8.00000i 0.266223i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −18.0000 −0.598010
$$907$$ 36.0000i 1.19536i 0.801735 + 0.597680i $$0.203911\pi$$
−0.801735 + 0.597680i $$0.796089\pi$$
$$908$$ 20.0000i 0.663723i
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ 0 0
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 0 0
$$918$$ − 8.00000i − 0.264039i
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ 28.0000 0.922631
$$922$$ 18.0000i 0.592798i
$$923$$ − 48.0000i − 1.57994i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 22.0000 0.722965
$$927$$ 16.0000i 0.525509i
$$928$$ 2.00000i 0.0656532i
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ − 24.0000i − 0.786146i
$$933$$ − 24.0000i − 0.785725i
$$934$$ 22.0000 0.719862
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ − 2.00000i − 0.0653372i −0.999466 0.0326686i $$-0.989599\pi$$
0.999466 0.0326686i $$-0.0104006\pi$$
$$938$$ 24.0000i 0.783628i
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ − 4.00000i − 0.130327i
$$943$$ 48.0000i 1.56310i
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 2.00000i − 0.0649913i −0.999472 0.0324956i $$-0.989654\pi$$
0.999472 0.0324956i $$-0.0103455\pi$$
$$948$$ − 14.0000i − 0.454699i
$$949$$ 60.0000 1.94768
$$950$$ 0 0
$$951$$ 2.00000 0.0648544
$$952$$ 16.0000i 0.518563i
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 12.0000i 0.386896i
$$963$$ − 16.0000i − 0.515593i
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ 34.0000i 1.09337i 0.837340 + 0.546683i $$0.184110\pi$$
−0.837340 + 0.546683i $$0.815890\pi$$
$$968$$ − 11.0000i − 0.353553i
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 24.0000i 0.769405i
$$974$$ −40.0000 −1.28168
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ 30.0000i 0.959785i 0.877327 + 0.479893i $$0.159324\pi$$
−0.877327 + 0.479893i $$0.840676\pi$$
$$978$$ − 8.00000i − 0.255812i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −12.0000 −0.383131
$$982$$ 16.0000i 0.510581i
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ 0 0
$$986$$ 16.0000 0.509544
$$987$$ − 24.0000i − 0.763928i
$$988$$ 6.00000i 0.190885i
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 50.0000 1.58830 0.794151 0.607720i $$-0.207916\pi$$
0.794151 + 0.607720i $$0.207916\pi$$
$$992$$ 2.00000i 0.0635001i
$$993$$ − 20.0000i − 0.634681i
$$994$$ −16.0000 −0.507489
$$995$$ 0 0
$$996$$ −2.00000 −0.0633724
$$997$$ − 44.0000i − 1.39349i −0.717317 0.696747i $$-0.754630\pi$$
0.717317 0.696747i $$-0.245370\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ −2.00000 −0.0632772
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 2850.2.d.e.799.1 2
5.2 odd 4 570.2.a.h.1.1 1
5.3 odd 4 2850.2.a.n.1.1 1
5.4 even 2 inner 2850.2.d.e.799.2 2
15.2 even 4 1710.2.a.c.1.1 1
15.8 even 4 8550.2.a.bh.1.1 1
20.7 even 4 4560.2.a.bc.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.h.1.1 1 5.2 odd 4
1710.2.a.c.1.1 1 15.2 even 4
2850.2.a.n.1.1 1 5.3 odd 4
2850.2.d.e.799.1 2 1.1 even 1 trivial
2850.2.d.e.799.2 2 5.4 even 2 inner
4560.2.a.bc.1.1 1 20.7 even 4
8550.2.a.bh.1.1 1 15.8 even 4