# Properties

 Label 2850.2.d.r.799.1 Level $2850$ Weight $2$ Character 2850.799 Analytic conductor $22.757$ Analytic rank $0$ 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: $$0$$ 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 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.r.799.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} +2.00000i q^{28} -6.00000 q^{29} -6.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} +10.0000i q^{37} +1.00000i q^{38} -6.00000 q^{39} +4.00000 q^{41} -2.00000i q^{42} -12.0000i q^{43} -4.00000 q^{44} -4.00000 q^{46} +4.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -4.00000 q^{51} -6.00000i q^{52} +10.0000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} +6.00000i q^{58} -10.0000 q^{59} +2.00000 q^{61} +6.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +12.0000i q^{67} -4.00000i q^{68} +4.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} +10.0000 q^{74} +1.00000 q^{76} -8.00000i q^{77} +6.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} -4.00000i q^{82} -2.00000i q^{83} -2.00000 q^{84} -12.0000 q^{86} -6.00000i q^{87} +4.00000i q^{88} +8.00000 q^{89} +12.0000 q^{91} +4.00000i q^{92} -6.00000i q^{93} +4.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} -3.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} + 12 q^{26} - 12 q^{29} - 12 q^{31} + 8 q^{34} + 2 q^{36} - 12 q^{39} + 8 q^{41} - 8 q^{44} - 8 q^{46} + 6 q^{49} - 8 q^{51} - 2 q^{54} + 4 q^{56} - 20 q^{59} + 4 q^{61} - 2 q^{64} + 8 q^{66} + 8 q^{69} + 16 q^{71} + 20 q^{74} + 2 q^{76} - 20 q^{79} + 2 q^{81} - 4 q^{84} - 24 q^{86} + 16 q^{89} + 24 q^{91} + 8 q^{94} + 2 q^{96} - 8 q^{99} + O(q^{100})$$

## 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 $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ − 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
$$17$$ 4.00000i 0.970143i 0.874475 + 0.485071i $$0.161206\pi$$
−0.874475 + 0.485071i $$0.838794\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ − 4.00000i − 0.852803i
$$23$$ − 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
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ − 12.0000i − 1.82998i −0.403473 0.914991i $$-0.632197\pi$$
0.403473 0.914991i $$-0.367803\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ − 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
$$58$$ 6.00000i 0.787839i
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 6.00000i 0.762001i
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ − 4.00000i − 0.485071i
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ − 8.00000i − 0.911685i
$$78$$ 6.00000i 0.679366i
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 4.00000i − 0.441726i
$$83$$ − 2.00000i − 0.219529i −0.993958 0.109764i $$-0.964990\pi$$
0.993958 0.109764i $$-0.0350096\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −12.0000 −1.29399
$$87$$ − 6.00000i − 0.643268i
$$88$$ 4.00000i 0.426401i
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 4.00000i 0.417029i
$$93$$ − 6.00000i − 0.622171i
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 4.00000i 0.396059i
$$103$$ 12.0000i 1.18240i 0.806527 + 0.591198i $$0.201345\pi$$
−0.806527 + 0.591198i $$0.798655\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 8.00000i 0.773389i 0.922208 + 0.386695i $$0.126383\pi$$
−0.922208 + 0.386695i $$0.873617\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ − 2.00000i − 0.188982i
$$113$$ 14.0000i 1.31701i 0.752577 + 0.658505i $$0.228811\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ − 6.00000i − 0.554700i
$$118$$ 10.0000i 0.920575i
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ 4.00000i 0.360668i
$$124$$ 6.00000 0.538816
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ − 20.0000i − 1.77471i −0.461084 0.887357i $$-0.652539\pi$$
0.461084 0.887357i $$-0.347461\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ 2.00000i 0.173422i
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ − 8.00000i − 0.671345i
$$143$$ 24.0000i 2.00698i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 3.00000i 0.247436i
$$148$$ − 10.0000i − 0.821995i
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 0 0
$$151$$ −2.00000 −0.162758 −0.0813788 0.996683i $$-0.525932\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ − 4.00000i − 0.323381i
$$154$$ −8.00000 −0.644658
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ 12.0000i 0.957704i 0.877896 + 0.478852i $$0.158947\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ 10.0000i 0.795557i
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 16.0000i 1.25322i 0.779334 + 0.626608i $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ 0 0
$$166$$ −2.00000 −0.155230
$$167$$ 24.0000i 1.85718i 0.371113 + 0.928588i $$0.378976\pi$$
−0.371113 + 0.928588i $$0.621024\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 12.0000i 0.914991i
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ − 10.0000i − 0.751646i
$$178$$ − 8.00000i − 0.599625i
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ − 12.0000i − 0.889499i
$$183$$ 2.00000i 0.147844i
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −6.00000 −0.439941
$$187$$ 16.0000i 1.17004i
$$188$$ − 4.00000i − 0.291730i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 22.0000i 1.56744i 0.621117 + 0.783718i $$0.286679\pi$$
−0.621117 + 0.783718i $$0.713321\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 10.0000i 0.703598i
$$203$$ 12.0000i 0.842235i
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ 12.0000 0.836080
$$207$$ 4.00000i 0.278019i
$$208$$ 6.00000i 0.416025i
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ − 10.0000i − 0.686803i
$$213$$ 8.00000i 0.548151i
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 12.0000i 0.814613i
$$218$$ − 16.0000i − 1.08366i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ 10.0000i 0.671156i
$$223$$ − 24.0000i − 1.60716i −0.595198 0.803579i $$-0.702926\pi$$
0.595198 0.803579i $$-0.297074\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ − 6.00000i − 0.393919i
$$233$$ 12.0000i 0.786146i 0.919507 + 0.393073i $$0.128588\pi$$
−0.919507 + 0.393073i $$0.871412\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 10.0000 0.650945
$$237$$ − 10.0000i − 0.649570i
$$238$$ − 8.00000i − 0.518563i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 4.00000 0.255031
$$247$$ − 6.00000i − 0.381771i
$$248$$ − 6.00000i − 0.381000i
$$249$$ 2.00000 0.126745
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ − 16.0000i − 1.00591i
$$254$$ −20.0000 −1.25491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ − 12.0000i − 0.747087i
$$259$$ 20.0000 1.24274
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ − 12.0000i − 0.741362i
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 8.00000i 0.489592i
$$268$$ − 12.0000i − 0.733017i
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 4.00000i 0.242536i
$$273$$ 12.0000i 0.726273i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ 28.0000i 1.68236i 0.540758 + 0.841178i $$0.318138\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 4.00000 0.238620 0.119310 0.992857i $$-0.461932\pi$$
0.119310 + 0.992857i $$0.461932\pi$$
$$282$$ 4.00000i 0.238197i
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 24.0000 1.41915
$$287$$ − 8.00000i − 0.472225i
$$288$$ 1.00000i 0.0589256i
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ − 2.00000i − 0.117041i
$$293$$ − 2.00000i − 0.116841i −0.998292 0.0584206i $$-0.981394\pi$$
0.998292 0.0584206i $$-0.0186065\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ − 4.00000i − 0.232104i
$$298$$ 14.0000i 0.810998i
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ 2.00000i 0.115087i
$$303$$ − 10.0000i − 0.574485i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ −4.00000 −0.228665
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 8.00000i 0.455842i
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ − 6.00000i − 0.339683i
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ 12.0000 0.677199
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 10.0000i 0.560772i
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 8.00000i 0.445823i
$$323$$ − 4.00000i − 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ 16.0000i 0.884802i
$$328$$ 4.00000i 0.220863i
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 2.00000i 0.109764i
$$333$$ − 10.0000i − 0.547997i
$$334$$ 24.0000 1.31322
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ − 34.0000i − 1.85210i −0.377403 0.926049i $$-0.623183\pi$$
0.377403 0.926049i $$-0.376817\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ −24.0000 −1.29967
$$342$$ − 1.00000i − 0.0540738i
$$343$$ − 20.0000i − 1.07990i
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ − 2.00000i − 0.107366i −0.998558 0.0536828i $$-0.982904\pi$$
0.998558 0.0536828i $$-0.0170960\pi$$
$$348$$ 6.00000i 0.321634i
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ − 4.00000i − 0.213201i
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ −10.0000 −0.531494
$$355$$ 0 0
$$356$$ −8.00000 −0.423999
$$357$$ 8.00000i 0.423405i
$$358$$ − 2.00000i − 0.105703i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 5.00000i 0.262432i
$$364$$ −12.0000 −0.628971
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ − 22.0000i − 1.14839i −0.818718 0.574195i $$-0.805315\pi$$
0.818718 0.574195i $$-0.194685\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ 20.0000 1.03835
$$372$$ 6.00000i 0.311086i
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 16.0000 0.827340
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ − 36.0000i − 1.85409i
$$378$$ 2.00000i 0.102869i
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ 20.0000 1.02463
$$382$$ 8.00000i 0.409316i
$$383$$ 16.0000i 0.817562i 0.912633 + 0.408781i $$0.134046\pi$$
−0.912633 + 0.408781i $$0.865954\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 12.0000i 0.609994i
$$388$$ − 2.00000i − 0.101535i
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 3.00000i 0.151523i
$$393$$ 12.0000i 0.605320i
$$394$$ 22.0000 1.10834
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ − 24.0000i − 1.20453i −0.798298 0.602263i $$-0.794266\pi$$
0.798298 0.602263i $$-0.205734\pi$$
$$398$$ − 4.00000i − 0.200502i
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 12.0000i 0.598506i
$$403$$ − 36.0000i − 1.79329i
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 40.0000i 1.98273i
$$408$$ − 4.00000i − 0.198030i
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 12.0000i − 0.591198i
$$413$$ 20.0000i 0.984136i
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 4.00000i 0.195881i
$$418$$ 4.00000i 0.195646i
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ − 4.00000i − 0.194487i
$$424$$ −10.0000 −0.485643
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ − 4.00000i − 0.193574i
$$428$$ − 8.00000i − 0.386695i
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 30.0000i − 1.44171i −0.693087 0.720854i $$-0.743750\pi$$
0.693087 0.720854i $$-0.256250\pi$$
$$434$$ 12.0000 0.576018
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ 4.00000i 0.191346i
$$438$$ 2.00000i 0.0955637i
$$439$$ 38.0000 1.81364 0.906821 0.421517i $$-0.138502\pi$$
0.906821 + 0.421517i $$0.138502\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 24.0000i 1.14156i
$$443$$ − 6.00000i − 0.285069i −0.989790 0.142534i $$-0.954475\pi$$
0.989790 0.142534i $$-0.0455251\pi$$
$$444$$ 10.0000 0.474579
$$445$$ 0 0
$$446$$ −24.0000 −1.13643
$$447$$ − 14.0000i − 0.662177i
$$448$$ 2.00000i 0.0944911i
$$449$$ −12.0000 −0.566315 −0.283158 0.959073i $$-0.591382\pi$$
−0.283158 + 0.959073i $$0.591382\pi$$
$$450$$ 0 0
$$451$$ 16.0000 0.753411
$$452$$ − 14.0000i − 0.658505i
$$453$$ − 2.00000i − 0.0939682i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ − 26.0000i − 1.21490i
$$459$$ 4.00000 0.186704
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ − 8.00000i − 0.372194i
$$463$$ 18.0000i 0.836531i 0.908325 + 0.418265i $$0.137362\pi$$
−0.908325 + 0.418265i $$0.862638\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ − 22.0000i − 1.01804i −0.860755 0.509019i $$-0.830008\pi$$
0.860755 0.509019i $$-0.169992\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ −12.0000 −0.552931
$$472$$ − 10.0000i − 0.460287i
$$473$$ − 48.0000i − 2.20704i
$$474$$ −10.0000 −0.459315
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ − 10.0000i − 0.457869i
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ − 10.0000i − 0.455488i
$$483$$ − 8.00000i − 0.364013i
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 20.0000i 0.906287i 0.891438 + 0.453143i $$0.149697\pi$$
−0.891438 + 0.453143i $$0.850303\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ − 4.00000i − 0.180334i
$$493$$ − 24.0000i − 1.08091i
$$494$$ −6.00000 −0.269953
$$495$$ 0 0
$$496$$ −6.00000 −0.269408
$$497$$ − 16.0000i − 0.717698i
$$498$$ − 2.00000i − 0.0896221i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ 0 0
$$503$$ 8.00000i 0.356702i 0.983967 + 0.178351i $$0.0570763\pi$$
−0.983967 + 0.178351i $$0.942924\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ −16.0000 −0.711287
$$507$$ − 23.0000i − 1.02147i
$$508$$ 20.0000i 0.887357i
$$509$$ −38.0000 −1.68432 −0.842160 0.539227i $$-0.818716\pi$$
−0.842160 + 0.539227i $$0.818716\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 1.00000i 0.0441511i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −12.0000 −0.528271
$$517$$ 16.0000i 0.703679i
$$518$$ − 20.0000i − 0.878750i
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ 40.0000 1.75243 0.876216 0.481919i $$-0.160060\pi$$
0.876216 + 0.481919i $$0.160060\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ − 24.0000i − 1.04546i
$$528$$ 4.00000i 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 10.0000 0.433963
$$532$$ − 2.00000i − 0.0867110i
$$533$$ 24.0000i 1.03956i
$$534$$ 8.00000 0.346194
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 2.00000i 0.0863064i
$$538$$ 14.0000i 0.603583i
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ − 20.0000i − 0.859074i
$$543$$ 0 0
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ 12.0000 0.513553
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 4.00000i 0.170251i
$$553$$ 20.0000i 0.850487i
$$554$$ 28.0000 1.18961
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 10.0000i 0.423714i 0.977301 + 0.211857i $$0.0679510\pi$$
−0.977301 + 0.211857i $$0.932049\pi$$
$$558$$ − 6.00000i − 0.254000i
$$559$$ 72.0000 3.04528
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ − 4.00000i − 0.168730i
$$563$$ − 24.0000i − 1.01148i −0.862686 0.505740i $$-0.831220\pi$$
0.862686 0.505740i $$-0.168780\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 8.00000i 0.335673i
$$569$$ −12.0000 −0.503066 −0.251533 0.967849i $$-0.580935\pi$$
−0.251533 + 0.967849i $$0.580935\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ − 24.0000i − 1.00349i
$$573$$ − 8.00000i − 0.334205i
$$574$$ −8.00000 −0.333914
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 10.0000i 0.416305i 0.978096 + 0.208153i $$0.0667451\pi$$
−0.978096 + 0.208153i $$0.933255\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ −4.00000 −0.165948
$$582$$ 2.00000i 0.0829027i
$$583$$ 40.0000i 1.65663i
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −2.00000 −0.0826192
$$587$$ 22.0000i 0.908037i 0.890992 + 0.454019i $$0.150010\pi$$
−0.890992 + 0.454019i $$0.849990\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ 6.00000 0.247226
$$590$$ 0 0
$$591$$ −22.0000 −0.904959
$$592$$ 10.0000i 0.410997i
$$593$$ 8.00000i 0.328521i 0.986417 + 0.164260i $$0.0525237\pi$$
−0.986417 + 0.164260i $$0.947476\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 14.0000 0.573462
$$597$$ 4.00000i 0.163709i
$$598$$ − 24.0000i − 0.981433i
$$599$$ 28.0000 1.14405 0.572024 0.820237i $$-0.306158\pi$$
0.572024 + 0.820237i $$0.306158\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 24.0000i 0.978167i
$$603$$ − 12.0000i − 0.488678i
$$604$$ 2.00000 0.0813788
$$605$$ 0 0
$$606$$ −10.0000 −0.406222
$$607$$ − 28.0000i − 1.13648i −0.822861 0.568242i $$-0.807624\pi$$
0.822861 0.568242i $$-0.192376\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ −12.0000 −0.486265
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 4.00000i 0.161690i
$$613$$ 16.0000i 0.646234i 0.946359 + 0.323117i $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 8.00000 0.322329
$$617$$ − 48.0000i − 1.93241i −0.257780 0.966204i $$-0.582991\pi$$
0.257780 0.966204i $$-0.417009\pi$$
$$618$$ 12.0000i 0.482711i
$$619$$ −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
$$622$$ − 16.0000i − 0.641542i
$$623$$ − 16.0000i − 0.641026i
$$624$$ −6.00000 −0.240192
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ − 4.00000i − 0.159745i
$$628$$ − 12.0000i − 0.478852i
$$629$$ −40.0000 −1.59490
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ − 10.0000i − 0.397779i
$$633$$ − 8.00000i − 0.317971i
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ 18.0000i 0.713186i
$$638$$ 24.0000i 0.950169i
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ −20.0000 −0.789953 −0.394976 0.918691i $$-0.629247\pi$$
−0.394976 + 0.918691i $$0.629247\pi$$
$$642$$ 8.00000i 0.315735i
$$643$$ 20.0000i 0.788723i 0.918955 + 0.394362i $$0.129034\pi$$
−0.918955 + 0.394362i $$0.870966\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −40.0000 −1.57014
$$650$$ 0 0
$$651$$ −12.0000 −0.470317
$$652$$ − 16.0000i − 0.626608i
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 16.0000 0.625650
$$655$$ 0 0
$$656$$ 4.00000 0.156174
$$657$$ − 2.00000i − 0.0780274i
$$658$$ − 8.00000i − 0.311872i
$$659$$ −22.0000 −0.856998 −0.428499 0.903542i $$-0.640958\pi$$
−0.428499 + 0.903542i $$0.640958\pi$$
$$660$$ 0 0
$$661$$ −8.00000 −0.311164 −0.155582 0.987823i $$-0.549725\pi$$
−0.155582 + 0.987823i $$0.549725\pi$$
$$662$$ 12.0000i 0.466393i
$$663$$ − 24.0000i − 0.932083i
$$664$$ 2.00000 0.0776151
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 24.0000i 0.929284i
$$668$$ − 24.0000i − 0.928588i
$$669$$ 24.0000 0.927894
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ − 2.00000i − 0.0771517i
$$673$$ − 26.0000i − 1.00223i −0.865382 0.501113i $$-0.832924\pi$$
0.865382 0.501113i $$-0.167076\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ − 6.00000i − 0.230599i −0.993331 0.115299i $$-0.963217\pi$$
0.993331 0.115299i $$-0.0367827\pi$$
$$678$$ 14.0000i 0.537667i
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 24.0000i 0.919007i
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 26.0000i 0.991962i
$$688$$ − 12.0000i − 0.457496i
$$689$$ −60.0000 −2.28582
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ − 18.0000i − 0.684257i
$$693$$ 8.00000i 0.303895i
$$694$$ −2.00000 −0.0759190
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 16.0000i 0.606043i
$$698$$ 22.0000i 0.832712i
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ 10.0000 0.377695 0.188847 0.982006i $$-0.439525\pi$$
0.188847 + 0.982006i $$0.439525\pi$$
$$702$$ − 6.00000i − 0.226455i
$$703$$ − 10.0000i − 0.377157i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 20.0000i 0.752177i
$$708$$ 10.0000i 0.375823i
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 8.00000i 0.299813i
$$713$$ 24.0000i 0.898807i
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ −2.00000 −0.0747435
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ 24.0000 0.893807
$$722$$ − 1.00000i − 0.0372161i
$$723$$ 10.0000i 0.371904i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ 26.0000i 0.964287i 0.876092 + 0.482143i $$0.160142\pi$$
−0.876092 + 0.482143i $$0.839858\pi$$
$$728$$ 12.0000i 0.444750i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ − 2.00000i − 0.0739221i
$$733$$ 8.00000i 0.295487i 0.989026 + 0.147743i $$0.0472010\pi$$
−0.989026 + 0.147743i $$0.952799\pi$$
$$734$$ −22.0000 −0.812035
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 48.0000i 1.76810i
$$738$$ 4.00000i 0.147242i
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 6.00000 0.220416
$$742$$ − 20.0000i − 0.734223i
$$743$$ − 48.0000i − 1.76095i −0.474093 0.880475i $$-0.657224\pi$$
0.474093 0.880475i $$-0.342776\pi$$
$$744$$ 6.00000 0.219971
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ 2.00000i 0.0731762i
$$748$$ − 16.0000i − 0.585018i
$$749$$ 16.0000 0.584627
$$750$$ 0 0
$$751$$ −38.0000 −1.38664 −0.693320 0.720630i $$-0.743853\pi$$
−0.693320 + 0.720630i $$0.743853\pi$$
$$752$$ 4.00000i 0.145865i
$$753$$ 0 0
$$754$$ −36.0000 −1.31104
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 8.00000i 0.290765i 0.989376 + 0.145382i $$0.0464413\pi$$
−0.989376 + 0.145382i $$0.953559\pi$$
$$758$$ 8.00000i 0.290573i
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ −34.0000 −1.23250 −0.616250 0.787551i $$-0.711349\pi$$
−0.616250 + 0.787551i $$0.711349\pi$$
$$762$$ − 20.0000i − 0.724524i
$$763$$ − 32.0000i − 1.15848i
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ − 60.0000i − 2.16647i
$$768$$ 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
$$771$$ 18.0000 0.648254
$$772$$ 14.0000i 0.503871i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 20.0000i 0.717496i
$$778$$ − 10.0000i − 0.358517i
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ − 16.0000i − 0.572159i
$$783$$ 6.00000i 0.214423i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ − 22.0000i − 0.783718i
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ 28.0000 0.995565
$$792$$ − 4.00000i − 0.142134i
$$793$$ 12.0000i 0.426132i
$$794$$ −24.0000 −0.851728
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 2.00000i 0.0708436i 0.999372 + 0.0354218i $$0.0112775\pi$$
−0.999372 + 0.0354218i $$0.988723\pi$$
$$798$$ 2.00000i 0.0707992i
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ 24.0000i 0.847469i
$$803$$ 8.00000i 0.282314i
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ −36.0000 −1.26805
$$807$$ − 14.0000i − 0.492823i
$$808$$ − 10.0000i − 0.351799i
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ 0 0
$$811$$ −8.00000 −0.280918 −0.140459 0.990086i $$-0.544858\pi$$
−0.140459 + 0.990086i $$0.544858\pi$$
$$812$$ − 12.0000i − 0.421117i
$$813$$ 20.0000i 0.701431i
$$814$$ 40.0000 1.40200
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ 12.0000i 0.419827i
$$818$$ − 6.00000i − 0.209785i
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 0 0
$$823$$ 26.0000i 0.906303i 0.891434 + 0.453152i $$0.149700\pi$$
−0.891434 + 0.453152i $$0.850300\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ 20.0000 0.695889
$$827$$ 48.0000i 1.66912i 0.550914 + 0.834562i $$0.314279\pi$$
−0.550914 + 0.834562i $$0.685721\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ 40.0000 1.38926 0.694629 0.719368i $$-0.255569\pi$$
0.694629 + 0.719368i $$0.255569\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ − 6.00000i − 0.208013i
$$833$$ 12.0000i 0.415775i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 6.00000i 0.207390i
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 8.00000i 0.275698i
$$843$$ 4.00000i 0.137767i
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ − 10.0000i − 0.343604i
$$848$$ 10.0000i 0.343401i
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ 40.0000 1.37118
$$852$$ − 8.00000i − 0.274075i
$$853$$ 36.0000i 1.23262i 0.787505 + 0.616308i $$0.211372\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ 30.0000i 1.02478i 0.858753 + 0.512390i $$0.171240\pi$$
−0.858753 + 0.512390i $$0.828760\pi$$
$$858$$ 24.0000i 0.819346i
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 0 0
$$861$$ 8.00000 0.272639
$$862$$ 40.0000i 1.36241i
$$863$$ − 32.0000i − 1.08929i −0.838666 0.544646i $$-0.816664\pi$$
0.838666 0.544646i $$-0.183336\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −30.0000 −1.01944
$$867$$ 1.00000i 0.0339618i
$$868$$ − 12.0000i − 0.407307i
$$869$$ −40.0000 −1.35691
$$870$$ 0 0
$$871$$ −72.0000 −2.43963
$$872$$ 16.0000i 0.541828i
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ − 34.0000i − 1.14810i −0.818821 0.574049i $$-0.805372\pi$$
0.818821 0.574049i $$-0.194628\pi$$
$$878$$ − 38.0000i − 1.28244i
$$879$$ 2.00000 0.0674583
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ − 36.0000i − 1.21150i −0.795656 0.605748i $$-0.792874\pi$$
0.795656 0.605748i $$-0.207126\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ −6.00000 −0.201574
$$887$$ 48.0000i 1.61168i 0.592132 + 0.805841i $$0.298286\pi$$
−0.592132 + 0.805841i $$0.701714\pi$$
$$888$$ − 10.0000i − 0.335578i
$$889$$ −40.0000 −1.34156
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 24.0000i 0.803579i
$$893$$ − 4.00000i − 0.133855i
$$894$$ −14.0000 −0.468230
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 24.0000i 0.801337i
$$898$$ 12.0000i 0.400445i
$$899$$ 36.0000 1.20067
$$900$$ 0 0
$$901$$ −40.0000 −1.33259
$$902$$ − 16.0000i − 0.532742i
$$903$$ − 24.0000i − 0.798670i
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ −2.00000 −0.0664455
$$907$$ − 20.0000i − 0.664089i −0.943264 0.332045i $$-0.892262\pi$$
0.943264 0.332045i $$-0.107738\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ − 8.00000i − 0.264761i
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ −26.0000 −0.859064
$$917$$ − 24.0000i − 0.792550i
$$918$$ − 4.00000i − 0.132020i
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ − 18.0000i − 0.592798i
$$923$$ 48.0000i 1.57994i
$$924$$ −8.00000 −0.263181
$$925$$ 0 0
$$926$$ 18.0000 0.591517
$$927$$ − 12.0000i − 0.394132i
$$928$$ 6.00000i 0.196960i
$$929$$ −42.0000 −1.37798 −0.688988 0.724773i $$-0.741945\pi$$
−0.688988 + 0.724773i $$0.741945\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ − 12.0000i − 0.393073i
$$933$$ 16.0000i 0.523816i
$$934$$ −22.0000 −0.719862
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 26.0000i 0.849383i 0.905338 + 0.424691i $$0.139617\pi$$
−0.905338 + 0.424691i $$0.860383\pi$$
$$938$$ − 24.0000i − 0.783628i
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 12.0000i 0.390981i
$$943$$ − 16.0000i − 0.521032i
$$944$$ −10.0000 −0.325472
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ − 38.0000i − 1.23483i −0.786636 0.617417i $$-0.788179\pi$$
0.786636 0.617417i $$-0.211821\pi$$
$$948$$ 10.0000i 0.324785i
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 8.00000i 0.259281i
$$953$$ − 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
$$956$$ 0 0
$$957$$ − 24.0000i − 0.775810i
$$958$$ − 24.0000i − 0.775405i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 60.0000i 1.93448i
$$963$$ − 8.00000i − 0.257796i
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ − 18.0000i − 0.578841i −0.957202 0.289420i $$-0.906537\pi$$
0.957202 0.289420i $$-0.0934626\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 4.00000 0.128499
$$970$$ 0 0
$$971$$ −42.0000 −1.34784 −0.673922 0.738802i $$-0.735392\pi$$
−0.673922 + 0.738802i $$0.735392\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ − 8.00000i − 0.256468i
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 16.0000i 0.511624i
$$979$$ 32.0000 1.02272
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ − 12.0000i − 0.382935i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ −4.00000 −0.127515
$$985$$ 0 0
$$986$$ −24.0000 −0.764316
$$987$$ 8.00000i 0.254643i
$$988$$ 6.00000i 0.190885i
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ −2.00000 −0.0635321 −0.0317660 0.999495i $$-0.510113\pi$$
−0.0317660 + 0.999495i $$0.510113\pi$$
$$992$$ 6.00000i 0.190500i
$$993$$ − 12.0000i − 0.380808i
$$994$$ −16.0000 −0.507489
$$995$$ 0 0
$$996$$ −2.00000 −0.0633724
$$997$$ − 52.0000i − 1.64686i −0.567420 0.823428i $$-0.692059\pi$$
0.567420 0.823428i $$-0.307941\pi$$
$$998$$ − 4.00000i − 0.126618i
$$999$$ 10.0000 0.316386
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.r.799.1 2
5.2 odd 4 2850.2.a.bb.1.1 1
5.3 odd 4 570.2.a.a.1.1 1
5.4 even 2 inner 2850.2.d.r.799.2 2
15.2 even 4 8550.2.a.n.1.1 1
15.8 even 4 1710.2.a.q.1.1 1
20.3 even 4 4560.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.a.1.1 1 5.3 odd 4
1710.2.a.q.1.1 1 15.8 even 4
2850.2.a.bb.1.1 1 5.2 odd 4
2850.2.d.r.799.1 2 1.1 even 1 trivial
2850.2.d.r.799.2 2 5.4 even 2 inner
4560.2.a.t.1.1 1 20.3 even 4
8550.2.a.n.1.1 1 15.2 even 4