# Properties

 Label 2850.2.d.i.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: $$x^{2} + 1$$ 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.i.799.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -4.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -6.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} -1.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} -4.00000 q^{44} +4.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -1.00000i q^{57} +6.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -4.00000i q^{67} +2.00000i q^{68} +4.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} -6.00000i q^{73} +6.00000 q^{74} -1.00000 q^{76} -2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -12.0000i q^{83} -4.00000 q^{86} +6.00000i q^{87} +4.00000i q^{88} -10.0000 q^{89} -4.00000i q^{92} -4.00000i q^{93} +12.0000 q^{94} -1.00000 q^{96} -2.00000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 2 q^{19} + 2 q^{24} + 4 q^{26} - 12 q^{29} + 8 q^{31} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} - 8 q^{44} + 8 q^{46} + 14 q^{49} - 4 q^{51} + 2 q^{54} + 24 q^{59} - 4 q^{61} - 2 q^{64} - 8 q^{66} + 8 q^{69} + 16 q^{71} + 12 q^{74} - 2 q^{76} + 8 q^{79} + 2 q^{81} - 8 q^{86} - 20 q^{89} + 24 q^{94} - 2 q^{96} - 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 2 * q^19 + 2 * q^24 + 4 * q^26 - 12 * q^29 + 8 * q^31 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 - 8 * q^44 + 8 * q^46 + 14 * q^49 - 4 * q^51 + 2 * q^54 + 24 * q^59 - 4 * q^61 - 2 * q^64 - 8 * q^66 + 8 * q^69 + 16 * q^71 + 12 * q^74 - 2 * q^76 + 8 * q^79 + 2 * q^81 - 8 * q^86 - 20 * q^89 + 24 * q^94 - 2 * q^96 - 8 * q^99

## 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
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 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
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 4.00000i − 0.696311i
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 12.0000i 1.75038i 0.483779 + 0.875190i $$0.339264\pi$$
−0.483779 + 0.875190i $$0.660736\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ − 2.00000i − 0.277350i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 1.00000i − 0.132453i
$$58$$ 6.00000i 0.787839i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ − 4.00000i − 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 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
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 10.0000i − 1.10432i
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 6.00000i 0.643268i
$$88$$ 4.00000i 0.426401i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 4.00000i − 0.417029i
$$93$$ − 4.00000i − 0.414781i
$$94$$ 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 $$-0.0323753\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ − 2.00000i − 0.184900i
$$118$$ − 12.0000i − 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 2.00000i 0.181071i
$$123$$ − 10.0000i − 0.901670i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 4.00000i 0.354943i 0.984126 + 0.177471i $$0.0567917\pi$$
−0.984126 + 0.177471i $$0.943208\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ − 8.00000i − 0.671345i
$$143$$ 8.00000i 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ − 7.00000i − 0.577350i
$$148$$ − 6.00000i − 0.493197i
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ 2.00000i 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ − 4.00000i − 0.318223i
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 4.00000i 0.304997i
$$173$$ − 10.0000i − 0.760286i −0.924928 0.380143i $$-0.875875\pi$$
0.924928 0.380143i $$-0.124125\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ − 12.0000i − 0.901975i
$$178$$ 10.0000i 0.749532i
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ 2.00000i 0.147844i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ − 8.00000i − 0.585018i
$$188$$ − 12.0000i − 0.875190i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 20.0000 1.44715 0.723575 0.690246i $$-0.242498\pi$$
0.723575 + 0.690246i $$0.242498\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ − 2.00000i − 0.142494i −0.997459 0.0712470i $$-0.977302\pi$$
0.997459 0.0712470i $$-0.0226979\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ − 10.0000i − 0.703598i
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ − 4.00000i − 0.278019i
$$208$$ 2.00000i 0.138675i
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ − 8.00000i − 0.548151i
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ − 6.00000i − 0.406371i
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ − 6.00000i − 0.402694i
$$223$$ 12.0000i 0.803579i 0.915732 + 0.401790i $$0.131612\pi$$
−0.915732 + 0.401790i $$0.868388\pi$$
$$224$$ 0 0
$$225$$ 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
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ − 4.00000i − 0.259828i
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ 2.00000i 0.127257i
$$248$$ 4.00000i 0.254000i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ 16.0000i 1.00591i
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 12.0000i 0.741362i
$$263$$ − 4.00000i − 0.246651i −0.992366 0.123325i $$-0.960644\pi$$
0.992366 0.123325i $$-0.0393559\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 10.0000i 0.611990i
$$268$$ 4.00000i 0.244339i
$$269$$ 2.00000 0.121942 0.0609711 0.998140i $$-0.480580\pi$$
0.0609711 + 0.998140i $$0.480580\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ − 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
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 6.00000i 0.351123i
$$293$$ − 2.00000i − 0.116841i −0.998292 0.0584206i $$-0.981394\pi$$
0.998292 0.0584206i $$-0.0186065\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 4.00000i 0.232104i
$$298$$ 18.0000i 1.04271i
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 12.0000i 0.690522i
$$303$$ − 10.0000i − 0.574485i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −28.0000 −1.58773 −0.793867 0.608091i $$-0.791935\pi$$
−0.793867 + 0.608091i $$0.791935\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 26.0000i 1.46031i 0.683284 + 0.730153i $$0.260551\pi$$
−0.683284 + 0.730153i $$0.739449\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ − 2.00000i − 0.111283i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ − 6.00000i − 0.331801i
$$328$$ 10.0000i 0.552158i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ − 6.00000i − 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 26.0000i − 1.41631i −0.706057 0.708155i $$-0.749528\pi$$
0.706057 0.708155i $$-0.250472\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 1.00000i 0.0540738i
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ −10.0000 −0.537603
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ − 4.00000i − 0.213201i
$$353$$ 26.0000i 1.38384i 0.721974 + 0.691920i $$0.243235\pi$$
−0.721974 + 0.691920i $$0.756765\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ 20.0000i 1.05703i
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 18.0000i − 0.946059i
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ − 24.0000i − 1.25279i −0.779506 0.626395i $$-0.784530\pi$$
0.779506 0.626395i $$-0.215470\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 4.00000i 0.207390i
$$373$$ − 22.0000i − 1.13912i −0.821951 0.569558i $$-0.807114\pi$$
0.821951 0.569558i $$-0.192886\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ − 12.0000i − 0.618031i
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ − 20.0000i − 1.02329i
$$383$$ 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
$$389$$ 22.0000 1.11544 0.557722 0.830028i $$-0.311675\pi$$
0.557722 + 0.830028i $$0.311675\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 7.00000i 0.353553i
$$393$$ 12.0000i 0.605320i
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 18.0000i 0.903394i 0.892171 + 0.451697i $$0.149181\pi$$
−0.892171 + 0.451697i $$0.850819\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ 8.00000i 0.398508i
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000i 1.18964i
$$408$$ − 2.00000i − 0.0990148i
$$409$$ −18.0000 −0.890043 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ 4.00000i 0.197066i
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ − 20.0000i − 0.979404i
$$418$$ − 4.00000i − 0.195646i
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ − 12.0000i − 0.583460i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ − 4.00000i − 0.193347i
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ 4.00000i 0.191346i
$$438$$ 6.00000i 0.286691i
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ − 4.00000i − 0.190261i
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ 12.0000 0.568216
$$447$$ 18.0000i 0.851371i
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 40.0000 1.88353
$$452$$ 14.0000i 0.658505i
$$453$$ 12.0000i 0.563809i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ 6.00000i 0.280668i 0.990104 + 0.140334i $$0.0448177\pi$$
−0.990104 + 0.140334i $$0.955182\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −38.0000 −1.76984 −0.884918 0.465746i $$-0.845786\pi$$
−0.884918 + 0.465746i $$0.845786\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ 28.0000i 1.29569i 0.761774 + 0.647843i $$0.224329\pi$$
−0.761774 + 0.647843i $$0.775671\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 12.0000i 0.552345i
$$473$$ − 16.0000i − 0.735681i
$$474$$ −4.00000 −0.183726
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 12.0000i 0.548867i
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ − 26.0000i − 1.18427i
$$483$$ 0 0
$$484$$ −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
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 10.0000i 0.450835i
$$493$$ 12.0000i 0.540453i
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 12.0000i 0.537733i
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 20.0000i 0.892644i
$$503$$ − 36.0000i − 1.60516i −0.596544 0.802580i $$-0.703460\pi$$
0.596544 0.802580i $$-0.296540\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ − 9.00000i − 0.399704i
$$508$$ − 4.00000i − 0.177471i
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 1.00000i 0.0441511i
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 48.0000i 2.11104i
$$518$$ 0 0
$$519$$ −10.0000 −0.438951
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ 28.0000i 1.22435i 0.790721 + 0.612177i $$0.209706\pi$$
−0.790721 + 0.612177i $$0.790294\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −4.00000 −0.174408
$$527$$ − 8.00000i − 0.348485i
$$528$$ − 4.00000i − 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 20.0000i 0.866296i
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 20.0000i 0.863064i
$$538$$ − 2.00000i − 0.0862261i
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ − 8.00000i − 0.343629i
$$543$$ − 18.0000i − 0.772454i
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ 10.0000i 0.427179i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 4.00000i 0.170251i
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ − 42.0000i − 1.77960i −0.456354 0.889799i $$-0.650845\pi$$
0.456354 0.889799i $$-0.349155\pi$$
$$558$$ 4.00000i 0.169334i
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 6.00000i 0.253095i
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 0 0
$$568$$ 8.00000i 0.335673i
$$569$$ 38.0000 1.59304 0.796521 0.604610i $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ − 8.00000i − 0.334497i
$$573$$ − 20.0000i − 0.835512i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000i 1.24892i 0.781058 + 0.624458i $$0.214680\pi$$
−0.781058 + 0.624458i $$0.785320\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ 2.00000 0.0831172
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 2.00000i 0.0829027i
$$583$$ 24.0000i 0.993978i
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −2.00000 −0.0826192
$$587$$ 28.0000i 1.15568i 0.816149 + 0.577842i $$0.196105\pi$$
−0.816149 + 0.577842i $$0.803895\pi$$
$$588$$ 7.00000i 0.288675i
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 6.00000i 0.246598i
$$593$$ 10.0000i 0.410651i 0.978694 + 0.205325i $$0.0658253\pi$$
−0.978694 + 0.205325i $$0.934175\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 0 0
$$598$$ 8.00000i 0.327144i
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 12.0000 0.488273
$$605$$ 0 0
$$606$$ −10.0000 −0.406222
$$607$$ − 4.00000i − 0.162355i −0.996700 0.0811775i $$-0.974132\pi$$
0.996700 0.0811775i $$-0.0258681\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ − 2.00000i − 0.0808452i
$$613$$ 22.0000i 0.888572i 0.895885 + 0.444286i $$0.146543\pi$$
−0.895885 + 0.444286i $$0.853457\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 14.0000i 0.563619i 0.959470 + 0.281809i $$0.0909346\pi$$
−0.959470 + 0.281809i $$0.909065\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ −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
$$622$$ 28.0000i 1.12270i
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ − 4.00000i − 0.159745i
$$628$$ − 2.00000i − 0.0798087i
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 4.00000i 0.159111i
$$633$$ 4.00000i 0.158986i
$$634$$ 26.0000 1.03259
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 14.0000i 0.554700i
$$638$$ 24.0000i 0.950169i
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ − 4.00000i − 0.157867i
$$643$$ − 4.00000i − 0.157745i −0.996885 0.0788723i $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −2.00000 −0.0786889
$$647$$ − 36.0000i − 1.41531i −0.706560 0.707653i $$-0.749754\pi$$
0.706560 0.707653i $$-0.250246\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ − 6.00000i − 0.234798i −0.993085 0.117399i $$-0.962544\pi$$
0.993085 0.117399i $$-0.0374557\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ − 4.00000i − 0.155347i
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ − 24.0000i − 0.929284i
$$668$$ 0 0
$$669$$ 12.0000 0.463947
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ − 14.0000i − 0.539660i −0.962908 0.269830i $$-0.913032\pi$$
0.962908 0.269830i $$-0.0869676\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 42.0000i 1.61419i 0.590421 + 0.807096i $$0.298962\pi$$
−0.590421 + 0.807096i $$0.701038\pi$$
$$678$$ 14.0000i 0.537667i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 20.0000 0.766402
$$682$$ − 16.0000i − 0.612672i
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 10.0000i − 0.381524i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ 10.0000i 0.380143i
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ − 20.0000i − 0.757554i
$$698$$ − 10.0000i − 0.378506i
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ 6.00000i 0.226294i
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ 26.0000 0.978523
$$707$$ 0 0
$$708$$ 12.0000i 0.450988i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ − 10.0000i − 0.374766i
$$713$$ 16.0000i 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ 12.0000i 0.448148i
$$718$$ 12.0000i 0.447836i
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 1.00000i − 0.0372161i
$$723$$ − 26.0000i − 0.966950i
$$724$$ −18.0000 −0.668965
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ − 40.0000i − 1.48352i −0.670667 0.741759i $$-0.733992\pi$$
0.670667 0.741759i $$-0.266008\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ − 2.00000i − 0.0739221i
$$733$$ 46.0000i 1.69905i 0.527549 + 0.849524i $$0.323111\pi$$
−0.527549 + 0.849524i $$0.676889\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ − 16.0000i − 0.589368i
$$738$$ 10.0000i 0.368105i
$$739$$ −52.0000 −1.91285 −0.956425 0.291977i $$-0.905687\pi$$
−0.956425 + 0.291977i $$0.905687\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ 16.0000i 0.586983i 0.955962 + 0.293492i $$0.0948173\pi$$
−0.955962 + 0.293492i $$0.905183\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 12.0000i 0.439057i
$$748$$ 8.00000i 0.292509i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 12.0000i 0.437595i
$$753$$ 20.0000i 0.728841i
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 38.0000i − 1.38113i −0.723269 0.690567i $$-0.757361\pi$$
0.723269 0.690567i $$-0.242639\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ − 4.00000i − 0.144905i
$$763$$ 0 0
$$764$$ −20.0000 −0.723575
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 24.0000i 0.866590i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 46.0000 1.65880 0.829401 0.558653i $$-0.188682\pi$$
0.829401 + 0.558653i $$0.188682\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ − 2.00000i − 0.0719816i
$$773$$ 30.0000i 1.07903i 0.841978 + 0.539513i $$0.181391\pi$$
−0.841978 + 0.539513i $$0.818609\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ − 22.0000i − 0.788738i
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ − 8.00000i − 0.286079i
$$783$$ − 6.00000i − 0.214423i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 52.0000i 1.85360i 0.375555 + 0.926800i $$0.377452\pi$$
−0.375555 + 0.926800i $$0.622548\pi$$
$$788$$ 2.00000i 0.0712470i
$$789$$ −4.00000 −0.142404
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 4.00000i − 0.142134i
$$793$$ − 4.00000i − 0.142044i
$$794$$ 18.0000 0.638796
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 54.0000i − 1.91278i −0.292096 0.956389i $$-0.594353\pi$$
0.292096 0.956389i $$-0.405647\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 14.0000i 0.494357i
$$803$$ − 24.0000i − 0.846942i
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ − 2.00000i − 0.0704033i
$$808$$ 10.0000i 0.351799i
$$809$$ −34.0000 −1.19538 −0.597688 0.801729i $$-0.703914\pi$$
−0.597688 + 0.801729i $$0.703914\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ − 4.00000i − 0.139942i
$$818$$ 18.0000i 0.629355i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ 10.0000i 0.348790i
$$823$$ 8.00000i 0.278862i 0.990232 + 0.139431i $$0.0445274\pi$$
−0.990232 + 0.139431i $$0.955473\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ − 2.00000i − 0.0693375i
$$833$$ − 14.0000i − 0.485071i
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ −4.00000 −0.138343
$$837$$ 4.00000i 0.138260i
$$838$$ 12.0000i 0.414533i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 26.0000i − 0.896019i
$$843$$ 6.00000i 0.206651i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 0 0
$$848$$ 6.00000i 0.206041i
$$849$$ −28.0000 −0.960958
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ 8.00000i 0.274075i
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ − 26.0000i − 0.888143i −0.895991 0.444072i $$-0.853534\pi$$
0.895991 0.444072i $$-0.146466\pi$$
$$858$$ − 8.00000i − 0.273115i
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 8.00000i − 0.272481i
$$863$$ 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
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 6.00000i 0.203186i
$$873$$ 2.00000i 0.0676897i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ 46.0000i 1.55331i 0.629926 + 0.776655i $$0.283085\pi$$
−0.629926 + 0.776655i $$0.716915\pi$$
$$878$$ 12.0000i 0.404980i
$$879$$ −2.00000 −0.0674583
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 48.0000i 1.61168i 0.592132 + 0.805841i $$0.298286\pi$$
−0.592132 + 0.805841i $$0.701714\pi$$
$$888$$ 6.00000i 0.201347i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ − 12.0000i − 0.401790i
$$893$$ 12.0000i 0.401565i
$$894$$ 18.0000 0.602010
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 8.00000i 0.267112i
$$898$$ − 30.0000i − 1.00111i
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ − 40.0000i − 1.33185i
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 12.0000 0.398673
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ − 20.0000i − 0.663723i
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ − 48.0000i − 1.58857i
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ − 2.00000i − 0.0660098i
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 38.0000i 1.25146i
$$923$$ 16.0000i 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 8.00000 0.262896
$$927$$ 4.00000i 0.131377i
$$928$$ 6.00000i 0.196960i
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 7.00000 0.229416
$$932$$ − 18.0000i − 0.589610i
$$933$$ 28.0000i 0.916679i
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ − 10.0000i − 0.326686i −0.986569 0.163343i $$-0.947772\pi$$
0.986569 0.163343i $$-0.0522277\pi$$
$$938$$ 0 0
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 40.0000i 1.30258i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 4.00000i 0.129914i
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 26.0000 0.843108
$$952$$ 0 0
$$953$$ 42.0000i 1.36051i 0.732974 + 0.680257i $$0.238132\pi$$
−0.732974 + 0.680257i $$0.761868\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 24.0000i 0.775810i
$$958$$ 20.0000i 0.646171i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 12.0000i 0.386896i
$$963$$ − 4.00000i − 0.128898i
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000i 1.02905i 0.857475 + 0.514525i $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 0 0
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 2.00000i − 0.0639857i −0.999488 0.0319928i $$-0.989815\pi$$
0.999488 0.0319928i $$-0.0101854\pi$$
$$978$$ 4.00000i 0.127906i
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ 36.0000i 1.14881i
$$983$$ 56.0000i 1.78612i 0.449935 + 0.893061i $$0.351447\pi$$
−0.449935 + 0.893061i $$0.648553\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ − 2.00000i − 0.0636285i
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ − 4.00000i − 0.127000i
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ − 38.0000i − 1.20347i −0.798695 0.601736i $$-0.794476\pi$$
0.798695 0.601736i $$-0.205524\pi$$
$$998$$ − 36.0000i − 1.13956i
$$999$$ −6.00000 −0.189832
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.i.799.1 2
5.2 odd 4 570.2.a.g.1.1 1
5.3 odd 4 2850.2.a.m.1.1 1
5.4 even 2 inner 2850.2.d.i.799.2 2
15.2 even 4 1710.2.a.i.1.1 1
15.8 even 4 8550.2.a.x.1.1 1
20.7 even 4 4560.2.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.g.1.1 1 5.2 odd 4
1710.2.a.i.1.1 1 15.2 even 4
2850.2.a.m.1.1 1 5.3 odd 4
2850.2.d.i.799.1 2 1.1 even 1 trivial
2850.2.d.i.799.2 2 5.4 even 2 inner
4560.2.a.s.1.1 1 20.7 even 4
8550.2.a.x.1.1 1 15.8 even 4