# Properties

 Label 1850.2.b.b.149.2 Level $1850$ Weight $2$ Character 1850.149 Analytic conductor $14.772$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ 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 370) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.149 Dual form 1850.2.b.b.149.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -2.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -4.00000 q^{21} +2.00000 q^{24} +2.00000 q^{26} +4.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} +1.00000i q^{37} -2.00000i q^{38} +4.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -6.00000i q^{47} +2.00000i q^{48} +3.00000 q^{49} -12.0000 q^{51} +2.00000i q^{52} -6.00000i q^{53} -4.00000 q^{54} +2.00000 q^{56} -4.00000i q^{57} -6.00000i q^{58} +6.00000 q^{59} -10.0000 q^{61} -10.0000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +2.00000i q^{67} -6.00000i q^{68} +1.00000i q^{72} -2.00000i q^{73} -1.00000 q^{74} +2.00000 q^{76} +4.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} -6.00000i q^{82} +6.00000i q^{83} +4.00000 q^{84} -4.00000 q^{86} -12.0000i q^{87} +6.00000 q^{89} +4.00000 q^{91} -20.0000i q^{93} +6.00000 q^{94} -2.00000 q^{96} +2.00000i q^{97} +3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 4q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 4q^{6} - 2q^{9} - 4q^{14} + 2q^{16} - 4q^{19} - 8q^{21} + 4q^{24} + 4q^{26} - 12q^{29} - 20q^{31} - 12q^{34} + 2q^{36} + 8q^{39} - 12q^{41} + 6q^{49} - 24q^{51} - 8q^{54} + 4q^{56} + 12q^{59} - 20q^{61} - 2q^{64} - 2q^{74} + 4q^{76} + 20q^{79} - 22q^{81} + 8q^{84} - 8q^{86} + 12q^{89} + 8q^{91} + 12q^{94} - 4q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ 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 $$0.5\pi$$
$$12$$ − 2.00000i − 0.577350i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 4.00000i 0.769800i
$$28$$ − 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
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.00000i 0.164399i
$$38$$ − 2.00000i − 0.324443i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 6.00000i − 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 2.00000i 0.288675i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −12.0000 −1.68034
$$52$$ 2.00000i 0.277350i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ − 4.00000i − 0.529813i
$$58$$ − 6.00000i − 0.787839i
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ − 10.0000i − 1.27000i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ 4.00000i 0.452911i
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ − 6.00000i − 0.662589i
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ − 12.0000i − 1.28654i
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ − 20.0000i − 2.07390i
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ −2.00000 −0.204124
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ − 12.0000i − 1.18818i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 6.00000i 0.580042i 0.957020 + 0.290021i $$0.0936623\pi$$
−0.957020 + 0.290021i $$0.906338\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 2.00000i 0.188982i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 2.00000i 0.184900i
$$118$$ 6.00000i 0.552345i
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 10.0000i − 0.905357i
$$123$$ − 12.0000i − 1.08200i
$$124$$ 10.0000 0.898027
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ − 4.00000i − 0.346844i
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 6.00000i 0.494872i
$$148$$ − 1.00000i − 0.0821995i
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 2.00000i 0.162221i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ − 10.0000i − 0.798087i −0.916932 0.399043i $$-0.869342\pi$$
0.916932 0.399043i $$-0.130658\pi$$
$$158$$ 10.0000i 0.795557i
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 11.0000i − 0.864242i
$$163$$ 16.0000i 1.25322i 0.779334 + 0.626608i $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 4.00000i 0.308607i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ − 4.00000i − 0.304997i
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ 12.0000 0.909718
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000i 0.901975i
$$178$$ 6.00000i 0.449719i
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 4.00000i 0.296500i
$$183$$ − 20.0000i − 1.47844i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 20.0000 1.46647
$$187$$ 0 0
$$188$$ 6.00000i 0.437595i
$$189$$ −8.00000 −0.581914
$$190$$ 0 0
$$191$$ −6.00000 −0.434145 −0.217072 0.976156i $$-0.569651\pi$$
−0.217072 + 0.976156i $$0.569651\pi$$
$$192$$ − 2.00000i − 0.144338i
$$193$$ − 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
$$199$$ 22.0000 1.55954 0.779769 0.626067i $$-0.215336\pi$$
0.779769 + 0.626067i $$0.215336\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 18.0000i 1.26648i
$$203$$ − 12.0000i − 0.842235i
$$204$$ 12.0000 0.840168
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ − 2.00000i − 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ −6.00000 −0.410152
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ − 20.0000i − 1.35769i
$$218$$ − 14.0000i − 0.948200i
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ − 2.00000i − 0.134231i
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 24.0000i 1.59294i 0.604681 + 0.796468i $$0.293301\pi$$
−0.604681 + 0.796468i $$0.706699\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 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 $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 20.0000i 1.29914i
$$238$$ − 12.0000i − 0.777844i
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ − 10.0000i − 0.641500i
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 12.0000 0.765092
$$247$$ 4.00000i 0.254514i
$$248$$ 10.0000i 0.635001i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 0 0
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ − 6.00000i − 0.370681i
$$263$$ 6.00000i 0.369976i 0.982741 + 0.184988i $$0.0592246\pi$$
−0.982741 + 0.184988i $$0.940775\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ 12.0000i 0.734388i
$$268$$ − 2.00000i − 0.122169i
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 8.00000i 0.484182i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 10.0000 0.598684
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 12.0000i 0.714590i
$$283$$ 16.0000i 0.951101i 0.879688 + 0.475551i $$0.157751\pi$$
−0.879688 + 0.475551i $$0.842249\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 12.0000i − 0.708338i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −4.00000 −0.234484
$$292$$ 2.00000i 0.117041i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ −6.00000 −0.349927
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ − 18.0000i − 1.04271i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 20.0000i 1.15087i
$$303$$ 36.0000i 2.06815i
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 26.0000i 1.48390i 0.670456 + 0.741949i $$0.266098\pi$$
−0.670456 + 0.741949i $$0.733902\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 12.0000i 0.672927i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ − 12.0000i − 0.667698i
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ − 28.0000i − 1.54840i
$$328$$ 6.00000i 0.331295i
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ − 6.00000i − 0.329293i
$$333$$ − 1.00000i − 0.0547997i
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 12.0000 0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 2.00000i 0.108148i
$$343$$ 20.0000i 1.07990i
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 12.0000i 0.643268i
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ 0 0
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ − 24.0000i − 1.27021i
$$358$$ − 6.00000i − 0.317110i
$$359$$ −36.0000 −1.90001 −0.950004 0.312239i $$-0.898921\pi$$
−0.950004 + 0.312239i $$0.898921\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ − 22.0000i − 1.15629i
$$363$$ − 22.0000i − 1.15470i
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 20.0000 1.04542
$$367$$ 2.00000i 0.104399i 0.998637 + 0.0521996i $$0.0166232\pi$$
−0.998637 + 0.0521996i $$0.983377\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 20.0000i 1.03695i
$$373$$ 34.0000i 1.76045i 0.474554 + 0.880227i $$0.342610\pi$$
−0.474554 + 0.880227i $$0.657390\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ 12.0000i 0.618031i
$$378$$ − 8.00000i − 0.411476i
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ − 6.00000i − 0.306987i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ − 4.00000i − 0.203331i
$$388$$ − 2.00000i − 0.101535i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 3.00000i − 0.151523i
$$393$$ − 12.0000i − 0.605320i
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 14.0000i 0.702640i 0.936255 + 0.351320i $$0.114267\pi$$
−0.936255 + 0.351320i $$0.885733\pi$$
$$398$$ 22.0000i 1.10276i
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ − 4.00000i − 0.199502i
$$403$$ 20.0000i 0.996271i
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 0 0
$$408$$ 12.0000i 0.594089i
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ − 4.00000i − 0.197066i
$$413$$ 12.0000i 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 8.00000i 0.391762i
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ 6.00000i 0.291730i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 20.0000i − 0.967868i
$$428$$ − 6.00000i − 0.290021i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ − 2.00000i − 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ 20.0000 0.960031
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ 4.00000i 0.191127i
$$439$$ 22.0000 1.05000 0.525001 0.851101i $$-0.324065\pi$$
0.525001 + 0.851101i $$0.324065\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 12.0000i 0.570782i
$$443$$ − 6.00000i − 0.285069i −0.989790 0.142534i $$-0.954475\pi$$
0.989790 0.142534i $$-0.0455251\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ − 36.0000i − 1.70274i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000i 0.282216i
$$453$$ 40.0000i 1.87936i
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 26.0000i 1.21623i 0.793849 + 0.608114i $$0.208074\pi$$
−0.793849 + 0.608114i $$0.791926\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ −24.0000 −1.12022
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ 40.0000i 1.85896i 0.368875 + 0.929479i $$0.379743\pi$$
−0.368875 + 0.929479i $$0.620257\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 20.0000 0.921551
$$472$$ − 6.00000i − 0.276172i
$$473$$ 0 0
$$474$$ −20.0000 −0.918630
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ 6.00000i 0.274721i
$$478$$ − 6.00000i − 0.274434i
$$479$$ 18.0000 0.822441 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ − 22.0000i − 1.00207i
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 10.0000 0.453609
$$487$$ 20.0000i 0.906287i 0.891438 + 0.453143i $$0.149697\pi$$
−0.891438 + 0.453143i $$0.850303\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ −32.0000 −1.44709
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 12.0000i 0.541002i
$$493$$ − 36.0000i − 1.62136i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ 0 0
$$498$$ − 12.0000i − 0.537733i
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ 18.0000i 0.803379i
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 18.0000i 0.799408i
$$508$$ − 2.00000i − 0.0887357i
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 1.00000i 0.0441942i
$$513$$ − 8.00000i − 0.353209i
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ − 2.00000i − 0.0878750i
$$519$$ −36.0000 −1.58022
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ −6.00000 −0.261612
$$527$$ − 60.0000i − 2.61364i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 4.00000i 0.173422i
$$533$$ 12.0000i 0.519778i
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ − 12.0000i − 0.517838i
$$538$$ 6.00000i 0.258678i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ − 44.0000i − 1.88822i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ 44.0000i 1.88130i 0.339372 + 0.940652i $$0.389785\pi$$
−0.339372 + 0.940652i $$0.610215\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 20.0000i 0.850487i
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 30.0000i − 1.27114i −0.772043 0.635570i $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ 10.0000i 0.423334i
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 18.0000i 0.759284i
$$563$$ − 36.0000i − 1.51722i −0.651546 0.758610i $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ − 22.0000i − 0.923913i
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ − 12.0000i − 0.501307i
$$574$$ 12.0000 0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 38.0000i 1.58196i 0.611842 + 0.790980i $$0.290429\pi$$
−0.611842 + 0.790980i $$0.709571\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ − 4.00000i − 0.165805i
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ − 6.00000i − 0.247436i
$$589$$ 20.0000 0.824086
$$590$$ 0 0
$$591$$ 36.0000 1.48084
$$592$$ 1.00000i 0.0410997i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ 44.0000i 1.80080i
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ − 8.00000i − 0.326056i
$$603$$ − 2.00000i − 0.0814463i
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ −36.0000 −1.46240
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ − 2.00000i − 0.0811107i
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ 6.00000i 0.242536i
$$613$$ 34.0000i 1.37325i 0.727013 + 0.686624i $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ −26.0000 −1.04927
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 18.0000i − 0.721734i
$$623$$ 12.0000i 0.480770i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ 10.0000i 0.399043i
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 14.0000 0.557331 0.278666 0.960388i $$-0.410108\pi$$
0.278666 + 0.960388i $$0.410108\pi$$
$$632$$ − 10.0000i − 0.397779i
$$633$$ 40.0000i 1.58986i
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ −12.0000 −0.475831
$$637$$ − 6.00000i − 0.237729i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ − 32.0000i − 1.26196i −0.775800 0.630978i $$-0.782654\pi$$
0.775800 0.630978i $$-0.217346\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ 48.0000i 1.88707i 0.331266 + 0.943537i $$0.392524\pi$$
−0.331266 + 0.943537i $$0.607476\pi$$
$$648$$ 11.0000i 0.432121i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 40.0000 1.56772
$$652$$ − 16.0000i − 0.626608i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 28.0000 1.09489
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 2.00000i 0.0780274i
$$658$$ 12.0000i 0.467809i
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ − 10.0000i − 0.388661i
$$663$$ 24.0000i 0.932083i
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 1.00000 0.0387492
$$667$$ 0 0
$$668$$ − 12.0000i − 0.464294i
$$669$$ 28.0000 1.08254
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 4.00000i − 0.154303i
$$673$$ − 26.0000i − 1.00223i −0.865382 0.501113i $$-0.832924\pi$$
0.865382 0.501113i $$-0.167076\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 42.0000i − 1.61419i −0.590421 0.807096i $$-0.701038\pi$$
0.590421 0.807096i $$-0.298962\pi$$
$$678$$ 12.0000i 0.460857i
$$679$$ −4.00000 −0.153506
$$680$$ 0 0
$$681$$ −48.0000 −1.83936
$$682$$ 0 0
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ 20.0000i 0.763048i
$$688$$ 4.00000i 0.152499i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ − 18.0000i − 0.684257i
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ −12.0000 −0.454859
$$697$$ − 36.0000i − 1.36360i
$$698$$ − 26.0000i − 0.984115i
$$699$$ 36.0000 1.36165
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 8.00000i 0.301941i
$$703$$ − 2.00000i − 0.0754314i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 36.0000i 1.35392i
$$708$$ − 12.0000i − 0.450988i
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ − 6.00000i − 0.224860i
$$713$$ 0 0
$$714$$ 24.0000 0.898177
$$715$$ 0 0
$$716$$ 6.00000 0.224231
$$717$$ − 12.0000i − 0.448148i
$$718$$ − 36.0000i − 1.34351i
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ − 15.0000i − 0.558242i
$$723$$ − 44.0000i − 1.63638i
$$724$$ 22.0000 0.817624
$$725$$ 0 0
$$726$$ 22.0000 0.816497
$$727$$ 32.0000i 1.18681i 0.804902 + 0.593407i $$0.202218\pi$$
−0.804902 + 0.593407i $$0.797782\pi$$
$$728$$ − 4.00000i − 0.148250i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 20.0000i 0.739221i
$$733$$ − 14.0000i − 0.517102i −0.965998 0.258551i $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ −2.00000 −0.0738213
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 6.00000i 0.220863i
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 12.0000i 0.440534i
$$743$$ − 18.0000i − 0.660356i −0.943919 0.330178i $$-0.892891\pi$$
0.943919 0.330178i $$-0.107109\pi$$
$$744$$ −20.0000 −0.733236
$$745$$ 0 0
$$746$$ −34.0000 −1.24483
$$747$$ − 6.00000i − 0.219529i
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ − 6.00000i − 0.218797i
$$753$$ 36.0000i 1.31191i
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 8.00000 0.290957
$$757$$ − 10.0000i − 0.363456i −0.983349 0.181728i $$-0.941831\pi$$
0.983349 0.181728i $$-0.0581691\pi$$
$$758$$ 16.0000i 0.581146i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ − 4.00000i − 0.144905i
$$763$$ − 28.0000i − 1.01367i
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 12.0000i − 0.433295i
$$768$$ 2.00000i 0.0721688i
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ 2.00000i 0.0719816i
$$773$$ 18.0000i 0.647415i 0.946157 + 0.323708i $$0.104929\pi$$
−0.946157 + 0.323708i $$0.895071\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ − 4.00000i − 0.143499i
$$778$$ − 6.00000i − 0.215110i
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ − 24.0000i − 0.857690i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ − 10.0000i − 0.356462i −0.983989 0.178231i $$-0.942963\pi$$
0.983989 0.178231i $$-0.0570374\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ 20.0000i 0.710221i
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ −22.0000 −0.779769
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ 8.00000i 0.283197i
$$799$$ 36.0000 1.27359
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 6.00000i − 0.211867i
$$803$$ 0 0
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ −20.0000 −0.704470
$$807$$ 12.0000i 0.422420i
$$808$$ − 18.0000i − 0.633238i
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 12.0000i 0.421117i
$$813$$ − 32.0000i − 1.12229i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −12.0000 −0.420084
$$817$$ − 8.00000i − 0.279885i
$$818$$ − 2.00000i − 0.0699284i
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ 34.0000i 1.18517i 0.805510 + 0.592583i $$0.201892\pi$$
−0.805510 + 0.592583i $$0.798108\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ 24.0000i 0.834562i 0.908778 + 0.417281i $$0.137017\pi$$
−0.908778 + 0.417281i $$0.862983\pi$$
$$828$$ 0 0
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ −52.0000 −1.80386
$$832$$ 2.00000i 0.0693375i
$$833$$ 18.0000i 0.623663i
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 40.0000i − 1.38260i
$$838$$ − 24.0000i − 0.829066i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 10.0000i − 0.344623i
$$843$$ 36.0000i 1.23991i
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ −6.00000 −0.206284
$$847$$ − 22.0000i − 0.755929i
$$848$$ − 6.00000i − 0.206041i
$$849$$ −32.0000 −1.09824
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000i 0.342393i 0.985237 + 0.171197i $$0.0547634\pi$$
−0.985237 + 0.171197i $$0.945237\pi$$
$$854$$ 20.0000 0.684386
$$855$$ 0 0
$$856$$ 6.00000 0.205076
$$857$$ 42.0000i 1.43469i 0.696717 + 0.717346i $$0.254643\pi$$
−0.696717 + 0.717346i $$0.745357\pi$$
$$858$$ 0 0
$$859$$ −50.0000 −1.70598 −0.852989 0.521929i $$-0.825213\pi$$
−0.852989 + 0.521929i $$0.825213\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ 30.0000i 1.02180i
$$863$$ − 54.0000i − 1.83818i −0.394046 0.919091i $$-0.628925\pi$$
0.394046 0.919091i $$-0.371075\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ − 38.0000i − 1.29055i
$$868$$ 20.0000i 0.678844i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 14.0000i 0.474100i
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ − 10.0000i − 0.337676i −0.985644 0.168838i $$-0.945999\pi$$
0.985644 0.168838i $$-0.0540015\pi$$
$$878$$ 22.0000i 0.742464i
$$879$$ 12.0000 0.404750
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ − 56.0000i − 1.88455i −0.334840 0.942275i $$-0.608682\pi$$
0.334840 0.942275i $$-0.391318\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 6.00000 0.201574
$$887$$ − 42.0000i − 1.41022i −0.709097 0.705111i $$-0.750897\pi$$
0.709097 0.705111i $$-0.249103\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ −4.00000 −0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 14.0000i 0.468755i
$$893$$ 12.0000i 0.401565i
$$894$$ 36.0000 1.20402
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ 30.0000i 1.00111i
$$899$$ 60.0000 2.00111
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ − 16.0000i − 0.532447i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −40.0000 −1.32891
$$907$$ 8.00000i 0.265636i 0.991140 + 0.132818i $$0.0424025\pi$$
−0.991140 + 0.132818i $$0.957597\pi$$
$$908$$ − 24.0000i − 0.796468i
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 0 0
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ − 12.0000i − 0.396275i
$$918$$ − 24.0000i − 0.792118i
$$919$$ 46.0000 1.51740 0.758700 0.651440i $$-0.225835\pi$$
0.758700 + 0.651440i $$0.225835\pi$$
$$920$$ 0 0
$$921$$ −52.0000 −1.71346
$$922$$ 6.00000i 0.197599i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −40.0000 −1.31448
$$927$$ − 4.00000i − 0.131377i
$$928$$ − 6.00000i − 0.196960i
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 18.0000i 0.589610i
$$933$$ − 36.0000i − 1.17859i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 26.0000i 0.849383i 0.905338 + 0.424691i $$0.139617\pi$$
−0.905338 + 0.424691i $$0.860383\pi$$
$$938$$ − 4.00000i − 0.130605i
$$939$$ −44.0000 −1.43589
$$940$$ 0 0
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ 20.0000i 0.651635i
$$943$$ 0 0
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 48.0000i 1.55979i 0.625910 + 0.779895i $$0.284728\pi$$
−0.625910 + 0.779895i $$0.715272\pi$$
$$948$$ − 20.0000i − 0.649570i
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 36.0000 1.16738
$$952$$ 12.0000i 0.388922i
$$953$$ − 42.0000i − 1.36051i −0.732974 0.680257i $$-0.761868\pi$$
0.732974 0.680257i $$-0.238132\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ 18.0000i 0.581554i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 2.00000i 0.0644826i
$$963$$ − 6.00000i − 0.193347i
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 4.00000i − 0.128631i −0.997930 0.0643157i $$-0.979514\pi$$
0.997930 0.0643157i $$-0.0204865\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 10.0000i 0.320750i
$$973$$ 8.00000i 0.256468i
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ − 32.0000i − 1.02325i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ 36.0000i 1.14881i
$$983$$ − 18.0000i − 0.574111i −0.957914 0.287055i $$-0.907324\pi$$
0.957914 0.287055i $$-0.0926764\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ 24.0000i 0.763928i
$$988$$ − 4.00000i − 0.127257i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ − 10.0000i − 0.317500i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ − 10.0000i − 0.316703i −0.987383 0.158352i $$-0.949382\pi$$
0.987383 0.158352i $$-0.0506179\pi$$
$$998$$ − 14.0000i − 0.443162i
$$999$$ −4.00000 −0.126554
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 1850.2.b.b.149.2 2
5.2 odd 4 1850.2.a.f.1.1 1
5.3 odd 4 370.2.a.d.1.1 1
5.4 even 2 inner 1850.2.b.b.149.1 2
15.8 even 4 3330.2.a.d.1.1 1
20.3 even 4 2960.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.d.1.1 1 5.3 odd 4
1850.2.a.f.1.1 1 5.2 odd 4
1850.2.b.b.149.1 2 5.4 even 2 inner
1850.2.b.b.149.2 2 1.1 even 1 trivial
2960.2.a.m.1.1 1 20.3 even 4
3330.2.a.d.1.1 1 15.8 even 4