# Properties

 Label 4400.2.b.i.4049.1 Level $4400$ Weight $2$ Character 4400.4049 Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.4049 Dual form 4400.2.b.i.4049.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -3.00000i q^{7} +2.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -3.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} -7.00000i q^{17} +5.00000 q^{19} -3.00000 q^{21} -6.00000i q^{23} -5.00000i q^{27} -5.00000 q^{29} +3.00000 q^{31} +1.00000i q^{33} +3.00000i q^{37} +6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} +2.00000i q^{47} -2.00000 q^{49} -7.00000 q^{51} +1.00000i q^{53} -5.00000i q^{57} -10.0000 q^{59} +7.00000 q^{61} -6.00000i q^{63} -8.00000i q^{67} -6.00000 q^{69} -7.00000 q^{71} -14.0000i q^{73} +3.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +5.00000i q^{87} +15.0000 q^{89} +18.0000 q^{91} -3.00000i q^{93} -12.0000i q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} - 2 q^{11} + 10 q^{19} - 6 q^{21} - 10 q^{29} + 6 q^{31} + 12 q^{39} + 4 q^{41} - 4 q^{49} - 14 q^{51} - 20 q^{59} + 14 q^{61} - 12 q^{69} - 14 q^{71} + 20 q^{79} + 2 q^{81} + 30 q^{89} + 36 q^{91} - 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 - 2 * q^11 + 10 * q^19 - 6 * q^21 - 10 * q^29 + 6 * q^31 + 12 * q^39 + 4 * q^41 - 4 * q^49 - 14 * q^51 - 20 * q^59 + 14 * q^61 - 12 * q^69 - 14 * q^71 + 20 * q^79 + 2 * q^81 + 30 * q^89 + 36 * q^91 - 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.00000i − 0.577350i −0.957427 0.288675i $$-0.906785\pi$$
0.957427 0.288675i $$-0.0932147\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 7.00000i − 1.69775i −0.528594 0.848875i $$-0.677281\pi$$
0.528594 0.848875i $$-0.322719\pi$$
$$18$$ 0 0
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ − 6.00000i − 1.25109i −0.780189 0.625543i $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.00000i − 0.962250i
$$28$$ 0 0
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ 1.00000i 0.174078i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000i 0.493197i 0.969118 + 0.246598i $$0.0793129\pi$$
−0.969118 + 0.246598i $$0.920687\pi$$
$$38$$ 0 0
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 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
$$47$$ 2.00000i 0.291730i 0.989305 + 0.145865i $$0.0465965\pi$$
−0.989305 + 0.145865i $$0.953403\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ −7.00000 −0.980196
$$52$$ 0 0
$$53$$ 1.00000i 0.137361i 0.997639 + 0.0686803i $$0.0218788\pi$$
−0.997639 + 0.0686803i $$0.978121\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 5.00000i − 0.662266i
$$58$$ 0 0
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 0 0
$$63$$ − 6.00000i − 0.755929i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −7.00000 −0.830747 −0.415374 0.909651i $$-0.636349\pi$$
−0.415374 + 0.909651i $$0.636349\pi$$
$$72$$ 0 0
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.00000i 0.341882i
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 5.00000i 0.536056i
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ 18.0000 1.88691
$$92$$ 0 0
$$93$$ − 3.00000i − 0.311086i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 12.0000i − 1.21842i −0.793011 0.609208i $$-0.791488\pi$$
0.793011 0.609208i $$-0.208512\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 8.00000i − 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 0 0
$$113$$ 16.0000i 1.50515i 0.658505 + 0.752577i $$0.271189\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 12.0000i 1.10940i
$$118$$ 0 0
$$119$$ −21.0000 −1.92507
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ − 2.00000i − 0.180334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −17.0000 −1.48530 −0.742648 0.669681i $$-0.766431\pi$$
−0.742648 + 0.669681i $$0.766431\pi$$
$$132$$ 0 0
$$133$$ − 15.0000i − 1.30066i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 2.00000 0.168430
$$142$$ 0 0
$$143$$ − 6.00000i − 0.501745i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.00000i 0.164957i
$$148$$ 0 0
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ 0 0
$$151$$ −2.00000 −0.162758 −0.0813788 0.996683i $$-0.525932\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ 0 0
$$153$$ − 14.0000i − 1.13183i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3.00000i 0.239426i 0.992809 + 0.119713i $$0.0381975\pi$$
−0.992809 + 0.119713i $$0.961803\pi$$
$$158$$ 0 0
$$159$$ 1.00000 0.0793052
$$160$$ 0 0
$$161$$ −18.0000 −1.41860
$$162$$ 0 0
$$163$$ 19.0000i 1.48819i 0.668071 + 0.744097i $$0.267120\pi$$
−0.668071 + 0.744097i $$0.732880\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 3.00000i − 0.232147i −0.993241 0.116073i $$-0.962969\pi$$
0.993241 0.116073i $$-0.0370308\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 10.0000 0.764719
$$172$$ 0 0
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 10.0000i 0.751646i
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ − 7.00000i − 0.517455i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 7.00000i 0.511891i
$$188$$ 0 0
$$189$$ −15.0000 −1.09109
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ 11.0000i 0.791797i 0.918294 + 0.395899i $$0.129567\pi$$
−0.918294 + 0.395899i $$0.870433\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 0 0
$$199$$ −25.0000 −1.77220 −0.886102 0.463491i $$-0.846597\pi$$
−0.886102 + 0.463491i $$0.846597\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 0 0
$$203$$ 15.0000i 1.05279i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 12.0000i − 0.834058i
$$208$$ 0 0
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ 0 0
$$213$$ 7.00000i 0.479632i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 9.00000i − 0.610960i
$$218$$ 0 0
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 42.0000 2.82523
$$222$$ 0 0
$$223$$ − 6.00000i − 0.401790i −0.979613 0.200895i $$-0.935615\pi$$
0.979613 0.200895i $$-0.0643850\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2.00000i 0.132745i 0.997795 + 0.0663723i $$0.0211425\pi$$
−0.997795 + 0.0663723i $$0.978857\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ 0 0
$$233$$ − 9.00000i − 0.589610i −0.955557 0.294805i $$-0.904745\pi$$
0.955557 0.294805i $$-0.0952546\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 10.0000i − 0.649570i
$$238$$ 0 0
$$239$$ 10.0000 0.646846 0.323423 0.946254i $$-0.395166\pi$$
0.323423 + 0.946254i $$0.395166\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 0 0
$$243$$ − 16.0000i − 1.02640i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 30.0000i 1.90885i
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ 0 0
$$253$$ 6.00000i 0.377217i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 0 0
$$259$$ 9.00000 0.559233
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ 0 0
$$263$$ 9.00000i 0.554964i 0.960731 + 0.277482i $$0.0894999\pi$$
−0.960731 + 0.277482i $$0.910500\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 15.0000i − 0.917985i
$$268$$ 0 0
$$269$$ 20.0000 1.21942 0.609711 0.792624i $$-0.291286\pi$$
0.609711 + 0.792624i $$0.291286\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ − 18.0000i − 1.08941i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 12.0000i − 0.721010i −0.932757 0.360505i $$-0.882604\pi$$
0.932757 0.360505i $$-0.117396\pi$$
$$278$$ 0 0
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ − 6.00000i − 0.356663i −0.983970 0.178331i $$-0.942930\pi$$
0.983970 0.178331i $$-0.0570699\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6.00000i − 0.354169i
$$288$$ 0 0
$$289$$ −32.0000 −1.88235
$$290$$ 0 0
$$291$$ −12.0000 −0.703452
$$292$$ 0 0
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.00000i 0.290129i
$$298$$ 0 0
$$299$$ 36.0000 2.08193
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ − 2.00000i − 0.114897i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.00000i 0.114146i 0.998370 + 0.0570730i $$0.0181768\pi$$
−0.998370 + 0.0570730i $$0.981823\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 3.00000 0.170114 0.0850572 0.996376i $$-0.472893\pi$$
0.0850572 + 0.996376i $$0.472893\pi$$
$$312$$ 0 0
$$313$$ 6.00000i 0.339140i 0.985518 + 0.169570i $$0.0542379\pi$$
−0.985518 + 0.169570i $$0.945762\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 7.00000i − 0.393159i −0.980488 0.196580i $$-0.937017\pi$$
0.980488 0.196580i $$-0.0629834\pi$$
$$318$$ 0 0
$$319$$ 5.00000 0.279946
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ 0 0
$$323$$ − 35.0000i − 1.94745i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 10.0000i − 0.553001i
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ 6.00000i 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 17.0000i − 0.926049i −0.886345 0.463025i $$-0.846764\pi$$
0.886345 0.463025i $$-0.153236\pi$$
$$338$$ 0 0
$$339$$ 16.0000 0.869001
$$340$$ 0 0
$$341$$ −3.00000 −0.162459
$$342$$ 0 0
$$343$$ − 15.0000i − 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 18.0000i − 0.966291i −0.875540 0.483145i $$-0.839494\pi$$
0.875540 0.483145i $$-0.160506\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ 30.0000 1.60128
$$352$$ 0 0
$$353$$ − 34.0000i − 1.80964i −0.425797 0.904819i $$-0.640006\pi$$
0.425797 0.904819i $$-0.359994\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 21.0000i 1.11144i
$$358$$ 0 0
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ − 1.00000i − 0.0524864i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 28.0000i − 1.46159i −0.682598 0.730794i $$-0.739150\pi$$
0.682598 0.730794i $$-0.260850\pi$$
$$368$$ 0 0
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ 3.00000 0.155752
$$372$$ 0 0
$$373$$ 6.00000i 0.310668i 0.987862 + 0.155334i $$0.0496454\pi$$
−0.987862 + 0.155334i $$0.950355\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 30.0000i − 1.54508i
$$378$$ 0 0
$$379$$ −30.0000 −1.54100 −0.770498 0.637442i $$-0.779993\pi$$
−0.770498 + 0.637442i $$0.779993\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ 34.0000i 1.73732i 0.495410 + 0.868659i $$0.335018\pi$$
−0.495410 + 0.868659i $$0.664982\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.00000i 0.406663i
$$388$$ 0 0
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −42.0000 −2.12403
$$392$$ 0 0
$$393$$ 17.0000i 0.857537i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 0 0
$$399$$ −15.0000 −0.750939
$$400$$ 0 0
$$401$$ −13.0000 −0.649189 −0.324595 0.945853i $$-0.605228\pi$$
−0.324595 + 0.945853i $$0.605228\pi$$
$$402$$ 0 0
$$403$$ 18.0000i 0.896644i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 3.00000i − 0.148704i
$$408$$ 0 0
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ 30.0000i 1.47620i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 20.0000i 0.979404i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ 0 0
$$423$$ 4.00000i 0.194487i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 21.0000i − 1.01626i
$$428$$ 0 0
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 16.0000i 0.768911i 0.923144 + 0.384455i $$0.125611\pi$$
−0.923144 + 0.384455i $$0.874389\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 30.0000i − 1.43509i
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 0 0
$$443$$ 4.00000i 0.190046i 0.995475 + 0.0950229i $$0.0302924\pi$$
−0.995475 + 0.0950229i $$0.969708\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 15.0000i 0.709476i
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −2.00000 −0.0941763
$$452$$ 0 0
$$453$$ 2.00000i 0.0939682i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.00000i 0.140334i 0.997535 + 0.0701670i $$0.0223532\pi$$
−0.997535 + 0.0701670i $$0.977647\pi$$
$$458$$ 0 0
$$459$$ −35.0000 −1.63366
$$460$$ 0 0
$$461$$ 27.0000 1.25752 0.628758 0.777601i $$-0.283564\pi$$
0.628758 + 0.777601i $$0.283564\pi$$
$$462$$ 0 0
$$463$$ 34.0000i 1.58011i 0.613033 + 0.790057i $$0.289949\pi$$
−0.613033 + 0.790057i $$0.710051\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 23.0000i − 1.06431i −0.846646 0.532157i $$-0.821382\pi$$
0.846646 0.532157i $$-0.178618\pi$$
$$468$$ 0 0
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ 3.00000 0.138233
$$472$$ 0 0
$$473$$ − 4.00000i − 0.183920i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2.00000i 0.0915737i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −18.0000 −0.820729
$$482$$ 0 0
$$483$$ 18.0000i 0.819028i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.0000i 0.543772i 0.962329 + 0.271886i $$0.0876473\pi$$
−0.962329 + 0.271886i $$0.912353\pi$$
$$488$$ 0 0
$$489$$ 19.0000 0.859210
$$490$$ 0 0
$$491$$ 3.00000 0.135388 0.0676941 0.997706i $$-0.478436\pi$$
0.0676941 + 0.997706i $$0.478436\pi$$
$$492$$ 0 0
$$493$$ 35.0000i 1.57632i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 21.0000i 0.941979i
$$498$$ 0 0
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ −3.00000 −0.134030
$$502$$ 0 0
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 23.0000i 1.02147i
$$508$$ 0 0
$$509$$ −20.0000 −0.886484 −0.443242 0.896402i $$-0.646172\pi$$
−0.443242 + 0.896402i $$0.646172\pi$$
$$510$$ 0 0
$$511$$ −42.0000 −1.85797
$$512$$ 0 0
$$513$$ − 25.0000i − 1.10378i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 2.00000i − 0.0879599i
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ − 16.0000i − 0.699631i −0.936819 0.349816i $$-0.886244\pi$$
0.936819 0.349816i $$-0.113756\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 21.0000i − 0.914774i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −20.0000 −0.867926
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ −23.0000 −0.988847 −0.494424 0.869221i $$-0.664621\pi$$
−0.494424 + 0.869221i $$0.664621\pi$$
$$542$$ 0 0
$$543$$ − 2.00000i − 0.0858282i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ 0 0
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ −25.0000 −1.06504
$$552$$ 0 0
$$553$$ − 30.0000i − 1.27573i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 7.00000 0.295540
$$562$$ 0 0
$$563$$ − 6.00000i − 0.252870i −0.991975 0.126435i $$-0.959647\pi$$
0.991975 0.126435i $$-0.0403535\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 3.00000i − 0.125988i
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −27.0000 −1.12991 −0.564957 0.825120i $$-0.691107\pi$$
−0.564957 + 0.825120i $$0.691107\pi$$
$$572$$ 0 0
$$573$$ 12.0000i 0.501307i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 38.0000i 1.58196i 0.611842 + 0.790980i $$0.290429\pi$$
−0.611842 + 0.790980i $$0.709571\pi$$
$$578$$ 0 0
$$579$$ 11.0000 0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ 0 0
$$583$$ − 1.00000i − 0.0414158i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 27.0000i 1.11441i 0.830375 + 0.557205i $$0.188126\pi$$
−0.830375 + 0.557205i $$0.811874\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 0 0
$$593$$ − 14.0000i − 0.574911i −0.957794 0.287456i $$-0.907191\pi$$
0.957794 0.287456i $$-0.0928094\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 25.0000i 1.02318i
$$598$$ 0 0
$$599$$ 45.0000 1.83865 0.919325 0.393499i $$-0.128735\pi$$
0.919325 + 0.393499i $$0.128735\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ 0 0
$$603$$ − 16.0000i − 0.651570i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 47.0000i 1.90767i 0.300329 + 0.953836i $$0.402903\pi$$
−0.300329 + 0.953836i $$0.597097\pi$$
$$608$$ 0 0
$$609$$ 15.0000 0.607831
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ 0 0
$$613$$ 26.0000i 1.05013i 0.851062 + 0.525065i $$0.175959\pi$$
−0.851062 + 0.525065i $$0.824041\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8.00000i 0.322068i 0.986949 + 0.161034i $$0.0514829\pi$$
−0.986949 + 0.161034i $$0.948517\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ −30.0000 −1.20386
$$622$$ 0 0
$$623$$ − 45.0000i − 1.80289i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 5.00000i 0.199681i
$$628$$ 0 0
$$629$$ 21.0000 0.837325
$$630$$ 0 0
$$631$$ 33.0000 1.31371 0.656855 0.754017i $$-0.271887\pi$$
0.656855 + 0.754017i $$0.271887\pi$$
$$632$$ 0 0
$$633$$ − 23.0000i − 0.914168i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 12.0000i − 0.475457i
$$638$$ 0 0
$$639$$ −14.0000 −0.553831
$$640$$ 0 0
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ 0 0
$$643$$ 19.0000i 0.749287i 0.927169 + 0.374643i $$0.122235\pi$$
−0.927169 + 0.374643i $$0.877765\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 42.0000i 1.65119i 0.564263 + 0.825595i $$0.309160\pi$$
−0.564263 + 0.825595i $$0.690840\pi$$
$$648$$ 0 0
$$649$$ 10.0000 0.392534
$$650$$ 0 0
$$651$$ −9.00000 −0.352738
$$652$$ 0 0
$$653$$ 31.0000i 1.21312i 0.795036 + 0.606562i $$0.207452\pi$$
−0.795036 + 0.606562i $$0.792548\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 28.0000i − 1.09238i
$$658$$ 0 0
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ 0 0
$$661$$ 2.00000 0.0777910 0.0388955 0.999243i $$-0.487616\pi$$
0.0388955 + 0.999243i $$0.487616\pi$$
$$662$$ 0 0
$$663$$ − 42.0000i − 1.63114i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 30.0000i 1.16160i
$$668$$ 0 0
$$669$$ −6.00000 −0.231973
$$670$$ 0 0
$$671$$ −7.00000 −0.270232
$$672$$ 0 0
$$673$$ − 29.0000i − 1.11787i −0.829212 0.558934i $$-0.811211\pi$$
0.829212 0.558934i $$-0.188789\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 28.0000i 1.07613i 0.842904 + 0.538064i $$0.180844\pi$$
−0.842904 + 0.538064i $$0.819156\pi$$
$$678$$ 0 0
$$679$$ −36.0000 −1.38155
$$680$$ 0 0
$$681$$ 2.00000 0.0766402
$$682$$ 0 0
$$683$$ − 31.0000i − 1.18618i −0.805135 0.593091i $$-0.797907\pi$$
0.805135 0.593091i $$-0.202093\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.0000i 0.381524i
$$688$$ 0 0
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ 38.0000 1.44559 0.722794 0.691063i $$-0.242858\pi$$
0.722794 + 0.691063i $$0.242858\pi$$
$$692$$ 0 0
$$693$$ 6.00000i 0.227921i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 14.0000i − 0.530288i
$$698$$ 0 0
$$699$$ −9.00000 −0.340411
$$700$$ 0 0
$$701$$ 7.00000 0.264386 0.132193 0.991224i $$-0.457798\pi$$
0.132193 + 0.991224i $$0.457798\pi$$
$$702$$ 0 0
$$703$$ 15.0000i 0.565736i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 6.00000i − 0.225653i
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 20.0000 0.750059
$$712$$ 0 0
$$713$$ − 18.0000i − 0.674105i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 10.0000i − 0.373457i
$$718$$ 0 0
$$719$$ 25.0000 0.932343 0.466171 0.884694i $$-0.345633\pi$$
0.466171 + 0.884694i $$0.345633\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 0 0
$$723$$ 18.0000i 0.669427i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 22.0000i 0.815935i 0.912996 + 0.407967i $$0.133762\pi$$
−0.912996 + 0.407967i $$0.866238\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 28.0000 1.03562
$$732$$ 0 0
$$733$$ − 24.0000i − 0.886460i −0.896408 0.443230i $$-0.853832\pi$$
0.896408 0.443230i $$-0.146168\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8.00000i 0.294684i
$$738$$ 0 0
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 0 0
$$741$$ 30.0000 1.10208
$$742$$ 0 0
$$743$$ − 21.0000i − 0.770415i −0.922830 0.385208i $$-0.874130\pi$$
0.922830 0.385208i $$-0.125870\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 12.0000i − 0.439057i
$$748$$ 0 0
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ −17.0000 −0.620339 −0.310169 0.950681i $$-0.600386\pi$$
−0.310169 + 0.950681i $$0.600386\pi$$
$$752$$ 0 0
$$753$$ 2.00000i 0.0728841i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38.0000i 1.38113i 0.723269 + 0.690567i $$0.242639\pi$$
−0.723269 + 0.690567i $$0.757361\pi$$
$$758$$ 0 0
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ − 30.0000i − 1.08607i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 60.0000i − 2.16647i
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ 0 0
$$773$$ − 19.0000i − 0.683383i −0.939812 0.341691i $$-0.889000\pi$$
0.939812 0.341691i $$-0.111000\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 9.00000i − 0.322873i
$$778$$ 0 0
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ 7.00000 0.250480
$$782$$ 0 0
$$783$$ 25.0000i 0.893427i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ 0 0
$$789$$ 9.00000 0.320408
$$790$$ 0 0
$$791$$ 48.0000 1.70668
$$792$$ 0 0
$$793$$ 42.0000i 1.49146i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 18.0000i 0.637593i 0.947823 + 0.318796i $$0.103279\pi$$
−0.947823 + 0.318796i $$0.896721\pi$$
$$798$$ 0 0
$$799$$ 14.0000 0.495284
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 0 0
$$803$$ 14.0000i 0.494049i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 20.0000i − 0.704033i
$$808$$ 0 0
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ −7.00000 −0.245803 −0.122902 0.992419i $$-0.539220\pi$$
−0.122902 + 0.992419i $$0.539220\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 20.0000i 0.699711i
$$818$$ 0 0
$$819$$ 36.0000 1.25794
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ − 16.0000i − 0.557725i −0.960331 0.278862i $$-0.910043\pi$$
0.960331 0.278862i $$-0.0899574\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 8.00000i − 0.278187i −0.990279 0.139094i $$-0.955581\pi$$
0.990279 0.139094i $$-0.0444189\pi$$
$$828$$ 0 0
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ 0 0
$$831$$ −12.0000 −0.416275
$$832$$ 0 0
$$833$$ 14.0000i 0.485071i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 15.0000i − 0.518476i
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 0 0
$$843$$ 18.0000i 0.619953i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 3.00000i − 0.103081i
$$848$$ 0 0
$$849$$ −6.00000 −0.205919
$$850$$ 0 0
$$851$$ 18.0000 0.617032
$$852$$ 0 0
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 7.00000i − 0.239115i −0.992827 0.119558i $$-0.961852\pi$$
0.992827 0.119558i $$-0.0381477\pi$$
$$858$$ 0 0
$$859$$ 30.0000 1.02359 0.511793 0.859109i $$-0.328981\pi$$
0.511793 + 0.859109i $$0.328981\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ − 6.00000i − 0.204242i −0.994772 0.102121i $$-0.967437\pi$$
0.994772 0.102121i $$-0.0325630\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 32.0000i 1.08678i
$$868$$ 0 0
$$869$$ −10.0000 −0.339227
$$870$$ 0 0
$$871$$ 48.0000 1.62642
$$872$$ 0 0
$$873$$ − 24.0000i − 0.812277i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 38.0000i 1.28317i 0.767052 + 0.641584i $$0.221723\pi$$
−0.767052 + 0.641584i $$0.778277\pi$$
$$878$$ 0 0
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ 9.00000i 0.302874i 0.988467 + 0.151437i $$0.0483901\pi$$
−0.988467 + 0.151437i $$0.951610\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 8.00000i − 0.268614i −0.990940 0.134307i $$-0.957119\pi$$
0.990940 0.134307i $$-0.0428808\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ 10.0000i 0.334637i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 36.0000i − 1.20201i
$$898$$ 0 0
$$899$$ −15.0000 −0.500278
$$900$$ 0 0
$$901$$ 7.00000 0.233204
$$902$$ 0 0
$$903$$ − 12.0000i − 0.399335i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 57.0000i 1.89265i 0.323211 + 0.946327i $$0.395238\pi$$
−0.323211 + 0.946327i $$0.604762\pi$$
$$908$$ 0 0
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ −27.0000 −0.894550 −0.447275 0.894397i $$-0.647605\pi$$
−0.447275 + 0.894397i $$0.647605\pi$$
$$912$$ 0 0
$$913$$ 6.00000i 0.198571i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 51.0000i 1.68417i
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 2.00000 0.0659022
$$922$$ 0 0
$$923$$ − 42.0000i − 1.38245i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 8.00000i 0.262754i
$$928$$ 0 0
$$929$$ −35.0000 −1.14831 −0.574156 0.818746i $$-0.694670\pi$$
−0.574156 + 0.818746i $$0.694670\pi$$
$$930$$ 0 0
$$931$$ −10.0000 −0.327737
$$932$$ 0 0
$$933$$ − 3.00000i − 0.0982156i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 38.0000i 1.24141i 0.784046 + 0.620703i $$0.213153\pi$$
−0.784046 + 0.620703i $$0.786847\pi$$
$$938$$ 0 0
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 17.0000 0.554184 0.277092 0.960843i $$-0.410629\pi$$
0.277092 + 0.960843i $$0.410629\pi$$
$$942$$ 0 0
$$943$$ − 12.0000i − 0.390774i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 27.0000i 0.877382i 0.898638 + 0.438691i $$0.144558\pi$$
−0.898638 + 0.438691i $$0.855442\pi$$
$$948$$ 0 0
$$949$$ 84.0000 2.72676
$$950$$ 0 0
$$951$$ −7.00000 −0.226991
$$952$$ 0 0
$$953$$ − 39.0000i − 1.26333i −0.775240 0.631667i $$-0.782371\pi$$
0.775240 0.631667i $$-0.217629\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 5.00000i − 0.161627i
$$958$$ 0 0
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ − 16.0000i − 0.515593i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 27.0000i 0.868261i 0.900850 + 0.434131i $$0.142944\pi$$
−0.900850 + 0.434131i $$0.857056\pi$$
$$968$$ 0 0
$$969$$ −35.0000 −1.12436
$$970$$ 0 0
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ 0 0
$$973$$ 60.0000i 1.92351i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 12.0000i − 0.383914i −0.981403 0.191957i $$-0.938517\pi$$
0.981403 0.191957i $$-0.0614834\pi$$
$$978$$ 0 0
$$979$$ −15.0000 −0.479402
$$980$$ 0 0
$$981$$ 20.0000 0.638551
$$982$$ 0 0
$$983$$ 54.0000i 1.72233i 0.508323 + 0.861166i $$0.330265\pi$$
−0.508323 + 0.861166i $$0.669735\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 6.00000i − 0.190982i
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 0 0
$$993$$ − 28.0000i − 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 32.0000i − 1.01345i −0.862108 0.506725i $$-0.830856\pi$$
0.862108 0.506725i $$-0.169144\pi$$
$$998$$ 0 0
$$999$$ 15.0000 0.474579
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 4400.2.b.i.4049.1 2
4.3 odd 2 550.2.b.a.199.2 2
5.2 odd 4 4400.2.a.l.1.1 1
5.3 odd 4 880.2.a.i.1.1 1
5.4 even 2 inner 4400.2.b.i.4049.2 2
12.11 even 2 4950.2.c.m.199.1 2
15.8 even 4 7920.2.a.d.1.1 1
20.3 even 4 110.2.a.b.1.1 1
20.7 even 4 550.2.a.f.1.1 1
20.19 odd 2 550.2.b.a.199.1 2
40.3 even 4 3520.2.a.y.1.1 1
40.13 odd 4 3520.2.a.h.1.1 1
55.43 even 4 9680.2.a.x.1.1 1
60.23 odd 4 990.2.a.d.1.1 1
60.47 odd 4 4950.2.a.bc.1.1 1
60.59 even 2 4950.2.c.m.199.2 2
140.83 odd 4 5390.2.a.bf.1.1 1
220.43 odd 4 1210.2.a.b.1.1 1
220.87 odd 4 6050.2.a.bj.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.b.1.1 1 20.3 even 4
550.2.a.f.1.1 1 20.7 even 4
550.2.b.a.199.1 2 20.19 odd 2
550.2.b.a.199.2 2 4.3 odd 2
880.2.a.i.1.1 1 5.3 odd 4
990.2.a.d.1.1 1 60.23 odd 4
1210.2.a.b.1.1 1 220.43 odd 4
3520.2.a.h.1.1 1 40.13 odd 4
3520.2.a.y.1.1 1 40.3 even 4
4400.2.a.l.1.1 1 5.2 odd 4
4400.2.b.i.4049.1 2 1.1 even 1 trivial
4400.2.b.i.4049.2 2 5.4 even 2 inner
4950.2.a.bc.1.1 1 60.47 odd 4
4950.2.c.m.199.1 2 12.11 even 2
4950.2.c.m.199.2 2 60.59 even 2
5390.2.a.bf.1.1 1 140.83 odd 4
6050.2.a.bj.1.1 1 220.87 odd 4
7920.2.a.d.1.1 1 15.8 even 4
9680.2.a.x.1.1 1 55.43 even 4