# Properties

 Label 3200.2.c.k.2049.1 Level $3200$ Weight $2$ Character 3200.2049 Analytic conductor $25.552$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2049.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3200.2049 Dual form 3200.2.c.k.2049.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} +2.00000i q^{13} +2.00000i q^{17} +2.00000 q^{19} -8.00000 q^{21} -4.00000i q^{23} -4.00000i q^{27} +6.00000 q^{29} -4.00000i q^{33} -10.0000i q^{37} +4.00000 q^{39} -6.00000 q^{41} -6.00000i q^{43} -8.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -6.00000i q^{53} -4.00000i q^{57} +14.0000 q^{59} +2.00000 q^{61} +4.00000i q^{63} +10.0000i q^{67} -8.00000 q^{69} -12.0000 q^{71} +14.0000i q^{73} -8.00000i q^{77} -8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -12.0000i q^{87} +2.00000 q^{89} +8.00000 q^{91} +2.00000i q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 4q^{11} + 4q^{19} - 16q^{21} + 12q^{29} + 8q^{39} - 12q^{41} - 18q^{49} + 8q^{51} + 28q^{59} + 4q^{61} - 16q^{69} - 24q^{71} - 16q^{79} - 22q^{81} + 4q^{89} + 16q^{91} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 2.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 4.00000i − 1.51186i −0.654654 0.755929i $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −8.00000 −1.74574
$$22$$ 0 0
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 4.00000i − 0.769800i
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ − 4.00000i − 0.696311i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 6.00000i − 0.914991i −0.889212 0.457496i $$-0.848747\pi$$
0.889212 0.457496i $$-0.151253\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ 0 0
$$59$$ 14.0000 1.82264 0.911322 0.411693i $$-0.135063\pi$$
0.911322 + 0.411693i $$0.135063\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 4.00000i 0.503953i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 10.0000i 1.22169i 0.791748 + 0.610847i $$0.209171\pi$$
−0.791748 + 0.610847i $$0.790829\pi$$
$$68$$ 0 0
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 14.0000i 1.63858i 0.573382 + 0.819288i $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 8.00000i − 0.911685i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 12.0000i − 1.28654i
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 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
$$107$$ − 2.00000i − 0.193347i −0.995316 0.0966736i $$-0.969180\pi$$
0.995316 0.0966736i $$-0.0308203\pi$$
$$108$$ 0 0
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 0 0
$$111$$ −20.0000 −1.89832
$$112$$ 0 0
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 12.0000i 1.08200i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 0 0
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ 0 0
$$139$$ −10.0000 −0.848189 −0.424094 0.905618i $$-0.639408\pi$$
−0.424094 + 0.905618i $$0.639408\pi$$
$$140$$ 0 0
$$141$$ −16.0000 −1.34744
$$142$$ 0 0
$$143$$ 4.00000i 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 18.0000i 1.48461i
$$148$$ 0 0
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ − 2.00000i − 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ −16.0000 −1.26098
$$162$$ 0 0
$$163$$ − 2.00000i − 0.156652i −0.996928 0.0783260i $$-0.975042\pi$$
0.996928 0.0783260i $$-0.0249575\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 20.0000i − 1.54765i −0.633402 0.773823i $$-0.718342\pi$$
0.633402 0.773823i $$-0.281658\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 28.0000i − 2.10461i
$$178$$ 0 0
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ − 4.00000i − 0.295689i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ 0 0
$$189$$ −16.0000 −1.16383
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.0000i 0.997459i 0.866758 + 0.498729i $$0.166200\pi$$
−0.866758 + 0.498729i $$0.833800\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 20.0000 1.41069
$$202$$ 0 0
$$203$$ − 24.0000i − 1.68447i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.00000i 0.278019i
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 22.0000 1.51454 0.757271 0.653101i $$-0.226532\pi$$
0.757271 + 0.653101i $$0.226532\pi$$
$$212$$ 0 0
$$213$$ 24.0000i 1.64445i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 28.0000 1.89206
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ 0 0
$$233$$ − 18.0000i − 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 16.0000i 1.03931i
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 10.0000i 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000i 0.254514i
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 0 0
$$253$$ − 8.00000i − 0.502956i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 0 0
$$259$$ −40.0000 −2.48548
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 4.00000i − 0.244796i
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 0 0
$$273$$ − 16.0000i − 0.968364i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6.00000i 0.360505i 0.983620 + 0.180253i $$0.0576915\pi$$
−0.983620 + 0.180253i $$0.942309\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 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
$$287$$ 24.0000i 1.41668i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 4.00000 0.234484
$$292$$ 0 0
$$293$$ − 14.0000i − 0.817889i −0.912559 0.408944i $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 8.00000i − 0.464207i
$$298$$ 0 0
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ 0 0
$$303$$ 12.0000i 0.689382i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 18.0000i 1.02731i 0.857996 + 0.513657i $$0.171710\pi$$
−0.857996 + 0.513657i $$0.828290\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 28.0000 1.58773 0.793867 0.608091i $$-0.208065\pi$$
0.793867 + 0.608091i $$0.208065\pi$$
$$312$$ 0 0
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 4.00000i 0.222566i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 12.0000i − 0.663602i
$$328$$ 0 0
$$329$$ −32.0000 −1.76422
$$330$$ 0 0
$$331$$ −14.0000 −0.769510 −0.384755 0.923019i $$-0.625714\pi$$
−0.384755 + 0.923019i $$0.625714\pi$$
$$332$$ 0 0
$$333$$ 10.0000i 0.547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 0 0
$$339$$ 4.00000 0.217250
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 8.00000i 0.431959i
$$344$$ 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
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ 0 0
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 16.0000i − 0.846810i
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 14.0000i 0.734809i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ 0 0
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ −2.00000 −0.102733 −0.0513665 0.998680i $$-0.516358\pi$$
−0.0513665 + 0.998680i $$0.516358\pi$$
$$380$$ 0 0
$$381$$ 32.0000 1.63941
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 6.00000i 0.304997i
$$388$$ 0 0
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ − 12.0000i − 0.605320i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6.00000i 0.301131i 0.988600 + 0.150566i $$0.0481095\pi$$
−0.988600 + 0.150566i $$0.951890\pi$$
$$398$$ 0 0
$$399$$ −16.0000 −0.801002
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 20.0000i − 0.991363i
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ −20.0000 −0.986527
$$412$$ 0 0
$$413$$ − 56.0000i − 2.75558i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 20.0000i 0.979404i
$$418$$ 0 0
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 0 0
$$423$$ 8.00000i 0.388973i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 8.00000i − 0.387147i
$$428$$ 0 0
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ 0 0
$$433$$ 30.0000i 1.44171i 0.693087 + 0.720854i $$0.256250\pi$$
−0.693087 + 0.720854i $$0.743750\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 8.00000i − 0.382692i
$$438$$ 0 0
$$439$$ 36.0000 1.71819 0.859093 0.511819i $$-0.171028\pi$$
0.859093 + 0.511819i $$0.171028\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ − 6.00000i − 0.285069i −0.989790 0.142534i $$-0.954475\pi$$
0.989790 0.142534i $$-0.0455251\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 36.0000i 1.70274i
$$448$$ 0 0
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 0 0
$$453$$ 8.00000i 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000i 0.280668i 0.990104 + 0.140334i $$0.0448177\pi$$
−0.990104 + 0.140334i $$0.955182\pi$$
$$458$$ 0 0
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 14.0000i − 0.647843i −0.946084 0.323921i $$-0.894999\pi$$
0.946084 0.323921i $$-0.105001\pi$$
$$468$$ 0 0
$$469$$ 40.0000 1.84703
$$470$$ 0 0
$$471$$ −36.0000 −1.65879
$$472$$ 0 0
$$473$$ − 12.0000i − 0.551761i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 0 0
$$483$$ 32.0000i 1.45605i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 20.0000i − 0.906287i −0.891438 0.453143i $$-0.850303\pi$$
0.891438 0.453143i $$-0.149697\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 10.0000 0.451294 0.225647 0.974209i $$-0.427550\pi$$
0.225647 + 0.974209i $$0.427550\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 48.0000i 2.15309i
$$498$$ 0 0
$$499$$ −22.0000 −0.984855 −0.492428 0.870353i $$-0.663890\pi$$
−0.492428 + 0.870353i $$0.663890\pi$$
$$500$$ 0 0
$$501$$ −40.0000 −1.78707
$$502$$ 0 0
$$503$$ − 20.0000i − 0.891756i −0.895094 0.445878i $$-0.852892\pi$$
0.895094 0.445878i $$-0.147108\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 18.0000i − 0.799408i
$$508$$ 0 0
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ 56.0000 2.47729
$$512$$ 0 0
$$513$$ − 8.00000i − 0.353209i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 16.0000i − 0.703679i
$$518$$ 0 0
$$519$$ 36.0000 1.58022
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ − 14.0000i − 0.612177i −0.952003 0.306089i $$-0.900980\pi$$
0.952003 0.306089i $$-0.0990204\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −14.0000 −0.607548
$$532$$ 0 0
$$533$$ − 12.0000i − 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 12.0000i 0.517838i
$$538$$ 0 0
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 34.0000 1.46177 0.730887 0.682498i $$-0.239107\pi$$
0.730887 + 0.682498i $$0.239107\pi$$
$$542$$ 0 0
$$543$$ − 4.00000i − 0.171656i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 38.0000i − 1.62476i −0.583127 0.812381i $$-0.698171\pi$$
0.583127 0.812381i $$-0.301829\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 32.0000i 1.36078i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ 0 0
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ − 18.0000i − 0.758610i −0.925272 0.379305i $$-0.876163\pi$$
0.925272 0.379305i $$-0.123837\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 44.0000i 1.84783i
$$568$$ 0 0
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ −38.0000 −1.59025 −0.795125 0.606445i $$-0.792595\pi$$
−0.795125 + 0.606445i $$0.792595\pi$$
$$572$$ 0 0
$$573$$ − 32.0000i − 1.33682i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 0 0
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ − 12.0000i − 0.496989i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 34.0000i − 1.40333i −0.712507 0.701665i $$-0.752440\pi$$
0.712507 0.701665i $$-0.247560\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 28.0000 1.15177
$$592$$ 0 0
$$593$$ 18.0000i 0.739171i 0.929197 + 0.369586i $$0.120500\pi$$
−0.929197 + 0.369586i $$0.879500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.00000i 0.327418i
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 0 0
$$603$$ − 10.0000i − 0.407231i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 16.0000i − 0.649420i −0.945814 0.324710i $$-0.894733\pi$$
0.945814 0.324710i $$-0.105267\pi$$
$$608$$ 0 0
$$609$$ −48.0000 −1.94506
$$610$$ 0 0
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ 34.0000i 1.37325i 0.727013 + 0.686624i $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.00000i 0.0805170i 0.999189 + 0.0402585i $$0.0128181\pi$$
−0.999189 + 0.0402585i $$0.987182\pi$$
$$618$$ 0 0
$$619$$ 46.0000 1.84890 0.924448 0.381308i $$-0.124526\pi$$
0.924448 + 0.381308i $$0.124526\pi$$
$$620$$ 0 0
$$621$$ −16.0000 −0.642058
$$622$$ 0 0
$$623$$ − 8.00000i − 0.320513i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 8.00000i − 0.319489i
$$628$$ 0 0
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ 44.0000 1.75161 0.875806 0.482663i $$-0.160330\pi$$
0.875806 + 0.482663i $$0.160330\pi$$
$$632$$ 0 0
$$633$$ − 44.0000i − 1.74884i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 18.0000i − 0.713186i
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ − 42.0000i − 1.65632i −0.560493 0.828159i $$-0.689388\pi$$
0.560493 0.828159i $$-0.310612\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000i 0.471769i 0.971781 + 0.235884i $$0.0757987\pi$$
−0.971781 + 0.235884i $$0.924201\pi$$
$$648$$ 0 0
$$649$$ 28.0000 1.09910
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 42.0000i 1.64359i 0.569785 + 0.821794i $$0.307026\pi$$
−0.569785 + 0.821794i $$0.692974\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 14.0000i − 0.546192i
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 34.0000 1.32245 0.661223 0.750189i $$-0.270038\pi$$
0.661223 + 0.750189i $$0.270038\pi$$
$$662$$ 0 0
$$663$$ 8.00000i 0.310694i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 24.0000i − 0.929284i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ − 2.00000i − 0.0770943i −0.999257 0.0385472i $$-0.987727\pi$$
0.999257 0.0385472i $$-0.0122730\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 22.0000i 0.845529i 0.906240 + 0.422764i $$0.138940\pi$$
−0.906240 + 0.422764i $$0.861060\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ 36.0000 1.37952
$$682$$ 0 0
$$683$$ 42.0000i 1.60709i 0.595247 + 0.803543i $$0.297054\pi$$
−0.595247 + 0.803543i $$0.702946\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 28.0000i − 1.06827i
$$688$$ 0 0
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 6.00000 0.228251 0.114125 0.993466i $$-0.463593\pi$$
0.114125 + 0.993466i $$0.463593\pi$$
$$692$$ 0 0
$$693$$ 8.00000i 0.303895i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 12.0000i − 0.454532i
$$698$$ 0 0
$$699$$ −36.0000 −1.36165
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ − 20.0000i − 0.754314i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 24.0000i 0.902613i
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 48.0000i 1.79259i
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 0 0
$$723$$ 4.00000i 0.148762i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 12.0000i − 0.445055i −0.974926 0.222528i $$-0.928569\pi$$
0.974926 0.222528i $$-0.0714308\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ − 6.00000i − 0.221615i −0.993842 0.110808i $$-0.964656\pi$$
0.993842 0.110808i $$-0.0353437\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 20.0000i 0.736709i
$$738$$ 0 0
$$739$$ 18.0000 0.662141 0.331070 0.943606i $$-0.392590\pi$$
0.331070 + 0.943606i $$0.392590\pi$$
$$740$$ 0 0
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ − 44.0000i − 1.61420i −0.590412 0.807102i $$-0.701035\pi$$
0.590412 0.807102i $$-0.298965\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 6.00000i − 0.219529i
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 0 0
$$753$$ − 36.0000i − 1.31191i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 46.0000i 1.67190i 0.548807 + 0.835949i $$0.315082\pi$$
−0.548807 + 0.835949i $$0.684918\pi$$
$$758$$ 0 0
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ − 24.0000i − 0.868858i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 28.0000i 1.01102i
$$768$$ 0 0
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 0 0
$$771$$ −36.0000 −1.29651
$$772$$ 0 0
$$773$$ − 54.0000i − 1.94225i −0.238581 0.971123i $$-0.576682\pi$$
0.238581 0.971123i $$-0.423318\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 80.0000i 2.86998i
$$778$$ 0 0
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ −24.0000 −0.858788
$$782$$ 0 0
$$783$$ − 24.0000i − 0.857690i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 22.0000i − 0.784215i −0.919919 0.392108i $$-0.871746\pi$$
0.919919 0.392108i $$-0.128254\pi$$
$$788$$ 0 0
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 8.00000 0.284447
$$792$$ 0 0
$$793$$ 4.00000i 0.142044i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 18.0000i − 0.637593i −0.947823 0.318796i $$-0.896721\pi$$
0.947823 0.318796i $$-0.103279\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ 0 0
$$803$$ 28.0000i 0.988099i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 20.0000i 0.704033i
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 18.0000 0.632065 0.316033 0.948748i $$-0.397649\pi$$
0.316033 + 0.948748i $$0.397649\pi$$
$$812$$ 0 0
$$813$$ 16.0000i 0.561144i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 12.0000i − 0.419827i
$$818$$ 0 0
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ 28.0000i 0.976019i 0.872838 + 0.488009i $$0.162277\pi$$
−0.872838 + 0.488009i $$0.837723\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 22.0000i 0.765015i 0.923952 + 0.382507i $$0.124939\pi$$
−0.923952 + 0.382507i $$0.875061\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 12.0000 0.416275
$$832$$ 0 0
$$833$$ − 18.0000i − 0.623663i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 36.0000i 1.23991i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 28.0000i 0.962091i
$$848$$ 0 0
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ −40.0000 −1.37118
$$852$$ 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
$$857$$ 6.00000i 0.204956i 0.994735 + 0.102478i $$0.0326771\pi$$
−0.994735 + 0.102478i $$0.967323\pi$$
$$858$$ 0 0
$$859$$ −50.0000 −1.70598 −0.852989 0.521929i $$-0.825213\pi$$
−0.852989 + 0.521929i $$0.825213\pi$$
$$860$$ 0 0
$$861$$ 48.0000 1.63584
$$862$$ 0 0
$$863$$ 32.0000i 1.08929i 0.838666 + 0.544646i $$0.183336\pi$$
−0.838666 + 0.544646i $$0.816664\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 26.0000i − 0.883006i
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ 0 0
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 0 0
$$879$$ −28.0000 −0.944417
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ 0 0
$$883$$ − 34.0000i − 1.14419i −0.820187 0.572096i $$-0.806131\pi$$
0.820187 0.572096i $$-0.193869\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36.0000i 1.20876i 0.796696 + 0.604381i $$0.206579\pi$$
−0.796696 + 0.604381i $$0.793421\pi$$
$$888$$ 0 0
$$889$$ 64.0000 2.14649
$$890$$ 0 0
$$891$$ −22.0000 −0.737028
$$892$$ 0 0
$$893$$ − 16.0000i − 0.535420i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 16.0000i − 0.534224i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ 48.0000i 1.59734i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 38.0000i 1.26177i 0.775877 + 0.630885i $$0.217308\pi$$
−0.775877 + 0.630885i $$0.782692\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ 12.0000i 0.397142i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 24.0000i − 0.792550i
$$918$$ 0 0
$$919$$ 36.0000 1.18753 0.593765 0.804638i $$-0.297641\pi$$
0.593765 + 0.804638i $$0.297641\pi$$
$$920$$ 0 0
$$921$$ 36.0000 1.18624
$$922$$ 0 0
$$923$$ − 24.0000i − 0.789970i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 4.00000i − 0.131377i
$$928$$ 0 0
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ 0 0
$$933$$ − 56.0000i − 1.83336i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 46.0000i − 1.50275i −0.659873 0.751377i $$-0.729390\pi$$
0.659873 0.751377i $$-0.270610\pi$$
$$938$$ 0 0
$$939$$ 20.0000 0.652675
$$940$$ 0 0
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ 0 0
$$943$$ 24.0000i 0.781548i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 14.0000i − 0.454939i −0.973785 0.227469i $$-0.926955\pi$$
0.973785 0.227469i $$-0.0730452\pi$$
$$948$$ 0 0
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ 58.0000i 1.87880i 0.342817 + 0.939402i $$0.388619\pi$$
−0.342817 + 0.939402i $$0.611381\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 24.0000i − 0.775810i
$$958$$ 0 0
$$959$$ −40.0000 −1.29167
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 2.00000i 0.0644491i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 28.0000i 0.900419i 0.892923 + 0.450210i $$0.148651\pi$$
−0.892923 + 0.450210i $$0.851349\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ −38.0000 −1.21948 −0.609739 0.792602i $$-0.708726\pi$$
−0.609739 + 0.792602i $$0.708726\pi$$
$$972$$ 0 0
$$973$$ 40.0000i 1.28234i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 30.0000i − 0.959785i −0.877327 0.479893i $$-0.840676\pi$$
0.877327 0.479893i $$-0.159324\pi$$
$$978$$ 0 0
$$979$$ 4.00000 0.127841
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ 0 0
$$983$$ − 20.0000i − 0.637901i −0.947771 0.318950i $$-0.896670\pi$$
0.947771 0.318950i $$-0.103330\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 64.0000i 2.03714i
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 28.0000i 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 54.0000i 1.71020i 0.518465 + 0.855099i $$0.326503\pi$$
−0.518465 + 0.855099i $$0.673497\pi$$
$$998$$ 0 0
$$999$$ −40.0000 −1.26554
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 3200.2.c.k.2049.1 2
4.3 odd 2 3200.2.c.e.2049.2 2
5.2 odd 4 128.2.a.b.1.1 yes 1
5.3 odd 4 3200.2.a.u.1.1 1
5.4 even 2 inner 3200.2.c.k.2049.2 2
8.3 odd 2 3200.2.c.l.2049.1 2
8.5 even 2 3200.2.c.f.2049.2 2
15.2 even 4 1152.2.a.h.1.1 1
20.3 even 4 3200.2.a.h.1.1 1
20.7 even 4 128.2.a.d.1.1 yes 1
20.19 odd 2 3200.2.c.e.2049.1 2
35.27 even 4 6272.2.a.g.1.1 1
40.3 even 4 3200.2.a.x.1.1 1
40.13 odd 4 3200.2.a.e.1.1 1
40.19 odd 2 3200.2.c.l.2049.2 2
40.27 even 4 128.2.a.a.1.1 1
40.29 even 2 3200.2.c.f.2049.1 2
40.37 odd 4 128.2.a.c.1.1 yes 1
60.47 odd 4 1152.2.a.c.1.1 1
80.27 even 4 256.2.b.c.129.1 2
80.37 odd 4 256.2.b.a.129.2 2
80.67 even 4 256.2.b.c.129.2 2
80.77 odd 4 256.2.b.a.129.1 2
120.77 even 4 1152.2.a.r.1.1 1
120.107 odd 4 1152.2.a.m.1.1 1
140.27 odd 4 6272.2.a.a.1.1 1
160.27 even 8 1024.2.e.i.769.1 4
160.37 odd 8 1024.2.e.m.769.2 4
160.67 even 8 1024.2.e.i.257.2 4
160.77 odd 8 1024.2.e.m.257.2 4
160.107 even 8 1024.2.e.i.769.2 4
160.117 odd 8 1024.2.e.m.769.1 4
160.147 even 8 1024.2.e.i.257.1 4
160.157 odd 8 1024.2.e.m.257.1 4
240.77 even 4 2304.2.d.b.1153.2 2
240.107 odd 4 2304.2.d.r.1153.1 2
240.197 even 4 2304.2.d.b.1153.1 2
240.227 odd 4 2304.2.d.r.1153.2 2
280.27 odd 4 6272.2.a.h.1.1 1
280.237 even 4 6272.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.a.a.1.1 1 40.27 even 4
128.2.a.b.1.1 yes 1 5.2 odd 4
128.2.a.c.1.1 yes 1 40.37 odd 4
128.2.a.d.1.1 yes 1 20.7 even 4
256.2.b.a.129.1 2 80.77 odd 4
256.2.b.a.129.2 2 80.37 odd 4
256.2.b.c.129.1 2 80.27 even 4
256.2.b.c.129.2 2 80.67 even 4
1024.2.e.i.257.1 4 160.147 even 8
1024.2.e.i.257.2 4 160.67 even 8
1024.2.e.i.769.1 4 160.27 even 8
1024.2.e.i.769.2 4 160.107 even 8
1024.2.e.m.257.1 4 160.157 odd 8
1024.2.e.m.257.2 4 160.77 odd 8
1024.2.e.m.769.1 4 160.117 odd 8
1024.2.e.m.769.2 4 160.37 odd 8
1152.2.a.c.1.1 1 60.47 odd 4
1152.2.a.h.1.1 1 15.2 even 4
1152.2.a.m.1.1 1 120.107 odd 4
1152.2.a.r.1.1 1 120.77 even 4
2304.2.d.b.1153.1 2 240.197 even 4
2304.2.d.b.1153.2 2 240.77 even 4
2304.2.d.r.1153.1 2 240.107 odd 4
2304.2.d.r.1153.2 2 240.227 odd 4
3200.2.a.e.1.1 1 40.13 odd 4
3200.2.a.h.1.1 1 20.3 even 4
3200.2.a.u.1.1 1 5.3 odd 4
3200.2.a.x.1.1 1 40.3 even 4
3200.2.c.e.2049.1 2 20.19 odd 2
3200.2.c.e.2049.2 2 4.3 odd 2
3200.2.c.f.2049.1 2 40.29 even 2
3200.2.c.f.2049.2 2 8.5 even 2
3200.2.c.k.2049.1 2 1.1 even 1 trivial
3200.2.c.k.2049.2 2 5.4 even 2 inner
3200.2.c.l.2049.1 2 8.3 odd 2
3200.2.c.l.2049.2 2 40.19 odd 2
6272.2.a.a.1.1 1 140.27 odd 4
6272.2.a.b.1.1 1 280.237 even 4
6272.2.a.g.1.1 1 35.27 even 4
6272.2.a.h.1.1 1 280.27 odd 4