# Properties

 Label 4400.2.b.y.4049.2 Level $4400$ Weight $2$ Character 4400.4049 Analytic conductor $35.134$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.2 Root $$-1.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.4049 Dual form 4400.2.b.y.4049.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.30278i q^{3} -4.30278i q^{7} +1.30278 q^{9} +O(q^{10})$$ $$q-1.30278i q^{3} -4.30278i q^{7} +1.30278 q^{9} +1.00000 q^{11} -5.00000i q^{13} -3.90833i q^{17} -1.00000 q^{19} -5.60555 q^{21} -3.69722i q^{23} -5.60555i q^{27} +9.90833 q^{29} +4.21110 q^{31} -1.30278i q^{33} +9.60555i q^{37} -6.51388 q^{39} +1.60555 q^{41} -7.21110i q^{43} +3.00000i q^{47} -11.5139 q^{49} -5.09167 q^{51} -2.30278i q^{53} +1.30278i q^{57} +0.211103 q^{59} +2.90833 q^{61} -5.60555i q^{63} +4.00000i q^{67} -4.81665 q^{69} -4.60555 q^{71} -2.90833i q^{73} -4.30278i q^{77} -0.0916731 q^{79} -3.39445 q^{81} +14.5139i q^{83} -12.9083i q^{87} -5.30278 q^{89} -21.5139 q^{91} -5.48612i q^{93} +11.6972i q^{97} +1.30278 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{9} + O(q^{10})$$ $$4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91} - 2 q^{99} + O(q^{100})$$

## 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
$$3$$ − 1.30278i − 0.752158i −0.926588 0.376079i $$-0.877272\pi$$
0.926588 0.376079i $$-0.122728\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 4.30278i − 1.62630i −0.582057 0.813148i $$-0.697752\pi$$
0.582057 0.813148i $$-0.302248\pi$$
$$8$$ 0 0
$$9$$ 1.30278 0.434259
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ − 5.00000i − 1.38675i −0.720577 0.693375i $$-0.756123\pi$$
0.720577 0.693375i $$-0.243877\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 3.90833i − 0.947909i −0.880549 0.473954i $$-0.842826\pi$$
0.880549 0.473954i $$-0.157174\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ −5.60555 −1.22323
$$22$$ 0 0
$$23$$ − 3.69722i − 0.770925i −0.922724 0.385462i $$-0.874042\pi$$
0.922724 0.385462i $$-0.125958\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.60555i − 1.07879i
$$28$$ 0 0
$$29$$ 9.90833 1.83993 0.919965 0.392000i $$-0.128217\pi$$
0.919965 + 0.392000i $$0.128217\pi$$
$$30$$ 0 0
$$31$$ 4.21110 0.756336 0.378168 0.925737i $$-0.376554\pi$$
0.378168 + 0.925737i $$0.376554\pi$$
$$32$$ 0 0
$$33$$ − 1.30278i − 0.226784i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.60555i 1.57914i 0.613659 + 0.789571i $$0.289697\pi$$
−0.613659 + 0.789571i $$0.710303\pi$$
$$38$$ 0 0
$$39$$ −6.51388 −1.04306
$$40$$ 0 0
$$41$$ 1.60555 0.250745 0.125372 0.992110i $$-0.459987\pi$$
0.125372 + 0.992110i $$0.459987\pi$$
$$42$$ 0 0
$$43$$ − 7.21110i − 1.09968i −0.835269 0.549841i $$-0.814688\pi$$
0.835269 0.549841i $$-0.185312\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.00000i 0.437595i 0.975770 + 0.218797i $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ 0 0
$$49$$ −11.5139 −1.64484
$$50$$ 0 0
$$51$$ −5.09167 −0.712977
$$52$$ 0 0
$$53$$ − 2.30278i − 0.316311i −0.987414 0.158155i $$-0.949445\pi$$
0.987414 0.158155i $$-0.0505547\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.30278i 0.172557i
$$58$$ 0 0
$$59$$ 0.211103 0.0274832 0.0137416 0.999906i $$-0.495626\pi$$
0.0137416 + 0.999906i $$0.495626\pi$$
$$60$$ 0 0
$$61$$ 2.90833 0.372373 0.186187 0.982514i $$-0.440387\pi$$
0.186187 + 0.982514i $$0.440387\pi$$
$$62$$ 0 0
$$63$$ − 5.60555i − 0.706233i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ −4.81665 −0.579857
$$70$$ 0 0
$$71$$ −4.60555 −0.546578 −0.273289 0.961932i $$-0.588112\pi$$
−0.273289 + 0.961932i $$0.588112\pi$$
$$72$$ 0 0
$$73$$ − 2.90833i − 0.340394i −0.985410 0.170197i $$-0.945560\pi$$
0.985410 0.170197i $$-0.0544404\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 4.30278i − 0.490347i
$$78$$ 0 0
$$79$$ −0.0916731 −0.0103140 −0.00515701 0.999987i $$-0.501642\pi$$
−0.00515701 + 0.999987i $$0.501642\pi$$
$$80$$ 0 0
$$81$$ −3.39445 −0.377161
$$82$$ 0 0
$$83$$ 14.5139i 1.59311i 0.604569 + 0.796553i $$0.293345\pi$$
−0.604569 + 0.796553i $$0.706655\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 12.9083i − 1.38392i
$$88$$ 0 0
$$89$$ −5.30278 −0.562093 −0.281047 0.959694i $$-0.590682\pi$$
−0.281047 + 0.959694i $$0.590682\pi$$
$$90$$ 0 0
$$91$$ −21.5139 −2.25527
$$92$$ 0 0
$$93$$ − 5.48612i − 0.568884i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 11.6972i 1.18767i 0.804586 + 0.593837i $$0.202387\pi$$
−0.804586 + 0.593837i $$0.797613\pi$$
$$98$$ 0 0
$$99$$ 1.30278 0.130934
$$100$$ 0 0
$$101$$ 17.5139 1.74270 0.871348 0.490666i $$-0.163246\pi$$
0.871348 + 0.490666i $$0.163246\pi$$
$$102$$ 0 0
$$103$$ − 7.90833i − 0.779231i −0.920978 0.389615i $$-0.872608\pi$$
0.920978 0.389615i $$-0.127392\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 3.00000i 0.290021i 0.989430 + 0.145010i $$0.0463216\pi$$
−0.989430 + 0.145010i $$0.953678\pi$$
$$108$$ 0 0
$$109$$ 6.51388 0.623916 0.311958 0.950096i $$-0.399015\pi$$
0.311958 + 0.950096i $$0.399015\pi$$
$$110$$ 0 0
$$111$$ 12.5139 1.18776
$$112$$ 0 0
$$113$$ − 10.8167i − 1.01755i −0.860901 0.508773i $$-0.830099\pi$$
0.860901 0.508773i $$-0.169901\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 6.51388i − 0.602208i
$$118$$ 0 0
$$119$$ −16.8167 −1.54158
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ − 2.09167i − 0.188600i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 17.1194i 1.51910i 0.650447 + 0.759552i $$0.274582\pi$$
−0.650447 + 0.759552i $$0.725418\pi$$
$$128$$ 0 0
$$129$$ −9.39445 −0.827135
$$130$$ 0 0
$$131$$ −0.908327 −0.0793609 −0.0396804 0.999212i $$-0.512634\pi$$
−0.0396804 + 0.999212i $$0.512634\pi$$
$$132$$ 0 0
$$133$$ 4.30278i 0.373098i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 2.09167i − 0.178704i −0.996000 0.0893518i $$-0.971520\pi$$
0.996000 0.0893518i $$-0.0284796\pi$$
$$138$$ 0 0
$$139$$ 8.21110 0.696457 0.348228 0.937410i $$-0.386783\pi$$
0.348228 + 0.937410i $$0.386783\pi$$
$$140$$ 0 0
$$141$$ 3.90833 0.329141
$$142$$ 0 0
$$143$$ − 5.00000i − 0.418121i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 15.0000i 1.23718i
$$148$$ 0 0
$$149$$ −2.78890 −0.228475 −0.114238 0.993453i $$-0.536443\pi$$
−0.114238 + 0.993453i $$0.536443\pi$$
$$150$$ 0 0
$$151$$ 20.8167 1.69404 0.847018 0.531565i $$-0.178396\pi$$
0.847018 + 0.531565i $$0.178396\pi$$
$$152$$ 0 0
$$153$$ − 5.09167i − 0.411637i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.78890i 0.382196i 0.981571 + 0.191098i $$0.0612048\pi$$
−0.981571 + 0.191098i $$0.938795\pi$$
$$158$$ 0 0
$$159$$ −3.00000 −0.237915
$$160$$ 0 0
$$161$$ −15.9083 −1.25375
$$162$$ 0 0
$$163$$ 5.69722i 0.446241i 0.974791 + 0.223121i $$0.0716243\pi$$
−0.974791 + 0.223121i $$0.928376\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 15.4222i 1.19341i 0.802462 + 0.596703i $$0.203523\pi$$
−0.802462 + 0.596703i $$0.796477\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −1.30278 −0.0996257
$$172$$ 0 0
$$173$$ − 16.8167i − 1.27855i −0.768980 0.639273i $$-0.779235\pi$$
0.768980 0.639273i $$-0.220765\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 0.275019i − 0.0206717i
$$178$$ 0 0
$$179$$ −5.51388 −0.412127 −0.206063 0.978539i $$-0.566065\pi$$
−0.206063 + 0.978539i $$0.566065\pi$$
$$180$$ 0 0
$$181$$ −9.09167 −0.675779 −0.337889 0.941186i $$-0.609713\pi$$
−0.337889 + 0.941186i $$0.609713\pi$$
$$182$$ 0 0
$$183$$ − 3.78890i − 0.280083i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 3.90833i − 0.285805i
$$188$$ 0 0
$$189$$ −24.1194 −1.75443
$$190$$ 0 0
$$191$$ −6.69722 −0.484594 −0.242297 0.970202i $$-0.577901\pi$$
−0.242297 + 0.970202i $$0.577901\pi$$
$$192$$ 0 0
$$193$$ 1.21110i 0.0871771i 0.999050 + 0.0435885i $$0.0138791\pi$$
−0.999050 + 0.0435885i $$0.986121\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 9.69722i 0.690899i 0.938437 + 0.345449i $$0.112273\pi$$
−0.938437 + 0.345449i $$0.887727\pi$$
$$198$$ 0 0
$$199$$ −24.5139 −1.73774 −0.868871 0.495038i $$-0.835154\pi$$
−0.868871 + 0.495038i $$0.835154\pi$$
$$200$$ 0 0
$$201$$ 5.21110 0.367563
$$202$$ 0 0
$$203$$ − 42.6333i − 2.99227i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 4.81665i − 0.334781i
$$208$$ 0 0
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −25.2389 −1.73751 −0.868757 0.495238i $$-0.835081\pi$$
−0.868757 + 0.495238i $$0.835081\pi$$
$$212$$ 0 0
$$213$$ 6.00000i 0.411113i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 18.1194i − 1.23003i
$$218$$ 0 0
$$219$$ −3.78890 −0.256030
$$220$$ 0 0
$$221$$ −19.5416 −1.31451
$$222$$ 0 0
$$223$$ 20.6333i 1.38171i 0.722994 + 0.690854i $$0.242765\pi$$
−0.722994 + 0.690854i $$0.757235\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 5.30278i − 0.351958i −0.984394 0.175979i $$-0.943691\pi$$
0.984394 0.175979i $$-0.0563090\pi$$
$$228$$ 0 0
$$229$$ −13.7250 −0.906972 −0.453486 0.891263i $$-0.649820\pi$$
−0.453486 + 0.891263i $$0.649820\pi$$
$$230$$ 0 0
$$231$$ −5.60555 −0.368818
$$232$$ 0 0
$$233$$ − 5.09167i − 0.333567i −0.985994 0.166783i $$-0.946662\pi$$
0.985994 0.166783i $$-0.0533380\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0.119429i 0.00775778i
$$238$$ 0 0
$$239$$ −4.11943 −0.266464 −0.133232 0.991085i $$-0.542535\pi$$
−0.133232 + 0.991085i $$0.542535\pi$$
$$240$$ 0 0
$$241$$ −24.9361 −1.60627 −0.803137 0.595794i $$-0.796837\pi$$
−0.803137 + 0.595794i $$0.796837\pi$$
$$242$$ 0 0
$$243$$ − 12.3944i − 0.795104i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000i 0.318142i
$$248$$ 0 0
$$249$$ 18.9083 1.19827
$$250$$ 0 0
$$251$$ 3.90833 0.246691 0.123346 0.992364i $$-0.460638\pi$$
0.123346 + 0.992364i $$0.460638\pi$$
$$252$$ 0 0
$$253$$ − 3.69722i − 0.232443i
$$254$$ 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
$$259$$ 41.3305 2.56815
$$260$$ 0 0
$$261$$ 12.9083 0.799005
$$262$$ 0 0
$$263$$ − 1.18335i − 0.0729683i −0.999334 0.0364841i $$-0.988384\pi$$
0.999334 0.0364841i $$-0.0116158\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.90833i 0.422783i
$$268$$ 0 0
$$269$$ 23.7250 1.44654 0.723269 0.690567i $$-0.242639\pi$$
0.723269 + 0.690567i $$0.242639\pi$$
$$270$$ 0 0
$$271$$ −14.2111 −0.863263 −0.431632 0.902050i $$-0.642062\pi$$
−0.431632 + 0.902050i $$0.642062\pi$$
$$272$$ 0 0
$$273$$ 28.0278i 1.69632i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 21.6056i 1.29815i 0.760724 + 0.649076i $$0.224844\pi$$
−0.760724 + 0.649076i $$0.775156\pi$$
$$278$$ 0 0
$$279$$ 5.48612 0.328446
$$280$$ 0 0
$$281$$ −22.8167 −1.36113 −0.680564 0.732689i $$-0.738265\pi$$
−0.680564 + 0.732689i $$0.738265\pi$$
$$282$$ 0 0
$$283$$ 2.69722i 0.160333i 0.996781 + 0.0801667i $$0.0255453\pi$$
−0.996781 + 0.0801667i $$0.974455\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6.90833i − 0.407786i
$$288$$ 0 0
$$289$$ 1.72498 0.101469
$$290$$ 0 0
$$291$$ 15.2389 0.893318
$$292$$ 0 0
$$293$$ − 15.2111i − 0.888642i −0.895868 0.444321i $$-0.853445\pi$$
0.895868 0.444321i $$-0.146555\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 5.60555i − 0.325267i
$$298$$ 0 0
$$299$$ −18.4861 −1.06908
$$300$$ 0 0
$$301$$ −31.0278 −1.78841
$$302$$ 0 0
$$303$$ − 22.8167i − 1.31078i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 6.09167i 0.347670i 0.984775 + 0.173835i $$0.0556160\pi$$
−0.984775 + 0.173835i $$0.944384\pi$$
$$308$$ 0 0
$$309$$ −10.3028 −0.586104
$$310$$ 0 0
$$311$$ 16.8167 0.953585 0.476792 0.879016i $$-0.341799\pi$$
0.476792 + 0.879016i $$0.341799\pi$$
$$312$$ 0 0
$$313$$ − 21.8167i − 1.23315i −0.787296 0.616575i $$-0.788520\pi$$
0.787296 0.616575i $$-0.211480\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9.90833i 0.556507i 0.960508 + 0.278254i $$0.0897555\pi$$
−0.960508 + 0.278254i $$0.910244\pi$$
$$318$$ 0 0
$$319$$ 9.90833 0.554760
$$320$$ 0 0
$$321$$ 3.90833 0.218142
$$322$$ 0 0
$$323$$ 3.90833i 0.217465i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 8.48612i − 0.469284i
$$328$$ 0 0
$$329$$ 12.9083 0.711659
$$330$$ 0 0
$$331$$ 14.3944 0.791190 0.395595 0.918425i $$-0.370538\pi$$
0.395595 + 0.918425i $$0.370538\pi$$
$$332$$ 0 0
$$333$$ 12.5139i 0.685756i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 26.8444i 1.46231i 0.682212 + 0.731154i $$0.261018\pi$$
−0.682212 + 0.731154i $$0.738982\pi$$
$$338$$ 0 0
$$339$$ −14.0917 −0.765355
$$340$$ 0 0
$$341$$ 4.21110 0.228044
$$342$$ 0 0
$$343$$ 19.4222i 1.04870i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 5.51388i − 0.296000i −0.988987 0.148000i $$-0.952716\pi$$
0.988987 0.148000i $$-0.0472836\pi$$
$$348$$ 0 0
$$349$$ 26.8167 1.43546 0.717731 0.696320i $$-0.245181\pi$$
0.717731 + 0.696320i $$0.245181\pi$$
$$350$$ 0 0
$$351$$ −28.0278 −1.49601
$$352$$ 0 0
$$353$$ 24.6333i 1.31110i 0.755152 + 0.655549i $$0.227563\pi$$
−0.755152 + 0.655549i $$0.772437\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 21.9083i 1.15951i
$$358$$ 0 0
$$359$$ 15.2111 0.802811 0.401406 0.915900i $$-0.368522\pi$$
0.401406 + 0.915900i $$0.368522\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ − 1.30278i − 0.0683780i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 24.3028i 1.26859i 0.773089 + 0.634297i $$0.218710\pi$$
−0.773089 + 0.634297i $$0.781290\pi$$
$$368$$ 0 0
$$369$$ 2.09167 0.108888
$$370$$ 0 0
$$371$$ −9.90833 −0.514415
$$372$$ 0 0
$$373$$ 1.42221i 0.0736390i 0.999322 + 0.0368195i $$0.0117227\pi$$
−0.999322 + 0.0368195i $$0.988277\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 49.5416i − 2.55152i
$$378$$ 0 0
$$379$$ 24.8167 1.27475 0.637373 0.770555i $$-0.280021\pi$$
0.637373 + 0.770555i $$0.280021\pi$$
$$380$$ 0 0
$$381$$ 22.3028 1.14261
$$382$$ 0 0
$$383$$ − 21.6333i − 1.10541i −0.833377 0.552705i $$-0.813596\pi$$
0.833377 0.552705i $$-0.186404\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 9.39445i − 0.477547i
$$388$$ 0 0
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ −14.4500 −0.730766
$$392$$ 0 0
$$393$$ 1.18335i 0.0596919i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 25.3028i 1.26991i 0.772549 + 0.634955i $$0.218981\pi$$
−0.772549 + 0.634955i $$0.781019\pi$$
$$398$$ 0 0
$$399$$ 5.60555 0.280629
$$400$$ 0 0
$$401$$ −27.2111 −1.35886 −0.679429 0.733741i $$-0.737772\pi$$
−0.679429 + 0.733741i $$0.737772\pi$$
$$402$$ 0 0
$$403$$ − 21.0555i − 1.04885i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9.60555i 0.476129i
$$408$$ 0 0
$$409$$ −8.21110 −0.406013 −0.203006 0.979177i $$-0.565071\pi$$
−0.203006 + 0.979177i $$0.565071\pi$$
$$410$$ 0 0
$$411$$ −2.72498 −0.134413
$$412$$ 0 0
$$413$$ − 0.908327i − 0.0446958i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 10.6972i − 0.523845i
$$418$$ 0 0
$$419$$ 13.6056 0.664675 0.332337 0.943161i $$-0.392163\pi$$
0.332337 + 0.943161i $$0.392163\pi$$
$$420$$ 0 0
$$421$$ 4.30278 0.209704 0.104852 0.994488i $$-0.466563\pi$$
0.104852 + 0.994488i $$0.466563\pi$$
$$422$$ 0 0
$$423$$ 3.90833i 0.190029i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 12.5139i − 0.605589i
$$428$$ 0 0
$$429$$ −6.51388 −0.314493
$$430$$ 0 0
$$431$$ −33.0000 −1.58955 −0.794777 0.606902i $$-0.792412\pi$$
−0.794777 + 0.606902i $$0.792412\pi$$
$$432$$ 0 0
$$433$$ − 5.00000i − 0.240285i −0.992757 0.120142i $$-0.961665\pi$$
0.992757 0.120142i $$-0.0383351\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.69722i 0.176862i
$$438$$ 0 0
$$439$$ 20.6972 0.987825 0.493912 0.869512i $$-0.335566\pi$$
0.493912 + 0.869512i $$0.335566\pi$$
$$440$$ 0 0
$$441$$ −15.0000 −0.714286
$$442$$ 0 0
$$443$$ − 1.39445i − 0.0662523i −0.999451 0.0331261i $$-0.989454\pi$$
0.999451 0.0331261i $$-0.0105463\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 3.63331i 0.171850i
$$448$$ 0 0
$$449$$ −41.5139 −1.95916 −0.979581 0.201052i $$-0.935564\pi$$
−0.979581 + 0.201052i $$0.935564\pi$$
$$450$$ 0 0
$$451$$ 1.60555 0.0756025
$$452$$ 0 0
$$453$$ − 27.1194i − 1.27418i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 24.3028i − 1.13684i −0.822740 0.568418i $$-0.807556\pi$$
0.822740 0.568418i $$-0.192444\pi$$
$$458$$ 0 0
$$459$$ −21.9083 −1.02259
$$460$$ 0 0
$$461$$ 17.7889 0.828512 0.414256 0.910161i $$-0.364042\pi$$
0.414256 + 0.910161i $$0.364042\pi$$
$$462$$ 0 0
$$463$$ 26.2111i 1.21813i 0.793119 + 0.609067i $$0.208456\pi$$
−0.793119 + 0.609067i $$0.791544\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 24.6333i − 1.13989i −0.821682 0.569947i $$-0.806964\pi$$
0.821682 0.569947i $$-0.193036\pi$$
$$468$$ 0 0
$$469$$ 17.2111 0.794735
$$470$$ 0 0
$$471$$ 6.23886 0.287471
$$472$$ 0 0
$$473$$ − 7.21110i − 0.331567i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 3.00000i − 0.137361i
$$478$$ 0 0
$$479$$ −13.1833 −0.602362 −0.301181 0.953567i $$-0.597381\pi$$
−0.301181 + 0.953567i $$0.597381\pi$$
$$480$$ 0 0
$$481$$ 48.0278 2.18988
$$482$$ 0 0
$$483$$ 20.7250i 0.943019i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 10.2111i 0.462709i 0.972869 + 0.231355i $$0.0743158\pi$$
−0.972869 + 0.231355i $$0.925684\pi$$
$$488$$ 0 0
$$489$$ 7.42221 0.335644
$$490$$ 0 0
$$491$$ 24.2111 1.09263 0.546316 0.837579i $$-0.316030\pi$$
0.546316 + 0.837579i $$0.316030\pi$$
$$492$$ 0 0
$$493$$ − 38.7250i − 1.74409i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 19.8167i 0.888898i
$$498$$ 0 0
$$499$$ −21.5139 −0.963093 −0.481547 0.876420i $$-0.659925\pi$$
−0.481547 + 0.876420i $$0.659925\pi$$
$$500$$ 0 0
$$501$$ 20.0917 0.897630
$$502$$ 0 0
$$503$$ 16.6056i 0.740405i 0.928951 + 0.370202i $$0.120712\pi$$
−0.928951 + 0.370202i $$0.879288\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 15.6333i 0.694300i
$$508$$ 0 0
$$509$$ 26.3028 1.16585 0.582925 0.812526i $$-0.301908\pi$$
0.582925 + 0.812526i $$0.301908\pi$$
$$510$$ 0 0
$$511$$ −12.5139 −0.553581
$$512$$ 0 0
$$513$$ 5.60555i 0.247491i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.00000i 0.131940i
$$518$$ 0 0
$$519$$ −21.9083 −0.961669
$$520$$ 0 0
$$521$$ 23.4500 1.02736 0.513681 0.857981i $$-0.328282\pi$$
0.513681 + 0.857981i $$0.328282\pi$$
$$522$$ 0 0
$$523$$ − 3.57779i − 0.156446i −0.996936 0.0782230i $$-0.975075\pi$$
0.996936 0.0782230i $$-0.0249246\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 16.4584i − 0.716938i
$$528$$ 0 0
$$529$$ 9.33053 0.405675
$$530$$ 0 0
$$531$$ 0.275019 0.0119348
$$532$$ 0 0
$$533$$ − 8.02776i − 0.347721i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 7.18335i 0.309984i
$$538$$ 0 0
$$539$$ −11.5139 −0.495938
$$540$$ 0 0
$$541$$ −6.72498 −0.289130 −0.144565 0.989495i $$-0.546178\pi$$
−0.144565 + 0.989495i $$0.546178\pi$$
$$542$$ 0 0
$$543$$ 11.8444i 0.508292i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 18.1194i − 0.774731i −0.921926 0.387365i $$-0.873385\pi$$
0.921926 0.387365i $$-0.126615\pi$$
$$548$$ 0 0
$$549$$ 3.78890 0.161706
$$550$$ 0 0
$$551$$ −9.90833 −0.422109
$$552$$ 0 0
$$553$$ 0.394449i 0.0167737i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 9.42221i − 0.399232i −0.979874 0.199616i $$-0.936031\pi$$
0.979874 0.199616i $$-0.0639694\pi$$
$$558$$ 0 0
$$559$$ −36.0555 −1.52499
$$560$$ 0 0
$$561$$ −5.09167 −0.214971
$$562$$ 0 0
$$563$$ 18.9083i 0.796891i 0.917192 + 0.398445i $$0.130450\pi$$
−0.917192 + 0.398445i $$0.869550\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 14.6056i 0.613375i
$$568$$ 0 0
$$569$$ −15.1472 −0.635003 −0.317502 0.948258i $$-0.602844\pi$$
−0.317502 + 0.948258i $$0.602844\pi$$
$$570$$ 0 0
$$571$$ 17.3305 0.725260 0.362630 0.931933i $$-0.381879\pi$$
0.362630 + 0.931933i $$0.381879\pi$$
$$572$$ 0 0
$$573$$ 8.72498i 0.364491i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 31.3583i − 1.30546i −0.757589 0.652731i $$-0.773623\pi$$
0.757589 0.652731i $$-0.226377\pi$$
$$578$$ 0 0
$$579$$ 1.57779 0.0655709
$$580$$ 0 0
$$581$$ 62.4500 2.59086
$$582$$ 0 0
$$583$$ − 2.30278i − 0.0953712i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 37.5416i 1.54951i 0.632262 + 0.774755i $$0.282127\pi$$
−0.632262 + 0.774755i $$0.717873\pi$$
$$588$$ 0 0
$$589$$ −4.21110 −0.173515
$$590$$ 0 0
$$591$$ 12.6333 0.519665
$$592$$ 0 0
$$593$$ 13.6056i 0.558713i 0.960187 + 0.279357i $$0.0901211\pi$$
−0.960187 + 0.279357i $$0.909879\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 31.9361i 1.30706i
$$598$$ 0 0
$$599$$ −14.0917 −0.575770 −0.287885 0.957665i $$-0.592952\pi$$
−0.287885 + 0.957665i $$0.592952\pi$$
$$600$$ 0 0
$$601$$ 8.90833 0.363378 0.181689 0.983356i $$-0.441844\pi$$
0.181689 + 0.983356i $$0.441844\pi$$
$$602$$ 0 0
$$603$$ 5.21110i 0.212213i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7.21110i 0.292690i 0.989234 + 0.146345i $$0.0467509\pi$$
−0.989234 + 0.146345i $$0.953249\pi$$
$$608$$ 0 0
$$609$$ −55.5416 −2.25066
$$610$$ 0 0
$$611$$ 15.0000 0.606835
$$612$$ 0 0
$$613$$ 41.1194i 1.66080i 0.557169 + 0.830399i $$0.311888\pi$$
−0.557169 + 0.830399i $$0.688112\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 10.6056i 0.426963i 0.976947 + 0.213482i $$0.0684804\pi$$
−0.976947 + 0.213482i $$0.931520\pi$$
$$618$$ 0 0
$$619$$ 17.4222 0.700258 0.350129 0.936702i $$-0.386138\pi$$
0.350129 + 0.936702i $$0.386138\pi$$
$$620$$ 0 0
$$621$$ −20.7250 −0.831665
$$622$$ 0 0
$$623$$ 22.8167i 0.914130i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 1.30278i 0.0520278i
$$628$$ 0 0
$$629$$ 37.5416 1.49688
$$630$$ 0 0
$$631$$ 39.9361 1.58983 0.794915 0.606721i $$-0.207515\pi$$
0.794915 + 0.606721i $$0.207515\pi$$
$$632$$ 0 0
$$633$$ 32.8806i 1.30689i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 57.5694i 2.28098i
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 42.2111 1.66724 0.833619 0.552340i $$-0.186265\pi$$
0.833619 + 0.552340i $$0.186265\pi$$
$$642$$ 0 0
$$643$$ − 22.0000i − 0.867595i −0.901010 0.433798i $$-0.857173\pi$$
0.901010 0.433798i $$-0.142827\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 17.2389i − 0.677729i −0.940835 0.338865i $$-0.889957\pi$$
0.940835 0.338865i $$-0.110043\pi$$
$$648$$ 0 0
$$649$$ 0.211103 0.00828650
$$650$$ 0 0
$$651$$ −23.6056 −0.925174
$$652$$ 0 0
$$653$$ 19.1194i 0.748201i 0.927388 + 0.374101i $$0.122049\pi$$
−0.927388 + 0.374101i $$0.877951\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 3.78890i − 0.147819i
$$658$$ 0 0
$$659$$ −20.0917 −0.782660 −0.391330 0.920250i $$-0.627985\pi$$
−0.391330 + 0.920250i $$0.627985\pi$$
$$660$$ 0 0
$$661$$ 12.8167 0.498510 0.249255 0.968438i $$-0.419814\pi$$
0.249255 + 0.968438i $$0.419814\pi$$
$$662$$ 0 0
$$663$$ 25.4584i 0.988721i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 36.6333i − 1.41845i
$$668$$ 0 0
$$669$$ 26.8806 1.03926
$$670$$ 0 0
$$671$$ 2.90833 0.112275
$$672$$ 0 0
$$673$$ 6.02776i 0.232353i 0.993229 + 0.116176i $$0.0370638\pi$$
−0.993229 + 0.116176i $$0.962936\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 26.2389i − 1.00844i −0.863575 0.504221i $$-0.831780\pi$$
0.863575 0.504221i $$-0.168220\pi$$
$$678$$ 0 0
$$679$$ 50.3305 1.93151
$$680$$ 0 0
$$681$$ −6.90833 −0.264728
$$682$$ 0 0
$$683$$ − 9.84441i − 0.376686i −0.982103 0.188343i $$-0.939688\pi$$
0.982103 0.188343i $$-0.0603116\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 17.8806i 0.682186i
$$688$$ 0 0
$$689$$ −11.5139 −0.438644
$$690$$ 0 0
$$691$$ 26.5416 1.00969 0.504846 0.863210i $$-0.331549\pi$$
0.504846 + 0.863210i $$0.331549\pi$$
$$692$$ 0 0
$$693$$ − 5.60555i − 0.212937i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 6.27502i − 0.237683i
$$698$$ 0 0
$$699$$ −6.63331 −0.250895
$$700$$ 0 0
$$701$$ −26.7889 −1.01180 −0.505901 0.862591i $$-0.668840\pi$$
−0.505901 + 0.862591i $$0.668840\pi$$
$$702$$ 0 0
$$703$$ − 9.60555i − 0.362280i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 75.3583i − 2.83414i
$$708$$ 0 0
$$709$$ −11.6333 −0.436898 −0.218449 0.975848i $$-0.570100\pi$$
−0.218449 + 0.975848i $$0.570100\pi$$
$$710$$ 0 0
$$711$$ −0.119429 −0.00447895
$$712$$ 0 0
$$713$$ − 15.5694i − 0.583078i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 5.36669i 0.200423i
$$718$$ 0 0
$$719$$ 28.8167 1.07468 0.537340 0.843366i $$-0.319429\pi$$
0.537340 + 0.843366i $$0.319429\pi$$
$$720$$ 0 0
$$721$$ −34.0278 −1.26726
$$722$$ 0 0
$$723$$ 32.4861i 1.20817i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 0.330532i − 0.0122588i −0.999981 0.00612938i $$-0.998049\pi$$
0.999981 0.00612938i $$-0.00195105\pi$$
$$728$$ 0 0
$$729$$ −26.3305 −0.975205
$$730$$ 0 0
$$731$$ −28.1833 −1.04240
$$732$$ 0 0
$$733$$ − 12.3944i − 0.457799i −0.973450 0.228900i $$-0.926487\pi$$
0.973450 0.228900i $$-0.0735128\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.00000i 0.147342i
$$738$$ 0 0
$$739$$ 9.88057 0.363463 0.181731 0.983348i $$-0.441830\pi$$
0.181731 + 0.983348i $$0.441830\pi$$
$$740$$ 0 0
$$741$$ 6.51388 0.239293
$$742$$ 0 0
$$743$$ 44.3028i 1.62531i 0.582744 + 0.812656i $$0.301979\pi$$
−0.582744 + 0.812656i $$0.698021\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 18.9083i 0.691820i
$$748$$ 0 0
$$749$$ 12.9083 0.471660
$$750$$ 0 0
$$751$$ 5.66947 0.206882 0.103441 0.994636i $$-0.467015\pi$$
0.103441 + 0.994636i $$0.467015\pi$$
$$752$$ 0 0
$$753$$ − 5.09167i − 0.185551i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 23.0555i − 0.837967i −0.907994 0.418983i $$-0.862387\pi$$
0.907994 0.418983i $$-0.137613\pi$$
$$758$$ 0 0
$$759$$ −4.81665 −0.174833
$$760$$ 0 0
$$761$$ −42.4222 −1.53780 −0.768902 0.639367i $$-0.779197\pi$$
−0.768902 + 0.639367i $$0.779197\pi$$
$$762$$ 0 0
$$763$$ − 28.0278i − 1.01467i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 1.05551i − 0.0381124i
$$768$$ 0 0
$$769$$ 26.8167 0.967033 0.483517 0.875335i $$-0.339359\pi$$
0.483517 + 0.875335i $$0.339359\pi$$
$$770$$ 0 0
$$771$$ −23.4500 −0.844530
$$772$$ 0 0
$$773$$ 22.1194i 0.795581i 0.917476 + 0.397790i $$0.130223\pi$$
−0.917476 + 0.397790i $$0.869777\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 53.8444i − 1.93166i
$$778$$ 0 0
$$779$$ −1.60555 −0.0575248
$$780$$ 0 0
$$781$$ −4.60555 −0.164800
$$782$$ 0 0
$$783$$ − 55.5416i − 1.98490i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4.21110i 0.150110i 0.997179 + 0.0750548i $$0.0239132\pi$$
−0.997179 + 0.0750548i $$0.976087\pi$$
$$788$$ 0 0
$$789$$ −1.54163 −0.0548836
$$790$$ 0 0
$$791$$ −46.5416 −1.65483
$$792$$ 0 0
$$793$$ − 14.5416i − 0.516389i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 14.5139i − 0.514108i −0.966397 0.257054i $$-0.917248\pi$$
0.966397 0.257054i $$-0.0827518\pi$$
$$798$$ 0 0
$$799$$ 11.7250 0.414800
$$800$$ 0 0
$$801$$ −6.90833 −0.244094
$$802$$ 0 0
$$803$$ − 2.90833i − 0.102633i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 30.9083i − 1.08802i
$$808$$ 0 0
$$809$$ 3.63331 0.127740 0.0638701 0.997958i $$-0.479656\pi$$
0.0638701 + 0.997958i $$0.479656\pi$$
$$810$$ 0 0
$$811$$ 54.8722 1.92682 0.963411 0.268028i $$-0.0863719\pi$$
0.963411 + 0.268028i $$0.0863719\pi$$
$$812$$ 0 0
$$813$$ 18.5139i 0.649310i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.21110i 0.252285i
$$818$$ 0 0
$$819$$ −28.0278 −0.979369
$$820$$ 0 0
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ 0 0
$$823$$ − 10.4222i − 0.363295i −0.983364 0.181648i $$-0.941857\pi$$
0.983364 0.181648i $$-0.0581430\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 7.81665i 0.271812i 0.990722 + 0.135906i $$0.0433945\pi$$
−0.990722 + 0.135906i $$0.956606\pi$$
$$828$$ 0 0
$$829$$ 38.7527 1.34594 0.672969 0.739671i $$-0.265019\pi$$
0.672969 + 0.739671i $$0.265019\pi$$
$$830$$ 0 0
$$831$$ 28.1472 0.976415
$$832$$ 0 0
$$833$$ 45.0000i 1.55916i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 23.6056i − 0.815927i
$$838$$ 0 0
$$839$$ 16.1194 0.556505 0.278252 0.960508i $$-0.410245\pi$$
0.278252 + 0.960508i $$0.410245\pi$$
$$840$$ 0 0
$$841$$ 69.1749 2.38534
$$842$$ 0 0
$$843$$ 29.7250i 1.02378i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 4.30278i − 0.147845i
$$848$$ 0 0
$$849$$ 3.51388 0.120596
$$850$$ 0 0
$$851$$ 35.5139 1.21740
$$852$$ 0 0
$$853$$ − 19.7250i − 0.675370i −0.941259 0.337685i $$-0.890356\pi$$
0.941259 0.337685i $$-0.109644\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 3.00000i − 0.102478i −0.998686 0.0512390i $$-0.983683\pi$$
0.998686 0.0512390i $$-0.0163170\pi$$
$$858$$ 0 0
$$859$$ 48.6056 1.65840 0.829200 0.558952i $$-0.188796\pi$$
0.829200 + 0.558952i $$0.188796\pi$$
$$860$$ 0 0
$$861$$ −9.00000 −0.306719
$$862$$ 0 0
$$863$$ − 19.6056i − 0.667381i −0.942683 0.333690i $$-0.891706\pi$$
0.942683 0.333690i $$-0.108294\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 2.24726i − 0.0763210i
$$868$$ 0 0
$$869$$ −0.0916731 −0.00310980
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 0 0
$$873$$ 15.2389i 0.515757i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20.0000i 0.675352i 0.941262 + 0.337676i $$0.109641\pi$$
−0.941262 + 0.337676i $$0.890359\pi$$
$$878$$ 0 0
$$879$$ −19.8167 −0.668399
$$880$$ 0 0
$$881$$ −34.5416 −1.16374 −0.581869 0.813283i $$-0.697678\pi$$
−0.581869 + 0.813283i $$0.697678\pi$$
$$882$$ 0 0
$$883$$ − 12.4500i − 0.418975i −0.977811 0.209487i $$-0.932821\pi$$
0.977811 0.209487i $$-0.0671795\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 47.2389i − 1.58613i −0.609140 0.793063i $$-0.708485\pi$$
0.609140 0.793063i $$-0.291515\pi$$
$$888$$ 0 0
$$889$$ 73.6611 2.47051
$$890$$ 0 0
$$891$$ −3.39445 −0.113718
$$892$$ 0 0
$$893$$ − 3.00000i − 0.100391i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 24.0833i 0.804117i
$$898$$ 0 0
$$899$$ 41.7250 1.39161
$$900$$ 0 0
$$901$$ −9.00000 −0.299833
$$902$$ 0 0
$$903$$ 40.4222i 1.34517i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ 0 0
$$909$$ 22.8167 0.756781
$$910$$ 0 0
$$911$$ 39.2111 1.29912 0.649561 0.760310i $$-0.274953\pi$$
0.649561 + 0.760310i $$0.274953\pi$$
$$912$$ 0 0
$$913$$ 14.5139i 0.480339i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 3.90833i 0.129064i
$$918$$ 0 0
$$919$$ 41.2111 1.35943 0.679714 0.733477i $$-0.262104\pi$$
0.679714 + 0.733477i $$0.262104\pi$$
$$920$$ 0 0
$$921$$ 7.93608 0.261503
$$922$$ 0 0
$$923$$ 23.0278i 0.757968i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 10.3028i − 0.338388i
$$928$$ 0 0
$$929$$ −46.3944 −1.52215 −0.761076 0.648662i $$-0.775329\pi$$
−0.761076 + 0.648662i $$0.775329\pi$$
$$930$$ 0 0
$$931$$ 11.5139 0.377352
$$932$$ 0 0
$$933$$ − 21.9083i − 0.717246i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 5.21110i 0.170239i 0.996371 + 0.0851196i $$0.0271273\pi$$
−0.996371 + 0.0851196i $$0.972873\pi$$
$$938$$ 0 0
$$939$$ −28.4222 −0.927524
$$940$$ 0 0
$$941$$ 52.3944 1.70801 0.854005 0.520265i $$-0.174167\pi$$
0.854005 + 0.520265i $$0.174167\pi$$
$$942$$ 0 0
$$943$$ − 5.93608i − 0.193305i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 36.6333i 1.19042i 0.803569 + 0.595211i $$0.202932\pi$$
−0.803569 + 0.595211i $$0.797068\pi$$
$$948$$ 0 0
$$949$$ −14.5416 −0.472041
$$950$$ 0 0
$$951$$ 12.9083 0.418581
$$952$$ 0 0
$$953$$ 49.2666i 1.59590i 0.602722 + 0.797951i $$0.294083\pi$$
−0.602722 + 0.797951i $$0.705917\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 12.9083i − 0.417267i
$$958$$ 0 0
$$959$$ −9.00000 −0.290625
$$960$$ 0 0
$$961$$ −13.2666 −0.427955
$$962$$ 0 0
$$963$$ 3.90833i 0.125944i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 4.09167i − 0.131579i −0.997834 0.0657897i $$-0.979043\pi$$
0.997834 0.0657897i $$-0.0209566\pi$$
$$968$$ 0 0
$$969$$ 5.09167 0.163568
$$970$$ 0 0
$$971$$ 30.3583 0.974244 0.487122 0.873334i $$-0.338047\pi$$
0.487122 + 0.873334i $$0.338047\pi$$
$$972$$ 0 0
$$973$$ − 35.3305i − 1.13264i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 15.9722i − 0.510997i −0.966809 0.255499i $$-0.917760\pi$$
0.966809 0.255499i $$-0.0822396\pi$$
$$978$$ 0 0
$$979$$ −5.30278 −0.169477
$$980$$ 0 0
$$981$$ 8.48612 0.270941
$$982$$ 0 0
$$983$$ − 48.8444i − 1.55789i −0.627089 0.778947i $$-0.715754\pi$$
0.627089 0.778947i $$-0.284246\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 16.8167i − 0.535280i
$$988$$ 0 0
$$989$$ −26.6611 −0.847773
$$990$$ 0 0
$$991$$ 6.09167 0.193508 0.0967542 0.995308i $$-0.469154\pi$$
0.0967542 + 0.995308i $$0.469154\pi$$
$$992$$ 0 0
$$993$$ − 18.7527i − 0.595100i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 14.2750i 0.452094i 0.974116 + 0.226047i $$0.0725804\pi$$
−0.974116 + 0.226047i $$0.927420\pi$$
$$998$$ 0 0
$$999$$ 53.8444 1.70356
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.y.4049.2 4
4.3 odd 2 275.2.b.c.199.4 4
5.2 odd 4 4400.2.a.bs.1.1 2
5.3 odd 4 4400.2.a.bh.1.2 2
5.4 even 2 inner 4400.2.b.y.4049.3 4
12.11 even 2 2475.2.c.k.199.1 4
20.3 even 4 275.2.a.f.1.2 yes 2
20.7 even 4 275.2.a.e.1.1 2
20.19 odd 2 275.2.b.c.199.1 4
60.23 odd 4 2475.2.a.o.1.1 2
60.47 odd 4 2475.2.a.t.1.2 2
60.59 even 2 2475.2.c.k.199.4 4
220.43 odd 4 3025.2.a.h.1.1 2
220.87 odd 4 3025.2.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 2 20.7 even 4
275.2.a.f.1.2 yes 2 20.3 even 4
275.2.b.c.199.1 4 20.19 odd 2
275.2.b.c.199.4 4 4.3 odd 2
2475.2.a.o.1.1 2 60.23 odd 4
2475.2.a.t.1.2 2 60.47 odd 4
2475.2.c.k.199.1 4 12.11 even 2
2475.2.c.k.199.4 4 60.59 even 2
3025.2.a.h.1.1 2 220.43 odd 4
3025.2.a.n.1.2 2 220.87 odd 4
4400.2.a.bh.1.2 2 5.3 odd 4
4400.2.a.bs.1.1 2 5.2 odd 4
4400.2.b.y.4049.2 4 1.1 even 1 trivial
4400.2.b.y.4049.3 4 5.4 even 2 inner