# Properties

 Label 1900.2.a.e.1.1 Level $1900$ Weight $2$ Character 1900.1 Self dual yes Analytic conductor $15.172$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 380) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1900.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.585786 q^{3} +4.82843 q^{7} -2.65685 q^{9} +O(q^{10})$$ $$q+0.585786 q^{3} +4.82843 q^{7} -2.65685 q^{9} -2.00000 q^{11} -2.24264 q^{13} +4.82843 q^{17} -1.00000 q^{19} +2.82843 q^{21} +6.00000 q^{23} -3.31371 q^{27} +10.4853 q^{29} -1.17157 q^{31} -1.17157 q^{33} +10.2426 q^{37} -1.31371 q^{39} -7.65685 q^{41} +0.828427 q^{43} -0.828427 q^{47} +16.3137 q^{49} +2.82843 q^{51} +5.07107 q^{53} -0.585786 q^{57} +2.34315 q^{59} -2.34315 q^{61} -12.8284 q^{63} +0.585786 q^{67} +3.51472 q^{69} +10.8284 q^{71} -14.4853 q^{73} -9.65685 q^{77} -9.65685 q^{79} +6.02944 q^{81} -9.31371 q^{83} +6.14214 q^{87} +10.4853 q^{89} -10.8284 q^{91} -0.686292 q^{93} +1.75736 q^{97} +5.31371 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 4 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 + 4 * q^7 + 6 * q^9 $$2 q + 4 q^{3} + 4 q^{7} + 6 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{17} - 2 q^{19} + 12 q^{23} + 16 q^{27} + 4 q^{29} - 8 q^{31} - 8 q^{33} + 12 q^{37} + 20 q^{39} - 4 q^{41} - 4 q^{43} + 4 q^{47} + 10 q^{49} - 4 q^{53} - 4 q^{57} + 16 q^{59} - 16 q^{61} - 20 q^{63} + 4 q^{67} + 24 q^{69} + 16 q^{71} - 12 q^{73} - 8 q^{77} - 8 q^{79} + 46 q^{81} + 4 q^{83} - 16 q^{87} + 4 q^{89} - 16 q^{91} - 24 q^{93} + 12 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + 4 * q^3 + 4 * q^7 + 6 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^17 - 2 * q^19 + 12 * q^23 + 16 * q^27 + 4 * q^29 - 8 * q^31 - 8 * q^33 + 12 * q^37 + 20 * q^39 - 4 * q^41 - 4 * q^43 + 4 * q^47 + 10 * q^49 - 4 * q^53 - 4 * q^57 + 16 * q^59 - 16 * q^61 - 20 * q^63 + 4 * q^67 + 24 * q^69 + 16 * q^71 - 12 * q^73 - 8 * q^77 - 8 * q^79 + 46 * q^81 + 4 * q^83 - 16 * q^87 + 4 * q^89 - 16 * q^91 - 24 * q^93 + 12 * q^97 - 12 * q^99

## 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
$$3$$ 0.585786 0.338204 0.169102 0.985599i $$-0.445913\pi$$
0.169102 + 0.985599i $$0.445913\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.82843 1.82497 0.912487 0.409106i $$-0.134159\pi$$
0.912487 + 0.409106i $$0.134159\pi$$
$$8$$ 0 0
$$9$$ −2.65685 −0.885618
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −2.24264 −0.621997 −0.310998 0.950410i $$-0.600663\pi$$
−0.310998 + 0.950410i $$0.600663\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.82843 1.17107 0.585533 0.810649i $$-0.300885\pi$$
0.585533 + 0.810649i $$0.300885\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.82843 0.617213
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −3.31371 −0.637723
$$28$$ 0 0
$$29$$ 10.4853 1.94707 0.973534 0.228543i $$-0.0733960\pi$$
0.973534 + 0.228543i $$0.0733960\pi$$
$$30$$ 0 0
$$31$$ −1.17157 −0.210421 −0.105210 0.994450i $$-0.533552\pi$$
−0.105210 + 0.994450i $$0.533552\pi$$
$$32$$ 0 0
$$33$$ −1.17157 −0.203945
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.2426 1.68388 0.841940 0.539571i $$-0.181414\pi$$
0.841940 + 0.539571i $$0.181414\pi$$
$$38$$ 0 0
$$39$$ −1.31371 −0.210362
$$40$$ 0 0
$$41$$ −7.65685 −1.19580 −0.597900 0.801571i $$-0.703998\pi$$
−0.597900 + 0.801571i $$0.703998\pi$$
$$42$$ 0 0
$$43$$ 0.828427 0.126334 0.0631670 0.998003i $$-0.479880\pi$$
0.0631670 + 0.998003i $$0.479880\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −0.828427 −0.120839 −0.0604193 0.998173i $$-0.519244\pi$$
−0.0604193 + 0.998173i $$0.519244\pi$$
$$48$$ 0 0
$$49$$ 16.3137 2.33053
$$50$$ 0 0
$$51$$ 2.82843 0.396059
$$52$$ 0 0
$$53$$ 5.07107 0.696565 0.348282 0.937390i $$-0.386765\pi$$
0.348282 + 0.937390i $$0.386765\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.585786 −0.0775893
$$58$$ 0 0
$$59$$ 2.34315 0.305052 0.152526 0.988299i $$-0.451259\pi$$
0.152526 + 0.988299i $$0.451259\pi$$
$$60$$ 0 0
$$61$$ −2.34315 −0.300009 −0.150005 0.988685i $$-0.547929\pi$$
−0.150005 + 0.988685i $$0.547929\pi$$
$$62$$ 0 0
$$63$$ −12.8284 −1.61623
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.585786 0.0715652 0.0357826 0.999360i $$-0.488608\pi$$
0.0357826 + 0.999360i $$0.488608\pi$$
$$68$$ 0 0
$$69$$ 3.51472 0.423122
$$70$$ 0 0
$$71$$ 10.8284 1.28510 0.642549 0.766245i $$-0.277877\pi$$
0.642549 + 0.766245i $$0.277877\pi$$
$$72$$ 0 0
$$73$$ −14.4853 −1.69537 −0.847687 0.530497i $$-0.822005\pi$$
−0.847687 + 0.530497i $$0.822005\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −9.65685 −1.10050
$$78$$ 0 0
$$79$$ −9.65685 −1.08648 −0.543240 0.839577i $$-0.682803\pi$$
−0.543240 + 0.839577i $$0.682803\pi$$
$$80$$ 0 0
$$81$$ 6.02944 0.669937
$$82$$ 0 0
$$83$$ −9.31371 −1.02231 −0.511156 0.859488i $$-0.670783\pi$$
−0.511156 + 0.859488i $$0.670783\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 6.14214 0.658506
$$88$$ 0 0
$$89$$ 10.4853 1.11144 0.555719 0.831370i $$-0.312443\pi$$
0.555719 + 0.831370i $$0.312443\pi$$
$$90$$ 0 0
$$91$$ −10.8284 −1.13513
$$92$$ 0 0
$$93$$ −0.686292 −0.0711651
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.75736 0.178433 0.0892164 0.996012i $$-0.471564\pi$$
0.0892164 + 0.996012i $$0.471564\pi$$
$$98$$ 0 0
$$99$$ 5.31371 0.534048
$$100$$ 0 0
$$101$$ 4.00000 0.398015 0.199007 0.979998i $$-0.436228\pi$$
0.199007 + 0.979998i $$0.436228\pi$$
$$102$$ 0 0
$$103$$ 15.8995 1.56662 0.783312 0.621629i $$-0.213529\pi$$
0.783312 + 0.621629i $$0.213529\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −4.58579 −0.443325 −0.221662 0.975123i $$-0.571148\pi$$
−0.221662 + 0.975123i $$0.571148\pi$$
$$108$$ 0 0
$$109$$ −8.82843 −0.845610 −0.422805 0.906221i $$-0.638954\pi$$
−0.422805 + 0.906221i $$0.638954\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ −5.07107 −0.477046 −0.238523 0.971137i $$-0.576663\pi$$
−0.238523 + 0.971137i $$0.576663\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 5.95837 0.550851
$$118$$ 0 0
$$119$$ 23.3137 2.13716
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ −4.48528 −0.404424
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 21.5563 1.91282 0.956408 0.292033i $$-0.0943316\pi$$
0.956408 + 0.292033i $$0.0943316\pi$$
$$128$$ 0 0
$$129$$ 0.485281 0.0427266
$$130$$ 0 0
$$131$$ −5.65685 −0.494242 −0.247121 0.968985i $$-0.579484\pi$$
−0.247121 + 0.968985i $$0.579484\pi$$
$$132$$ 0 0
$$133$$ −4.82843 −0.418678
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 20.1421 1.72086 0.860429 0.509570i $$-0.170195\pi$$
0.860429 + 0.509570i $$0.170195\pi$$
$$138$$ 0 0
$$139$$ −17.3137 −1.46853 −0.734265 0.678863i $$-0.762473\pi$$
−0.734265 + 0.678863i $$0.762473\pi$$
$$140$$ 0 0
$$141$$ −0.485281 −0.0408681
$$142$$ 0 0
$$143$$ 4.48528 0.375078
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 9.55635 0.788194
$$148$$ 0 0
$$149$$ −8.00000 −0.655386 −0.327693 0.944784i $$-0.606271\pi$$
−0.327693 + 0.944784i $$0.606271\pi$$
$$150$$ 0 0
$$151$$ 18.1421 1.47639 0.738193 0.674590i $$-0.235679\pi$$
0.738193 + 0.674590i $$0.235679\pi$$
$$152$$ 0 0
$$153$$ −12.8284 −1.03712
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3.17157 −0.253119 −0.126560 0.991959i $$-0.540393\pi$$
−0.126560 + 0.991959i $$0.540393\pi$$
$$158$$ 0 0
$$159$$ 2.97056 0.235581
$$160$$ 0 0
$$161$$ 28.9706 2.28320
$$162$$ 0 0
$$163$$ 2.48528 0.194662 0.0973311 0.995252i $$-0.468969\pi$$
0.0973311 + 0.995252i $$0.468969\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.7279 0.830152 0.415076 0.909787i $$-0.363755\pi$$
0.415076 + 0.909787i $$0.363755\pi$$
$$168$$ 0 0
$$169$$ −7.97056 −0.613120
$$170$$ 0 0
$$171$$ 2.65685 0.203175
$$172$$ 0 0
$$173$$ 3.89949 0.296473 0.148237 0.988952i $$-0.452640\pi$$
0.148237 + 0.988952i $$0.452640\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1.37258 0.103170
$$178$$ 0 0
$$179$$ 21.6569 1.61871 0.809355 0.587320i $$-0.199817\pi$$
0.809355 + 0.587320i $$0.199817\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ −1.37258 −0.101464
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −9.65685 −0.706179
$$188$$ 0 0
$$189$$ −16.0000 −1.16383
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ −0.585786 −0.0421658 −0.0210829 0.999778i $$-0.506711\pi$$
−0.0210829 + 0.999778i $$0.506711\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.31371 0.378586 0.189293 0.981921i $$-0.439380\pi$$
0.189293 + 0.981921i $$0.439380\pi$$
$$198$$ 0 0
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 0 0
$$201$$ 0.343146 0.0242036
$$202$$ 0 0
$$203$$ 50.6274 3.55335
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −15.9411 −1.10798
$$208$$ 0 0
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ −11.5147 −0.792706 −0.396353 0.918098i $$-0.629724\pi$$
−0.396353 + 0.918098i $$0.629724\pi$$
$$212$$ 0 0
$$213$$ 6.34315 0.434625
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −5.65685 −0.384012
$$218$$ 0 0
$$219$$ −8.48528 −0.573382
$$220$$ 0 0
$$221$$ −10.8284 −0.728399
$$222$$ 0 0
$$223$$ −19.2132 −1.28661 −0.643306 0.765609i $$-0.722438\pi$$
−0.643306 + 0.765609i $$0.722438\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −20.5858 −1.36633 −0.683163 0.730266i $$-0.739396\pi$$
−0.683163 + 0.730266i $$0.739396\pi$$
$$228$$ 0 0
$$229$$ −21.6569 −1.43113 −0.715563 0.698549i $$-0.753830\pi$$
−0.715563 + 0.698549i $$0.753830\pi$$
$$230$$ 0 0
$$231$$ −5.65685 −0.372194
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −5.65685 −0.367452
$$238$$ 0 0
$$239$$ 10.3431 0.669042 0.334521 0.942388i $$-0.391425\pi$$
0.334521 + 0.942388i $$0.391425\pi$$
$$240$$ 0 0
$$241$$ −30.2843 −1.95078 −0.975391 0.220484i $$-0.929236\pi$$
−0.975391 + 0.220484i $$0.929236\pi$$
$$242$$ 0 0
$$243$$ 13.4731 0.864299
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.24264 0.142696
$$248$$ 0 0
$$249$$ −5.45584 −0.345750
$$250$$ 0 0
$$251$$ −13.6569 −0.862013 −0.431006 0.902349i $$-0.641841\pi$$
−0.431006 + 0.902349i $$0.641841\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.24264 0.139892 0.0699460 0.997551i $$-0.477717\pi$$
0.0699460 + 0.997551i $$0.477717\pi$$
$$258$$ 0 0
$$259$$ 49.4558 3.07304
$$260$$ 0 0
$$261$$ −27.8579 −1.72436
$$262$$ 0 0
$$263$$ −3.65685 −0.225491 −0.112746 0.993624i $$-0.535965\pi$$
−0.112746 + 0.993624i $$0.535965\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.14214 0.375893
$$268$$ 0 0
$$269$$ −22.4853 −1.37095 −0.685476 0.728095i $$-0.740406\pi$$
−0.685476 + 0.728095i $$0.740406\pi$$
$$270$$ 0 0
$$271$$ 23.6569 1.43705 0.718526 0.695500i $$-0.244817\pi$$
0.718526 + 0.695500i $$0.244817\pi$$
$$272$$ 0 0
$$273$$ −6.34315 −0.383905
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −16.8284 −1.01112 −0.505561 0.862791i $$-0.668714\pi$$
−0.505561 + 0.862791i $$0.668714\pi$$
$$278$$ 0 0
$$279$$ 3.11270 0.186352
$$280$$ 0 0
$$281$$ 18.9706 1.13169 0.565844 0.824512i $$-0.308550\pi$$
0.565844 + 0.824512i $$0.308550\pi$$
$$282$$ 0 0
$$283$$ 2.68629 0.159683 0.0798417 0.996808i $$-0.474559\pi$$
0.0798417 + 0.996808i $$0.474559\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −36.9706 −2.18230
$$288$$ 0 0
$$289$$ 6.31371 0.371395
$$290$$ 0 0
$$291$$ 1.02944 0.0603467
$$292$$ 0 0
$$293$$ 23.4142 1.36787 0.683936 0.729542i $$-0.260267\pi$$
0.683936 + 0.729542i $$0.260267\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 6.62742 0.384562
$$298$$ 0 0
$$299$$ −13.4558 −0.778172
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ 2.34315 0.134610
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −7.89949 −0.450848 −0.225424 0.974261i $$-0.572377\pi$$
−0.225424 + 0.974261i $$0.572377\pi$$
$$308$$ 0 0
$$309$$ 9.31371 0.529838
$$310$$ 0 0
$$311$$ −14.0000 −0.793867 −0.396934 0.917847i $$-0.629926\pi$$
−0.396934 + 0.917847i $$0.629926\pi$$
$$312$$ 0 0
$$313$$ −18.0000 −1.01742 −0.508710 0.860938i $$-0.669877\pi$$
−0.508710 + 0.860938i $$0.669877\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5.75736 0.323366 0.161683 0.986843i $$-0.448308\pi$$
0.161683 + 0.986843i $$0.448308\pi$$
$$318$$ 0 0
$$319$$ −20.9706 −1.17413
$$320$$ 0 0
$$321$$ −2.68629 −0.149934
$$322$$ 0 0
$$323$$ −4.82843 −0.268661
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −5.17157 −0.285989
$$328$$ 0 0
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ 1.17157 0.0643955 0.0321977 0.999482i $$-0.489749\pi$$
0.0321977 + 0.999482i $$0.489749\pi$$
$$332$$ 0 0
$$333$$ −27.2132 −1.49127
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −17.7574 −0.967305 −0.483652 0.875260i $$-0.660690\pi$$
−0.483652 + 0.875260i $$0.660690\pi$$
$$338$$ 0 0
$$339$$ −2.97056 −0.161339
$$340$$ 0 0
$$341$$ 2.34315 0.126888
$$342$$ 0 0
$$343$$ 44.9706 2.42818
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −22.9706 −1.23312 −0.616562 0.787306i $$-0.711475\pi$$
−0.616562 + 0.787306i $$0.711475\pi$$
$$348$$ 0 0
$$349$$ −0.343146 −0.0183682 −0.00918409 0.999958i $$-0.502923\pi$$
−0.00918409 + 0.999958i $$0.502923\pi$$
$$350$$ 0 0
$$351$$ 7.43146 0.396662
$$352$$ 0 0
$$353$$ 3.85786 0.205333 0.102667 0.994716i $$-0.467262\pi$$
0.102667 + 0.994716i $$0.467262\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 13.6569 0.722797
$$358$$ 0 0
$$359$$ −21.3137 −1.12489 −0.562447 0.826833i $$-0.690140\pi$$
−0.562447 + 0.826833i $$0.690140\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −4.10051 −0.215221
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 28.1421 1.46901 0.734504 0.678605i $$-0.237415\pi$$
0.734504 + 0.678605i $$0.237415\pi$$
$$368$$ 0 0
$$369$$ 20.3431 1.05902
$$370$$ 0 0
$$371$$ 24.4853 1.27121
$$372$$ 0 0
$$373$$ −9.55635 −0.494809 −0.247405 0.968912i $$-0.579578\pi$$
−0.247405 + 0.968912i $$0.579578\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −23.5147 −1.21107
$$378$$ 0 0
$$379$$ 22.3431 1.14769 0.573845 0.818964i $$-0.305451\pi$$
0.573845 + 0.818964i $$0.305451\pi$$
$$380$$ 0 0
$$381$$ 12.6274 0.646922
$$382$$ 0 0
$$383$$ −2.92893 −0.149661 −0.0748307 0.997196i $$-0.523842\pi$$
−0.0748307 + 0.997196i $$0.523842\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2.20101 −0.111884
$$388$$ 0 0
$$389$$ −28.6274 −1.45147 −0.725734 0.687976i $$-0.758500\pi$$
−0.725734 + 0.687976i $$0.758500\pi$$
$$390$$ 0 0
$$391$$ 28.9706 1.46510
$$392$$ 0 0
$$393$$ −3.31371 −0.167154
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 32.1421 1.61317 0.806584 0.591120i $$-0.201314\pi$$
0.806584 + 0.591120i $$0.201314\pi$$
$$398$$ 0 0
$$399$$ −2.82843 −0.141598
$$400$$ 0 0
$$401$$ −6.68629 −0.333897 −0.166949 0.985966i $$-0.553391\pi$$
−0.166949 + 0.985966i $$0.553391\pi$$
$$402$$ 0 0
$$403$$ 2.62742 0.130881
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −20.4853 −1.01542
$$408$$ 0 0
$$409$$ 9.51472 0.470473 0.235236 0.971938i $$-0.424414\pi$$
0.235236 + 0.971938i $$0.424414\pi$$
$$410$$ 0 0
$$411$$ 11.7990 0.582001
$$412$$ 0 0
$$413$$ 11.3137 0.556711
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −10.1421 −0.496663
$$418$$ 0 0
$$419$$ 9.65685 0.471768 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$420$$ 0 0
$$421$$ −8.34315 −0.406620 −0.203310 0.979114i $$-0.565170\pi$$
−0.203310 + 0.979114i $$0.565170\pi$$
$$422$$ 0 0
$$423$$ 2.20101 0.107017
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −11.3137 −0.547509
$$428$$ 0 0
$$429$$ 2.62742 0.126853
$$430$$ 0 0
$$431$$ 33.4558 1.61151 0.805756 0.592248i $$-0.201759\pi$$
0.805756 + 0.592248i $$0.201759\pi$$
$$432$$ 0 0
$$433$$ 15.2132 0.731100 0.365550 0.930792i $$-0.380881\pi$$
0.365550 + 0.930792i $$0.380881\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ −29.6569 −1.41544 −0.707722 0.706491i $$-0.750277\pi$$
−0.707722 + 0.706491i $$0.750277\pi$$
$$440$$ 0 0
$$441$$ −43.3431 −2.06396
$$442$$ 0 0
$$443$$ −34.2843 −1.62889 −0.814447 0.580237i $$-0.802960\pi$$
−0.814447 + 0.580237i $$0.802960\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −4.68629 −0.221654
$$448$$ 0 0
$$449$$ −13.5147 −0.637799 −0.318900 0.947789i $$-0.603313\pi$$
−0.318900 + 0.947789i $$0.603313\pi$$
$$450$$ 0 0
$$451$$ 15.3137 0.721094
$$452$$ 0 0
$$453$$ 10.6274 0.499320
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18.9706 −0.887405 −0.443703 0.896174i $$-0.646335\pi$$
−0.443703 + 0.896174i $$0.646335\pi$$
$$458$$ 0 0
$$459$$ −16.0000 −0.746816
$$460$$ 0 0
$$461$$ −5.31371 −0.247484 −0.123742 0.992314i $$-0.539490\pi$$
−0.123742 + 0.992314i $$0.539490\pi$$
$$462$$ 0 0
$$463$$ 9.31371 0.432845 0.216422 0.976300i $$-0.430561\pi$$
0.216422 + 0.976300i $$0.430561\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −9.31371 −0.430987 −0.215494 0.976505i $$-0.569136\pi$$
−0.215494 + 0.976505i $$0.569136\pi$$
$$468$$ 0 0
$$469$$ 2.82843 0.130605
$$470$$ 0 0
$$471$$ −1.85786 −0.0856059
$$472$$ 0 0
$$473$$ −1.65685 −0.0761822
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −13.4731 −0.616890
$$478$$ 0 0
$$479$$ −10.0000 −0.456912 −0.228456 0.973554i $$-0.573368\pi$$
−0.228456 + 0.973554i $$0.573368\pi$$
$$480$$ 0 0
$$481$$ −22.9706 −1.04737
$$482$$ 0 0
$$483$$ 16.9706 0.772187
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −24.3848 −1.10498 −0.552490 0.833520i $$-0.686322\pi$$
−0.552490 + 0.833520i $$0.686322\pi$$
$$488$$ 0 0
$$489$$ 1.45584 0.0658355
$$490$$ 0 0
$$491$$ 35.3137 1.59369 0.796843 0.604187i $$-0.206502\pi$$
0.796843 + 0.604187i $$0.206502\pi$$
$$492$$ 0 0
$$493$$ 50.6274 2.28014
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 52.2843 2.34527
$$498$$ 0 0
$$499$$ −26.9706 −1.20737 −0.603684 0.797224i $$-0.706301\pi$$
−0.603684 + 0.797224i $$0.706301\pi$$
$$500$$ 0 0
$$501$$ 6.28427 0.280761
$$502$$ 0 0
$$503$$ −2.97056 −0.132451 −0.0662254 0.997805i $$-0.521096\pi$$
−0.0662254 + 0.997805i $$0.521096\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −4.66905 −0.207360
$$508$$ 0 0
$$509$$ −33.7990 −1.49811 −0.749057 0.662506i $$-0.769493\pi$$
−0.749057 + 0.662506i $$0.769493\pi$$
$$510$$ 0 0
$$511$$ −69.9411 −3.09401
$$512$$ 0 0
$$513$$ 3.31371 0.146304
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.65685 0.0728684
$$518$$ 0 0
$$519$$ 2.28427 0.100268
$$520$$ 0 0
$$521$$ 28.3431 1.24174 0.620868 0.783915i $$-0.286780\pi$$
0.620868 + 0.783915i $$0.286780\pi$$
$$522$$ 0 0
$$523$$ 6.72792 0.294191 0.147096 0.989122i $$-0.453007\pi$$
0.147096 + 0.989122i $$0.453007\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −5.65685 −0.246416
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −6.22540 −0.270159
$$532$$ 0 0
$$533$$ 17.1716 0.743783
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 12.6863 0.547454
$$538$$ 0 0
$$539$$ −32.6274 −1.40536
$$540$$ 0 0
$$541$$ 11.3137 0.486414 0.243207 0.969974i $$-0.421801\pi$$
0.243207 + 0.969974i $$0.421801\pi$$
$$542$$ 0 0
$$543$$ −10.5442 −0.452493
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −36.3848 −1.55570 −0.777850 0.628450i $$-0.783690\pi$$
−0.777850 + 0.628450i $$0.783690\pi$$
$$548$$ 0 0
$$549$$ 6.22540 0.265693
$$550$$ 0 0
$$551$$ −10.4853 −0.446688
$$552$$ 0 0
$$553$$ −46.6274 −1.98280
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −41.7990 −1.77108 −0.885540 0.464563i $$-0.846211\pi$$
−0.885540 + 0.464563i $$0.846211\pi$$
$$558$$ 0 0
$$559$$ −1.85786 −0.0785793
$$560$$ 0 0
$$561$$ −5.65685 −0.238833
$$562$$ 0 0
$$563$$ 11.4142 0.481052 0.240526 0.970643i $$-0.422680\pi$$
0.240526 + 0.970643i $$0.422680\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 29.1127 1.22262
$$568$$ 0 0
$$569$$ 0.828427 0.0347295 0.0173647 0.999849i $$-0.494472\pi$$
0.0173647 + 0.999849i $$0.494472\pi$$
$$570$$ 0 0
$$571$$ −14.6863 −0.614602 −0.307301 0.951612i $$-0.599426\pi$$
−0.307301 + 0.951612i $$0.599426\pi$$
$$572$$ 0 0
$$573$$ 2.34315 0.0978863
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6.00000 0.249783 0.124892 0.992170i $$-0.460142\pi$$
0.124892 + 0.992170i $$0.460142\pi$$
$$578$$ 0 0
$$579$$ −0.343146 −0.0142607
$$580$$ 0 0
$$581$$ −44.9706 −1.86569
$$582$$ 0 0
$$583$$ −10.1421 −0.420044
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14.9706 0.617901 0.308951 0.951078i $$-0.400022\pi$$
0.308951 + 0.951078i $$0.400022\pi$$
$$588$$ 0 0
$$589$$ 1.17157 0.0482738
$$590$$ 0 0
$$591$$ 3.11270 0.128039
$$592$$ 0 0
$$593$$ −8.62742 −0.354286 −0.177143 0.984185i $$-0.556685\pi$$
−0.177143 + 0.984185i $$0.556685\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −6.05887 −0.247973
$$598$$ 0 0
$$599$$ −4.68629 −0.191477 −0.0957383 0.995407i $$-0.530521\pi$$
−0.0957383 + 0.995407i $$0.530521\pi$$
$$600$$ 0 0
$$601$$ −18.0000 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$602$$ 0 0
$$603$$ −1.55635 −0.0633794
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −5.07107 −0.205828 −0.102914 0.994690i $$-0.532817\pi$$
−0.102914 + 0.994690i $$0.532817\pi$$
$$608$$ 0 0
$$609$$ 29.6569 1.20176
$$610$$ 0 0
$$611$$ 1.85786 0.0751611
$$612$$ 0 0
$$613$$ −17.5147 −0.707413 −0.353706 0.935356i $$-0.615079\pi$$
−0.353706 + 0.935356i $$0.615079\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.20101 0.249643 0.124822 0.992179i $$-0.460164\pi$$
0.124822 + 0.992179i $$0.460164\pi$$
$$618$$ 0 0
$$619$$ 18.0000 0.723481 0.361741 0.932279i $$-0.382183\pi$$
0.361741 + 0.932279i $$0.382183\pi$$
$$620$$ 0 0
$$621$$ −19.8823 −0.797847
$$622$$ 0 0
$$623$$ 50.6274 2.02834
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 1.17157 0.0467881
$$628$$ 0 0
$$629$$ 49.4558 1.97193
$$630$$ 0 0
$$631$$ −3.65685 −0.145577 −0.0727885 0.997347i $$-0.523190\pi$$
−0.0727885 + 0.997347i $$0.523190\pi$$
$$632$$ 0 0
$$633$$ −6.74517 −0.268096
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −36.5858 −1.44958
$$638$$ 0 0
$$639$$ −28.7696 −1.13811
$$640$$ 0 0
$$641$$ −21.3137 −0.841841 −0.420920 0.907098i $$-0.638293\pi$$
−0.420920 + 0.907098i $$0.638293\pi$$
$$642$$ 0 0
$$643$$ −24.6274 −0.971211 −0.485605 0.874178i $$-0.661401\pi$$
−0.485605 + 0.874178i $$0.661401\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −6.68629 −0.262865 −0.131433 0.991325i $$-0.541958\pi$$
−0.131433 + 0.991325i $$0.541958\pi$$
$$648$$ 0 0
$$649$$ −4.68629 −0.183953
$$650$$ 0 0
$$651$$ −3.31371 −0.129874
$$652$$ 0 0
$$653$$ −13.7990 −0.539996 −0.269998 0.962861i $$-0.587023\pi$$
−0.269998 + 0.962861i $$0.587023\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 38.4853 1.50145
$$658$$ 0 0
$$659$$ −34.6274 −1.34889 −0.674446 0.738324i $$-0.735618\pi$$
−0.674446 + 0.738324i $$0.735618\pi$$
$$660$$ 0 0
$$661$$ −12.6274 −0.491150 −0.245575 0.969378i $$-0.578977\pi$$
−0.245575 + 0.969378i $$0.578977\pi$$
$$662$$ 0 0
$$663$$ −6.34315 −0.246347
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 62.9117 2.43595
$$668$$ 0 0
$$669$$ −11.2548 −0.435137
$$670$$ 0 0
$$671$$ 4.68629 0.180912
$$672$$ 0 0
$$673$$ −10.2426 −0.394825 −0.197412 0.980321i $$-0.563254\pi$$
−0.197412 + 0.980321i $$0.563254\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −25.5563 −0.982210 −0.491105 0.871100i $$-0.663407\pi$$
−0.491105 + 0.871100i $$0.663407\pi$$
$$678$$ 0 0
$$679$$ 8.48528 0.325635
$$680$$ 0 0
$$681$$ −12.0589 −0.462097
$$682$$ 0 0
$$683$$ 38.7279 1.48188 0.740941 0.671570i $$-0.234380\pi$$
0.740941 + 0.671570i $$0.234380\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −12.6863 −0.484012
$$688$$ 0 0
$$689$$ −11.3726 −0.433261
$$690$$ 0 0
$$691$$ −15.6569 −0.595615 −0.297807 0.954626i $$-0.596255\pi$$
−0.297807 + 0.954626i $$0.596255\pi$$
$$692$$ 0 0
$$693$$ 25.6569 0.974623
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −36.9706 −1.40036
$$698$$ 0 0
$$699$$ −5.85786 −0.221565
$$700$$ 0 0
$$701$$ 49.6569 1.87551 0.937757 0.347293i $$-0.112899\pi$$
0.937757 + 0.347293i $$0.112899\pi$$
$$702$$ 0 0
$$703$$ −10.2426 −0.386309
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 19.3137 0.726367
$$708$$ 0 0
$$709$$ 18.9706 0.712454 0.356227 0.934399i $$-0.384063\pi$$
0.356227 + 0.934399i $$0.384063\pi$$
$$710$$ 0 0
$$711$$ 25.6569 0.962207
$$712$$ 0 0
$$713$$ −7.02944 −0.263254
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.05887 0.226273
$$718$$ 0 0
$$719$$ 38.2843 1.42776 0.713881 0.700267i $$-0.246936\pi$$
0.713881 + 0.700267i $$0.246936\pi$$
$$720$$ 0 0
$$721$$ 76.7696 2.85905
$$722$$ 0 0
$$723$$ −17.7401 −0.659762
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 29.5147 1.09464 0.547320 0.836923i $$-0.315648\pi$$
0.547320 + 0.836923i $$0.315648\pi$$
$$728$$ 0 0
$$729$$ −10.1960 −0.377628
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ −39.6569 −1.46476 −0.732380 0.680896i $$-0.761590\pi$$
−0.732380 + 0.680896i $$0.761590\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.17157 −0.0431554
$$738$$ 0 0
$$739$$ 34.6274 1.27379 0.636895 0.770951i $$-0.280218\pi$$
0.636895 + 0.770951i $$0.280218\pi$$
$$740$$ 0 0
$$741$$ 1.31371 0.0482603
$$742$$ 0 0
$$743$$ −14.0416 −0.515137 −0.257569 0.966260i $$-0.582921\pi$$
−0.257569 + 0.966260i $$0.582921\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 24.7452 0.905378
$$748$$ 0 0
$$749$$ −22.1421 −0.809056
$$750$$ 0 0
$$751$$ 14.1421 0.516054 0.258027 0.966138i $$-0.416928\pi$$
0.258027 + 0.966138i $$0.416928\pi$$
$$752$$ 0 0
$$753$$ −8.00000 −0.291536
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −19.6569 −0.714441 −0.357220 0.934020i $$-0.616275\pi$$
−0.357220 + 0.934020i $$0.616275\pi$$
$$758$$ 0 0
$$759$$ −7.02944 −0.255152
$$760$$ 0 0
$$761$$ 30.6274 1.11024 0.555121 0.831769i $$-0.312672\pi$$
0.555121 + 0.831769i $$0.312672\pi$$
$$762$$ 0 0
$$763$$ −42.6274 −1.54322
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −5.25483 −0.189741
$$768$$ 0 0
$$769$$ 38.6274 1.39294 0.696470 0.717586i $$-0.254753\pi$$
0.696470 + 0.717586i $$0.254753\pi$$
$$770$$ 0 0
$$771$$ 1.31371 0.0473121
$$772$$ 0 0
$$773$$ 0.100505 0.00361492 0.00180746 0.999998i $$-0.499425\pi$$
0.00180746 + 0.999998i $$0.499425\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 28.9706 1.03931
$$778$$ 0 0
$$779$$ 7.65685 0.274335
$$780$$ 0 0
$$781$$ −21.6569 −0.774943
$$782$$ 0 0
$$783$$ −34.7452 −1.24169
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 41.8406 1.49146 0.745729 0.666250i $$-0.232102\pi$$
0.745729 + 0.666250i $$0.232102\pi$$
$$788$$ 0 0
$$789$$ −2.14214 −0.0762620
$$790$$ 0 0
$$791$$ −24.4853 −0.870596
$$792$$ 0 0
$$793$$ 5.25483 0.186605
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −11.8995 −0.421502 −0.210751 0.977540i $$-0.567591\pi$$
−0.210751 + 0.977540i $$0.567591\pi$$
$$798$$ 0 0
$$799$$ −4.00000 −0.141510
$$800$$ 0 0
$$801$$ −27.8579 −0.984309
$$802$$ 0 0
$$803$$ 28.9706 1.02235
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −13.1716 −0.463661
$$808$$ 0 0
$$809$$ −39.2548 −1.38013 −0.690063 0.723749i $$-0.742417\pi$$
−0.690063 + 0.723749i $$0.742417\pi$$
$$810$$ 0 0
$$811$$ 19.7990 0.695237 0.347618 0.937636i $$-0.386991\pi$$
0.347618 + 0.937636i $$0.386991\pi$$
$$812$$ 0 0
$$813$$ 13.8579 0.486017
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −0.828427 −0.0289830
$$818$$ 0 0
$$819$$ 28.7696 1.00529
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ 15.1716 0.528848 0.264424 0.964407i $$-0.414818\pi$$
0.264424 + 0.964407i $$0.414818\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 29.0711 1.01090 0.505450 0.862856i $$-0.331326\pi$$
0.505450 + 0.862856i $$0.331326\pi$$
$$828$$ 0 0
$$829$$ −40.4264 −1.40407 −0.702034 0.712144i $$-0.747724\pi$$
−0.702034 + 0.712144i $$0.747724\pi$$
$$830$$ 0 0
$$831$$ −9.85786 −0.341966
$$832$$ 0 0
$$833$$ 78.7696 2.72920
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 3.88225 0.134190
$$838$$ 0 0
$$839$$ 43.5980 1.50517 0.752585 0.658495i $$-0.228807\pi$$
0.752585 + 0.658495i $$0.228807\pi$$
$$840$$ 0 0
$$841$$ 80.9411 2.79107
$$842$$ 0 0
$$843$$ 11.1127 0.382742
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −33.7990 −1.16135
$$848$$ 0 0
$$849$$ 1.57359 0.0540056
$$850$$ 0 0
$$851$$ 61.4558 2.10668
$$852$$ 0 0
$$853$$ −42.0000 −1.43805 −0.719026 0.694983i $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 27.2132 0.929585 0.464793 0.885420i $$-0.346129\pi$$
0.464793 + 0.885420i $$0.346129\pi$$
$$858$$ 0 0
$$859$$ −24.2843 −0.828569 −0.414284 0.910148i $$-0.635968\pi$$
−0.414284 + 0.910148i $$0.635968\pi$$
$$860$$ 0 0
$$861$$ −21.6569 −0.738064
$$862$$ 0 0
$$863$$ 3.89949 0.132740 0.0663702 0.997795i $$-0.478858\pi$$
0.0663702 + 0.997795i $$0.478858\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 3.69848 0.125607
$$868$$ 0 0
$$869$$ 19.3137 0.655173
$$870$$ 0 0
$$871$$ −1.31371 −0.0445133
$$872$$ 0 0
$$873$$ −4.66905 −0.158023
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 37.0711 1.25180 0.625901 0.779903i $$-0.284732\pi$$
0.625901 + 0.779903i $$0.284732\pi$$
$$878$$ 0 0
$$879$$ 13.7157 0.462620
$$880$$ 0 0
$$881$$ 44.2843 1.49198 0.745988 0.665960i $$-0.231978\pi$$
0.745988 + 0.665960i $$0.231978\pi$$
$$882$$ 0 0
$$883$$ 39.1716 1.31823 0.659114 0.752043i $$-0.270931\pi$$
0.659114 + 0.752043i $$0.270931\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 49.0711 1.64765 0.823823 0.566848i $$-0.191837\pi$$
0.823823 + 0.566848i $$0.191837\pi$$
$$888$$ 0 0
$$889$$ 104.083 3.49084
$$890$$ 0 0
$$891$$ −12.0589 −0.403987
$$892$$ 0 0
$$893$$ 0.828427 0.0277223
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −7.88225 −0.263181
$$898$$ 0 0
$$899$$ −12.2843 −0.409703
$$900$$ 0 0
$$901$$ 24.4853 0.815723
$$902$$ 0 0
$$903$$ 2.34315 0.0779750
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 20.3848 0.676865 0.338433 0.940991i $$-0.390103\pi$$
0.338433 + 0.940991i $$0.390103\pi$$
$$908$$ 0 0
$$909$$ −10.6274 −0.352489
$$910$$ 0 0
$$911$$ 4.20101 0.139186 0.0695928 0.997575i $$-0.477830\pi$$
0.0695928 + 0.997575i $$0.477830\pi$$
$$912$$ 0 0
$$913$$ 18.6274 0.616478
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −27.3137 −0.901978
$$918$$ 0 0
$$919$$ 35.3137 1.16489 0.582446 0.812869i $$-0.302096\pi$$
0.582446 + 0.812869i $$0.302096\pi$$
$$920$$ 0 0
$$921$$ −4.62742 −0.152479
$$922$$ 0 0
$$923$$ −24.2843 −0.799327
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −42.2426 −1.38743
$$928$$ 0 0
$$929$$ 6.68629 0.219370 0.109685 0.993966i $$-0.465016\pi$$
0.109685 + 0.993966i $$0.465016\pi$$
$$930$$ 0 0
$$931$$ −16.3137 −0.534660
$$932$$ 0 0
$$933$$ −8.20101 −0.268489
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 40.1421 1.31139 0.655693 0.755027i $$-0.272376\pi$$
0.655693 + 0.755027i $$0.272376\pi$$
$$938$$ 0 0
$$939$$ −10.5442 −0.344096
$$940$$ 0 0
$$941$$ −30.2843 −0.987239 −0.493620 0.869678i $$-0.664326\pi$$
−0.493620 + 0.869678i $$0.664326\pi$$
$$942$$ 0 0
$$943$$ −45.9411 −1.49605
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8.34315 0.271116 0.135558 0.990769i $$-0.456717\pi$$
0.135558 + 0.990769i $$0.456717\pi$$
$$948$$ 0 0
$$949$$ 32.4853 1.05452
$$950$$ 0 0
$$951$$ 3.37258 0.109363
$$952$$ 0 0
$$953$$ −5.75736 −0.186499 −0.0932496 0.995643i $$-0.529725\pi$$
−0.0932496 + 0.995643i $$0.529725\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −12.2843 −0.397094
$$958$$ 0 0
$$959$$ 97.2548 3.14052
$$960$$ 0 0
$$961$$ −29.6274 −0.955723
$$962$$ 0 0
$$963$$ 12.1838 0.392616
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −20.6274 −0.663333 −0.331667 0.943397i $$-0.607611\pi$$
−0.331667 + 0.943397i $$0.607611\pi$$
$$968$$ 0 0
$$969$$ −2.82843 −0.0908622
$$970$$ 0 0
$$971$$ −35.1127 −1.12682 −0.563410 0.826177i $$-0.690511\pi$$
−0.563410 + 0.826177i $$0.690511\pi$$
$$972$$ 0 0
$$973$$ −83.5980 −2.68003
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −9.27208 −0.296640 −0.148320 0.988939i $$-0.547387\pi$$
−0.148320 + 0.988939i $$0.547387\pi$$
$$978$$ 0 0
$$979$$ −20.9706 −0.670222
$$980$$ 0 0
$$981$$ 23.4558 0.748887
$$982$$ 0 0
$$983$$ 23.4142 0.746797 0.373399 0.927671i $$-0.378192\pi$$
0.373399 + 0.927671i $$0.378192\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −2.34315 −0.0745832
$$988$$ 0 0
$$989$$ 4.97056 0.158055
$$990$$ 0 0
$$991$$ −51.1127 −1.62365 −0.811824 0.583902i $$-0.801525\pi$$
−0.811824 + 0.583902i $$0.801525\pi$$
$$992$$ 0 0
$$993$$ 0.686292 0.0217788
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 13.5147 0.428015 0.214008 0.976832i $$-0.431348\pi$$
0.214008 + 0.976832i $$0.431348\pi$$
$$998$$ 0 0
$$999$$ −33.9411 −1.07385
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 1900.2.a.e.1.1 2
4.3 odd 2 7600.2.a.u.1.2 2
5.2 odd 4 1900.2.c.d.1749.2 4
5.3 odd 4 1900.2.c.d.1749.3 4
5.4 even 2 380.2.a.c.1.2 2
15.14 odd 2 3420.2.a.g.1.1 2
20.19 odd 2 1520.2.a.o.1.1 2
40.19 odd 2 6080.2.a.y.1.2 2
40.29 even 2 6080.2.a.bl.1.1 2
95.94 odd 2 7220.2.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.c.1.2 2 5.4 even 2
1520.2.a.o.1.1 2 20.19 odd 2
1900.2.a.e.1.1 2 1.1 even 1 trivial
1900.2.c.d.1749.2 4 5.2 odd 4
1900.2.c.d.1749.3 4 5.3 odd 4
3420.2.a.g.1.1 2 15.14 odd 2
6080.2.a.y.1.2 2 40.19 odd 2
6080.2.a.bl.1.1 2 40.29 even 2
7220.2.a.m.1.1 2 95.94 odd 2
7600.2.a.u.1.2 2 4.3 odd 2