# Properties

 Label 3200.2.a.bt.1.1 Level $3200$ Weight $2$ Character 3200.1 Self dual yes Analytic conductor $25.552$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.5521286468$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 640) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 3200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.70928 q^{3} -2.63090 q^{7} -0.0783777 q^{9} +O(q^{10})$$ $$q-1.70928 q^{3} -2.63090 q^{7} -0.0783777 q^{9} +5.41855 q^{11} -6.34017 q^{13} +3.41855 q^{17} +3.26180 q^{19} +4.49693 q^{21} -1.36910 q^{23} +5.26180 q^{27} -2.00000 q^{29} +4.68035 q^{31} -9.26180 q^{33} +5.75872 q^{37} +10.8371 q^{39} -7.75872 q^{41} +4.44748 q^{43} -4.78765 q^{47} -0.0783777 q^{49} -5.84324 q^{51} -1.65983 q^{53} -5.57531 q^{57} -3.26180 q^{59} +2.49693 q^{61} +0.206204 q^{63} +7.86603 q^{67} +2.34017 q^{69} +6.15676 q^{71} -13.5753 q^{73} -14.2557 q^{77} -12.6803 q^{79} -8.75872 q^{81} +14.9711 q^{83} +3.41855 q^{87} -8.52359 q^{89} +16.6803 q^{91} -8.00000 q^{93} -4.58145 q^{97} -0.424694 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 4 * q^7 + 3 * q^9 $$3 q + 2 q^{3} - 4 q^{7} + 3 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{17} + 2 q^{19} - 4 q^{21} - 8 q^{23} + 8 q^{27} - 6 q^{29} - 8 q^{31} - 20 q^{33} - 8 q^{37} + 4 q^{39} + 2 q^{41} + 14 q^{43} - 4 q^{47} + 3 q^{49} - 24 q^{51} - 16 q^{53} + 4 q^{57} - 2 q^{59} - 10 q^{61} - 24 q^{63} + 10 q^{67} - 4 q^{69} + 12 q^{71} - 20 q^{73} - 16 q^{79} - q^{81} + 30 q^{83} - 4 q^{87} - 10 q^{89} + 28 q^{91} - 24 q^{93} - 28 q^{97} - 22 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 4 * q^7 + 3 * q^9 + 2 * q^11 - 8 * q^13 - 4 * q^17 + 2 * q^19 - 4 * q^21 - 8 * q^23 + 8 * q^27 - 6 * q^29 - 8 * q^31 - 20 * q^33 - 8 * q^37 + 4 * q^39 + 2 * q^41 + 14 * q^43 - 4 * q^47 + 3 * q^49 - 24 * q^51 - 16 * q^53 + 4 * q^57 - 2 * q^59 - 10 * q^61 - 24 * q^63 + 10 * q^67 - 4 * q^69 + 12 * q^71 - 20 * q^73 - 16 * q^79 - q^81 + 30 * q^83 - 4 * q^87 - 10 * q^89 + 28 * q^91 - 24 * q^93 - 28 * q^97 - 22 * 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
$$3$$ −1.70928 −0.986851 −0.493425 0.869788i $$-0.664255\pi$$
−0.493425 + 0.869788i $$0.664255\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.63090 −0.994386 −0.497193 0.867640i $$-0.665636\pi$$
−0.497193 + 0.867640i $$0.665636\pi$$
$$8$$ 0 0
$$9$$ −0.0783777 −0.0261259
$$10$$ 0 0
$$11$$ 5.41855 1.63375 0.816877 0.576812i $$-0.195703\pi$$
0.816877 + 0.576812i $$0.195703\pi$$
$$12$$ 0 0
$$13$$ −6.34017 −1.75845 −0.879224 0.476409i $$-0.841938\pi$$
−0.879224 + 0.476409i $$0.841938\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.41855 0.829120 0.414560 0.910022i $$-0.363935\pi$$
0.414560 + 0.910022i $$0.363935\pi$$
$$18$$ 0 0
$$19$$ 3.26180 0.748307 0.374154 0.927367i $$-0.377933\pi$$
0.374154 + 0.927367i $$0.377933\pi$$
$$20$$ 0 0
$$21$$ 4.49693 0.981310
$$22$$ 0 0
$$23$$ −1.36910 −0.285478 −0.142739 0.989760i $$-0.545591\pi$$
−0.142739 + 0.989760i $$0.545591\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.26180 1.01263
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 4.68035 0.840615 0.420307 0.907382i $$-0.361922\pi$$
0.420307 + 0.907382i $$0.361922\pi$$
$$32$$ 0 0
$$33$$ −9.26180 −1.61227
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.75872 0.946728 0.473364 0.880867i $$-0.343039\pi$$
0.473364 + 0.880867i $$0.343039\pi$$
$$38$$ 0 0
$$39$$ 10.8371 1.73533
$$40$$ 0 0
$$41$$ −7.75872 −1.21171 −0.605855 0.795575i $$-0.707169\pi$$
−0.605855 + 0.795575i $$0.707169\pi$$
$$42$$ 0 0
$$43$$ 4.44748 0.678234 0.339117 0.940744i $$-0.389872\pi$$
0.339117 + 0.940744i $$0.389872\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.78765 −0.698351 −0.349175 0.937057i $$-0.613538\pi$$
−0.349175 + 0.937057i $$0.613538\pi$$
$$48$$ 0 0
$$49$$ −0.0783777 −0.0111968
$$50$$ 0 0
$$51$$ −5.84324 −0.818218
$$52$$ 0 0
$$53$$ −1.65983 −0.227995 −0.113997 0.993481i $$-0.536366\pi$$
−0.113997 + 0.993481i $$0.536366\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −5.57531 −0.738467
$$58$$ 0 0
$$59$$ −3.26180 −0.424650 −0.212325 0.977199i $$-0.568103\pi$$
−0.212325 + 0.977199i $$0.568103\pi$$
$$60$$ 0 0
$$61$$ 2.49693 0.319699 0.159849 0.987141i $$-0.448899\pi$$
0.159849 + 0.987141i $$0.448899\pi$$
$$62$$ 0 0
$$63$$ 0.206204 0.0259792
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.86603 0.960989 0.480494 0.876998i $$-0.340457\pi$$
0.480494 + 0.876998i $$0.340457\pi$$
$$68$$ 0 0
$$69$$ 2.34017 0.281724
$$70$$ 0 0
$$71$$ 6.15676 0.730672 0.365336 0.930876i $$-0.380954\pi$$
0.365336 + 0.930876i $$0.380954\pi$$
$$72$$ 0 0
$$73$$ −13.5753 −1.58887 −0.794435 0.607350i $$-0.792233\pi$$
−0.794435 + 0.607350i $$0.792233\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −14.2557 −1.62458
$$78$$ 0 0
$$79$$ −12.6803 −1.42665 −0.713325 0.700833i $$-0.752812\pi$$
−0.713325 + 0.700833i $$0.752812\pi$$
$$80$$ 0 0
$$81$$ −8.75872 −0.973192
$$82$$ 0 0
$$83$$ 14.9711 1.64329 0.821644 0.570001i $$-0.193057\pi$$
0.821644 + 0.570001i $$0.193057\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 3.41855 0.366507
$$88$$ 0 0
$$89$$ −8.52359 −0.903499 −0.451749 0.892145i $$-0.649200\pi$$
−0.451749 + 0.892145i $$0.649200\pi$$
$$90$$ 0 0
$$91$$ 16.6803 1.74858
$$92$$ 0 0
$$93$$ −8.00000 −0.829561
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −4.58145 −0.465176 −0.232588 0.972575i $$-0.574719\pi$$
−0.232588 + 0.972575i $$0.574719\pi$$
$$98$$ 0 0
$$99$$ −0.424694 −0.0426833
$$100$$ 0 0
$$101$$ −2.31351 −0.230203 −0.115101 0.993354i $$-0.536719\pi$$
−0.115101 + 0.993354i $$0.536719\pi$$
$$102$$ 0 0
$$103$$ −16.2062 −1.59684 −0.798422 0.602098i $$-0.794332\pi$$
−0.798422 + 0.602098i $$0.794332\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −15.6514 −1.51308 −0.756540 0.653948i $$-0.773112\pi$$
−0.756540 + 0.653948i $$0.773112\pi$$
$$108$$ 0 0
$$109$$ 12.3402 1.18197 0.590987 0.806681i $$-0.298738\pi$$
0.590987 + 0.806681i $$0.298738\pi$$
$$110$$ 0 0
$$111$$ −9.84324 −0.934279
$$112$$ 0 0
$$113$$ 9.36069 0.880580 0.440290 0.897856i $$-0.354876\pi$$
0.440290 + 0.897856i $$0.354876\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.496928 0.0459411
$$118$$ 0 0
$$119$$ −8.99386 −0.824466
$$120$$ 0 0
$$121$$ 18.3607 1.66915
$$122$$ 0 0
$$123$$ 13.2618 1.19578
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1.95055 −0.173083 −0.0865417 0.996248i $$-0.527582\pi$$
−0.0865417 + 0.996248i $$0.527582\pi$$
$$128$$ 0 0
$$129$$ −7.60197 −0.669316
$$130$$ 0 0
$$131$$ −15.5753 −1.36082 −0.680410 0.732831i $$-0.738198\pi$$
−0.680410 + 0.732831i $$0.738198\pi$$
$$132$$ 0 0
$$133$$ −8.58145 −0.744106
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −15.2039 −1.29896 −0.649480 0.760379i $$-0.725013\pi$$
−0.649480 + 0.760379i $$0.725013\pi$$
$$138$$ 0 0
$$139$$ 2.58145 0.218956 0.109478 0.993989i $$-0.465082\pi$$
0.109478 + 0.993989i $$0.465082\pi$$
$$140$$ 0 0
$$141$$ 8.18342 0.689168
$$142$$ 0 0
$$143$$ −34.3545 −2.87287
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0.133969 0.0110496
$$148$$ 0 0
$$149$$ 1.81658 0.148820 0.0744101 0.997228i $$-0.476293\pi$$
0.0744101 + 0.997228i $$0.476293\pi$$
$$150$$ 0 0
$$151$$ 5.16290 0.420151 0.210075 0.977685i $$-0.432629\pi$$
0.210075 + 0.977685i $$0.432629\pi$$
$$152$$ 0 0
$$153$$ −0.267938 −0.0216615
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −13.7587 −1.09807 −0.549033 0.835801i $$-0.685004\pi$$
−0.549033 + 0.835801i $$0.685004\pi$$
$$158$$ 0 0
$$159$$ 2.83710 0.224997
$$160$$ 0 0
$$161$$ 3.60197 0.283875
$$162$$ 0 0
$$163$$ 6.29072 0.492728 0.246364 0.969177i $$-0.420764\pi$$
0.246364 + 0.969177i $$0.420764\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.89269 0.301226 0.150613 0.988593i $$-0.451875\pi$$
0.150613 + 0.988593i $$0.451875\pi$$
$$168$$ 0 0
$$169$$ 27.1978 2.09214
$$170$$ 0 0
$$171$$ −0.255652 −0.0195502
$$172$$ 0 0
$$173$$ 1.44521 0.109877 0.0549387 0.998490i $$-0.482504\pi$$
0.0549387 + 0.998490i $$0.482504\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 5.57531 0.419066
$$178$$ 0 0
$$179$$ −11.9421 −0.892598 −0.446299 0.894884i $$-0.647258\pi$$
−0.446299 + 0.894884i $$0.647258\pi$$
$$180$$ 0 0
$$181$$ −15.3607 −1.14175 −0.570876 0.821037i $$-0.693396\pi$$
−0.570876 + 0.821037i $$0.693396\pi$$
$$182$$ 0 0
$$183$$ −4.26794 −0.315495
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 18.5236 1.35458
$$188$$ 0 0
$$189$$ −13.8432 −1.00695
$$190$$ 0 0
$$191$$ −25.3607 −1.83504 −0.917518 0.397695i $$-0.869810\pi$$
−0.917518 + 0.397695i $$0.869810\pi$$
$$192$$ 0 0
$$193$$ −4.58145 −0.329780 −0.164890 0.986312i $$-0.552727\pi$$
−0.164890 + 0.986312i $$0.552727\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −16.8638 −1.20149 −0.600747 0.799439i $$-0.705130\pi$$
−0.600747 + 0.799439i $$0.705130\pi$$
$$198$$ 0 0
$$199$$ 9.84324 0.697769 0.348885 0.937166i $$-0.386561\pi$$
0.348885 + 0.937166i $$0.386561\pi$$
$$200$$ 0 0
$$201$$ −13.4452 −0.948352
$$202$$ 0 0
$$203$$ 5.26180 0.369306
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0.107307 0.00745836
$$208$$ 0 0
$$209$$ 17.6742 1.22255
$$210$$ 0 0
$$211$$ −15.5753 −1.07225 −0.536124 0.844139i $$-0.680112\pi$$
−0.536124 + 0.844139i $$0.680112\pi$$
$$212$$ 0 0
$$213$$ −10.5236 −0.721065
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −12.3135 −0.835896
$$218$$ 0 0
$$219$$ 23.2039 1.56798
$$220$$ 0 0
$$221$$ −21.6742 −1.45796
$$222$$ 0 0
$$223$$ 16.9854 1.13743 0.568715 0.822535i $$-0.307441\pi$$
0.568715 + 0.822535i $$0.307441\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −19.9649 −1.32512 −0.662559 0.749009i $$-0.730530\pi$$
−0.662559 + 0.749009i $$0.730530\pi$$
$$228$$ 0 0
$$229$$ −23.6742 −1.56444 −0.782218 0.623005i $$-0.785912\pi$$
−0.782218 + 0.623005i $$0.785912\pi$$
$$230$$ 0 0
$$231$$ 24.3668 1.60322
$$232$$ 0 0
$$233$$ −13.5753 −0.889348 −0.444674 0.895693i $$-0.646680\pi$$
−0.444674 + 0.895693i $$0.646680\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 21.6742 1.40789
$$238$$ 0 0
$$239$$ 8.36683 0.541206 0.270603 0.962691i $$-0.412777\pi$$
0.270603 + 0.962691i $$0.412777\pi$$
$$240$$ 0 0
$$241$$ 9.91548 0.638712 0.319356 0.947635i $$-0.396533\pi$$
0.319356 + 0.947635i $$0.396533\pi$$
$$242$$ 0 0
$$243$$ −0.814315 −0.0522383
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −20.6803 −1.31586
$$248$$ 0 0
$$249$$ −25.5897 −1.62168
$$250$$ 0 0
$$251$$ −17.6163 −1.11193 −0.555967 0.831204i $$-0.687652\pi$$
−0.555967 + 0.831204i $$0.687652\pi$$
$$252$$ 0 0
$$253$$ −7.41855 −0.466400
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 3.68649 0.229957 0.114978 0.993368i $$-0.463320\pi$$
0.114978 + 0.993368i $$0.463320\pi$$
$$258$$ 0 0
$$259$$ −15.1506 −0.941413
$$260$$ 0 0
$$261$$ 0.156755 0.00970292
$$262$$ 0 0
$$263$$ 0.107307 0.00661684 0.00330842 0.999995i $$-0.498947\pi$$
0.00330842 + 0.999995i $$0.498947\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 14.5692 0.891618
$$268$$ 0 0
$$269$$ −3.85762 −0.235203 −0.117602 0.993061i $$-0.537521\pi$$
−0.117602 + 0.993061i $$0.537521\pi$$
$$270$$ 0 0
$$271$$ −21.3074 −1.29433 −0.647165 0.762350i $$-0.724046\pi$$
−0.647165 + 0.762350i $$0.724046\pi$$
$$272$$ 0 0
$$273$$ −28.5113 −1.72558
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1.44521 0.0868344 0.0434172 0.999057i $$-0.486176\pi$$
0.0434172 + 0.999057i $$0.486176\pi$$
$$278$$ 0 0
$$279$$ −0.366835 −0.0219618
$$280$$ 0 0
$$281$$ 12.4391 0.742053 0.371026 0.928622i $$-0.379006\pi$$
0.371026 + 0.928622i $$0.379006\pi$$
$$282$$ 0 0
$$283$$ 6.29072 0.373945 0.186972 0.982365i $$-0.440133\pi$$
0.186972 + 0.982365i $$0.440133\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 20.4124 1.20491
$$288$$ 0 0
$$289$$ −5.31351 −0.312559
$$290$$ 0 0
$$291$$ 7.83096 0.459059
$$292$$ 0 0
$$293$$ 1.07838 0.0629995 0.0314998 0.999504i $$-0.489972\pi$$
0.0314998 + 0.999504i $$0.489972\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 28.5113 1.65439
$$298$$ 0 0
$$299$$ 8.68035 0.501997
$$300$$ 0 0
$$301$$ −11.7009 −0.674427
$$302$$ 0 0
$$303$$ 3.95443 0.227176
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 25.2267 1.43977 0.719883 0.694096i $$-0.244196\pi$$
0.719883 + 0.694096i $$0.244196\pi$$
$$308$$ 0 0
$$309$$ 27.7009 1.57585
$$310$$ 0 0
$$311$$ 18.8371 1.06815 0.534077 0.845436i $$-0.320659\pi$$
0.534077 + 0.845436i $$0.320659\pi$$
$$312$$ 0 0
$$313$$ −15.2039 −0.859377 −0.429689 0.902977i $$-0.641377\pi$$
−0.429689 + 0.902977i $$0.641377\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 16.8638 0.947163 0.473582 0.880750i $$-0.342961\pi$$
0.473582 + 0.880750i $$0.342961\pi$$
$$318$$ 0 0
$$319$$ −10.8371 −0.606761
$$320$$ 0 0
$$321$$ 26.7526 1.49318
$$322$$ 0 0
$$323$$ 11.1506 0.620437
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −21.0928 −1.16643
$$328$$ 0 0
$$329$$ 12.5958 0.694430
$$330$$ 0 0
$$331$$ 23.5753 1.29582 0.647908 0.761719i $$-0.275644\pi$$
0.647908 + 0.761719i $$0.275644\pi$$
$$332$$ 0 0
$$333$$ −0.451356 −0.0247341
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −14.8371 −0.808228 −0.404114 0.914709i $$-0.632420\pi$$
−0.404114 + 0.914709i $$0.632420\pi$$
$$338$$ 0 0
$$339$$ −16.0000 −0.869001
$$340$$ 0 0
$$341$$ 25.3607 1.37336
$$342$$ 0 0
$$343$$ 18.6225 1.00552
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −0.133969 −0.00719184 −0.00359592 0.999994i $$-0.501145\pi$$
−0.00359592 + 0.999994i $$0.501145\pi$$
$$348$$ 0 0
$$349$$ −22.3135 −1.19441 −0.597207 0.802087i $$-0.703723\pi$$
−0.597207 + 0.802087i $$0.703723\pi$$
$$350$$ 0 0
$$351$$ −33.3607 −1.78066
$$352$$ 0 0
$$353$$ 22.8371 1.21550 0.607748 0.794130i $$-0.292073\pi$$
0.607748 + 0.794130i $$0.292073\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 15.3730 0.813624
$$358$$ 0 0
$$359$$ −31.8843 −1.68279 −0.841394 0.540422i $$-0.818265\pi$$
−0.841394 + 0.540422i $$0.818265\pi$$
$$360$$ 0 0
$$361$$ −8.36069 −0.440036
$$362$$ 0 0
$$363$$ −31.3835 −1.64721
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −30.4619 −1.59010 −0.795048 0.606547i $$-0.792554\pi$$
−0.795048 + 0.606547i $$0.792554\pi$$
$$368$$ 0 0
$$369$$ 0.608111 0.0316570
$$370$$ 0 0
$$371$$ 4.36683 0.226715
$$372$$ 0 0
$$373$$ −10.0722 −0.521521 −0.260760 0.965404i $$-0.583973\pi$$
−0.260760 + 0.965404i $$0.583973\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.6803 0.653071
$$378$$ 0 0
$$379$$ −14.7792 −0.759159 −0.379579 0.925159i $$-0.623931\pi$$
−0.379579 + 0.925159i $$0.623931\pi$$
$$380$$ 0 0
$$381$$ 3.33403 0.170808
$$382$$ 0 0
$$383$$ 14.4619 0.738966 0.369483 0.929237i $$-0.379535\pi$$
0.369483 + 0.929237i $$0.379535\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −0.348583 −0.0177195
$$388$$ 0 0
$$389$$ 3.17727 0.161094 0.0805471 0.996751i $$-0.474333\pi$$
0.0805471 + 0.996751i $$0.474333\pi$$
$$390$$ 0 0
$$391$$ −4.68035 −0.236695
$$392$$ 0 0
$$393$$ 26.6225 1.34293
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −10.8227 −0.543177 −0.271589 0.962413i $$-0.587549\pi$$
−0.271589 + 0.962413i $$0.587549\pi$$
$$398$$ 0 0
$$399$$ 14.6681 0.734321
$$400$$ 0 0
$$401$$ −12.5236 −0.625398 −0.312699 0.949852i $$-0.601233\pi$$
−0.312699 + 0.949852i $$0.601233\pi$$
$$402$$ 0 0
$$403$$ −29.6742 −1.47818
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 31.2039 1.54672
$$408$$ 0 0
$$409$$ −13.2885 −0.657072 −0.328536 0.944491i $$-0.606555\pi$$
−0.328536 + 0.944491i $$0.606555\pi$$
$$410$$ 0 0
$$411$$ 25.9877 1.28188
$$412$$ 0 0
$$413$$ 8.58145 0.422266
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −4.41241 −0.216077
$$418$$ 0 0
$$419$$ −0.255652 −0.0124894 −0.00624471 0.999981i $$-0.501988\pi$$
−0.00624471 + 0.999981i $$0.501988\pi$$
$$420$$ 0 0
$$421$$ 2.49693 0.121693 0.0608464 0.998147i $$-0.480620\pi$$
0.0608464 + 0.998147i $$0.480620\pi$$
$$422$$ 0 0
$$423$$ 0.375245 0.0182451
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −6.56916 −0.317904
$$428$$ 0 0
$$429$$ 58.7214 2.83510
$$430$$ 0 0
$$431$$ 0.993857 0.0478724 0.0239362 0.999713i $$-0.492380\pi$$
0.0239362 + 0.999713i $$0.492380\pi$$
$$432$$ 0 0
$$433$$ −17.6286 −0.847178 −0.423589 0.905855i $$-0.639230\pi$$
−0.423589 + 0.905855i $$0.639230\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.46573 −0.213625
$$438$$ 0 0
$$439$$ 40.5113 1.93350 0.966750 0.255725i $$-0.0823142\pi$$
0.966750 + 0.255725i $$0.0823142\pi$$
$$440$$ 0 0
$$441$$ 0.00614307 0.000292527 0
$$442$$ 0 0
$$443$$ −17.7093 −0.841393 −0.420697 0.907201i $$-0.638214\pi$$
−0.420697 + 0.907201i $$0.638214\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −3.10504 −0.146863
$$448$$ 0 0
$$449$$ −1.28846 −0.0608061 −0.0304030 0.999538i $$-0.509679\pi$$
−0.0304030 + 0.999538i $$0.509679\pi$$
$$450$$ 0 0
$$451$$ −42.0410 −1.97964
$$452$$ 0 0
$$453$$ −8.82482 −0.414626
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 26.3545 1.23281 0.616407 0.787428i $$-0.288588\pi$$
0.616407 + 0.787428i $$0.288588\pi$$
$$458$$ 0 0
$$459$$ 17.9877 0.839595
$$460$$ 0 0
$$461$$ 41.0349 1.91119 0.955593 0.294690i $$-0.0952165\pi$$
0.955593 + 0.294690i $$0.0952165\pi$$
$$462$$ 0 0
$$463$$ −28.7708 −1.33709 −0.668547 0.743670i $$-0.733083\pi$$
−0.668547 + 0.743670i $$0.733083\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −5.12783 −0.237287 −0.118644 0.992937i $$-0.537855\pi$$
−0.118644 + 0.992937i $$0.537855\pi$$
$$468$$ 0 0
$$469$$ −20.6947 −0.955593
$$470$$ 0 0
$$471$$ 23.5174 1.08363
$$472$$ 0 0
$$473$$ 24.0989 1.10807
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0.130094 0.00595657
$$478$$ 0 0
$$479$$ −13.6742 −0.624790 −0.312395 0.949952i $$-0.601131\pi$$
−0.312395 + 0.949952i $$0.601131\pi$$
$$480$$ 0 0
$$481$$ −36.5113 −1.66477
$$482$$ 0 0
$$483$$ −6.15676 −0.280142
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 8.51971 0.386065 0.193033 0.981192i $$-0.438168\pi$$
0.193033 + 0.981192i $$0.438168\pi$$
$$488$$ 0 0
$$489$$ −10.7526 −0.486249
$$490$$ 0 0
$$491$$ −4.73820 −0.213832 −0.106916 0.994268i $$-0.534098\pi$$
−0.106916 + 0.994268i $$0.534098\pi$$
$$492$$ 0 0
$$493$$ −6.83710 −0.307928
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −16.1978 −0.726570
$$498$$ 0 0
$$499$$ 11.0928 0.496580 0.248290 0.968686i $$-0.420131\pi$$
0.248290 + 0.968686i $$0.420131\pi$$
$$500$$ 0 0
$$501$$ −6.65368 −0.297265
$$502$$ 0 0
$$503$$ 8.42082 0.375466 0.187733 0.982220i $$-0.439886\pi$$
0.187733 + 0.982220i $$0.439886\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −46.4885 −2.06463
$$508$$ 0 0
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ 35.7152 1.57995
$$512$$ 0 0
$$513$$ 17.1629 0.757760
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −25.9421 −1.14093
$$518$$ 0 0
$$519$$ −2.47027 −0.108433
$$520$$ 0 0
$$521$$ −10.3135 −0.451843 −0.225922 0.974145i $$-0.572539\pi$$
−0.225922 + 0.974145i $$0.572539\pi$$
$$522$$ 0 0
$$523$$ 16.2784 0.711806 0.355903 0.934523i $$-0.384173\pi$$
0.355903 + 0.934523i $$0.384173\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −21.1256 −0.918503
$$530$$ 0 0
$$531$$ 0.255652 0.0110944
$$532$$ 0 0
$$533$$ 49.1917 2.13073
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 20.4124 0.880860
$$538$$ 0 0
$$539$$ −0.424694 −0.0182929
$$540$$ 0 0
$$541$$ 3.36069 0.144487 0.0722437 0.997387i $$-0.476984\pi$$
0.0722437 + 0.997387i $$0.476984\pi$$
$$542$$ 0 0
$$543$$ 26.2557 1.12674
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1.51148 0.0646263 0.0323132 0.999478i $$-0.489713\pi$$
0.0323132 + 0.999478i $$0.489713\pi$$
$$548$$ 0 0
$$549$$ −0.195704 −0.00835243
$$550$$ 0 0
$$551$$ −6.52359 −0.277914
$$552$$ 0 0
$$553$$ 33.3607 1.41864
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 36.2290 1.53507 0.767536 0.641006i $$-0.221483\pi$$
0.767536 + 0.641006i $$0.221483\pi$$
$$558$$ 0 0
$$559$$ −28.1978 −1.19264
$$560$$ 0 0
$$561$$ −31.6619 −1.33677
$$562$$ 0 0
$$563$$ 2.17501 0.0916656 0.0458328 0.998949i $$-0.485406\pi$$
0.0458328 + 0.998949i $$0.485406\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 23.0433 0.967728
$$568$$ 0 0
$$569$$ 33.1194 1.38844 0.694219 0.719764i $$-0.255750\pi$$
0.694219 + 0.719764i $$0.255750\pi$$
$$570$$ 0 0
$$571$$ 18.4657 0.772767 0.386383 0.922338i $$-0.373724\pi$$
0.386383 + 0.922338i $$0.373724\pi$$
$$572$$ 0 0
$$573$$ 43.3484 1.81091
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −14.2101 −0.591573 −0.295787 0.955254i $$-0.595582\pi$$
−0.295787 + 0.955254i $$0.595582\pi$$
$$578$$ 0 0
$$579$$ 7.83096 0.325444
$$580$$ 0 0
$$581$$ −39.3874 −1.63406
$$582$$ 0 0
$$583$$ −8.99386 −0.372487
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 11.7503 0.484987 0.242494 0.970153i $$-0.422035\pi$$
0.242494 + 0.970153i $$0.422035\pi$$
$$588$$ 0 0
$$589$$ 15.2663 0.629038
$$590$$ 0 0
$$591$$ 28.8248 1.18569
$$592$$ 0 0
$$593$$ 8.00000 0.328521 0.164260 0.986417i $$-0.447476\pi$$
0.164260 + 0.986417i $$0.447476\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −16.8248 −0.688594
$$598$$ 0 0
$$599$$ 20.5646 0.840248 0.420124 0.907467i $$-0.361987\pi$$
0.420124 + 0.907467i $$0.361987\pi$$
$$600$$ 0 0
$$601$$ −8.60811 −0.351132 −0.175566 0.984468i $$-0.556176\pi$$
−0.175566 + 0.984468i $$0.556176\pi$$
$$602$$ 0 0
$$603$$ −0.616522 −0.0251067
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0.259528 0.0105339 0.00526695 0.999986i $$-0.498323\pi$$
0.00526695 + 0.999986i $$0.498323\pi$$
$$608$$ 0 0
$$609$$ −8.99386 −0.364449
$$610$$ 0 0
$$611$$ 30.3545 1.22801
$$612$$ 0 0
$$613$$ 47.1650 1.90498 0.952488 0.304576i $$-0.0985148\pi$$
0.952488 + 0.304576i $$0.0985148\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −40.2967 −1.62228 −0.811142 0.584849i $$-0.801154\pi$$
−0.811142 + 0.584849i $$0.801154\pi$$
$$618$$ 0 0
$$619$$ −30.6102 −1.23033 −0.615164 0.788399i $$-0.710910\pi$$
−0.615164 + 0.788399i $$0.710910\pi$$
$$620$$ 0 0
$$621$$ −7.20394 −0.289084
$$622$$ 0 0
$$623$$ 22.4247 0.898426
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −30.2101 −1.20647
$$628$$ 0 0
$$629$$ 19.6865 0.784952
$$630$$ 0 0
$$631$$ −2.21008 −0.0879819 −0.0439909 0.999032i $$-0.514007\pi$$
−0.0439909 + 0.999032i $$0.514007\pi$$
$$632$$ 0 0
$$633$$ 26.6225 1.05815
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.496928 0.0196890
$$638$$ 0 0
$$639$$ −0.482553 −0.0190895
$$640$$ 0 0
$$641$$ 7.92777 0.313128 0.156564 0.987668i $$-0.449958\pi$$
0.156564 + 0.987668i $$0.449958\pi$$
$$642$$ 0 0
$$643$$ 5.18115 0.204325 0.102162 0.994768i $$-0.467424\pi$$
0.102162 + 0.994768i $$0.467424\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.2062 0.479875 0.239938 0.970788i $$-0.422873\pi$$
0.239938 + 0.970788i $$0.422873\pi$$
$$648$$ 0 0
$$649$$ −17.6742 −0.693773
$$650$$ 0 0
$$651$$ 21.0472 0.824904
$$652$$ 0 0
$$653$$ 32.4969 1.27170 0.635852 0.771811i $$-0.280649\pi$$
0.635852 + 0.771811i $$0.280649\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 1.06400 0.0415107
$$658$$ 0 0
$$659$$ 29.4186 1.14598 0.572992 0.819561i $$-0.305783\pi$$
0.572992 + 0.819561i $$0.305783\pi$$
$$660$$ 0 0
$$661$$ 26.0677 1.01392 0.506958 0.861971i $$-0.330770\pi$$
0.506958 + 0.861971i $$0.330770\pi$$
$$662$$ 0 0
$$663$$ 37.0472 1.43879
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.73820 0.106024
$$668$$ 0 0
$$669$$ −29.0328 −1.12247
$$670$$ 0 0
$$671$$ 13.5297 0.522310
$$672$$ 0 0
$$673$$ 9.62863 0.371156 0.185578 0.982629i $$-0.440584\pi$$
0.185578 + 0.982629i $$0.440584\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 19.0205 0.731018 0.365509 0.930808i $$-0.380895\pi$$
0.365509 + 0.930808i $$0.380895\pi$$
$$678$$ 0 0
$$679$$ 12.0533 0.462564
$$680$$ 0 0
$$681$$ 34.1256 1.30769
$$682$$ 0 0
$$683$$ 17.4947 0.669415 0.334707 0.942322i $$-0.391363\pi$$
0.334707 + 0.942322i $$0.391363\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 40.4657 1.54386
$$688$$ 0 0
$$689$$ 10.5236 0.400917
$$690$$ 0 0
$$691$$ −16.3090 −0.620423 −0.310211 0.950668i $$-0.600400\pi$$
−0.310211 + 0.950668i $$0.600400\pi$$
$$692$$ 0 0
$$693$$ 1.11733 0.0424437
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −26.5236 −1.00465
$$698$$ 0 0
$$699$$ 23.2039 0.877653
$$700$$ 0 0
$$701$$ −12.9672 −0.489764 −0.244882 0.969553i $$-0.578749\pi$$
−0.244882 + 0.969553i $$0.578749\pi$$
$$702$$ 0 0
$$703$$ 18.7838 0.708444
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6.08661 0.228911
$$708$$ 0 0
$$709$$ 7.36069 0.276437 0.138218 0.990402i $$-0.455862\pi$$
0.138218 + 0.990402i $$0.455862\pi$$
$$710$$ 0 0
$$711$$ 0.993857 0.0372725
$$712$$ 0 0
$$713$$ −6.40787 −0.239977
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −14.3012 −0.534089
$$718$$ 0 0
$$719$$ 19.3197 0.720502 0.360251 0.932856i $$-0.382691\pi$$
0.360251 + 0.932856i $$0.382691\pi$$
$$720$$ 0 0
$$721$$ 42.6369 1.58788
$$722$$ 0 0
$$723$$ −16.9483 −0.630313
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −21.1545 −0.784577 −0.392288 0.919842i $$-0.628316\pi$$
−0.392288 + 0.919842i $$0.628316\pi$$
$$728$$ 0 0
$$729$$ 27.6681 1.02474
$$730$$ 0 0
$$731$$ 15.2039 0.562338
$$732$$ 0 0
$$733$$ −22.7526 −0.840386 −0.420193 0.907435i $$-0.638038\pi$$
−0.420193 + 0.907435i $$0.638038\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 42.6225 1.57002
$$738$$ 0 0
$$739$$ 28.4534 1.04668 0.523338 0.852125i $$-0.324686\pi$$
0.523338 + 0.852125i $$0.324686\pi$$
$$740$$ 0 0
$$741$$ 35.3484 1.29856
$$742$$ 0 0
$$743$$ 3.79380 0.139181 0.0695904 0.997576i $$-0.477831\pi$$
0.0695904 + 0.997576i $$0.477831\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −1.17340 −0.0429324
$$748$$ 0 0
$$749$$ 41.1773 1.50458
$$750$$ 0 0
$$751$$ −17.3607 −0.633501 −0.316750 0.948509i $$-0.602592\pi$$
−0.316750 + 0.948509i $$0.602592\pi$$
$$752$$ 0 0
$$753$$ 30.1112 1.09731
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −4.76487 −0.173182 −0.0865910 0.996244i $$-0.527597\pi$$
−0.0865910 + 0.996244i $$0.527597\pi$$
$$758$$ 0 0
$$759$$ 12.6803 0.460267
$$760$$ 0 0
$$761$$ −14.1978 −0.514670 −0.257335 0.966322i $$-0.582844\pi$$
−0.257335 + 0.966322i $$0.582844\pi$$
$$762$$ 0 0
$$763$$ −32.4657 −1.17534
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 20.6803 0.746724
$$768$$ 0 0
$$769$$ 54.7091 1.97286 0.986430 0.164181i $$-0.0524981\pi$$
0.986430 + 0.164181i $$0.0524981\pi$$
$$770$$ 0 0
$$771$$ −6.30122 −0.226933
$$772$$ 0 0
$$773$$ −11.0205 −0.396381 −0.198190 0.980164i $$-0.563506\pi$$
−0.198190 + 0.980164i $$0.563506\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 25.8966 0.929034
$$778$$ 0 0
$$779$$ −25.3074 −0.906731
$$780$$ 0 0
$$781$$ 33.3607 1.19374
$$782$$ 0 0
$$783$$ −10.5236 −0.376082
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −33.6925 −1.20101 −0.600503 0.799622i $$-0.705033\pi$$
−0.600503 + 0.799622i $$0.705033\pi$$
$$788$$ 0 0
$$789$$ −0.183417 −0.00652984
$$790$$ 0 0
$$791$$ −24.6270 −0.875636
$$792$$ 0 0
$$793$$ −15.8310 −0.562174
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 23.7009 0.839528 0.419764 0.907633i $$-0.362113\pi$$
0.419764 + 0.907633i $$0.362113\pi$$
$$798$$ 0 0
$$799$$ −16.3668 −0.579017
$$800$$ 0 0
$$801$$ 0.668060 0.0236047
$$802$$ 0 0
$$803$$ −73.5585 −2.59582
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.59374 0.232110
$$808$$ 0 0
$$809$$ −33.0349 −1.16145 −0.580723 0.814102i $$-0.697230\pi$$
−0.580723 + 0.814102i $$0.697230\pi$$
$$810$$ 0 0
$$811$$ −6.26794 −0.220097 −0.110049 0.993926i $$-0.535101\pi$$
−0.110049 + 0.993926i $$0.535101\pi$$
$$812$$ 0 0
$$813$$ 36.4202 1.27731
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 14.5068 0.507528
$$818$$ 0 0
$$819$$ −1.30737 −0.0456831
$$820$$ 0 0
$$821$$ −15.0616 −0.525652 −0.262826 0.964843i $$-0.584655\pi$$
−0.262826 + 0.964843i $$0.584655\pi$$
$$822$$ 0 0
$$823$$ −33.5669 −1.17007 −0.585034 0.811009i $$-0.698919\pi$$
−0.585034 + 0.811009i $$0.698919\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −10.6576 −0.370600 −0.185300 0.982682i $$-0.559326\pi$$
−0.185300 + 0.982682i $$0.559326\pi$$
$$828$$ 0 0
$$829$$ −52.4846 −1.82287 −0.911433 0.411447i $$-0.865023\pi$$
−0.911433 + 0.411447i $$0.865023\pi$$
$$830$$ 0 0
$$831$$ −2.47027 −0.0856926
$$832$$ 0 0
$$833$$ −0.267938 −0.00928351
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 24.6270 0.851234
$$838$$ 0 0
$$839$$ −18.2101 −0.628682 −0.314341 0.949310i $$-0.601783\pi$$
−0.314341 + 0.949310i $$0.601783\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −21.2618 −0.732295
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −48.3051 −1.65978
$$848$$ 0 0
$$849$$ −10.7526 −0.369028
$$850$$ 0 0
$$851$$ −7.88428 −0.270270
$$852$$ 0 0
$$853$$ −19.7542 −0.676371 −0.338185 0.941080i $$-0.609813\pi$$
−0.338185 + 0.941080i $$0.609813\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 48.9939 1.67360 0.836799 0.547510i $$-0.184424\pi$$
0.836799 + 0.547510i $$0.184424\pi$$
$$858$$ 0 0
$$859$$ −24.3090 −0.829412 −0.414706 0.909956i $$-0.636115\pi$$
−0.414706 + 0.909956i $$0.636115\pi$$
$$860$$ 0 0
$$861$$ −34.8904 −1.18906
$$862$$ 0 0
$$863$$ 44.3584 1.50998 0.754989 0.655737i $$-0.227642\pi$$
0.754989 + 0.655737i $$0.227642\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 9.08225 0.308449
$$868$$ 0 0
$$869$$ −68.7091 −2.33080
$$870$$ 0 0
$$871$$ −49.8720 −1.68985
$$872$$ 0 0
$$873$$ 0.359084 0.0121531
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −37.9565 −1.28170 −0.640850 0.767666i $$-0.721418\pi$$
−0.640850 + 0.767666i $$0.721418\pi$$
$$878$$ 0 0
$$879$$ −1.84324 −0.0621711
$$880$$ 0 0
$$881$$ 25.0661 0.844498 0.422249 0.906480i $$-0.361241\pi$$
0.422249 + 0.906480i $$0.361241\pi$$
$$882$$ 0 0
$$883$$ 21.1278 0.711008 0.355504 0.934675i $$-0.384309\pi$$
0.355504 + 0.934675i $$0.384309\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −36.1894 −1.21512 −0.607560 0.794274i $$-0.707852\pi$$
−0.607560 + 0.794274i $$0.707852\pi$$
$$888$$ 0 0
$$889$$ 5.13170 0.172112
$$890$$ 0 0
$$891$$ −47.4596 −1.58996
$$892$$ 0 0
$$893$$ −15.6163 −0.522581
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −14.8371 −0.495396
$$898$$ 0 0
$$899$$ −9.36069 −0.312197
$$900$$ 0 0
$$901$$ −5.67420 −0.189035
$$902$$ 0 0
$$903$$ 20.0000 0.665558
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 35.2678 1.17105 0.585523 0.810656i $$-0.300889\pi$$
0.585523 + 0.810656i $$0.300889\pi$$
$$908$$ 0 0
$$909$$ 0.181328 0.00601426
$$910$$ 0 0
$$911$$ 15.0061 0.497176 0.248588 0.968609i $$-0.420034\pi$$
0.248588 + 0.968609i $$0.420034\pi$$
$$912$$ 0 0
$$913$$ 81.1215 2.68473
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 40.9770 1.35318
$$918$$ 0 0
$$919$$ −32.8781 −1.08455 −0.542275 0.840201i $$-0.682437\pi$$
−0.542275 + 0.840201i $$0.682437\pi$$
$$920$$ 0 0
$$921$$ −43.1194 −1.42083
$$922$$ 0 0
$$923$$ −39.0349 −1.28485
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 1.27021 0.0417190
$$928$$ 0 0
$$929$$ −30.2290 −0.991781 −0.495890 0.868385i $$-0.665158\pi$$
−0.495890 + 0.868385i $$0.665158\pi$$
$$930$$ 0 0
$$931$$ −0.255652 −0.00837866
$$932$$ 0 0
$$933$$ −32.1978 −1.05411
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −28.4124 −0.928193 −0.464096 0.885785i $$-0.653621\pi$$
−0.464096 + 0.885785i $$0.653621\pi$$
$$938$$ 0 0
$$939$$ 25.9877 0.848077
$$940$$ 0 0
$$941$$ −42.0821 −1.37184 −0.685918 0.727679i $$-0.740599\pi$$
−0.685918 + 0.727679i $$0.740599\pi$$
$$942$$ 0 0
$$943$$ 10.6225 0.345916
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 22.2739 0.723805 0.361902 0.932216i $$-0.382127\pi$$
0.361902 + 0.932216i $$0.382127\pi$$
$$948$$ 0 0
$$949$$ 86.0698 2.79394
$$950$$ 0 0
$$951$$ −28.8248 −0.934709
$$952$$ 0 0
$$953$$ −14.6681 −0.475145 −0.237573 0.971370i $$-0.576352\pi$$
−0.237573 + 0.971370i $$0.576352\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 18.5236 0.598783
$$958$$ 0 0
$$959$$ 40.0000 1.29167
$$960$$ 0 0
$$961$$ −9.09436 −0.293367
$$962$$ 0 0
$$963$$ 1.22672 0.0395306
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0.403997 0.0129917 0.00649584 0.999979i $$-0.497932\pi$$
0.00649584 + 0.999979i $$0.497932\pi$$
$$968$$ 0 0
$$969$$ −19.0595 −0.612278
$$970$$ 0 0
$$971$$ −46.8326 −1.50293 −0.751464 0.659774i $$-0.770652\pi$$
−0.751464 + 0.659774i $$0.770652\pi$$
$$972$$ 0 0
$$973$$ −6.79153 −0.217726
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 53.6041 1.71495 0.857473 0.514529i $$-0.172033\pi$$
0.857473 + 0.514529i $$0.172033\pi$$
$$978$$ 0 0
$$979$$ −46.1855 −1.47610
$$980$$ 0 0
$$981$$ −0.967195 −0.0308802
$$982$$ 0 0
$$983$$ 19.9916 0.637633 0.318816 0.947817i $$-0.396715\pi$$
0.318816 + 0.947817i $$0.396715\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −21.5297 −0.685299
$$988$$ 0 0
$$989$$ −6.08906 −0.193621
$$990$$ 0 0
$$991$$ 24.6270 0.782303 0.391152 0.920326i $$-0.372077\pi$$
0.391152 + 0.920326i $$0.372077\pi$$
$$992$$ 0 0
$$993$$ −40.2967 −1.27878
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −20.9795 −0.664427 −0.332213 0.943204i $$-0.607795\pi$$
−0.332213 + 0.943204i $$0.607795\pi$$
$$998$$ 0 0
$$999$$ 30.3012 0.958688
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.a.bt.1.1 3
4.3 odd 2 3200.2.a.bq.1.3 3
5.2 odd 4 640.2.c.b.129.5 yes 6
5.3 odd 4 640.2.c.b.129.2 yes 6
5.4 even 2 3200.2.a.br.1.3 3
8.3 odd 2 3200.2.a.bv.1.1 3
8.5 even 2 3200.2.a.bo.1.3 3
20.3 even 4 640.2.c.a.129.5 yes 6
20.7 even 4 640.2.c.a.129.2 6
20.19 odd 2 3200.2.a.bs.1.1 3
40.3 even 4 640.2.c.d.129.2 yes 6
40.13 odd 4 640.2.c.c.129.5 yes 6
40.19 odd 2 3200.2.a.bp.1.3 3
40.27 even 4 640.2.c.d.129.5 yes 6
40.29 even 2 3200.2.a.bu.1.1 3
40.37 odd 4 640.2.c.c.129.2 yes 6
80.3 even 4 1280.2.f.l.129.2 6
80.13 odd 4 1280.2.f.j.129.6 6
80.27 even 4 1280.2.f.l.129.1 6
80.37 odd 4 1280.2.f.j.129.5 6
80.43 even 4 1280.2.f.i.129.5 6
80.53 odd 4 1280.2.f.k.129.1 6
80.67 even 4 1280.2.f.i.129.6 6
80.77 odd 4 1280.2.f.k.129.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.c.a.129.2 6 20.7 even 4
640.2.c.a.129.5 yes 6 20.3 even 4
640.2.c.b.129.2 yes 6 5.3 odd 4
640.2.c.b.129.5 yes 6 5.2 odd 4
640.2.c.c.129.2 yes 6 40.37 odd 4
640.2.c.c.129.5 yes 6 40.13 odd 4
640.2.c.d.129.2 yes 6 40.3 even 4
640.2.c.d.129.5 yes 6 40.27 even 4
1280.2.f.i.129.5 6 80.43 even 4
1280.2.f.i.129.6 6 80.67 even 4
1280.2.f.j.129.5 6 80.37 odd 4
1280.2.f.j.129.6 6 80.13 odd 4
1280.2.f.k.129.1 6 80.53 odd 4
1280.2.f.k.129.2 6 80.77 odd 4
1280.2.f.l.129.1 6 80.27 even 4
1280.2.f.l.129.2 6 80.3 even 4
3200.2.a.bo.1.3 3 8.5 even 2
3200.2.a.bp.1.3 3 40.19 odd 2
3200.2.a.bq.1.3 3 4.3 odd 2
3200.2.a.br.1.3 3 5.4 even 2
3200.2.a.bs.1.1 3 20.19 odd 2
3200.2.a.bt.1.1 3 1.1 even 1 trivial
3200.2.a.bu.1.1 3 40.29 even 2
3200.2.a.bv.1.1 3 8.3 odd 2