# Properties

 Label 637.2.a.l.1.1 Level $637$ Weight $2$ Character 637.1 Self dual yes Analytic conductor $5.086$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.746052.1 Defining polynomial: $$x^{5} - x^{4} - 7x^{3} + 8x + 2$$ x^5 - x^4 - 7*x^3 + 8*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.00852$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00852 q^{2} +1.75906 q^{3} +2.03417 q^{4} +0.905722 q^{5} -3.53311 q^{6} -0.0686323 q^{8} +0.0942784 q^{9} +O(q^{10})$$ $$q-2.00852 q^{2} +1.75906 q^{3} +2.03417 q^{4} +0.905722 q^{5} -3.53311 q^{6} -0.0686323 q^{8} +0.0942784 q^{9} -1.81916 q^{10} +0.716361 q^{11} +3.57822 q^{12} +1.00000 q^{13} +1.59322 q^{15} -3.93049 q^{16} +2.35227 q^{17} -0.189360 q^{18} +6.63591 q^{19} +1.84239 q^{20} -1.43883 q^{22} +3.75906 q^{23} -0.120728 q^{24} -4.17967 q^{25} -2.00852 q^{26} -5.11133 q^{27} +3.25799 q^{29} -3.20001 q^{30} +1.57050 q^{31} +8.03175 q^{32} +1.26012 q^{33} -4.72459 q^{34} +0.191778 q^{36} +5.20883 q^{37} -13.3284 q^{38} +1.75906 q^{39} -0.0621618 q^{40} +4.92168 q^{41} -9.43766 q^{43} +1.45720 q^{44} +0.0853900 q^{45} -7.55016 q^{46} -8.31986 q^{47} -6.91395 q^{48} +8.39497 q^{50} +4.13778 q^{51} +2.03417 q^{52} +14.0833 q^{53} +10.2662 q^{54} +0.648824 q^{55} +11.6729 q^{57} -6.54376 q^{58} +0.716361 q^{59} +3.24087 q^{60} -11.6527 q^{61} -3.15439 q^{62} -8.27099 q^{64} +0.905722 q^{65} -2.53098 q^{66} +9.39174 q^{67} +4.78492 q^{68} +6.61239 q^{69} +10.9914 q^{71} -0.00647055 q^{72} -3.47300 q^{73} -10.4621 q^{74} -7.35227 q^{75} +13.4986 q^{76} -3.53311 q^{78} +13.0082 q^{79} -3.55993 q^{80} -9.27395 q^{81} -9.88531 q^{82} +3.54083 q^{83} +2.13050 q^{85} +18.9558 q^{86} +5.73099 q^{87} -0.0491655 q^{88} +12.0501 q^{89} -0.171508 q^{90} +7.64656 q^{92} +2.76260 q^{93} +16.7106 q^{94} +6.01029 q^{95} +14.1283 q^{96} +7.43766 q^{97} +0.0675374 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 $$5 q + 4 q^{2} + 8 q^{4} + 2 q^{5} - 5 q^{6} + 9 q^{8} + 3 q^{9} - 5 q^{10} + 11 q^{11} + 5 q^{12} + 5 q^{13} + 10 q^{16} - 5 q^{17} + 9 q^{18} + 9 q^{19} + q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} + 4 q^{26} - 3 q^{29} - 13 q^{30} - 6 q^{31} + 22 q^{32} + 8 q^{33} - 22 q^{34} + 7 q^{36} + 4 q^{37} - 10 q^{38} + 28 q^{40} + 14 q^{41} + 2 q^{43} - 32 q^{45} + 3 q^{46} + q^{47} - 23 q^{48} + 9 q^{50} - 8 q^{51} + 8 q^{52} + 17 q^{53} + 23 q^{54} - 16 q^{57} - 27 q^{58} + 11 q^{59} - 29 q^{60} - 11 q^{61} - 23 q^{62} + 9 q^{64} + 2 q^{65} + 21 q^{66} + 13 q^{67} - 32 q^{68} + 18 q^{69} + 15 q^{71} - 19 q^{72} - 33 q^{74} - 20 q^{75} + 8 q^{76} - 5 q^{78} + 2 q^{79} + 55 q^{80} - 19 q^{81} + 34 q^{82} + 6 q^{83} - 22 q^{85} + 28 q^{86} - 8 q^{87} - 3 q^{88} - 4 q^{89} - 34 q^{90} + 21 q^{92} + 18 q^{93} + 20 q^{94} - 12 q^{95} - 37 q^{96} - 12 q^{97} + 11 q^{99}+O(q^{100})$$ 5 * q + 4 * q^2 + 8 * q^4 + 2 * q^5 - 5 * q^6 + 9 * q^8 + 3 * q^9 - 5 * q^10 + 11 * q^11 + 5 * q^12 + 5 * q^13 + 10 * q^16 - 5 * q^17 + 9 * q^18 + 9 * q^19 + q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 + 4 * q^26 - 3 * q^29 - 13 * q^30 - 6 * q^31 + 22 * q^32 + 8 * q^33 - 22 * q^34 + 7 * q^36 + 4 * q^37 - 10 * q^38 + 28 * q^40 + 14 * q^41 + 2 * q^43 - 32 * q^45 + 3 * q^46 + q^47 - 23 * q^48 + 9 * q^50 - 8 * q^51 + 8 * q^52 + 17 * q^53 + 23 * q^54 - 16 * q^57 - 27 * q^58 + 11 * q^59 - 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 2 * q^65 + 21 * q^66 + 13 * q^67 - 32 * q^68 + 18 * q^69 + 15 * q^71 - 19 * q^72 - 33 * q^74 - 20 * q^75 + 8 * q^76 - 5 * q^78 + 2 * q^79 + 55 * q^80 - 19 * q^81 + 34 * q^82 + 6 * q^83 - 22 * q^85 + 28 * q^86 - 8 * q^87 - 3 * q^88 - 4 * q^89 - 34 * q^90 + 21 * q^92 + 18 * q^93 + 20 * q^94 - 12 * q^95 - 37 * q^96 - 12 * q^97 + 11 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00852 −1.42024 −0.710121 0.704080i $$-0.751360\pi$$
−0.710121 + 0.704080i $$0.751360\pi$$
$$3$$ 1.75906 1.01559 0.507796 0.861477i $$-0.330460\pi$$
0.507796 + 0.861477i $$0.330460\pi$$
$$4$$ 2.03417 1.01709
$$5$$ 0.905722 0.405051 0.202526 0.979277i $$-0.435085\pi$$
0.202526 + 0.979277i $$0.435085\pi$$
$$6$$ −3.53311 −1.44238
$$7$$ 0 0
$$8$$ −0.0686323 −0.0242652
$$9$$ 0.0942784 0.0314261
$$10$$ −1.81916 −0.575270
$$11$$ 0.716361 0.215991 0.107996 0.994151i $$-0.465557\pi$$
0.107996 + 0.994151i $$0.465557\pi$$
$$12$$ 3.57822 1.03294
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 1.59322 0.411366
$$16$$ −3.93049 −0.982623
$$17$$ 2.35227 0.570510 0.285255 0.958452i $$-0.407922\pi$$
0.285255 + 0.958452i $$0.407922\pi$$
$$18$$ −0.189360 −0.0446327
$$19$$ 6.63591 1.52238 0.761191 0.648528i $$-0.224615\pi$$
0.761191 + 0.648528i $$0.224615\pi$$
$$20$$ 1.84239 0.411971
$$21$$ 0 0
$$22$$ −1.43883 −0.306759
$$23$$ 3.75906 0.783817 0.391909 0.920004i $$-0.371815\pi$$
0.391909 + 0.920004i $$0.371815\pi$$
$$24$$ −0.120728 −0.0246435
$$25$$ −4.17967 −0.835934
$$26$$ −2.00852 −0.393904
$$27$$ −5.11133 −0.983675
$$28$$ 0 0
$$29$$ 3.25799 0.604994 0.302497 0.953150i $$-0.402180\pi$$
0.302497 + 0.953150i $$0.402180\pi$$
$$30$$ −3.20001 −0.584239
$$31$$ 1.57050 0.282070 0.141035 0.990005i $$-0.454957\pi$$
0.141035 + 0.990005i $$0.454957\pi$$
$$32$$ 8.03175 1.41983
$$33$$ 1.26012 0.219359
$$34$$ −4.72459 −0.810261
$$35$$ 0 0
$$36$$ 0.191778 0.0319631
$$37$$ 5.20883 0.856326 0.428163 0.903702i $$-0.359161\pi$$
0.428163 + 0.903702i $$0.359161\pi$$
$$38$$ −13.3284 −2.16215
$$39$$ 1.75906 0.281674
$$40$$ −0.0621618 −0.00982864
$$41$$ 4.92168 0.768637 0.384318 0.923201i $$-0.374437\pi$$
0.384318 + 0.923201i $$0.374437\pi$$
$$42$$ 0 0
$$43$$ −9.43766 −1.43923 −0.719615 0.694373i $$-0.755682\pi$$
−0.719615 + 0.694373i $$0.755682\pi$$
$$44$$ 1.45720 0.219681
$$45$$ 0.0853900 0.0127292
$$46$$ −7.55016 −1.11321
$$47$$ −8.31986 −1.21358 −0.606788 0.794863i $$-0.707542\pi$$
−0.606788 + 0.794863i $$0.707542\pi$$
$$48$$ −6.91395 −0.997943
$$49$$ 0 0
$$50$$ 8.39497 1.18723
$$51$$ 4.13778 0.579405
$$52$$ 2.03417 0.282089
$$53$$ 14.0833 1.93449 0.967243 0.253854i $$-0.0816983\pi$$
0.967243 + 0.253854i $$0.0816983\pi$$
$$54$$ 10.2662 1.39706
$$55$$ 0.648824 0.0874874
$$56$$ 0 0
$$57$$ 11.6729 1.54612
$$58$$ −6.54376 −0.859238
$$59$$ 0.716361 0.0932623 0.0466311 0.998912i $$-0.485151\pi$$
0.0466311 + 0.998912i $$0.485151\pi$$
$$60$$ 3.24087 0.418395
$$61$$ −11.6527 −1.49197 −0.745986 0.665962i $$-0.768021\pi$$
−0.745986 + 0.665962i $$0.768021\pi$$
$$62$$ −3.15439 −0.400607
$$63$$ 0 0
$$64$$ −8.27099 −1.03387
$$65$$ 0.905722 0.112341
$$66$$ −2.53098 −0.311542
$$67$$ 9.39174 1.14738 0.573692 0.819071i $$-0.305511\pi$$
0.573692 + 0.819071i $$0.305511\pi$$
$$68$$ 4.78492 0.580257
$$69$$ 6.61239 0.796038
$$70$$ 0 0
$$71$$ 10.9914 1.30444 0.652220 0.758030i $$-0.273838\pi$$
0.652220 + 0.758030i $$0.273838\pi$$
$$72$$ −0.00647055 −0.000762561 0
$$73$$ −3.47300 −0.406484 −0.203242 0.979129i $$-0.565148\pi$$
−0.203242 + 0.979129i $$0.565148\pi$$
$$74$$ −10.4621 −1.21619
$$75$$ −7.35227 −0.848967
$$76$$ 13.4986 1.54839
$$77$$ 0 0
$$78$$ −3.53311 −0.400046
$$79$$ 13.0082 1.46353 0.731766 0.681556i $$-0.238696\pi$$
0.731766 + 0.681556i $$0.238696\pi$$
$$80$$ −3.55993 −0.398012
$$81$$ −9.27395 −1.03044
$$82$$ −9.88531 −1.09165
$$83$$ 3.54083 0.388656 0.194328 0.980937i $$-0.437747\pi$$
0.194328 + 0.980937i $$0.437747\pi$$
$$84$$ 0 0
$$85$$ 2.13050 0.231085
$$86$$ 18.9558 2.04405
$$87$$ 5.73099 0.614427
$$88$$ −0.0491655 −0.00524106
$$89$$ 12.0501 1.27730 0.638651 0.769496i $$-0.279493\pi$$
0.638651 + 0.769496i $$0.279493\pi$$
$$90$$ −0.171508 −0.0180785
$$91$$ 0 0
$$92$$ 7.64656 0.797209
$$93$$ 2.76260 0.286468
$$94$$ 16.7106 1.72357
$$95$$ 6.01029 0.616642
$$96$$ 14.1283 1.44196
$$97$$ 7.43766 0.755180 0.377590 0.925973i $$-0.376753\pi$$
0.377590 + 0.925973i $$0.376753\pi$$
$$98$$ 0 0
$$99$$ 0.0675374 0.00678776
$$100$$ −8.50216 −0.850216
$$101$$ −1.19905 −0.119310 −0.0596551 0.998219i $$-0.519000\pi$$
−0.0596551 + 0.998219i $$0.519000\pi$$
$$102$$ −8.31083 −0.822894
$$103$$ −14.4123 −1.42009 −0.710043 0.704158i $$-0.751325\pi$$
−0.710043 + 0.704158i $$0.751325\pi$$
$$104$$ −0.0686323 −0.00672995
$$105$$ 0 0
$$106$$ −28.2866 −2.74744
$$107$$ 13.5932 1.31411 0.657053 0.753845i $$-0.271803\pi$$
0.657053 + 0.753845i $$0.271803\pi$$
$$108$$ −10.3973 −1.00048
$$109$$ −13.7248 −1.31460 −0.657299 0.753630i $$-0.728301\pi$$
−0.657299 + 0.753630i $$0.728301\pi$$
$$110$$ −1.30318 −0.124253
$$111$$ 9.16262 0.869677
$$112$$ 0 0
$$113$$ −3.25799 −0.306486 −0.153243 0.988189i $$-0.548972\pi$$
−0.153243 + 0.988189i $$0.548972\pi$$
$$114$$ −23.4454 −2.19586
$$115$$ 3.40466 0.317486
$$116$$ 6.62731 0.615331
$$117$$ 0.0942784 0.00871604
$$118$$ −1.43883 −0.132455
$$119$$ 0 0
$$120$$ −0.109346 −0.00998189
$$121$$ −10.4868 −0.953348
$$122$$ 23.4047 2.11896
$$123$$ 8.65750 0.780621
$$124$$ 3.19466 0.286889
$$125$$ −8.31422 −0.743647
$$126$$ 0 0
$$127$$ −0.950834 −0.0843729 −0.0421865 0.999110i $$-0.513432\pi$$
−0.0421865 + 0.999110i $$0.513432\pi$$
$$128$$ 0.548979 0.0485233
$$129$$ −16.6014 −1.46167
$$130$$ −1.81916 −0.159551
$$131$$ −18.8196 −1.64428 −0.822138 0.569288i $$-0.807219\pi$$
−0.822138 + 0.569288i $$0.807219\pi$$
$$132$$ 2.56330 0.223106
$$133$$ 0 0
$$134$$ −18.8635 −1.62956
$$135$$ −4.62944 −0.398439
$$136$$ −0.161442 −0.0138435
$$137$$ 6.18179 0.528146 0.264073 0.964503i $$-0.414934\pi$$
0.264073 + 0.964503i $$0.414934\pi$$
$$138$$ −13.2811 −1.13057
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −14.6351 −1.23250
$$142$$ −22.0765 −1.85262
$$143$$ 0.716361 0.0599051
$$144$$ −0.370560 −0.0308800
$$145$$ 2.95083 0.245053
$$146$$ 6.97560 0.577305
$$147$$ 0 0
$$148$$ 10.5956 0.870956
$$149$$ −21.0771 −1.72670 −0.863351 0.504604i $$-0.831639\pi$$
−0.863351 + 0.504604i $$0.831639\pi$$
$$150$$ 14.7672 1.20574
$$151$$ −15.7234 −1.27955 −0.639777 0.768560i $$-0.720973\pi$$
−0.639777 + 0.768560i $$0.720973\pi$$
$$152$$ −0.455438 −0.0369409
$$153$$ 0.221768 0.0179289
$$154$$ 0 0
$$155$$ 1.42244 0.114253
$$156$$ 3.57822 0.286487
$$157$$ −7.78499 −0.621310 −0.310655 0.950523i $$-0.600548\pi$$
−0.310655 + 0.950523i $$0.600548\pi$$
$$158$$ −26.1272 −2.07857
$$159$$ 24.7733 1.96465
$$160$$ 7.27453 0.575102
$$161$$ 0 0
$$162$$ 18.6269 1.46347
$$163$$ 1.68991 0.132364 0.0661820 0.997808i $$-0.478918\pi$$
0.0661820 + 0.997808i $$0.478918\pi$$
$$164$$ 10.0115 0.781769
$$165$$ 1.14132 0.0888514
$$166$$ −7.11184 −0.551986
$$167$$ −21.8667 −1.69210 −0.846049 0.533105i $$-0.821025\pi$$
−0.846049 + 0.533105i $$0.821025\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −4.27917 −0.328197
$$171$$ 0.625623 0.0478426
$$172$$ −19.1978 −1.46382
$$173$$ 5.84122 0.444100 0.222050 0.975035i $$-0.428725\pi$$
0.222050 + 0.975035i $$0.428725\pi$$
$$174$$ −11.5108 −0.872634
$$175$$ 0 0
$$176$$ −2.81565 −0.212238
$$177$$ 1.26012 0.0947164
$$178$$ −24.2028 −1.81408
$$179$$ 2.53427 0.189421 0.0947103 0.995505i $$-0.469808\pi$$
0.0947103 + 0.995505i $$0.469808\pi$$
$$180$$ 0.173698 0.0129467
$$181$$ −10.7248 −0.797169 −0.398585 0.917132i $$-0.630498\pi$$
−0.398585 + 0.917132i $$0.630498\pi$$
$$182$$ 0 0
$$183$$ −20.4977 −1.51523
$$184$$ −0.257993 −0.0190195
$$185$$ 4.71775 0.346856
$$186$$ −5.54874 −0.406854
$$187$$ 1.68508 0.123225
$$188$$ −16.9240 −1.23431
$$189$$ 0 0
$$190$$ −12.0718 −0.875781
$$191$$ −1.67861 −0.121460 −0.0607298 0.998154i $$-0.519343\pi$$
−0.0607298 + 0.998154i $$0.519343\pi$$
$$192$$ −14.5491 −1.04999
$$193$$ −6.44816 −0.464148 −0.232074 0.972698i $$-0.574551\pi$$
−0.232074 + 0.972698i $$0.574551\pi$$
$$194$$ −14.9387 −1.07254
$$195$$ 1.59322 0.114093
$$196$$ 0 0
$$197$$ 1.87251 0.133411 0.0667054 0.997773i $$-0.478751\pi$$
0.0667054 + 0.997773i $$0.478751\pi$$
$$198$$ −0.135650 −0.00964026
$$199$$ −11.3967 −0.807888 −0.403944 0.914784i $$-0.632361\pi$$
−0.403944 + 0.914784i $$0.632361\pi$$
$$200$$ 0.286860 0.0202841
$$201$$ 16.5206 1.16527
$$202$$ 2.40833 0.169449
$$203$$ 0 0
$$204$$ 8.41694 0.589304
$$205$$ 4.45767 0.311337
$$206$$ 28.9475 2.01687
$$207$$ 0.354398 0.0246324
$$208$$ −3.93049 −0.272531
$$209$$ 4.75371 0.328821
$$210$$ 0 0
$$211$$ 7.53599 0.518799 0.259400 0.965770i $$-0.416475\pi$$
0.259400 + 0.965770i $$0.416475\pi$$
$$212$$ 28.6478 1.96754
$$213$$ 19.3345 1.32478
$$214$$ −27.3023 −1.86635
$$215$$ −8.54789 −0.582962
$$216$$ 0.350802 0.0238691
$$217$$ 0 0
$$218$$ 27.5666 1.86705
$$219$$ −6.10920 −0.412822
$$220$$ 1.31982 0.0889821
$$221$$ 2.35227 0.158231
$$222$$ −18.4033 −1.23515
$$223$$ 17.6349 1.18092 0.590459 0.807067i $$-0.298947\pi$$
0.590459 + 0.807067i $$0.298947\pi$$
$$224$$ 0 0
$$225$$ −0.394052 −0.0262702
$$226$$ 6.54376 0.435284
$$227$$ −5.32904 −0.353701 −0.176851 0.984238i $$-0.556591\pi$$
−0.176851 + 0.984238i $$0.556591\pi$$
$$228$$ 23.7447 1.57253
$$229$$ −8.51900 −0.562951 −0.281476 0.959568i $$-0.590824\pi$$
−0.281476 + 0.959568i $$0.590824\pi$$
$$230$$ −6.83834 −0.450907
$$231$$ 0 0
$$232$$ −0.223604 −0.0146803
$$233$$ 4.75371 0.311426 0.155713 0.987802i $$-0.450233\pi$$
0.155713 + 0.987802i $$0.450233\pi$$
$$234$$ −0.189360 −0.0123789
$$235$$ −7.53548 −0.491561
$$236$$ 1.45720 0.0948557
$$237$$ 22.8821 1.48635
$$238$$ 0 0
$$239$$ 14.8314 0.959365 0.479682 0.877442i $$-0.340752\pi$$
0.479682 + 0.877442i $$0.340752\pi$$
$$240$$ −6.26212 −0.404218
$$241$$ −6.12131 −0.394308 −0.197154 0.980373i $$-0.563170\pi$$
−0.197154 + 0.980373i $$0.563170\pi$$
$$242$$ 21.0630 1.35398
$$243$$ −0.979411 −0.0628292
$$244$$ −23.7035 −1.51746
$$245$$ 0 0
$$246$$ −17.3888 −1.10867
$$247$$ 6.63591 0.422233
$$248$$ −0.107787 −0.00684448
$$249$$ 6.22852 0.394716
$$250$$ 16.6993 1.05616
$$251$$ −13.9708 −0.881832 −0.440916 0.897548i $$-0.645346\pi$$
−0.440916 + 0.897548i $$0.645346\pi$$
$$252$$ 0 0
$$253$$ 2.69284 0.169298
$$254$$ 1.90977 0.119830
$$255$$ 3.74767 0.234688
$$256$$ 15.4393 0.964959
$$257$$ 17.2651 1.07696 0.538482 0.842637i $$-0.318998\pi$$
0.538482 + 0.842637i $$0.318998\pi$$
$$258$$ 33.3443 2.07592
$$259$$ 0 0
$$260$$ 1.84239 0.114260
$$261$$ 0.307158 0.0190126
$$262$$ 37.7996 2.33527
$$263$$ −2.60672 −0.160737 −0.0803687 0.996765i $$-0.525610\pi$$
−0.0803687 + 0.996765i $$0.525610\pi$$
$$264$$ −0.0864849 −0.00532278
$$265$$ 12.7555 0.783565
$$266$$ 0 0
$$267$$ 21.1967 1.29722
$$268$$ 19.1044 1.16699
$$269$$ −14.4895 −0.883443 −0.441721 0.897152i $$-0.645632\pi$$
−0.441721 + 0.897152i $$0.645632\pi$$
$$270$$ 9.29834 0.565879
$$271$$ −8.63591 −0.524594 −0.262297 0.964987i $$-0.584480\pi$$
−0.262297 + 0.964987i $$0.584480\pi$$
$$272$$ −9.24558 −0.560596
$$273$$ 0 0
$$274$$ −12.4163 −0.750095
$$275$$ −2.99415 −0.180554
$$276$$ 13.4507 0.809639
$$277$$ 12.2270 0.734647 0.367324 0.930093i $$-0.380274\pi$$
0.367324 + 0.930093i $$0.380274\pi$$
$$278$$ 8.03410 0.481853
$$279$$ 0.148064 0.00886437
$$280$$ 0 0
$$281$$ −24.1822 −1.44259 −0.721293 0.692630i $$-0.756452\pi$$
−0.721293 + 0.692630i $$0.756452\pi$$
$$282$$ 29.3950 1.75044
$$283$$ −30.7683 −1.82899 −0.914493 0.404601i $$-0.867410\pi$$
−0.914493 + 0.404601i $$0.867410\pi$$
$$284$$ 22.3584 1.32673
$$285$$ 10.5724 0.626257
$$286$$ −1.43883 −0.0850797
$$287$$ 0 0
$$288$$ 0.757221 0.0446197
$$289$$ −11.4668 −0.674519
$$290$$ −5.92682 −0.348035
$$291$$ 13.0833 0.766954
$$292$$ −7.06467 −0.413429
$$293$$ −31.8295 −1.85950 −0.929749 0.368193i $$-0.879976\pi$$
−0.929749 + 0.368193i $$0.879976\pi$$
$$294$$ 0 0
$$295$$ 0.648824 0.0377760
$$296$$ −0.357494 −0.0207789
$$297$$ −3.66156 −0.212465
$$298$$ 42.3338 2.45233
$$299$$ 3.75906 0.217392
$$300$$ −14.9558 −0.863472
$$301$$ 0 0
$$302$$ 31.5809 1.81728
$$303$$ −2.10920 −0.121170
$$304$$ −26.0824 −1.49593
$$305$$ −10.5541 −0.604324
$$306$$ −0.445427 −0.0254634
$$307$$ 28.7884 1.64304 0.821520 0.570179i $$-0.193126\pi$$
0.821520 + 0.570179i $$0.193126\pi$$
$$308$$ 0 0
$$309$$ −25.3521 −1.44223
$$310$$ −2.85700 −0.162266
$$311$$ 5.51862 0.312932 0.156466 0.987683i $$-0.449990\pi$$
0.156466 + 0.987683i $$0.449990\pi$$
$$312$$ −0.120728 −0.00683488
$$313$$ −4.84799 −0.274024 −0.137012 0.990569i $$-0.543750\pi$$
−0.137012 + 0.990569i $$0.543750\pi$$
$$314$$ 15.6363 0.882410
$$315$$ 0 0
$$316$$ 26.4608 1.48854
$$317$$ −7.65511 −0.429954 −0.214977 0.976619i $$-0.568968\pi$$
−0.214977 + 0.976619i $$0.568968\pi$$
$$318$$ −49.7577 −2.79027
$$319$$ 2.33390 0.130673
$$320$$ −7.49121 −0.418772
$$321$$ 23.9112 1.33459
$$322$$ 0 0
$$323$$ 15.6095 0.868534
$$324$$ −18.8648 −1.04804
$$325$$ −4.17967 −0.231846
$$326$$ −3.39423 −0.187989
$$327$$ −24.1427 −1.33510
$$328$$ −0.337786 −0.0186511
$$329$$ 0 0
$$330$$ −2.29236 −0.126190
$$331$$ 11.3432 0.623477 0.311739 0.950168i $$-0.399089\pi$$
0.311739 + 0.950168i $$0.399089\pi$$
$$332$$ 7.20265 0.395297
$$333$$ 0.491080 0.0269110
$$334$$ 43.9199 2.40319
$$335$$ 8.50631 0.464749
$$336$$ 0 0
$$337$$ 1.74149 0.0948649 0.0474324 0.998874i $$-0.484896\pi$$
0.0474324 + 0.998874i $$0.484896\pi$$
$$338$$ −2.00852 −0.109249
$$339$$ −5.73099 −0.311265
$$340$$ 4.33381 0.235034
$$341$$ 1.12504 0.0609246
$$342$$ −1.25658 −0.0679480
$$343$$ 0 0
$$344$$ 0.647729 0.0349232
$$345$$ 5.98898 0.322436
$$346$$ −11.7322 −0.630729
$$347$$ 21.0503 1.13004 0.565019 0.825078i $$-0.308869\pi$$
0.565019 + 0.825078i $$0.308869\pi$$
$$348$$ 11.6578 0.624925
$$349$$ −8.35601 −0.447287 −0.223643 0.974671i $$-0.571795\pi$$
−0.223643 + 0.974671i $$0.571795\pi$$
$$350$$ 0 0
$$351$$ −5.11133 −0.272822
$$352$$ 5.75364 0.306670
$$353$$ −8.53355 −0.454195 −0.227097 0.973872i $$-0.572924\pi$$
−0.227097 + 0.973872i $$0.572924\pi$$
$$354$$ −2.53098 −0.134520
$$355$$ 9.95515 0.528365
$$356$$ 24.5119 1.29913
$$357$$ 0 0
$$358$$ −5.09015 −0.269023
$$359$$ −16.1713 −0.853488 −0.426744 0.904372i $$-0.640339\pi$$
−0.426744 + 0.904372i $$0.640339\pi$$
$$360$$ −0.00586051 −0.000308876 0
$$361$$ 25.0353 1.31765
$$362$$ 21.5410 1.13217
$$363$$ −18.4469 −0.968212
$$364$$ 0 0
$$365$$ −3.14557 −0.164647
$$366$$ 41.1701 2.15200
$$367$$ 28.1540 1.46963 0.734813 0.678269i $$-0.237270\pi$$
0.734813 + 0.678269i $$0.237270\pi$$
$$368$$ −14.7749 −0.770197
$$369$$ 0.464008 0.0241553
$$370$$ −9.47571 −0.492619
$$371$$ 0 0
$$372$$ 5.61959 0.291362
$$373$$ −28.5037 −1.47586 −0.737932 0.674875i $$-0.764197\pi$$
−0.737932 + 0.674875i $$0.764197\pi$$
$$374$$ −3.38452 −0.175009
$$375$$ −14.6252 −0.755241
$$376$$ 0.571012 0.0294477
$$377$$ 3.25799 0.167795
$$378$$ 0 0
$$379$$ −7.26263 −0.373056 −0.186528 0.982450i $$-0.559724\pi$$
−0.186528 + 0.982450i $$0.559724\pi$$
$$380$$ 12.2259 0.627178
$$381$$ −1.67257 −0.0856884
$$382$$ 3.37152 0.172502
$$383$$ 12.9325 0.660822 0.330411 0.943837i $$-0.392813\pi$$
0.330411 + 0.943837i $$0.392813\pi$$
$$384$$ 0.965684 0.0492799
$$385$$ 0 0
$$386$$ 12.9513 0.659203
$$387$$ −0.889768 −0.0452294
$$388$$ 15.1295 0.768083
$$389$$ −21.1357 −1.07162 −0.535811 0.844338i $$-0.679994\pi$$
−0.535811 + 0.844338i $$0.679994\pi$$
$$390$$ −3.20001 −0.162039
$$391$$ 8.84232 0.447175
$$392$$ 0 0
$$393$$ −33.1047 −1.66991
$$394$$ −3.76098 −0.189476
$$395$$ 11.7818 0.592805
$$396$$ 0.137383 0.00690373
$$397$$ 19.2073 0.963988 0.481994 0.876175i $$-0.339913\pi$$
0.481994 + 0.876175i $$0.339913\pi$$
$$398$$ 22.8905 1.14740
$$399$$ 0 0
$$400$$ 16.4282 0.821408
$$401$$ 16.6692 0.832420 0.416210 0.909268i $$-0.363358\pi$$
0.416210 + 0.909268i $$0.363358\pi$$
$$402$$ −33.1820 −1.65497
$$403$$ 1.57050 0.0782321
$$404$$ −2.43908 −0.121349
$$405$$ −8.39961 −0.417380
$$406$$ 0 0
$$407$$ 3.73140 0.184959
$$408$$ −0.283985 −0.0140594
$$409$$ 12.3483 0.610585 0.305293 0.952259i $$-0.401246\pi$$
0.305293 + 0.952259i $$0.401246\pi$$
$$410$$ −8.95333 −0.442174
$$411$$ 10.8741 0.536381
$$412$$ −29.3171 −1.44435
$$413$$ 0 0
$$414$$ −0.711817 −0.0349839
$$415$$ 3.20700 0.157426
$$416$$ 8.03175 0.393789
$$417$$ −7.03622 −0.344565
$$418$$ −9.54794 −0.467005
$$419$$ 4.35934 0.212968 0.106484 0.994314i $$-0.466041\pi$$
0.106484 + 0.994314i $$0.466041\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −15.1362 −0.736820
$$423$$ −0.784383 −0.0381380
$$424$$ −0.966567 −0.0469407
$$425$$ −9.83171 −0.476908
$$426$$ −38.8338 −1.88150
$$427$$ 0 0
$$428$$ 27.6509 1.33656
$$429$$ 1.26012 0.0608391
$$430$$ 17.1687 0.827946
$$431$$ −23.3626 −1.12533 −0.562667 0.826683i $$-0.690225\pi$$
−0.562667 + 0.826683i $$0.690225\pi$$
$$432$$ 20.0900 0.966582
$$433$$ −2.71285 −0.130371 −0.0651856 0.997873i $$-0.520764\pi$$
−0.0651856 + 0.997873i $$0.520764\pi$$
$$434$$ 0 0
$$435$$ 5.19068 0.248874
$$436$$ −27.9186 −1.33706
$$437$$ 24.9448 1.19327
$$438$$ 12.2705 0.586306
$$439$$ −8.83519 −0.421681 −0.210840 0.977521i $$-0.567620\pi$$
−0.210840 + 0.977521i $$0.567620\pi$$
$$440$$ −0.0445303 −0.00212290
$$441$$ 0 0
$$442$$ −4.72459 −0.224726
$$443$$ −2.90558 −0.138048 −0.0690240 0.997615i $$-0.521989\pi$$
−0.0690240 + 0.997615i $$0.521989\pi$$
$$444$$ 18.6383 0.884536
$$445$$ 10.9140 0.517373
$$446$$ −35.4201 −1.67719
$$447$$ −37.0758 −1.75362
$$448$$ 0 0
$$449$$ −15.2777 −0.720998 −0.360499 0.932760i $$-0.617394\pi$$
−0.360499 + 0.932760i $$0.617394\pi$$
$$450$$ 0.791464 0.0373100
$$451$$ 3.52570 0.166019
$$452$$ −6.62731 −0.311723
$$453$$ −27.6584 −1.29950
$$454$$ 10.7035 0.502341
$$455$$ 0 0
$$456$$ −0.801141 −0.0375169
$$457$$ 23.6600 1.10677 0.553384 0.832926i $$-0.313336\pi$$
0.553384 + 0.832926i $$0.313336\pi$$
$$458$$ 17.1106 0.799527
$$459$$ −12.0232 −0.561196
$$460$$ 6.92566 0.322910
$$461$$ −26.6170 −1.23968 −0.619839 0.784729i $$-0.712802\pi$$
−0.619839 + 0.784729i $$0.712802\pi$$
$$462$$ 0 0
$$463$$ −1.44250 −0.0670385 −0.0335193 0.999438i $$-0.510672\pi$$
−0.0335193 + 0.999438i $$0.510672\pi$$
$$464$$ −12.8055 −0.594481
$$465$$ 2.50214 0.116034
$$466$$ −9.54794 −0.442300
$$467$$ −8.38959 −0.388224 −0.194112 0.980979i $$-0.562183\pi$$
−0.194112 + 0.980979i $$0.562183\pi$$
$$468$$ 0.191778 0.00886496
$$469$$ 0 0
$$470$$ 15.1352 0.698135
$$471$$ −13.6942 −0.630997
$$472$$ −0.0491655 −0.00226303
$$473$$ −6.76077 −0.310861
$$474$$ −45.9592 −2.11098
$$475$$ −27.7359 −1.27261
$$476$$ 0 0
$$477$$ 1.32775 0.0607934
$$478$$ −29.7893 −1.36253
$$479$$ 12.6122 0.576265 0.288132 0.957591i $$-0.406966\pi$$
0.288132 + 0.957591i $$0.406966\pi$$
$$480$$ 12.7963 0.584069
$$481$$ 5.20883 0.237502
$$482$$ 12.2948 0.560013
$$483$$ 0 0
$$484$$ −21.3320 −0.969636
$$485$$ 6.73645 0.305886
$$486$$ 1.96717 0.0892327
$$487$$ 21.5680 0.977341 0.488671 0.872468i $$-0.337482\pi$$
0.488671 + 0.872468i $$0.337482\pi$$
$$488$$ 0.799750 0.0362030
$$489$$ 2.97265 0.134428
$$490$$ 0 0
$$491$$ 39.2347 1.77064 0.885318 0.464987i $$-0.153941\pi$$
0.885318 + 0.464987i $$0.153941\pi$$
$$492$$ 17.6108 0.793958
$$493$$ 7.66368 0.345155
$$494$$ −13.3284 −0.599673
$$495$$ 0.0611701 0.00274939
$$496$$ −6.17283 −0.277168
$$497$$ 0 0
$$498$$ −12.5101 −0.560592
$$499$$ 9.16814 0.410422 0.205211 0.978718i $$-0.434212\pi$$
0.205211 + 0.978718i $$0.434212\pi$$
$$500$$ −16.9125 −0.756352
$$501$$ −38.4648 −1.71848
$$502$$ 28.0608 1.25241
$$503$$ 24.9370 1.11188 0.555942 0.831221i $$-0.312358\pi$$
0.555942 + 0.831221i $$0.312358\pi$$
$$504$$ 0 0
$$505$$ −1.08601 −0.0483267
$$506$$ −5.40864 −0.240443
$$507$$ 1.75906 0.0781224
$$508$$ −1.93416 −0.0858144
$$509$$ 5.89807 0.261428 0.130714 0.991420i $$-0.458273\pi$$
0.130714 + 0.991420i $$0.458273\pi$$
$$510$$ −7.52730 −0.333314
$$511$$ 0 0
$$512$$ −32.1083 −1.41900
$$513$$ −33.9183 −1.49753
$$514$$ −34.6773 −1.52955
$$515$$ −13.0535 −0.575208
$$516$$ −33.7700 −1.48664
$$517$$ −5.96003 −0.262122
$$518$$ 0 0
$$519$$ 10.2750 0.451024
$$520$$ −0.0621618 −0.00272597
$$521$$ −37.1895 −1.62930 −0.814652 0.579950i $$-0.803072\pi$$
−0.814652 + 0.579950i $$0.803072\pi$$
$$522$$ −0.616935 −0.0270025
$$523$$ −5.09080 −0.222605 −0.111303 0.993787i $$-0.535502\pi$$
−0.111303 + 0.993787i $$0.535502\pi$$
$$524$$ −38.2823 −1.67237
$$525$$ 0 0
$$526$$ 5.23567 0.228286
$$527$$ 3.69424 0.160924
$$528$$ −4.95289 −0.215547
$$529$$ −8.86950 −0.385630
$$530$$ −25.6198 −1.11285
$$531$$ 0.0675374 0.00293087
$$532$$ 0 0
$$533$$ 4.92168 0.213181
$$534$$ −42.5741 −1.84236
$$535$$ 12.3117 0.532280
$$536$$ −0.644577 −0.0278415
$$537$$ 4.45793 0.192374
$$538$$ 29.1026 1.25470
$$539$$ 0 0
$$540$$ −9.41707 −0.405246
$$541$$ −0.766850 −0.0329694 −0.0164847 0.999864i $$-0.505247\pi$$
−0.0164847 + 0.999864i $$0.505247\pi$$
$$542$$ 17.3454 0.745050
$$543$$ −18.8655 −0.809598
$$544$$ 18.8929 0.810025
$$545$$ −12.4309 −0.532480
$$546$$ 0 0
$$547$$ 14.1428 0.604702 0.302351 0.953197i $$-0.402229\pi$$
0.302351 + 0.953197i $$0.402229\pi$$
$$548$$ 12.5748 0.537170
$$549$$ −1.09859 −0.0468869
$$550$$ 6.01383 0.256430
$$551$$ 21.6197 0.921032
$$552$$ −0.453824 −0.0193160
$$553$$ 0 0
$$554$$ −24.5582 −1.04338
$$555$$ 8.29878 0.352264
$$556$$ −8.13668 −0.345072
$$557$$ −24.8627 −1.05347 −0.526733 0.850031i $$-0.676583\pi$$
−0.526733 + 0.850031i $$0.676583\pi$$
$$558$$ −0.297391 −0.0125895
$$559$$ −9.43766 −0.399171
$$560$$ 0 0
$$561$$ 2.96414 0.125146
$$562$$ 48.5705 2.04882
$$563$$ 44.0094 1.85478 0.927388 0.374101i $$-0.122049\pi$$
0.927388 + 0.374101i $$0.122049\pi$$
$$564$$ −29.7703 −1.25356
$$565$$ −2.95083 −0.124143
$$566$$ 61.7989 2.59760
$$567$$ 0 0
$$568$$ −0.754366 −0.0316525
$$569$$ −33.2616 −1.39440 −0.697199 0.716877i $$-0.745571\pi$$
−0.697199 + 0.716877i $$0.745571\pi$$
$$570$$ −21.2350 −0.889436
$$571$$ −12.3540 −0.516998 −0.258499 0.966011i $$-0.583228\pi$$
−0.258499 + 0.966011i $$0.583228\pi$$
$$572$$ 1.45720 0.0609286
$$573$$ −2.95276 −0.123353
$$574$$ 0 0
$$575$$ −15.7116 −0.655219
$$576$$ −0.779776 −0.0324907
$$577$$ −25.9659 −1.08097 −0.540486 0.841353i $$-0.681760\pi$$
−0.540486 + 0.841353i $$0.681760\pi$$
$$578$$ 23.0314 0.957979
$$579$$ −11.3427 −0.471385
$$580$$ 6.00250 0.249240
$$581$$ 0 0
$$582$$ −26.2781 −1.08926
$$583$$ 10.0887 0.417831
$$584$$ 0.238360 0.00986341
$$585$$ 0.0853900 0.00353044
$$586$$ 63.9303 2.64094
$$587$$ −23.9747 −0.989543 −0.494771 0.869023i $$-0.664748\pi$$
−0.494771 + 0.869023i $$0.664748\pi$$
$$588$$ 0 0
$$589$$ 10.4217 0.429418
$$590$$ −1.30318 −0.0536510
$$591$$ 3.29385 0.135491
$$592$$ −20.4733 −0.841445
$$593$$ 47.0480 1.93203 0.966015 0.258484i $$-0.0832230\pi$$
0.966015 + 0.258484i $$0.0832230\pi$$
$$594$$ 7.35433 0.301752
$$595$$ 0 0
$$596$$ −42.8744 −1.75620
$$597$$ −20.0474 −0.820484
$$598$$ −7.55016 −0.308749
$$599$$ 20.1736 0.824271 0.412135 0.911123i $$-0.364783\pi$$
0.412135 + 0.911123i $$0.364783\pi$$
$$600$$ 0.504604 0.0206004
$$601$$ 29.5773 1.20648 0.603242 0.797558i $$-0.293875\pi$$
0.603242 + 0.797558i $$0.293875\pi$$
$$602$$ 0 0
$$603$$ 0.885439 0.0360579
$$604$$ −31.9841 −1.30142
$$605$$ −9.49815 −0.386155
$$606$$ 4.23638 0.172091
$$607$$ −15.4420 −0.626771 −0.313385 0.949626i $$-0.601463\pi$$
−0.313385 + 0.949626i $$0.601463\pi$$
$$608$$ 53.2980 2.16152
$$609$$ 0 0
$$610$$ 21.1981 0.858287
$$611$$ −8.31986 −0.336586
$$612$$ 0.451115 0.0182352
$$613$$ 1.99485 0.0805711 0.0402855 0.999188i $$-0.487173\pi$$
0.0402855 + 0.999188i $$0.487173\pi$$
$$614$$ −57.8222 −2.33351
$$615$$ 7.84129 0.316191
$$616$$ 0 0
$$617$$ −2.85584 −0.114972 −0.0574858 0.998346i $$-0.518308\pi$$
−0.0574858 + 0.998346i $$0.518308\pi$$
$$618$$ 50.9202 2.04831
$$619$$ 31.9823 1.28548 0.642738 0.766086i $$-0.277798\pi$$
0.642738 + 0.766086i $$0.277798\pi$$
$$620$$ 2.89348 0.116205
$$621$$ −19.2138 −0.771022
$$622$$ −11.0843 −0.444440
$$623$$ 0 0
$$624$$ −6.91395 −0.276780
$$625$$ 13.3680 0.534719
$$626$$ 9.73730 0.389181
$$627$$ 8.36204 0.333948
$$628$$ −15.8360 −0.631925
$$629$$ 12.2526 0.488542
$$630$$ 0 0
$$631$$ 32.1115 1.27834 0.639169 0.769066i $$-0.279278\pi$$
0.639169 + 0.769066i $$0.279278\pi$$
$$632$$ −0.892781 −0.0355129
$$633$$ 13.2562 0.526888
$$634$$ 15.3755 0.610638
$$635$$ −0.861191 −0.0341753
$$636$$ 50.3930 1.99821
$$637$$ 0 0
$$638$$ −4.68769 −0.185588
$$639$$ 1.03625 0.0409935
$$640$$ 0.497222 0.0196544
$$641$$ 33.0248 1.30440 0.652200 0.758047i $$-0.273846\pi$$
0.652200 + 0.758047i $$0.273846\pi$$
$$642$$ −48.0263 −1.89545
$$643$$ 15.7942 0.622863 0.311432 0.950269i $$-0.399192\pi$$
0.311432 + 0.950269i $$0.399192\pi$$
$$644$$ 0 0
$$645$$ −15.0362 −0.592051
$$646$$ −31.3520 −1.23353
$$647$$ −4.64072 −0.182445 −0.0912227 0.995831i $$-0.529078\pi$$
−0.0912227 + 0.995831i $$0.529078\pi$$
$$648$$ 0.636493 0.0250038
$$649$$ 0.513173 0.0201438
$$650$$ 8.39497 0.329278
$$651$$ 0 0
$$652$$ 3.43757 0.134626
$$653$$ 26.8285 1.04988 0.524941 0.851139i $$-0.324087\pi$$
0.524941 + 0.851139i $$0.324087\pi$$
$$654$$ 48.4912 1.89616
$$655$$ −17.0453 −0.666016
$$656$$ −19.3446 −0.755280
$$657$$ −0.327429 −0.0127742
$$658$$ 0 0
$$659$$ −42.9889 −1.67461 −0.837306 0.546735i $$-0.815871\pi$$
−0.837306 + 0.546735i $$0.815871\pi$$
$$660$$ 2.32163 0.0903695
$$661$$ −29.4698 −1.14624 −0.573122 0.819470i $$-0.694268\pi$$
−0.573122 + 0.819470i $$0.694268\pi$$
$$662$$ −22.7830 −0.885488
$$663$$ 4.13778 0.160698
$$664$$ −0.243015 −0.00943082
$$665$$ 0 0
$$666$$ −0.986346 −0.0382201
$$667$$ 12.2470 0.474205
$$668$$ −44.4806 −1.72101
$$669$$ 31.0207 1.19933
$$670$$ −17.0851 −0.660056
$$671$$ −8.34752 −0.322252
$$672$$ 0 0
$$673$$ −20.1702 −0.777504 −0.388752 0.921342i $$-0.627094\pi$$
−0.388752 + 0.921342i $$0.627094\pi$$
$$674$$ −3.49782 −0.134731
$$675$$ 21.3637 0.822287
$$676$$ 2.03417 0.0782373
$$677$$ −6.20481 −0.238470 −0.119235 0.992866i $$-0.538044\pi$$
−0.119235 + 0.992866i $$0.538044\pi$$
$$678$$ 11.5108 0.442071
$$679$$ 0 0
$$680$$ −0.146221 −0.00560733
$$681$$ −9.37409 −0.359216
$$682$$ −2.25968 −0.0865276
$$683$$ −1.76952 −0.0677087 −0.0338543 0.999427i $$-0.510778\pi$$
−0.0338543 + 0.999427i $$0.510778\pi$$
$$684$$ 1.27262 0.0486600
$$685$$ 5.59899 0.213926
$$686$$ 0 0
$$687$$ −14.9854 −0.571729
$$688$$ 37.0946 1.41422
$$689$$ 14.0833 0.536530
$$690$$ −12.0290 −0.457937
$$691$$ 44.9317 1.70928 0.854641 0.519219i $$-0.173777\pi$$
0.854641 + 0.519219i $$0.173777\pi$$
$$692$$ 11.8820 0.451688
$$693$$ 0 0
$$694$$ −42.2800 −1.60493
$$695$$ −3.62289 −0.137424
$$696$$ −0.393331 −0.0149092
$$697$$ 11.5771 0.438515
$$698$$ 16.7832 0.635255
$$699$$ 8.36204 0.316281
$$700$$ 0 0
$$701$$ 38.5707 1.45679 0.728397 0.685156i $$-0.240266\pi$$
0.728397 + 0.685156i $$0.240266\pi$$
$$702$$ 10.2662 0.387474
$$703$$ 34.5653 1.30366
$$704$$ −5.92501 −0.223307
$$705$$ −13.2553 −0.499225
$$706$$ 17.1398 0.645066
$$707$$ 0 0
$$708$$ 2.56330 0.0963346
$$709$$ 8.77731 0.329639 0.164819 0.986324i $$-0.447296\pi$$
0.164819 + 0.986324i $$0.447296\pi$$
$$710$$ −19.9952 −0.750405
$$711$$ 1.22639 0.0459932
$$712$$ −0.827023 −0.0309940
$$713$$ 5.90360 0.221091
$$714$$ 0 0
$$715$$ 0.648824 0.0242646
$$716$$ 5.15515 0.192657
$$717$$ 26.0893 0.974323
$$718$$ 32.4804 1.21216
$$719$$ 4.20437 0.156796 0.0783982 0.996922i $$-0.475019\pi$$
0.0783982 + 0.996922i $$0.475019\pi$$
$$720$$ −0.335625 −0.0125080
$$721$$ 0 0
$$722$$ −50.2840 −1.87138
$$723$$ −10.7677 −0.400456
$$724$$ −21.8161 −0.810789
$$725$$ −13.6173 −0.505735
$$726$$ 37.0511 1.37509
$$727$$ −28.9856 −1.07502 −0.537509 0.843258i $$-0.680634\pi$$
−0.537509 + 0.843258i $$0.680634\pi$$
$$728$$ 0 0
$$729$$ 26.0990 0.966630
$$730$$ 6.31796 0.233838
$$731$$ −22.1999 −0.821094
$$732$$ −41.6958 −1.54112
$$733$$ −24.0345 −0.887733 −0.443867 0.896093i $$-0.646394\pi$$
−0.443867 + 0.896093i $$0.646394\pi$$
$$734$$ −56.5480 −2.08722
$$735$$ 0 0
$$736$$ 30.1918 1.11288
$$737$$ 6.72788 0.247825
$$738$$ −0.931971 −0.0343063
$$739$$ 11.8055 0.434273 0.217136 0.976141i $$-0.430328\pi$$
0.217136 + 0.976141i $$0.430328\pi$$
$$740$$ 9.59670 0.352782
$$741$$ 11.6729 0.428816
$$742$$ 0 0
$$743$$ 47.2786 1.73448 0.867241 0.497888i $$-0.165891\pi$$
0.867241 + 0.497888i $$0.165891\pi$$
$$744$$ −0.189603 −0.00695120
$$745$$ −19.0900 −0.699402
$$746$$ 57.2503 2.09608
$$747$$ 0.333824 0.0122140
$$748$$ 3.42773 0.125330
$$749$$ 0 0
$$750$$ 29.3750 1.07262
$$751$$ 5.47700 0.199859 0.0999294 0.994995i $$-0.468138\pi$$
0.0999294 + 0.994995i $$0.468138\pi$$
$$752$$ 32.7012 1.19249
$$753$$ −24.5755 −0.895581
$$754$$ −6.54376 −0.238310
$$755$$ −14.2410 −0.518285
$$756$$ 0 0
$$757$$ 10.7453 0.390546 0.195273 0.980749i $$-0.437441\pi$$
0.195273 + 0.980749i $$0.437441\pi$$
$$758$$ 14.5872 0.529830
$$759$$ 4.73686 0.171937
$$760$$ −0.412500 −0.0149630
$$761$$ −33.0399 −1.19770 −0.598848 0.800863i $$-0.704375\pi$$
−0.598848 + 0.800863i $$0.704375\pi$$
$$762$$ 3.35940 0.121698
$$763$$ 0 0
$$764$$ −3.41457 −0.123535
$$765$$ 0.200860 0.00726212
$$766$$ −25.9753 −0.938527
$$767$$ 0.716361 0.0258663
$$768$$ 27.1587 0.980004
$$769$$ 2.98332 0.107581 0.0537907 0.998552i $$-0.482870\pi$$
0.0537907 + 0.998552i $$0.482870\pi$$
$$770$$ 0 0
$$771$$ 30.3702 1.09376
$$772$$ −13.1167 −0.472079
$$773$$ −21.9085 −0.787995 −0.393998 0.919111i $$-0.628908\pi$$
−0.393998 + 0.919111i $$0.628908\pi$$
$$774$$ 1.78712 0.0642367
$$775$$ −6.56417 −0.235792
$$776$$ −0.510464 −0.0183246
$$777$$ 0 0
$$778$$ 42.4516 1.52196
$$779$$ 32.6598 1.17016
$$780$$ 3.24087 0.116042
$$781$$ 7.87381 0.281747
$$782$$ −17.7600 −0.635097
$$783$$ −16.6527 −0.595118
$$784$$ 0 0
$$785$$ −7.05104 −0.251662
$$786$$ 66.4917 2.37168
$$787$$ −13.3632 −0.476347 −0.238174 0.971223i $$-0.576549\pi$$
−0.238174 + 0.971223i $$0.576549\pi$$
$$788$$ 3.80900 0.135690
$$789$$ −4.58537 −0.163244
$$790$$ −23.6640 −0.841927
$$791$$ 0 0
$$792$$ −0.00463525 −0.000164706 0
$$793$$ −11.6527 −0.413798
$$794$$ −38.5784 −1.36910
$$795$$ 22.4377 0.795782
$$796$$ −23.1827 −0.821691
$$797$$ 32.5388 1.15258 0.576292 0.817244i $$-0.304499\pi$$
0.576292 + 0.817244i $$0.304499\pi$$
$$798$$ 0 0
$$799$$ −19.5706 −0.692357
$$800$$ −33.5701 −1.18688
$$801$$ 1.13606 0.0401407
$$802$$ −33.4805 −1.18224
$$803$$ −2.48792 −0.0877969
$$804$$ 33.6057 1.18518
$$805$$ 0 0
$$806$$ −3.15439 −0.111109
$$807$$ −25.4879 −0.897217
$$808$$ 0.0822938 0.00289509
$$809$$ 7.68827 0.270305 0.135153 0.990825i $$-0.456848\pi$$
0.135153 + 0.990825i $$0.456848\pi$$
$$810$$ 16.8708 0.592781
$$811$$ −48.3178 −1.69667 −0.848334 0.529461i $$-0.822394\pi$$
−0.848334 + 0.529461i $$0.822394\pi$$
$$812$$ 0 0
$$813$$ −15.1911 −0.532773
$$814$$ −7.49461 −0.262686
$$815$$ 1.53059 0.0536142
$$816$$ −16.2635 −0.569336
$$817$$ −62.6275 −2.19106
$$818$$ −24.8019 −0.867178
$$819$$ 0 0
$$820$$ 9.06766 0.316656
$$821$$ −3.73442 −0.130332 −0.0651661 0.997874i $$-0.520758\pi$$
−0.0651661 + 0.997874i $$0.520758\pi$$
$$822$$ −21.8409 −0.761790
$$823$$ 14.2318 0.496089 0.248045 0.968749i $$-0.420212\pi$$
0.248045 + 0.968749i $$0.420212\pi$$
$$824$$ 0.989150 0.0344587
$$825$$ −5.26688 −0.183369
$$826$$ 0 0
$$827$$ 48.3016 1.67961 0.839805 0.542888i $$-0.182669\pi$$
0.839805 + 0.542888i $$0.182669\pi$$
$$828$$ 0.720906 0.0250532
$$829$$ 11.5101 0.399763 0.199882 0.979820i $$-0.435944\pi$$
0.199882 + 0.979820i $$0.435944\pi$$
$$830$$ −6.44135 −0.223582
$$831$$ 21.5079 0.746102
$$832$$ −8.27099 −0.286745
$$833$$ 0 0
$$834$$ 14.1324 0.489366
$$835$$ −19.8052 −0.685386
$$836$$ 9.66985 0.334439
$$837$$ −8.02734 −0.277465
$$838$$ −8.75583 −0.302465
$$839$$ 13.1103 0.452616 0.226308 0.974056i $$-0.427334\pi$$
0.226308 + 0.974056i $$0.427334\pi$$
$$840$$ 0 0
$$841$$ −18.3855 −0.633982
$$842$$ 20.0852 0.692183
$$843$$ −42.5378 −1.46508
$$844$$ 15.3295 0.527663
$$845$$ 0.905722 0.0311578
$$846$$ 1.57545 0.0541652
$$847$$ 0 0
$$848$$ −55.3541 −1.90087
$$849$$ −54.1232 −1.85750
$$850$$ 19.7472 0.677325
$$851$$ 19.5803 0.671203
$$852$$ 39.3297 1.34741
$$853$$ 8.80346 0.301425 0.150712 0.988578i $$-0.451843\pi$$
0.150712 + 0.988578i $$0.451843\pi$$
$$854$$ 0 0
$$855$$ 0.566640 0.0193787
$$856$$ −0.932934 −0.0318870
$$857$$ 16.9651 0.579516 0.289758 0.957100i $$-0.406425\pi$$
0.289758 + 0.957100i $$0.406425\pi$$
$$858$$ −2.53098 −0.0864063
$$859$$ 14.5410 0.496132 0.248066 0.968743i $$-0.420205\pi$$
0.248066 + 0.968743i $$0.420205\pi$$
$$860$$ −17.3879 −0.592922
$$861$$ 0 0
$$862$$ 46.9243 1.59825
$$863$$ 39.0444 1.32909 0.664544 0.747249i $$-0.268626\pi$$
0.664544 + 0.747249i $$0.268626\pi$$
$$864$$ −41.0529 −1.39665
$$865$$ 5.29052 0.179883
$$866$$ 5.44882 0.185159
$$867$$ −20.1708 −0.685036
$$868$$ 0 0
$$869$$ 9.31854 0.316110
$$870$$ −10.4256 −0.353461
$$871$$ 9.39174 0.318227
$$872$$ 0.941966 0.0318990
$$873$$ 0.701211 0.0237324
$$874$$ −50.1022 −1.69473
$$875$$ 0 0
$$876$$ −12.4272 −0.419875
$$877$$ 32.5941 1.10062 0.550312 0.834959i $$-0.314509\pi$$
0.550312 + 0.834959i $$0.314509\pi$$
$$878$$ 17.7457 0.598888
$$879$$ −55.9899 −1.88849
$$880$$ −2.55020 −0.0859671
$$881$$ −43.4141 −1.46266 −0.731330 0.682024i $$-0.761100\pi$$
−0.731330 + 0.682024i $$0.761100\pi$$
$$882$$ 0 0
$$883$$ 28.2902 0.952040 0.476020 0.879434i $$-0.342079\pi$$
0.476020 + 0.879434i $$0.342079\pi$$
$$884$$ 4.78492 0.160934
$$885$$ 1.14132 0.0383650
$$886$$ 5.83592 0.196062
$$887$$ −50.2650 −1.68773 −0.843866 0.536554i $$-0.819726\pi$$
−0.843866 + 0.536554i $$0.819726\pi$$
$$888$$ −0.628852 −0.0211029
$$889$$ 0 0
$$890$$ −21.9210 −0.734794
$$891$$ −6.64349 −0.222565
$$892$$ 35.8724 1.20110
$$893$$ −55.2099 −1.84753
$$894$$ 74.4676 2.49057
$$895$$ 2.29535 0.0767250
$$896$$ 0 0
$$897$$ 6.61239 0.220781
$$898$$ 30.6856 1.02399
$$899$$ 5.11668 0.170651
$$900$$ −0.801570 −0.0267190
$$901$$ 33.1277 1.10364
$$902$$ −7.08145 −0.235786
$$903$$ 0 0
$$904$$ 0.223604 0.00743695
$$905$$ −9.71369 −0.322894
$$906$$ 55.5526 1.84561
$$907$$ 26.8277 0.890798 0.445399 0.895332i $$-0.353062\pi$$
0.445399 + 0.895332i $$0.353062\pi$$
$$908$$ −10.8402 −0.359744
$$909$$ −0.113045 −0.00374946
$$910$$ 0 0
$$911$$ −22.3560 −0.740687 −0.370344 0.928895i $$-0.620760\pi$$
−0.370344 + 0.928895i $$0.620760\pi$$
$$912$$ −45.8804 −1.51925
$$913$$ 2.53651 0.0839463
$$914$$ −47.5217 −1.57188
$$915$$ −18.5652 −0.613747
$$916$$ −17.3291 −0.572569
$$917$$ 0 0
$$918$$ 24.1489 0.797034
$$919$$ −8.62244 −0.284428 −0.142214 0.989836i $$-0.545422\pi$$
−0.142214 + 0.989836i $$0.545422\pi$$
$$920$$ −0.233670 −0.00770386
$$921$$ 50.6404 1.66866
$$922$$ 53.4609 1.76064
$$923$$ 10.9914 0.361786
$$924$$ 0 0
$$925$$ −21.7712 −0.715832
$$926$$ 2.89729 0.0952109
$$927$$ −1.35877 −0.0446278
$$928$$ 26.1674 0.858987
$$929$$ 41.3861 1.35783 0.678916 0.734216i $$-0.262450\pi$$
0.678916 + 0.734216i $$0.262450\pi$$
$$930$$ −5.02562 −0.164796
$$931$$ 0 0
$$932$$ 9.66985 0.316747
$$933$$ 9.70757 0.317812
$$934$$ 16.8507 0.551372
$$935$$ 1.52621 0.0499124
$$936$$ −0.00647055 −0.000211496 0
$$937$$ 21.3818 0.698514 0.349257 0.937027i $$-0.386434\pi$$
0.349257 + 0.937027i $$0.386434\pi$$
$$938$$ 0 0
$$939$$ −8.52788 −0.278297
$$940$$ −15.3285 −0.499959
$$941$$ −53.1480 −1.73258 −0.866288 0.499545i $$-0.833501\pi$$
−0.866288 + 0.499545i $$0.833501\pi$$
$$942$$ 27.5052 0.896168
$$943$$ 18.5009 0.602471
$$944$$ −2.81565 −0.0916416
$$945$$ 0 0
$$946$$ 13.5792 0.441497
$$947$$ −8.86936 −0.288216 −0.144108 0.989562i $$-0.546031\pi$$
−0.144108 + 0.989562i $$0.546031\pi$$
$$948$$ 46.5461 1.51175
$$949$$ −3.47300 −0.112738
$$950$$ 55.7082 1.80741
$$951$$ −13.4658 −0.436658
$$952$$ 0 0
$$953$$ −39.8167 −1.28979 −0.644894 0.764272i $$-0.723099\pi$$
−0.644894 + 0.764272i $$0.723099\pi$$
$$954$$ −2.66681 −0.0863413
$$955$$ −1.52035 −0.0491973
$$956$$ 30.1696 0.975756
$$957$$ 4.10546 0.132711
$$958$$ −25.3319 −0.818435
$$959$$ 0 0
$$960$$ −13.1775 −0.425301
$$961$$ −28.5335 −0.920437
$$962$$ −10.4621 −0.337310
$$963$$ 1.28155 0.0412973
$$964$$ −12.4518 −0.401045
$$965$$ −5.84024 −0.188004
$$966$$ 0 0
$$967$$ −22.1611 −0.712652 −0.356326 0.934362i $$-0.615971\pi$$
−0.356326 + 0.934362i $$0.615971\pi$$
$$968$$ 0.719735 0.0231332
$$969$$ 27.4579 0.882075
$$970$$ −13.5303 −0.434433
$$971$$ 36.0423 1.15665 0.578327 0.815805i $$-0.303706\pi$$
0.578327 + 0.815805i $$0.303706\pi$$
$$972$$ −1.99229 −0.0639027
$$973$$ 0 0
$$974$$ −43.3199 −1.38806
$$975$$ −7.35227 −0.235461
$$976$$ 45.8007 1.46604
$$977$$ 32.9416 1.05389 0.526947 0.849898i $$-0.323336\pi$$
0.526947 + 0.849898i $$0.323336\pi$$
$$978$$ −5.97064 −0.190920
$$979$$ 8.63219 0.275886
$$980$$ 0 0
$$981$$ −1.29395 −0.0413128
$$982$$ −78.8038 −2.51473
$$983$$ −4.19945 −0.133942 −0.0669709 0.997755i $$-0.521333\pi$$
−0.0669709 + 0.997755i $$0.521333\pi$$
$$984$$ −0.594185 −0.0189419
$$985$$ 1.69597 0.0540382
$$986$$ −15.3927 −0.490203
$$987$$ 0 0
$$988$$ 13.4986 0.429447
$$989$$ −35.4767 −1.12809
$$990$$ −0.122862 −0.00390480
$$991$$ −13.4139 −0.426105 −0.213053 0.977041i $$-0.568341\pi$$
−0.213053 + 0.977041i $$0.568341\pi$$
$$992$$ 12.6139 0.400491
$$993$$ 19.9533 0.633198
$$994$$ 0 0
$$995$$ −10.3222 −0.327236
$$996$$ 12.6699 0.401460
$$997$$ −47.8868 −1.51659 −0.758295 0.651911i $$-0.773967\pi$$
−0.758295 + 0.651911i $$0.773967\pi$$
$$998$$ −18.4144 −0.582899
$$999$$ −26.6240 −0.842347
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 637.2.a.l.1.1 5
3.2 odd 2 5733.2.a.bl.1.5 5
7.2 even 3 91.2.e.c.53.5 10
7.3 odd 6 637.2.e.m.79.5 10
7.4 even 3 91.2.e.c.79.5 yes 10
7.5 odd 6 637.2.e.m.508.5 10
7.6 odd 2 637.2.a.k.1.1 5
13.12 even 2 8281.2.a.bw.1.5 5
21.2 odd 6 819.2.j.h.235.1 10
21.11 odd 6 819.2.j.h.352.1 10
21.20 even 2 5733.2.a.bm.1.5 5
28.11 odd 6 1456.2.r.p.625.4 10
28.23 odd 6 1456.2.r.p.417.4 10
91.25 even 6 1183.2.e.f.170.1 10
91.51 even 6 1183.2.e.f.508.1 10
91.90 odd 2 8281.2.a.bx.1.5 5

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.5 10 7.2 even 3
91.2.e.c.79.5 yes 10 7.4 even 3
637.2.a.k.1.1 5 7.6 odd 2
637.2.a.l.1.1 5 1.1 even 1 trivial
637.2.e.m.79.5 10 7.3 odd 6
637.2.e.m.508.5 10 7.5 odd 6
819.2.j.h.235.1 10 21.2 odd 6
819.2.j.h.352.1 10 21.11 odd 6
1183.2.e.f.170.1 10 91.25 even 6
1183.2.e.f.508.1 10 91.51 even 6
1456.2.r.p.417.4 10 28.23 odd 6
1456.2.r.p.625.4 10 28.11 odd 6
5733.2.a.bl.1.5 5 3.2 odd 2
5733.2.a.bm.1.5 5 21.20 even 2
8281.2.a.bw.1.5 5 13.12 even 2
8281.2.a.bx.1.5 5 91.90 odd 2