# Properties

 Label 637.2.a.k.1.3 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 1.3 Root $$-0.265608$$ of defining polynomial Character $$\chi$$ $$=$$ 637.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.26561 q^{2} +2.62728 q^{3} -0.398235 q^{4} +2.90260 q^{5} +3.32511 q^{6} -3.03523 q^{8} +3.90260 q^{9} +O(q^{10})$$ $$q+1.26561 q^{2} +2.62728 q^{3} -0.398235 q^{4} +2.90260 q^{5} +3.32511 q^{6} -3.03523 q^{8} +3.90260 q^{9} +3.67356 q^{10} +2.03656 q^{11} -1.04628 q^{12} -1.00000 q^{13} +7.62594 q^{15} -3.04494 q^{16} -3.99866 q^{17} +4.93916 q^{18} -6.96210 q^{19} -1.15592 q^{20} +2.57749 q^{22} -0.627280 q^{23} -7.97439 q^{24} +3.42509 q^{25} -1.26561 q^{26} +2.37138 q^{27} +1.09606 q^{29} +9.65146 q^{30} +10.4325 q^{31} +2.21675 q^{32} +5.35062 q^{33} -5.06074 q^{34} -1.55415 q^{36} -3.08537 q^{37} -8.81129 q^{38} -2.62728 q^{39} -8.81005 q^{40} +0.521150 q^{41} +0.329024 q^{43} -0.811031 q^{44} +11.3277 q^{45} -0.793891 q^{46} -10.5457 q^{47} -7.99991 q^{48} +4.33482 q^{50} -10.5056 q^{51} +0.398235 q^{52} +7.11900 q^{53} +3.00124 q^{54} +5.91133 q^{55} -18.2914 q^{57} +1.38719 q^{58} -2.03656 q^{59} -3.03692 q^{60} -2.40081 q^{61} +13.2034 q^{62} +8.89542 q^{64} -2.90260 q^{65} +6.77179 q^{66} +14.6942 q^{67} +1.59241 q^{68} -1.64804 q^{69} +3.60141 q^{71} -11.8453 q^{72} -2.97573 q^{73} -3.90487 q^{74} +8.99866 q^{75} +2.77255 q^{76} -3.32511 q^{78} -8.76150 q^{79} -8.83824 q^{80} -5.47751 q^{81} +0.659572 q^{82} -12.8039 q^{83} -11.6065 q^{85} +0.416416 q^{86} +2.87966 q^{87} -6.18143 q^{88} +2.68098 q^{89} +14.3364 q^{90} +0.249805 q^{92} +27.4090 q^{93} -13.3467 q^{94} -20.2082 q^{95} +5.82403 q^{96} +2.32902 q^{97} +7.94789 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.26561 0.894920 0.447460 0.894304i $$-0.352329\pi$$
0.447460 + 0.894304i $$0.352329\pi$$
$$3$$ 2.62728 1.51686 0.758430 0.651754i $$-0.225967\pi$$
0.758430 + 0.651754i $$0.225967\pi$$
$$4$$ −0.398235 −0.199118
$$5$$ 2.90260 1.29808 0.649041 0.760753i $$-0.275170\pi$$
0.649041 + 0.760753i $$0.275170\pi$$
$$6$$ 3.32511 1.35747
$$7$$ 0 0
$$8$$ −3.03523 −1.07311
$$9$$ 3.90260 1.30087
$$10$$ 3.67356 1.16168
$$11$$ 2.03656 0.614047 0.307024 0.951702i $$-0.400667\pi$$
0.307024 + 0.951702i $$0.400667\pi$$
$$12$$ −1.04628 −0.302034
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 7.62594 1.96901
$$16$$ −3.04494 −0.761235
$$17$$ −3.99866 −0.969818 −0.484909 0.874565i $$-0.661147\pi$$
−0.484909 + 0.874565i $$0.661147\pi$$
$$18$$ 4.93916 1.16417
$$19$$ −6.96210 −1.59722 −0.798608 0.601852i $$-0.794430\pi$$
−0.798608 + 0.601852i $$0.794430\pi$$
$$20$$ −1.15592 −0.258471
$$21$$ 0 0
$$22$$ 2.57749 0.549523
$$23$$ −0.627280 −0.130797 −0.0653985 0.997859i $$-0.520832\pi$$
−0.0653985 + 0.997859i $$0.520832\pi$$
$$24$$ −7.97439 −1.62777
$$25$$ 3.42509 0.685017
$$26$$ −1.26561 −0.248206
$$27$$ 2.37138 0.456373
$$28$$ 0 0
$$29$$ 1.09606 0.203534 0.101767 0.994808i $$-0.467550\pi$$
0.101767 + 0.994808i $$0.467550\pi$$
$$30$$ 9.65146 1.76211
$$31$$ 10.4325 1.87373 0.936864 0.349693i $$-0.113714\pi$$
0.936864 + 0.349693i $$0.113714\pi$$
$$32$$ 2.21675 0.391870
$$33$$ 5.35062 0.931424
$$34$$ −5.06074 −0.867910
$$35$$ 0 0
$$36$$ −1.55415 −0.259025
$$37$$ −3.08537 −0.507232 −0.253616 0.967305i $$-0.581620\pi$$
−0.253616 + 0.967305i $$0.581620\pi$$
$$38$$ −8.81129 −1.42938
$$39$$ −2.62728 −0.420701
$$40$$ −8.81005 −1.39299
$$41$$ 0.521150 0.0813900 0.0406950 0.999172i $$-0.487043\pi$$
0.0406950 + 0.999172i $$0.487043\pi$$
$$42$$ 0 0
$$43$$ 0.329024 0.0501757 0.0250879 0.999685i $$-0.492013\pi$$
0.0250879 + 0.999685i $$0.492013\pi$$
$$44$$ −0.811031 −0.122268
$$45$$ 11.3277 1.68863
$$46$$ −0.793891 −0.117053
$$47$$ −10.5457 −1.53825 −0.769123 0.639101i $$-0.779307\pi$$
−0.769123 + 0.639101i $$0.779307\pi$$
$$48$$ −7.99991 −1.15469
$$49$$ 0 0
$$50$$ 4.33482 0.613036
$$51$$ −10.5056 −1.47108
$$52$$ 0.398235 0.0552253
$$53$$ 7.11900 0.977870 0.488935 0.872320i $$-0.337386\pi$$
0.488935 + 0.872320i $$0.337386\pi$$
$$54$$ 3.00124 0.408417
$$55$$ 5.91133 0.797084
$$56$$ 0 0
$$57$$ −18.2914 −2.42275
$$58$$ 1.38719 0.182147
$$59$$ −2.03656 −0.265138 −0.132569 0.991174i $$-0.542323\pi$$
−0.132569 + 0.991174i $$0.542323\pi$$
$$60$$ −3.03692 −0.392065
$$61$$ −2.40081 −0.307393 −0.153696 0.988118i $$-0.549118\pi$$
−0.153696 + 0.988118i $$0.549118\pi$$
$$62$$ 13.2034 1.67684
$$63$$ 0 0
$$64$$ 8.89542 1.11193
$$65$$ −2.90260 −0.360023
$$66$$ 6.77179 0.833550
$$67$$ 14.6942 1.79518 0.897589 0.440832i $$-0.145317\pi$$
0.897589 + 0.440832i $$0.145317\pi$$
$$68$$ 1.59241 0.193108
$$69$$ −1.64804 −0.198401
$$70$$ 0 0
$$71$$ 3.60141 0.427409 0.213704 0.976898i $$-0.431447\pi$$
0.213704 + 0.976898i $$0.431447\pi$$
$$72$$ −11.8453 −1.39598
$$73$$ −2.97573 −0.348283 −0.174141 0.984721i $$-0.555715\pi$$
−0.174141 + 0.984721i $$0.555715\pi$$
$$74$$ −3.90487 −0.453932
$$75$$ 8.99866 1.03908
$$76$$ 2.77255 0.318034
$$77$$ 0 0
$$78$$ −3.32511 −0.376494
$$79$$ −8.76150 −0.985746 −0.492873 0.870101i $$-0.664053\pi$$
−0.492873 + 0.870101i $$0.664053\pi$$
$$80$$ −8.83824 −0.988145
$$81$$ −5.47751 −0.608613
$$82$$ 0.659572 0.0728376
$$83$$ −12.8039 −1.40541 −0.702703 0.711483i $$-0.748024\pi$$
−0.702703 + 0.711483i $$0.748024\pi$$
$$84$$ 0 0
$$85$$ −11.6065 −1.25890
$$86$$ 0.416416 0.0449033
$$87$$ 2.87966 0.308732
$$88$$ −6.18143 −0.658943
$$89$$ 2.68098 0.284184 0.142092 0.989853i $$-0.454617\pi$$
0.142092 + 0.989853i $$0.454617\pi$$
$$90$$ 14.3364 1.51119
$$91$$ 0 0
$$92$$ 0.249805 0.0260440
$$93$$ 27.4090 2.84219
$$94$$ −13.3467 −1.37661
$$95$$ −20.2082 −2.07332
$$96$$ 5.82403 0.594413
$$97$$ 2.32902 0.236477 0.118238 0.992985i $$-0.462275\pi$$
0.118238 + 0.992985i $$0.462275\pi$$
$$98$$ 0 0
$$99$$ 7.94789 0.798793
$$100$$ −1.36399 −0.136399
$$101$$ −1.45324 −0.144603 −0.0723014 0.997383i $$-0.523034\pi$$
−0.0723014 + 0.997383i $$0.523034\pi$$
$$102$$ −13.2960 −1.31650
$$103$$ 11.6353 1.14646 0.573230 0.819394i $$-0.305690\pi$$
0.573230 + 0.819394i $$0.305690\pi$$
$$104$$ 3.03523 0.297628
$$105$$ 0 0
$$106$$ 9.00987 0.875115
$$107$$ 19.6259 1.89731 0.948656 0.316310i $$-0.102444\pi$$
0.948656 + 0.316310i $$0.102444\pi$$
$$108$$ −0.944368 −0.0908719
$$109$$ −1.10676 −0.106008 −0.0530040 0.998594i $$-0.516880\pi$$
−0.0530040 + 0.998594i $$0.516880\pi$$
$$110$$ 7.48143 0.713326
$$111$$ −8.10613 −0.769400
$$112$$ 0 0
$$113$$ −1.09606 −0.103109 −0.0515545 0.998670i $$-0.516418\pi$$
−0.0515545 + 0.998670i $$0.516418\pi$$
$$114$$ −23.1497 −2.16817
$$115$$ −1.82074 −0.169785
$$116$$ −0.436491 −0.0405272
$$117$$ −3.90260 −0.360796
$$118$$ −2.57749 −0.237277
$$119$$ 0 0
$$120$$ −23.1465 −2.11297
$$121$$ −6.85241 −0.622946
$$122$$ −3.03849 −0.275092
$$123$$ 1.36921 0.123457
$$124$$ −4.15458 −0.373092
$$125$$ −4.57134 −0.408873
$$126$$ 0 0
$$127$$ 5.18143 0.459778 0.229889 0.973217i $$-0.426164\pi$$
0.229889 + 0.973217i $$0.426164\pi$$
$$128$$ 6.82461 0.603216
$$129$$ 0.864439 0.0761096
$$130$$ −3.67356 −0.322192
$$131$$ −10.5667 −0.923217 −0.461609 0.887084i $$-0.652728\pi$$
−0.461609 + 0.887084i $$0.652728\pi$$
$$132$$ −2.13081 −0.185463
$$133$$ 0 0
$$134$$ 18.5971 1.60654
$$135$$ 6.88318 0.592410
$$136$$ 12.1368 1.04073
$$137$$ −5.87177 −0.501659 −0.250830 0.968031i $$-0.580703\pi$$
−0.250830 + 0.968031i $$0.580703\pi$$
$$138$$ −2.08577 −0.177553
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −27.7065 −2.33331
$$142$$ 4.55797 0.382497
$$143$$ −2.03656 −0.170306
$$144$$ −11.8832 −0.990265
$$145$$ 3.18143 0.264204
$$146$$ −3.76611 −0.311685
$$147$$ 0 0
$$148$$ 1.22870 0.100999
$$149$$ −10.1054 −0.827868 −0.413934 0.910307i $$-0.635846\pi$$
−0.413934 + 0.910307i $$0.635846\pi$$
$$150$$ 11.3888 0.929890
$$151$$ −0.187726 −0.0152769 −0.00763847 0.999971i $$-0.502431\pi$$
−0.00763847 + 0.999971i $$0.502431\pi$$
$$152$$ 21.1316 1.71400
$$153$$ −15.6052 −1.26160
$$154$$ 0 0
$$155$$ 30.2813 2.43225
$$156$$ 1.04628 0.0837691
$$157$$ 12.0718 0.963434 0.481717 0.876327i $$-0.340013\pi$$
0.481717 + 0.876327i $$0.340013\pi$$
$$158$$ −11.0886 −0.882164
$$159$$ 18.7036 1.48329
$$160$$ 6.43435 0.508680
$$161$$ 0 0
$$162$$ −6.93239 −0.544660
$$163$$ 14.9136 1.16812 0.584060 0.811711i $$-0.301463\pi$$
0.584060 + 0.811711i $$0.301463\pi$$
$$164$$ −0.207540 −0.0162062
$$165$$ 15.5307 1.20906
$$166$$ −16.2047 −1.25773
$$167$$ −5.05664 −0.391294 −0.195647 0.980674i $$-0.562681\pi$$
−0.195647 + 0.980674i $$0.562681\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −14.6893 −1.12662
$$171$$ −27.1703 −2.07776
$$172$$ −0.131029 −0.00999087
$$173$$ 0.595615 0.0452837 0.0226419 0.999744i $$-0.492792\pi$$
0.0226419 + 0.999744i $$0.492792\pi$$
$$174$$ 3.64453 0.276291
$$175$$ 0 0
$$176$$ −6.20121 −0.467434
$$177$$ −5.35062 −0.402177
$$178$$ 3.39308 0.254322
$$179$$ 8.07664 0.603676 0.301838 0.953359i $$-0.402400\pi$$
0.301838 + 0.953359i $$0.402400\pi$$
$$180$$ −4.51108 −0.336236
$$181$$ −1.89324 −0.140724 −0.0703618 0.997522i $$-0.522415\pi$$
−0.0703618 + 0.997522i $$0.522415\pi$$
$$182$$ 0 0
$$183$$ −6.30761 −0.466272
$$184$$ 1.90394 0.140360
$$185$$ −8.95559 −0.658428
$$186$$ 34.6891 2.54353
$$187$$ −8.14353 −0.595514
$$188$$ 4.19966 0.306292
$$189$$ 0 0
$$190$$ −25.5757 −1.85545
$$191$$ 3.70174 0.267849 0.133924 0.990992i $$-0.457242\pi$$
0.133924 + 0.990992i $$0.457242\pi$$
$$192$$ 23.3708 1.68664
$$193$$ 13.5875 0.978047 0.489024 0.872271i $$-0.337353\pi$$
0.489024 + 0.872271i $$0.337353\pi$$
$$194$$ 2.94763 0.211628
$$195$$ −7.62594 −0.546105
$$196$$ 0 0
$$197$$ −9.70258 −0.691280 −0.345640 0.938367i $$-0.612338\pi$$
−0.345640 + 0.938367i $$0.612338\pi$$
$$198$$ 10.0589 0.714856
$$199$$ −26.2720 −1.86237 −0.931185 0.364547i $$-0.881224\pi$$
−0.931185 + 0.364547i $$0.881224\pi$$
$$200$$ −10.3959 −0.735102
$$201$$ 38.6057 2.72304
$$202$$ −1.83923 −0.129408
$$203$$ 0 0
$$204$$ 4.18370 0.292918
$$205$$ 1.51269 0.105651
$$206$$ 14.7257 1.02599
$$207$$ −2.44802 −0.170149
$$208$$ 3.04494 0.211128
$$209$$ −14.1788 −0.980765
$$210$$ 0 0
$$211$$ 10.0338 0.690758 0.345379 0.938463i $$-0.387750\pi$$
0.345379 + 0.938463i $$0.387750\pi$$
$$212$$ −2.83504 −0.194711
$$213$$ 9.46191 0.648320
$$214$$ 24.8388 1.69794
$$215$$ 0.955026 0.0651322
$$216$$ −7.19769 −0.489740
$$217$$ 0 0
$$218$$ −1.40072 −0.0948688
$$219$$ −7.81807 −0.528296
$$220$$ −2.35410 −0.158713
$$221$$ 3.99866 0.268979
$$222$$ −10.2592 −0.688552
$$223$$ −17.4961 −1.17163 −0.585813 0.810446i $$-0.699225\pi$$
−0.585813 + 0.810446i $$0.699225\pi$$
$$224$$ 0 0
$$225$$ 13.3667 0.891116
$$226$$ −1.38719 −0.0922743
$$227$$ 9.51630 0.631619 0.315810 0.948823i $$-0.397724\pi$$
0.315810 + 0.948823i $$0.397724\pi$$
$$228$$ 7.28427 0.482413
$$229$$ −21.1170 −1.39545 −0.697725 0.716366i $$-0.745804\pi$$
−0.697725 + 0.716366i $$0.745804\pi$$
$$230$$ −2.30435 −0.151944
$$231$$ 0 0
$$232$$ −3.32680 −0.218415
$$233$$ 14.1788 0.928881 0.464441 0.885604i $$-0.346255\pi$$
0.464441 + 0.885604i $$0.346255\pi$$
$$234$$ −4.93916 −0.322883
$$235$$ −30.6099 −1.99677
$$236$$ 0.811031 0.0527936
$$237$$ −23.0189 −1.49524
$$238$$ 0 0
$$239$$ −16.5275 −1.06907 −0.534536 0.845145i $$-0.679514\pi$$
−0.534536 + 0.845145i $$0.679514\pi$$
$$240$$ −23.2205 −1.49888
$$241$$ 13.6890 0.881786 0.440893 0.897560i $$-0.354662\pi$$
0.440893 + 0.897560i $$0.354662\pi$$
$$242$$ −8.67247 −0.557487
$$243$$ −21.5051 −1.37955
$$244$$ 0.956089 0.0612073
$$245$$ 0 0
$$246$$ 1.73288 0.110484
$$247$$ 6.96210 0.442988
$$248$$ −31.6649 −2.01073
$$249$$ −33.6393 −2.13181
$$250$$ −5.78553 −0.365909
$$251$$ 14.6603 0.925349 0.462674 0.886528i $$-0.346890\pi$$
0.462674 + 0.886528i $$0.346890\pi$$
$$252$$ 0 0
$$253$$ −1.27750 −0.0803155
$$254$$ 6.55767 0.411464
$$255$$ −30.4936 −1.90958
$$256$$ −9.15355 −0.572097
$$257$$ 1.75277 0.109335 0.0546675 0.998505i $$-0.482590\pi$$
0.0546675 + 0.998505i $$0.482590\pi$$
$$258$$ 1.09404 0.0681120
$$259$$ 0 0
$$260$$ 1.15592 0.0716870
$$261$$ 4.27750 0.264770
$$262$$ −13.3733 −0.826206
$$263$$ 26.9416 1.66129 0.830645 0.556802i $$-0.187972\pi$$
0.830645 + 0.556802i $$0.187972\pi$$
$$264$$ −16.2404 −0.999525
$$265$$ 20.6636 1.26936
$$266$$ 0 0
$$267$$ 7.04370 0.431067
$$268$$ −5.85174 −0.357452
$$269$$ 22.0691 1.34558 0.672789 0.739835i $$-0.265096\pi$$
0.672789 + 0.739835i $$0.265096\pi$$
$$270$$ 8.71141 0.530159
$$271$$ 8.96210 0.544409 0.272204 0.962239i $$-0.412247\pi$$
0.272204 + 0.962239i $$0.412247\pi$$
$$272$$ 12.1757 0.738259
$$273$$ 0 0
$$274$$ −7.43137 −0.448945
$$275$$ 6.97541 0.420633
$$276$$ 0.656308 0.0395051
$$277$$ −7.52925 −0.452389 −0.226194 0.974082i $$-0.572628\pi$$
−0.226194 + 0.974082i $$0.572628\pi$$
$$278$$ 5.06243 0.303625
$$279$$ 40.7138 2.43747
$$280$$ 0 0
$$281$$ 29.7762 1.77630 0.888151 0.459553i $$-0.151990\pi$$
0.888151 + 0.459553i $$0.151990\pi$$
$$282$$ −35.0655 −2.08812
$$283$$ −0.301451 −0.0179194 −0.00895970 0.999960i $$-0.502852\pi$$
−0.00895970 + 0.999960i $$0.502852\pi$$
$$284$$ −1.43421 −0.0851046
$$285$$ −53.0926 −3.14493
$$286$$ −2.57749 −0.152410
$$287$$ 0 0
$$288$$ 8.65110 0.509771
$$289$$ −1.01069 −0.0594526
$$290$$ 4.02645 0.236441
$$291$$ 6.11900 0.358702
$$292$$ 1.18504 0.0693492
$$293$$ 19.2471 1.12443 0.562214 0.826992i $$-0.309950\pi$$
0.562214 + 0.826992i $$0.309950\pi$$
$$294$$ 0 0
$$295$$ −5.91133 −0.344171
$$296$$ 9.36480 0.544318
$$297$$ 4.82947 0.280234
$$298$$ −12.7895 −0.740876
$$299$$ 0.627280 0.0362765
$$300$$ −3.58358 −0.206898
$$301$$ 0 0
$$302$$ −0.237588 −0.0136716
$$303$$ −3.81807 −0.219342
$$304$$ 21.1992 1.21586
$$305$$ −6.96860 −0.399021
$$306$$ −19.7501 −1.12904
$$307$$ 3.57779 0.204195 0.102098 0.994774i $$-0.467445\pi$$
0.102098 + 0.994774i $$0.467445\pi$$
$$308$$ 0 0
$$309$$ 30.5692 1.73902
$$310$$ 38.3243 2.17667
$$311$$ 23.8306 1.35131 0.675655 0.737218i $$-0.263861\pi$$
0.675655 + 0.737218i $$0.263861\pi$$
$$312$$ 7.97439 0.451461
$$313$$ 18.0814 1.02202 0.511009 0.859575i $$-0.329272\pi$$
0.511009 + 0.859575i $$0.329272\pi$$
$$314$$ 15.2782 0.862196
$$315$$ 0 0
$$316$$ 3.48914 0.196279
$$317$$ −27.5482 −1.54726 −0.773630 0.633638i $$-0.781561\pi$$
−0.773630 + 0.633638i $$0.781561\pi$$
$$318$$ 23.6714 1.32743
$$319$$ 2.23220 0.124979
$$320$$ 25.8198 1.44337
$$321$$ 51.5628 2.87796
$$322$$ 0 0
$$323$$ 27.8391 1.54901
$$324$$ 2.18134 0.121185
$$325$$ −3.42509 −0.189990
$$326$$ 18.8747 1.04537
$$327$$ −2.90776 −0.160799
$$328$$ −1.58181 −0.0873408
$$329$$ 0 0
$$330$$ 19.6558 1.08202
$$331$$ −18.1814 −0.999339 −0.499669 0.866216i $$-0.666545\pi$$
−0.499669 + 0.866216i $$0.666545\pi$$
$$332$$ 5.09895 0.279841
$$333$$ −12.0410 −0.659841
$$334$$ −6.39972 −0.350177
$$335$$ 42.6513 2.33029
$$336$$ 0 0
$$337$$ −17.1381 −0.933572 −0.466786 0.884370i $$-0.654588\pi$$
−0.466786 + 0.884370i $$0.654588\pi$$
$$338$$ 1.26561 0.0688400
$$339$$ −2.87966 −0.156402
$$340$$ 4.62212 0.250670
$$341$$ 21.2464 1.15056
$$342$$ −34.3869 −1.85943
$$343$$ 0 0
$$344$$ −0.998663 −0.0538443
$$345$$ −4.78360 −0.257540
$$346$$ 0.753815 0.0405253
$$347$$ 22.2688 1.19545 0.597725 0.801701i $$-0.296071\pi$$
0.597725 + 0.801701i $$0.296071\pi$$
$$348$$ −1.14678 −0.0614741
$$349$$ −19.9368 −1.06719 −0.533595 0.845740i $$-0.679159\pi$$
−0.533595 + 0.845740i $$0.679159\pi$$
$$350$$ 0 0
$$351$$ −2.37138 −0.126575
$$352$$ 4.51456 0.240627
$$353$$ −22.9152 −1.21965 −0.609825 0.792536i $$-0.708760\pi$$
−0.609825 + 0.792536i $$0.708760\pi$$
$$354$$ −6.77179 −0.359917
$$355$$ 10.4535 0.554812
$$356$$ −1.06766 −0.0565860
$$357$$ 0 0
$$358$$ 10.2219 0.540242
$$359$$ 27.2314 1.43722 0.718610 0.695413i $$-0.244779\pi$$
0.718610 + 0.695413i $$0.244779\pi$$
$$360$$ −34.3821 −1.81210
$$361$$ 29.4708 1.55110
$$362$$ −2.39611 −0.125936
$$363$$ −18.0032 −0.944923
$$364$$ 0 0
$$365$$ −8.63735 −0.452099
$$366$$ −7.98297 −0.417276
$$367$$ 10.8564 0.566702 0.283351 0.959016i $$-0.408554\pi$$
0.283351 + 0.959016i $$0.408554\pi$$
$$368$$ 1.91003 0.0995671
$$369$$ 2.03384 0.105878
$$370$$ −11.3343 −0.589241
$$371$$ 0 0
$$372$$ −10.9152 −0.565929
$$373$$ −2.37144 −0.122789 −0.0613943 0.998114i $$-0.519555\pi$$
−0.0613943 + 0.998114i $$0.519555\pi$$
$$374$$ −10.3065 −0.532938
$$375$$ −12.0102 −0.620204
$$376$$ 32.0085 1.65071
$$377$$ −1.09606 −0.0564501
$$378$$ 0 0
$$379$$ −29.2197 −1.50092 −0.750458 0.660918i $$-0.770167\pi$$
−0.750458 + 0.660918i $$0.770167\pi$$
$$380$$ 8.04761 0.412834
$$381$$ 13.6131 0.697419
$$382$$ 4.68496 0.239703
$$383$$ 3.06595 0.156663 0.0783313 0.996927i $$-0.475041\pi$$
0.0783313 + 0.996927i $$0.475041\pi$$
$$384$$ 17.9302 0.914995
$$385$$ 0 0
$$386$$ 17.1964 0.875274
$$387$$ 1.28405 0.0652719
$$388$$ −0.927499 −0.0470866
$$389$$ 27.7410 1.40652 0.703261 0.710932i $$-0.251726\pi$$
0.703261 + 0.710932i $$0.251726\pi$$
$$390$$ −9.65146 −0.488721
$$391$$ 2.50828 0.126849
$$392$$ 0 0
$$393$$ −27.7617 −1.40039
$$394$$ −12.2797 −0.618641
$$395$$ −25.4311 −1.27958
$$396$$ −3.16513 −0.159054
$$397$$ −17.2312 −0.864808 −0.432404 0.901680i $$-0.642335\pi$$
−0.432404 + 0.901680i $$0.642335\pi$$
$$398$$ −33.2500 −1.66667
$$399$$ 0 0
$$400$$ −10.4292 −0.521459
$$401$$ 16.6440 0.831163 0.415582 0.909556i $$-0.363578\pi$$
0.415582 + 0.909556i $$0.363578\pi$$
$$402$$ 48.8597 2.43690
$$403$$ −10.4325 −0.519679
$$404$$ 0.578731 0.0287930
$$405$$ −15.8990 −0.790029
$$406$$ 0 0
$$407$$ −6.28355 −0.311464
$$408$$ 31.8869 1.57864
$$409$$ 13.6338 0.674147 0.337073 0.941478i $$-0.390563\pi$$
0.337073 + 0.941478i $$0.390563\pi$$
$$410$$ 1.91447 0.0945491
$$411$$ −15.4268 −0.760948
$$412$$ −4.63359 −0.228280
$$413$$ 0 0
$$414$$ −3.09824 −0.152270
$$415$$ −37.1645 −1.82433
$$416$$ −2.21675 −0.108685
$$417$$ 10.5091 0.514634
$$418$$ −17.9448 −0.877707
$$419$$ 10.8502 0.530066 0.265033 0.964239i $$-0.414617\pi$$
0.265033 + 0.964239i $$0.414617\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 12.6989 0.618173
$$423$$ −41.1556 −2.00105
$$424$$ −21.6078 −1.04937
$$425$$ −13.6958 −0.664342
$$426$$ 11.9751 0.580194
$$427$$ 0 0
$$428$$ −7.81574 −0.377788
$$429$$ −5.35062 −0.258331
$$430$$ 1.20869 0.0582881
$$431$$ 1.20953 0.0582609 0.0291304 0.999576i $$-0.490726\pi$$
0.0291304 + 0.999576i $$0.490726\pi$$
$$432$$ −7.22072 −0.347407
$$433$$ 5.56422 0.267399 0.133700 0.991022i $$-0.457314\pi$$
0.133700 + 0.991022i $$0.457314\pi$$
$$434$$ 0 0
$$435$$ 8.35851 0.400760
$$436$$ 0.440749 0.0211081
$$437$$ 4.36719 0.208911
$$438$$ −9.89461 −0.472783
$$439$$ 19.7192 0.941146 0.470573 0.882361i $$-0.344047\pi$$
0.470573 + 0.882361i $$0.344047\pi$$
$$440$$ −17.9422 −0.855362
$$441$$ 0 0
$$442$$ 5.06074 0.240715
$$443$$ 22.2310 1.05623 0.528113 0.849174i $$-0.322900\pi$$
0.528113 + 0.849174i $$0.322900\pi$$
$$444$$ 3.22815 0.153201
$$445$$ 7.78182 0.368894
$$446$$ −22.1432 −1.04851
$$447$$ −26.5498 −1.25576
$$448$$ 0 0
$$449$$ 18.4579 0.871082 0.435541 0.900169i $$-0.356557\pi$$
0.435541 + 0.900169i $$0.356557\pi$$
$$450$$ 16.9171 0.797478
$$451$$ 1.06136 0.0499773
$$452$$ 0.436491 0.0205308
$$453$$ −0.493209 −0.0231730
$$454$$ 12.0439 0.565249
$$455$$ 0 0
$$456$$ 55.5185 2.59989
$$457$$ −29.9819 −1.40250 −0.701248 0.712917i $$-0.747373\pi$$
−0.701248 + 0.712917i $$0.747373\pi$$
$$458$$ −26.7258 −1.24882
$$459$$ −9.48236 −0.442599
$$460$$ 0.725084 0.0338072
$$461$$ −29.1498 −1.35764 −0.678821 0.734304i $$-0.737509\pi$$
−0.678821 + 0.734304i $$0.737509\pi$$
$$462$$ 0 0
$$463$$ 1.55900 0.0724530 0.0362265 0.999344i $$-0.488466\pi$$
0.0362265 + 0.999344i $$0.488466\pi$$
$$464$$ −3.33744 −0.154937
$$465$$ 79.5575 3.68939
$$466$$ 17.9448 0.831275
$$467$$ −12.4231 −0.574874 −0.287437 0.957800i $$-0.592803\pi$$
−0.287437 + 0.957800i $$0.592803\pi$$
$$468$$ 1.55415 0.0718407
$$469$$ 0 0
$$470$$ −38.7402 −1.78695
$$471$$ 31.7160 1.46139
$$472$$ 6.18143 0.284523
$$473$$ 0.670079 0.0308103
$$474$$ −29.1329 −1.33812
$$475$$ −23.8458 −1.09412
$$476$$ 0 0
$$477$$ 27.7826 1.27208
$$478$$ −20.9173 −0.956735
$$479$$ −36.0558 −1.64743 −0.823716 0.567003i $$-0.808103\pi$$
−0.823716 + 0.567003i $$0.808103\pi$$
$$480$$ 16.9048 0.771597
$$481$$ 3.08537 0.140681
$$482$$ 17.3249 0.789128
$$483$$ 0 0
$$484$$ 2.72887 0.124040
$$485$$ 6.76023 0.306966
$$486$$ −27.2170 −1.23459
$$487$$ 7.30004 0.330796 0.165398 0.986227i $$-0.447109\pi$$
0.165398 + 0.986227i $$0.447109\pi$$
$$488$$ 7.28702 0.329868
$$489$$ 39.1821 1.77188
$$490$$ 0 0
$$491$$ 4.49178 0.202711 0.101356 0.994850i $$-0.467682\pi$$
0.101356 + 0.994850i $$0.467682\pi$$
$$492$$ −0.545267 −0.0245825
$$493$$ −4.38279 −0.197391
$$494$$ 8.81129 0.396439
$$495$$ 23.0696 1.03690
$$496$$ −31.7663 −1.42635
$$497$$ 0 0
$$498$$ −42.5742 −1.90780
$$499$$ 11.3674 0.508873 0.254437 0.967089i $$-0.418110\pi$$
0.254437 + 0.967089i $$0.418110\pi$$
$$500$$ 1.82047 0.0814139
$$501$$ −13.2852 −0.593539
$$502$$ 18.5542 0.828113
$$503$$ −17.1080 −0.762806 −0.381403 0.924409i $$-0.624559\pi$$
−0.381403 + 0.924409i $$0.624559\pi$$
$$504$$ 0 0
$$505$$ −4.21818 −0.187706
$$506$$ −1.61681 −0.0718759
$$507$$ 2.62728 0.116682
$$508$$ −2.06343 −0.0915498
$$509$$ −3.28284 −0.145509 −0.0727547 0.997350i $$-0.523179\pi$$
−0.0727547 + 0.997350i $$0.523179\pi$$
$$510$$ −38.5929 −1.70892
$$511$$ 0 0
$$512$$ −25.2340 −1.11520
$$513$$ −16.5098 −0.728926
$$514$$ 2.21833 0.0978462
$$515$$ 33.7726 1.48820
$$516$$ −0.344250 −0.0151548
$$517$$ −21.4770 −0.944555
$$518$$ 0 0
$$519$$ 1.56485 0.0686891
$$520$$ 8.81005 0.386346
$$521$$ 4.77061 0.209004 0.104502 0.994525i $$-0.466675\pi$$
0.104502 + 0.994525i $$0.466675\pi$$
$$522$$ 5.41363 0.236948
$$523$$ −25.5124 −1.11558 −0.557789 0.829983i $$-0.688350\pi$$
−0.557789 + 0.829983i $$0.688350\pi$$
$$524$$ 4.20803 0.183829
$$525$$ 0 0
$$526$$ 34.0975 1.48672
$$527$$ −41.7160 −1.81718
$$528$$ −16.2923 −0.709032
$$529$$ −22.6065 −0.982892
$$530$$ 26.1520 1.13597
$$531$$ −7.94789 −0.344909
$$532$$ 0 0
$$533$$ −0.521150 −0.0225735
$$534$$ 8.91456 0.385771
$$535$$ 56.9663 2.46287
$$536$$ −44.6001 −1.92643
$$537$$ 21.2196 0.915693
$$538$$ 27.9309 1.20418
$$539$$ 0 0
$$540$$ −2.74112 −0.117959
$$541$$ −16.5157 −0.710064 −0.355032 0.934854i $$-0.615530\pi$$
−0.355032 + 0.934854i $$0.615530\pi$$
$$542$$ 11.3425 0.487202
$$543$$ −4.97408 −0.213458
$$544$$ −8.86405 −0.380043
$$545$$ −3.21247 −0.137607
$$546$$ 0 0
$$547$$ 23.3317 0.997591 0.498796 0.866720i $$-0.333776\pi$$
0.498796 + 0.866720i $$0.333776\pi$$
$$548$$ 2.33835 0.0998892
$$549$$ −9.36942 −0.399877
$$550$$ 8.82814 0.376433
$$551$$ −7.63090 −0.325087
$$552$$ 5.00218 0.212907
$$553$$ 0 0
$$554$$ −9.52909 −0.404852
$$555$$ −23.5289 −0.998744
$$556$$ −1.59294 −0.0675557
$$557$$ −20.0471 −0.849422 −0.424711 0.905329i $$-0.639624\pi$$
−0.424711 + 0.905329i $$0.639624\pi$$
$$558$$ 51.5277 2.18134
$$559$$ −0.329024 −0.0139162
$$560$$ 0 0
$$561$$ −21.3953 −0.903312
$$562$$ 37.6851 1.58965
$$563$$ 40.5284 1.70807 0.854034 0.520218i $$-0.174149\pi$$
0.854034 + 0.520218i $$0.174149\pi$$
$$564$$ 11.0337 0.464602
$$565$$ −3.18143 −0.133844
$$566$$ −0.381519 −0.0160364
$$567$$ 0 0
$$568$$ −10.9311 −0.458659
$$569$$ −21.4504 −0.899246 −0.449623 0.893219i $$-0.648442\pi$$
−0.449623 + 0.893219i $$0.648442\pi$$
$$570$$ −67.1944 −2.81446
$$571$$ −10.9559 −0.458489 −0.229244 0.973369i $$-0.573625\pi$$
−0.229244 + 0.973369i $$0.573625\pi$$
$$572$$ 0.811031 0.0339109
$$573$$ 9.72552 0.406289
$$574$$ 0 0
$$575$$ −2.14849 −0.0895982
$$576$$ 34.7153 1.44647
$$577$$ −34.7415 −1.44631 −0.723154 0.690687i $$-0.757308\pi$$
−0.723154 + 0.690687i $$0.757308\pi$$
$$578$$ −1.27914 −0.0532053
$$579$$ 35.6981 1.48356
$$580$$ −1.26696 −0.0526076
$$581$$ 0 0
$$582$$ 7.74426 0.321010
$$583$$ 14.4983 0.600458
$$584$$ 9.03201 0.373747
$$585$$ −11.3277 −0.468342
$$586$$ 24.3593 1.00627
$$587$$ 22.8463 0.942967 0.471483 0.881875i $$-0.343719\pi$$
0.471483 + 0.881875i $$0.343719\pi$$
$$588$$ 0 0
$$589$$ −72.6320 −2.99275
$$590$$ −7.48143 −0.308006
$$591$$ −25.4914 −1.04858
$$592$$ 9.39476 0.386122
$$593$$ −17.5935 −0.722480 −0.361240 0.932473i $$-0.617646\pi$$
−0.361240 + 0.932473i $$0.617646\pi$$
$$594$$ 6.11222 0.250787
$$595$$ 0 0
$$596$$ 4.02433 0.164843
$$597$$ −69.0238 −2.82496
$$598$$ 0.793891 0.0324646
$$599$$ 31.0073 1.26692 0.633461 0.773774i $$-0.281634\pi$$
0.633461 + 0.773774i $$0.281634\pi$$
$$600$$ −27.3130 −1.11505
$$601$$ 1.43754 0.0586385 0.0293193 0.999570i $$-0.490666\pi$$
0.0293193 + 0.999570i $$0.490666\pi$$
$$602$$ 0 0
$$603$$ 57.3455 2.33529
$$604$$ 0.0747591 0.00304191
$$605$$ −19.8898 −0.808635
$$606$$ −4.83218 −0.196294
$$607$$ 33.0171 1.34012 0.670061 0.742306i $$-0.266268\pi$$
0.670061 + 0.742306i $$0.266268\pi$$
$$608$$ −15.4333 −0.625901
$$609$$ 0 0
$$610$$ −8.81953 −0.357092
$$611$$ 10.5457 0.426633
$$612$$ 6.21453 0.251208
$$613$$ −43.1657 −1.74345 −0.871723 0.489999i $$-0.836997\pi$$
−0.871723 + 0.489999i $$0.836997\pi$$
$$614$$ 4.52808 0.182738
$$615$$ 3.97426 0.160258
$$616$$ 0 0
$$617$$ 2.45772 0.0989441 0.0494721 0.998776i $$-0.484246\pi$$
0.0494721 + 0.998776i $$0.484246\pi$$
$$618$$ 38.6886 1.55628
$$619$$ −37.7789 −1.51846 −0.759231 0.650822i $$-0.774425\pi$$
−0.759231 + 0.650822i $$0.774425\pi$$
$$620$$ −12.0591 −0.484305
$$621$$ −1.48752 −0.0596922
$$622$$ 30.1602 1.20932
$$623$$ 0 0
$$624$$ 7.99991 0.320253
$$625$$ −30.3942 −1.21577
$$626$$ 22.8839 0.914625
$$627$$ −37.2516 −1.48768
$$628$$ −4.80741 −0.191837
$$629$$ 12.3374 0.491922
$$630$$ 0 0
$$631$$ −28.4828 −1.13388 −0.566942 0.823758i $$-0.691874\pi$$
−0.566942 + 0.823758i $$0.691874\pi$$
$$632$$ 26.5932 1.05782
$$633$$ 26.3617 1.04778
$$634$$ −34.8652 −1.38467
$$635$$ 15.0396 0.596829
$$636$$ −7.44843 −0.295350
$$637$$ 0 0
$$638$$ 2.82509 0.111847
$$639$$ 14.0549 0.556002
$$640$$ 19.8091 0.783024
$$641$$ −27.1922 −1.07403 −0.537014 0.843573i $$-0.680448\pi$$
−0.537014 + 0.843573i $$0.680448\pi$$
$$642$$ 65.2584 2.57554
$$643$$ 37.1664 1.46570 0.732849 0.680391i $$-0.238190\pi$$
0.732849 + 0.680391i $$0.238190\pi$$
$$644$$ 0 0
$$645$$ 2.50912 0.0987965
$$646$$ 35.2334 1.38624
$$647$$ −18.8319 −0.740357 −0.370178 0.928961i $$-0.620703\pi$$
−0.370178 + 0.928961i $$0.620703\pi$$
$$648$$ 16.6255 0.653111
$$649$$ −4.14759 −0.162807
$$650$$ −4.33482 −0.170026
$$651$$ 0 0
$$652$$ −5.93910 −0.232593
$$653$$ −26.0185 −1.01818 −0.509090 0.860713i $$-0.670018\pi$$
−0.509090 + 0.860713i $$0.670018\pi$$
$$654$$ −3.68008 −0.143903
$$655$$ −30.6709 −1.19841
$$656$$ −1.58687 −0.0619569
$$657$$ −11.6131 −0.453069
$$658$$ 0 0
$$659$$ −33.3339 −1.29851 −0.649253 0.760573i $$-0.724918\pi$$
−0.649253 + 0.760573i $$0.724918\pi$$
$$660$$ −6.18488 −0.240746
$$661$$ −6.29841 −0.244980 −0.122490 0.992470i $$-0.539088\pi$$
−0.122490 + 0.992470i $$0.539088\pi$$
$$662$$ −23.0105 −0.894329
$$663$$ 10.5056 0.408004
$$664$$ 38.8626 1.50816
$$665$$ 0 0
$$666$$ −15.2391 −0.590505
$$667$$ −0.687538 −0.0266216
$$668$$ 2.01373 0.0779136
$$669$$ −45.9672 −1.77719
$$670$$ 53.9799 2.08542
$$671$$ −4.88941 −0.188754
$$672$$ 0 0
$$673$$ 18.3188 0.706137 0.353068 0.935598i $$-0.385138\pi$$
0.353068 + 0.935598i $$0.385138\pi$$
$$674$$ −21.6901 −0.835473
$$675$$ 8.12219 0.312623
$$676$$ −0.398235 −0.0153167
$$677$$ −24.3392 −0.935430 −0.467715 0.883879i $$-0.654923\pi$$
−0.467715 + 0.883879i $$0.654923\pi$$
$$678$$ −3.64453 −0.139967
$$679$$ 0 0
$$680$$ 35.2284 1.35095
$$681$$ 25.0020 0.958078
$$682$$ 26.8896 1.02966
$$683$$ 11.7682 0.450297 0.225149 0.974324i $$-0.427713\pi$$
0.225149 + 0.974324i $$0.427713\pi$$
$$684$$ 10.8202 0.413719
$$685$$ −17.0434 −0.651195
$$686$$ 0 0
$$687$$ −55.4802 −2.11670
$$688$$ −1.00186 −0.0381955
$$689$$ −7.11900 −0.271212
$$690$$ −6.05417 −0.230478
$$691$$ −1.17785 −0.0448074 −0.0224037 0.999749i $$-0.507132\pi$$
−0.0224037 + 0.999749i $$0.507132\pi$$
$$692$$ −0.237195 −0.00901679
$$693$$ 0 0
$$694$$ 28.1835 1.06983
$$695$$ 11.6104 0.440408
$$696$$ −8.74043 −0.331305
$$697$$ −2.08390 −0.0789335
$$698$$ −25.2321 −0.955050
$$699$$ 37.2516 1.40898
$$700$$ 0 0
$$701$$ −31.2867 −1.18168 −0.590841 0.806788i $$-0.701204\pi$$
−0.590841 + 0.806788i $$0.701204\pi$$
$$702$$ −3.00124 −0.113275
$$703$$ 21.4806 0.810158
$$704$$ 18.1161 0.682776
$$705$$ −80.4208 −3.02882
$$706$$ −29.0016 −1.09149
$$707$$ 0 0
$$708$$ 2.13081 0.0800806
$$709$$ 15.3748 0.577411 0.288706 0.957418i $$-0.406775\pi$$
0.288706 + 0.957418i $$0.406775\pi$$
$$710$$ 13.2300 0.496512
$$711$$ −34.1926 −1.28232
$$712$$ −8.13739 −0.304962
$$713$$ −6.54409 −0.245078
$$714$$ 0 0
$$715$$ −5.91133 −0.221071
$$716$$ −3.21640 −0.120203
$$717$$ −43.4223 −1.62163
$$718$$ 34.4643 1.28620
$$719$$ 11.1417 0.415517 0.207759 0.978180i $$-0.433383\pi$$
0.207759 + 0.978180i $$0.433383\pi$$
$$720$$ −34.4921 −1.28545
$$721$$ 0 0
$$722$$ 37.2985 1.38811
$$723$$ 35.9648 1.33755
$$724$$ 0.753956 0.0280206
$$725$$ 3.75411 0.139424
$$726$$ −22.7850 −0.845631
$$727$$ 6.24735 0.231702 0.115851 0.993267i $$-0.463041\pi$$
0.115851 + 0.993267i $$0.463041\pi$$
$$728$$ 0 0
$$729$$ −40.0674 −1.48398
$$730$$ −10.9315 −0.404593
$$731$$ −1.31566 −0.0486613
$$732$$ 2.51191 0.0928430
$$733$$ −30.9669 −1.14379 −0.571894 0.820327i $$-0.693791\pi$$
−0.571894 + 0.820327i $$0.693791\pi$$
$$734$$ 13.7400 0.507153
$$735$$ 0 0
$$736$$ −1.39053 −0.0512554
$$737$$ 29.9256 1.10232
$$738$$ 2.57405 0.0947520
$$739$$ 2.33744 0.0859843 0.0429921 0.999075i $$-0.486311\pi$$
0.0429921 + 0.999075i $$0.486311\pi$$
$$740$$ 3.56643 0.131105
$$741$$ 18.2914 0.671951
$$742$$ 0 0
$$743$$ 24.3612 0.893726 0.446863 0.894603i $$-0.352541\pi$$
0.446863 + 0.894603i $$0.352541\pi$$
$$744$$ −83.1927 −3.04999
$$745$$ −29.3320 −1.07464
$$746$$ −3.00132 −0.109886
$$747$$ −49.9684 −1.82825
$$748$$ 3.24304 0.118577
$$749$$ 0 0
$$750$$ −15.2002 −0.555033
$$751$$ −12.0253 −0.438810 −0.219405 0.975634i $$-0.570412\pi$$
−0.219405 + 0.975634i $$0.570412\pi$$
$$752$$ 32.1110 1.17097
$$753$$ 38.5167 1.40363
$$754$$ −1.38719 −0.0505184
$$755$$ −0.544894 −0.0198307
$$756$$ 0 0
$$757$$ 25.9905 0.944641 0.472321 0.881427i $$-0.343416\pi$$
0.472321 + 0.881427i $$0.343416\pi$$
$$758$$ −36.9807 −1.34320
$$759$$ −3.35634 −0.121827
$$760$$ 61.3364 2.22491
$$761$$ −13.3270 −0.483103 −0.241552 0.970388i $$-0.577656\pi$$
−0.241552 + 0.970388i $$0.577656\pi$$
$$762$$ 17.2288 0.624134
$$763$$ 0 0
$$764$$ −1.47416 −0.0533334
$$765$$ −45.2956 −1.63767
$$766$$ 3.88029 0.140200
$$767$$ 2.03656 0.0735361
$$768$$ −24.0489 −0.867792
$$769$$ −9.24486 −0.333378 −0.166689 0.986010i $$-0.553308\pi$$
−0.166689 + 0.986010i $$0.553308\pi$$
$$770$$ 0 0
$$771$$ 4.60503 0.165846
$$772$$ −5.41101 −0.194746
$$773$$ 10.1419 0.364780 0.182390 0.983226i $$-0.441617\pi$$
0.182390 + 0.983226i $$0.441617\pi$$
$$774$$ 1.62510 0.0584132
$$775$$ 35.7322 1.28354
$$776$$ −7.06912 −0.253766
$$777$$ 0 0
$$778$$ 35.1092 1.25873
$$779$$ −3.62830 −0.129997
$$780$$ 3.03692 0.108739
$$781$$ 7.33450 0.262449
$$782$$ 3.17450 0.113520
$$783$$ 2.59919 0.0928873
$$784$$ 0 0
$$785$$ 35.0396 1.25062
$$786$$ −35.1354 −1.25324
$$787$$ −45.2823 −1.61414 −0.807070 0.590456i $$-0.798948\pi$$
−0.807070 + 0.590456i $$0.798948\pi$$
$$788$$ 3.86391 0.137646
$$789$$ 70.7831 2.51995
$$790$$ −32.1859 −1.14512
$$791$$ 0 0
$$792$$ −24.1237 −0.857197
$$793$$ 2.40081 0.0852554
$$794$$ −21.8079 −0.773935
$$795$$ 54.2891 1.92544
$$796$$ 10.4624 0.370831
$$797$$ −27.0784 −0.959165 −0.479583 0.877497i $$-0.659212\pi$$
−0.479583 + 0.877497i $$0.659212\pi$$
$$798$$ 0 0
$$799$$ 42.1686 1.49182
$$800$$ 7.59257 0.268438
$$801$$ 10.4628 0.369685
$$802$$ 21.0648 0.743825
$$803$$ −6.06026 −0.213862
$$804$$ −15.3741 −0.542204
$$805$$ 0 0
$$806$$ −13.2034 −0.465071
$$807$$ 57.9817 2.04105
$$808$$ 4.41091 0.155175
$$809$$ −25.7798 −0.906370 −0.453185 0.891417i $$-0.649712\pi$$
−0.453185 + 0.891417i $$0.649712\pi$$
$$810$$ −20.1219 −0.707013
$$811$$ 25.7829 0.905362 0.452681 0.891673i $$-0.350468\pi$$
0.452681 + 0.891673i $$0.350468\pi$$
$$812$$ 0 0
$$813$$ 23.5459 0.825792
$$814$$ −7.95252 −0.278736
$$815$$ 43.2881 1.51632
$$816$$ 31.9889 1.11984
$$817$$ −2.29070 −0.0801414
$$818$$ 17.2550 0.603308
$$819$$ 0 0
$$820$$ −0.602407 −0.0210370
$$821$$ −1.71073 −0.0597050 −0.0298525 0.999554i $$-0.509504\pi$$
−0.0298525 + 0.999554i $$0.509504\pi$$
$$822$$ −19.5243 −0.680987
$$823$$ −40.3773 −1.40747 −0.703733 0.710465i $$-0.748485\pi$$
−0.703733 + 0.710465i $$0.748485\pi$$
$$824$$ −35.3158 −1.23028
$$825$$ 18.3264 0.638042
$$826$$ 0 0
$$827$$ 19.5698 0.680509 0.340254 0.940333i $$-0.389487\pi$$
0.340254 + 0.940333i $$0.389487\pi$$
$$828$$ 0.974889 0.0338797
$$829$$ −41.5742 −1.44393 −0.721966 0.691928i $$-0.756761\pi$$
−0.721966 + 0.691928i $$0.756761\pi$$
$$830$$ −47.0357 −1.63263
$$831$$ −19.7815 −0.686211
$$832$$ −8.89542 −0.308393
$$833$$ 0 0
$$834$$ 13.3004 0.460556
$$835$$ −14.6774 −0.507932
$$836$$ 5.64648 0.195288
$$837$$ 24.7394 0.855119
$$838$$ 13.7321 0.474367
$$839$$ −45.8480 −1.58285 −0.791425 0.611266i $$-0.790660\pi$$
−0.791425 + 0.611266i $$0.790660\pi$$
$$840$$ 0 0
$$841$$ −27.7986 −0.958574
$$842$$ −12.6561 −0.436157
$$843$$ 78.2305 2.69440
$$844$$ −3.99583 −0.137542
$$845$$ 2.90260 0.0998525
$$846$$ −52.0869 −1.79078
$$847$$ 0 0
$$848$$ −21.6769 −0.744388
$$849$$ −0.791997 −0.0271812
$$850$$ −17.3335 −0.594534
$$851$$ 1.93539 0.0663443
$$852$$ −3.76807 −0.129092
$$853$$ −40.0236 −1.37038 −0.685191 0.728364i $$-0.740281\pi$$
−0.685191 + 0.728364i $$0.740281\pi$$
$$854$$ 0 0
$$855$$ −78.8645 −2.69711
$$856$$ −59.5692 −2.03603
$$857$$ −32.8702 −1.12282 −0.561412 0.827536i $$-0.689742\pi$$
−0.561412 + 0.827536i $$0.689742\pi$$
$$858$$ −6.77179 −0.231185
$$859$$ 34.0503 1.16178 0.580891 0.813981i $$-0.302704\pi$$
0.580891 + 0.813981i $$0.302704\pi$$
$$860$$ −0.380325 −0.0129690
$$861$$ 0 0
$$862$$ 1.53079 0.0521388
$$863$$ −14.0642 −0.478749 −0.239375 0.970927i $$-0.576942\pi$$
−0.239375 + 0.970927i $$0.576942\pi$$
$$864$$ 5.25677 0.178839
$$865$$ 1.72883 0.0587820
$$866$$ 7.04212 0.239301
$$867$$ −2.65537 −0.0901813
$$868$$ 0 0
$$869$$ −17.8434 −0.605295
$$870$$ 10.5786 0.358648
$$871$$ −14.6942 −0.497893
$$872$$ 3.35926 0.113759
$$873$$ 9.08925 0.307625
$$874$$ 5.52715 0.186959
$$875$$ 0 0
$$876$$ 3.11343 0.105193
$$877$$ 51.0669 1.72441 0.862204 0.506562i $$-0.169084\pi$$
0.862204 + 0.506562i $$0.169084\pi$$
$$878$$ 24.9568 0.842251
$$879$$ 50.5675 1.70560
$$880$$ −17.9996 −0.606768
$$881$$ 18.4203 0.620597 0.310298 0.950639i $$-0.399571\pi$$
0.310298 + 0.950639i $$0.399571\pi$$
$$882$$ 0 0
$$883$$ 0.126678 0.00426305 0.00213153 0.999998i $$-0.499322\pi$$
0.00213153 + 0.999998i $$0.499322\pi$$
$$884$$ −1.59241 −0.0535585
$$885$$ −15.5307 −0.522059
$$886$$ 28.1357 0.945239
$$887$$ −3.87470 −0.130100 −0.0650498 0.997882i $$-0.520721\pi$$
−0.0650498 + 0.997882i $$0.520721\pi$$
$$888$$ 24.6039 0.825654
$$889$$ 0 0
$$890$$ 9.84874 0.330131
$$891$$ −11.1553 −0.373717
$$892$$ 6.96757 0.233291
$$893$$ 73.4201 2.45691
$$894$$ −33.6016 −1.12381
$$895$$ 23.4433 0.783622
$$896$$ 0 0
$$897$$ 1.64804 0.0550265
$$898$$ 23.3605 0.779549
$$899$$ 11.4347 0.381367
$$900$$ −5.32311 −0.177437
$$901$$ −28.4665 −0.948356
$$902$$ 1.34326 0.0447257
$$903$$ 0 0
$$904$$ 3.32680 0.110648
$$905$$ −5.49533 −0.182671
$$906$$ −0.624210 −0.0207380
$$907$$ −46.7741 −1.55311 −0.776555 0.630050i $$-0.783035\pi$$
−0.776555 + 0.630050i $$0.783035\pi$$
$$908$$ −3.78973 −0.125766
$$909$$ −5.67142 −0.188109
$$910$$ 0 0
$$911$$ 5.93675 0.196693 0.0983467 0.995152i $$-0.468645\pi$$
0.0983467 + 0.995152i $$0.468645\pi$$
$$912$$ 55.6961 1.84428
$$913$$ −26.0759 −0.862986
$$914$$ −37.9454 −1.25512
$$915$$ −18.3085 −0.605260
$$916$$ 8.40952 0.277858
$$917$$ 0 0
$$918$$ −12.0010 −0.396091
$$919$$ −8.58701 −0.283259 −0.141630 0.989920i $$-0.545234\pi$$
−0.141630 + 0.989920i $$0.545234\pi$$
$$920$$ 5.52637 0.182199
$$921$$ 9.39985 0.309736
$$922$$ −36.8922 −1.21498
$$923$$ −3.60141 −0.118542
$$924$$ 0 0
$$925$$ −10.5677 −0.347463
$$926$$ 1.97309 0.0648396
$$927$$ 45.4079 1.49139
$$928$$ 2.42970 0.0797589
$$929$$ 8.45945 0.277546 0.138773 0.990324i $$-0.455684\pi$$
0.138773 + 0.990324i $$0.455684\pi$$
$$930$$ 100.689 3.30171
$$931$$ 0 0
$$932$$ −5.64648 −0.184957
$$933$$ 62.6097 2.04975
$$934$$ −15.7228 −0.514466
$$935$$ −23.6374 −0.773026
$$936$$ 11.8453 0.387175
$$937$$ 33.3596 1.08981 0.544905 0.838498i $$-0.316566\pi$$
0.544905 + 0.838498i $$0.316566\pi$$
$$938$$ 0 0
$$939$$ 47.5048 1.55026
$$940$$ 12.1899 0.397592
$$941$$ 13.4037 0.436949 0.218475 0.975843i $$-0.429892\pi$$
0.218475 + 0.975843i $$0.429892\pi$$
$$942$$ 40.1400 1.30783
$$943$$ −0.326907 −0.0106456
$$944$$ 6.20121 0.201832
$$945$$ 0 0
$$946$$ 0.848057 0.0275727
$$947$$ 43.0794 1.39989 0.699946 0.714196i $$-0.253207\pi$$
0.699946 + 0.714196i $$0.253207\pi$$
$$948$$ 9.16695 0.297729
$$949$$ 2.97573 0.0965962
$$950$$ −30.1794 −0.979150
$$951$$ −72.3768 −2.34698
$$952$$ 0 0
$$953$$ 16.7332 0.542040 0.271020 0.962574i $$-0.412639\pi$$
0.271020 + 0.962574i $$0.412639\pi$$
$$954$$ 35.1619 1.13841
$$955$$ 10.7447 0.347690
$$956$$ 6.58182 0.212871
$$957$$ 5.86462 0.189576
$$958$$ −45.6325 −1.47432
$$959$$ 0 0
$$960$$ 67.8360 2.18940
$$961$$ 77.8366 2.51086
$$962$$ 3.90487 0.125898
$$963$$ 76.5922 2.46815
$$964$$ −5.45144 −0.175579
$$965$$ 39.4390 1.26959
$$966$$ 0 0
$$967$$ 44.7594 1.43937 0.719683 0.694303i $$-0.244287\pi$$
0.719683 + 0.694303i $$0.244287\pi$$
$$968$$ 20.7986 0.668493
$$969$$ 73.1411 2.34963
$$970$$ 8.55580 0.274710
$$971$$ 4.20259 0.134867 0.0674337 0.997724i $$-0.478519\pi$$
0.0674337 + 0.997724i $$0.478519\pi$$
$$972$$ 8.56409 0.274693
$$973$$ 0 0
$$974$$ 9.23899 0.296036
$$975$$ −8.99866 −0.288188
$$976$$ 7.31033 0.233998
$$977$$ −25.6899 −0.821892 −0.410946 0.911660i $$-0.634801\pi$$
−0.410946 + 0.911660i $$0.634801\pi$$
$$978$$ 49.5892 1.58569
$$979$$ 5.45999 0.174502
$$980$$ 0 0
$$981$$ −4.31923 −0.137902
$$982$$ 5.68483 0.181410
$$983$$ 31.8244 1.01504 0.507520 0.861640i $$-0.330562\pi$$
0.507520 + 0.861640i $$0.330562\pi$$
$$984$$ −4.15586 −0.132484
$$985$$ −28.1627 −0.897339
$$986$$ −5.54689 −0.176649
$$987$$ 0 0
$$988$$ −2.77255 −0.0882067
$$989$$ −0.206390 −0.00656283
$$990$$ 29.1970 0.927942
$$991$$ 9.47478 0.300976 0.150488 0.988612i $$-0.451915\pi$$
0.150488 + 0.988612i $$0.451915\pi$$
$$992$$ 23.1262 0.734259
$$993$$ −47.7676 −1.51586
$$994$$ 0 0
$$995$$ −76.2570 −2.41751
$$996$$ 13.3964 0.424480
$$997$$ −21.9511 −0.695198 −0.347599 0.937643i $$-0.613003\pi$$
−0.347599 + 0.937643i $$0.613003\pi$$
$$998$$ 14.3866 0.455401
$$999$$ −7.31659 −0.231487
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.k.1.3 5
3.2 odd 2 5733.2.a.bm.1.3 5
7.2 even 3 637.2.e.m.508.3 10
7.3 odd 6 91.2.e.c.79.3 yes 10
7.4 even 3 637.2.e.m.79.3 10
7.5 odd 6 91.2.e.c.53.3 10
7.6 odd 2 637.2.a.l.1.3 5
13.12 even 2 8281.2.a.bx.1.3 5
21.5 even 6 819.2.j.h.235.3 10
21.17 even 6 819.2.j.h.352.3 10
21.20 even 2 5733.2.a.bl.1.3 5
28.3 even 6 1456.2.r.p.625.1 10
28.19 even 6 1456.2.r.p.417.1 10
91.12 odd 6 1183.2.e.f.508.3 10
91.38 odd 6 1183.2.e.f.170.3 10
91.90 odd 2 8281.2.a.bw.1.3 5

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.3 10 7.5 odd 6
91.2.e.c.79.3 yes 10 7.3 odd 6
637.2.a.k.1.3 5 1.1 even 1 trivial
637.2.a.l.1.3 5 7.6 odd 2
637.2.e.m.79.3 10 7.4 even 3
637.2.e.m.508.3 10 7.2 even 3
819.2.j.h.235.3 10 21.5 even 6
819.2.j.h.352.3 10 21.17 even 6
1183.2.e.f.170.3 10 91.38 odd 6
1183.2.e.f.508.3 10 91.12 odd 6
1456.2.r.p.417.1 10 28.19 even 6
1456.2.r.p.625.1 10 28.3 even 6
5733.2.a.bl.1.3 5 21.20 even 2
5733.2.a.bm.1.3 5 3.2 odd 2
8281.2.a.bw.1.3 5 91.90 odd 2
8281.2.a.bx.1.3 5 13.12 even 2