# Properties

 Label 644.2.a.d.1.5 Level $644$ Weight $2$ Character 644.1 Self dual yes Analytic conductor $5.142$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$644 = 2^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 644.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.14236589017$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.6963152.1 Defining polynomial: $$x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10$$ x^5 - 2*x^4 - 10*x^3 + 10*x^2 + 29*x + 10 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$-2.25688$$ of defining polynomial Character $$\chi$$ $$=$$ 644.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.25688 q^{3} -3.04771 q^{5} -1.00000 q^{7} +7.60729 q^{9} +O(q^{10})$$ $$q+3.25688 q^{3} -3.04771 q^{5} -1.00000 q^{7} +7.60729 q^{9} +5.22523 q^{11} +3.38206 q^{13} -9.92604 q^{15} -2.63895 q^{17} +3.39461 q^{19} -3.25688 q^{21} -1.00000 q^{23} +4.28854 q^{25} +15.0054 q^{27} -4.00190 q^{29} +6.97185 q^{31} +17.0180 q^{33} +3.04771 q^{35} -5.11916 q^{37} +11.0150 q^{39} -5.89583 q^{41} -7.90838 q^{43} -23.1848 q^{45} +9.54893 q^{47} +1.00000 q^{49} -8.59475 q^{51} -11.8200 q^{53} -15.9250 q^{55} +11.0558 q^{57} +1.51979 q^{59} +1.13934 q^{61} -7.60729 q^{63} -10.3076 q^{65} -8.80688 q^{67} -3.25688 q^{69} -14.5333 q^{71} -13.2271 q^{73} +13.9673 q^{75} -5.22523 q^{77} +14.4398 q^{79} +26.0490 q^{81} -0.693795 q^{83} +8.04275 q^{85} -13.0337 q^{87} +10.8511 q^{89} -3.38206 q^{91} +22.7065 q^{93} -10.3458 q^{95} -3.23476 q^{97} +39.7498 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10})$$ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9 $$5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9} + 2 q^{11} + 13 q^{13} + 4 q^{15} + 4 q^{17} + 12 q^{19} - 3 q^{21} - 5 q^{23} + 19 q^{25} + 15 q^{27} + 13 q^{29} - 3 q^{31} + 24 q^{33} - 2 q^{35} - 4 q^{37} + 3 q^{39} + q^{41} - 8 q^{43} - 16 q^{45} + 5 q^{47} + 5 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 12 q^{59} + 20 q^{61} - 10 q^{63} - 12 q^{65} - 12 q^{67} - 3 q^{69} + 9 q^{71} - 9 q^{73} + 35 q^{75} - 2 q^{77} - 8 q^{79} - 11 q^{81} - 28 q^{83} + 16 q^{85} - 15 q^{87} + 32 q^{89} - 13 q^{91} - 15 q^{93} - 36 q^{95} + 4 q^{97} + 28 q^{99}+O(q^{100})$$ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9 + 2 * q^11 + 13 * q^13 + 4 * q^15 + 4 * q^17 + 12 * q^19 - 3 * q^21 - 5 * q^23 + 19 * q^25 + 15 * q^27 + 13 * q^29 - 3 * q^31 + 24 * q^33 - 2 * q^35 - 4 * q^37 + 3 * q^39 + q^41 - 8 * q^43 - 16 * q^45 + 5 * q^47 + 5 * q^49 - 16 * q^51 - 8 * q^53 - 2 * q^55 + 12 * q^57 + 12 * q^59 + 20 * q^61 - 10 * q^63 - 12 * q^65 - 12 * q^67 - 3 * q^69 + 9 * q^71 - 9 * q^73 + 35 * q^75 - 2 * q^77 - 8 * q^79 - 11 * q^81 - 28 * q^83 + 16 * q^85 - 15 * q^87 + 32 * q^89 - 13 * q^91 - 15 * q^93 - 36 * q^95 + 4 * q^97 + 28 * 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
$$3$$ 3.25688 1.88036 0.940181 0.340675i $$-0.110655\pi$$
0.940181 + 0.340675i $$0.110655\pi$$
$$4$$ 0 0
$$5$$ −3.04771 −1.36298 −0.681489 0.731828i $$-0.738667\pi$$
−0.681489 + 0.731828i $$0.738667\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 7.60729 2.53576
$$10$$ 0 0
$$11$$ 5.22523 1.57546 0.787732 0.616018i $$-0.211255\pi$$
0.787732 + 0.616018i $$0.211255\pi$$
$$12$$ 0 0
$$13$$ 3.38206 0.938016 0.469008 0.883194i $$-0.344612\pi$$
0.469008 + 0.883194i $$0.344612\pi$$
$$14$$ 0 0
$$15$$ −9.92604 −2.56289
$$16$$ 0 0
$$17$$ −2.63895 −0.640039 −0.320019 0.947411i $$-0.603689\pi$$
−0.320019 + 0.947411i $$0.603689\pi$$
$$18$$ 0 0
$$19$$ 3.39461 0.778777 0.389388 0.921074i $$-0.372686\pi$$
0.389388 + 0.921074i $$0.372686\pi$$
$$20$$ 0 0
$$21$$ −3.25688 −0.710710
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 4.28854 0.857708
$$26$$ 0 0
$$27$$ 15.0054 2.88779
$$28$$ 0 0
$$29$$ −4.00190 −0.743134 −0.371567 0.928406i $$-0.621179\pi$$
−0.371567 + 0.928406i $$0.621179\pi$$
$$30$$ 0 0
$$31$$ 6.97185 1.25218 0.626091 0.779750i $$-0.284654\pi$$
0.626091 + 0.779750i $$0.284654\pi$$
$$32$$ 0 0
$$33$$ 17.0180 2.96245
$$34$$ 0 0
$$35$$ 3.04771 0.515157
$$36$$ 0 0
$$37$$ −5.11916 −0.841585 −0.420792 0.907157i $$-0.638248\pi$$
−0.420792 + 0.907157i $$0.638248\pi$$
$$38$$ 0 0
$$39$$ 11.0150 1.76381
$$40$$ 0 0
$$41$$ −5.89583 −0.920774 −0.460387 0.887718i $$-0.652289\pi$$
−0.460387 + 0.887718i $$0.652289\pi$$
$$42$$ 0 0
$$43$$ −7.90838 −1.20602 −0.603008 0.797735i $$-0.706031\pi$$
−0.603008 + 0.797735i $$0.706031\pi$$
$$44$$ 0 0
$$45$$ −23.1848 −3.45619
$$46$$ 0 0
$$47$$ 9.54893 1.39286 0.696428 0.717627i $$-0.254772\pi$$
0.696428 + 0.717627i $$0.254772\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −8.59475 −1.20351
$$52$$ 0 0
$$53$$ −11.8200 −1.62360 −0.811799 0.583937i $$-0.801512\pi$$
−0.811799 + 0.583937i $$0.801512\pi$$
$$54$$ 0 0
$$55$$ −15.9250 −2.14732
$$56$$ 0 0
$$57$$ 11.0558 1.46438
$$58$$ 0 0
$$59$$ 1.51979 0.197860 0.0989299 0.995094i $$-0.468458\pi$$
0.0989299 + 0.995094i $$0.468458\pi$$
$$60$$ 0 0
$$61$$ 1.13934 0.145877 0.0729385 0.997336i $$-0.476762\pi$$
0.0729385 + 0.997336i $$0.476762\pi$$
$$62$$ 0 0
$$63$$ −7.60729 −0.958428
$$64$$ 0 0
$$65$$ −10.3076 −1.27849
$$66$$ 0 0
$$67$$ −8.80688 −1.07593 −0.537966 0.842967i $$-0.680807\pi$$
−0.537966 + 0.842967i $$0.680807\pi$$
$$68$$ 0 0
$$69$$ −3.25688 −0.392083
$$70$$ 0 0
$$71$$ −14.5333 −1.72479 −0.862394 0.506237i $$-0.831036\pi$$
−0.862394 + 0.506237i $$0.831036\pi$$
$$72$$ 0 0
$$73$$ −13.2271 −1.54812 −0.774059 0.633114i $$-0.781777\pi$$
−0.774059 + 0.633114i $$0.781777\pi$$
$$74$$ 0 0
$$75$$ 13.9673 1.61280
$$76$$ 0 0
$$77$$ −5.22523 −0.595470
$$78$$ 0 0
$$79$$ 14.4398 1.62461 0.812303 0.583236i $$-0.198214\pi$$
0.812303 + 0.583236i $$0.198214\pi$$
$$80$$ 0 0
$$81$$ 26.0490 2.89433
$$82$$ 0 0
$$83$$ −0.693795 −0.0761538 −0.0380769 0.999275i $$-0.512123\pi$$
−0.0380769 + 0.999275i $$0.512123\pi$$
$$84$$ 0 0
$$85$$ 8.04275 0.872359
$$86$$ 0 0
$$87$$ −13.0337 −1.39736
$$88$$ 0 0
$$89$$ 10.8511 1.15021 0.575106 0.818079i $$-0.304961\pi$$
0.575106 + 0.818079i $$0.304961\pi$$
$$90$$ 0 0
$$91$$ −3.38206 −0.354537
$$92$$ 0 0
$$93$$ 22.7065 2.35456
$$94$$ 0 0
$$95$$ −10.3458 −1.06146
$$96$$ 0 0
$$97$$ −3.23476 −0.328440 −0.164220 0.986424i $$-0.552511\pi$$
−0.164220 + 0.986424i $$0.552511\pi$$
$$98$$ 0 0
$$99$$ 39.7498 3.99501
$$100$$ 0 0
$$101$$ −8.32672 −0.828540 −0.414270 0.910154i $$-0.635963\pi$$
−0.414270 + 0.910154i $$0.635963\pi$$
$$102$$ 0 0
$$103$$ 0.418345 0.0412208 0.0206104 0.999788i $$-0.493439\pi$$
0.0206104 + 0.999788i $$0.493439\pi$$
$$104$$ 0 0
$$105$$ 9.92604 0.968682
$$106$$ 0 0
$$107$$ −11.9084 −1.15123 −0.575613 0.817722i $$-0.695237\pi$$
−0.575613 + 0.817722i $$0.695237\pi$$
$$108$$ 0 0
$$109$$ 17.9154 1.71598 0.857992 0.513663i $$-0.171712\pi$$
0.857992 + 0.513663i $$0.171712\pi$$
$$110$$ 0 0
$$111$$ −16.6725 −1.58248
$$112$$ 0 0
$$113$$ −12.6375 −1.18884 −0.594418 0.804156i $$-0.702617\pi$$
−0.594418 + 0.804156i $$0.702617\pi$$
$$114$$ 0 0
$$115$$ 3.04771 0.284201
$$116$$ 0 0
$$117$$ 25.7283 2.37859
$$118$$ 0 0
$$119$$ 2.63895 0.241912
$$120$$ 0 0
$$121$$ 16.3030 1.48209
$$122$$ 0 0
$$123$$ −19.2020 −1.73139
$$124$$ 0 0
$$125$$ 2.16832 0.193940
$$126$$ 0 0
$$127$$ −3.22333 −0.286024 −0.143012 0.989721i $$-0.545679\pi$$
−0.143012 + 0.989721i $$0.545679\pi$$
$$128$$ 0 0
$$129$$ −25.7567 −2.26775
$$130$$ 0 0
$$131$$ −14.0910 −1.23114 −0.615569 0.788083i $$-0.711074\pi$$
−0.615569 + 0.788083i $$0.711074\pi$$
$$132$$ 0 0
$$133$$ −3.39461 −0.294350
$$134$$ 0 0
$$135$$ −45.7321 −3.93600
$$136$$ 0 0
$$137$$ 2.67250 0.228327 0.114164 0.993462i $$-0.463581\pi$$
0.114164 + 0.993462i $$0.463581\pi$$
$$138$$ 0 0
$$139$$ 6.23315 0.528689 0.264344 0.964428i $$-0.414844\pi$$
0.264344 + 0.964428i $$0.414844\pi$$
$$140$$ 0 0
$$141$$ 31.0998 2.61907
$$142$$ 0 0
$$143$$ 17.6721 1.47781
$$144$$ 0 0
$$145$$ 12.1966 1.01287
$$146$$ 0 0
$$147$$ 3.25688 0.268623
$$148$$ 0 0
$$149$$ −5.58165 −0.457267 −0.228633 0.973513i $$-0.573426\pi$$
−0.228633 + 0.973513i $$0.573426\pi$$
$$150$$ 0 0
$$151$$ −11.8325 −0.962916 −0.481458 0.876469i $$-0.659893\pi$$
−0.481458 + 0.876469i $$0.659893\pi$$
$$152$$ 0 0
$$153$$ −20.0752 −1.62299
$$154$$ 0 0
$$155$$ −21.2482 −1.70670
$$156$$ 0 0
$$157$$ 17.3015 1.38081 0.690406 0.723422i $$-0.257432\pi$$
0.690406 + 0.723422i $$0.257432\pi$$
$$158$$ 0 0
$$159$$ −38.4963 −3.05295
$$160$$ 0 0
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ 14.3212 1.12172 0.560861 0.827910i $$-0.310470\pi$$
0.560861 + 0.827910i $$0.310470\pi$$
$$164$$ 0 0
$$165$$ −51.8658 −4.03775
$$166$$ 0 0
$$167$$ 6.28358 0.486238 0.243119 0.969996i $$-0.421829\pi$$
0.243119 + 0.969996i $$0.421829\pi$$
$$168$$ 0 0
$$169$$ −1.56164 −0.120126
$$170$$ 0 0
$$171$$ 25.8238 1.97479
$$172$$ 0 0
$$173$$ −2.60539 −0.198084 −0.0990421 0.995083i $$-0.531578\pi$$
−0.0990421 + 0.995083i $$0.531578\pi$$
$$174$$ 0 0
$$175$$ −4.28854 −0.324183
$$176$$ 0 0
$$177$$ 4.94978 0.372048
$$178$$ 0 0
$$179$$ 25.2341 1.88609 0.943044 0.332667i $$-0.107949\pi$$
0.943044 + 0.332667i $$0.107949\pi$$
$$180$$ 0 0
$$181$$ −1.36485 −0.101448 −0.0507242 0.998713i $$-0.516153\pi$$
−0.0507242 + 0.998713i $$0.516153\pi$$
$$182$$ 0 0
$$183$$ 3.71068 0.274302
$$184$$ 0 0
$$185$$ 15.6017 1.14706
$$186$$ 0 0
$$187$$ −13.7891 −1.00836
$$188$$ 0 0
$$189$$ −15.0054 −1.09148
$$190$$ 0 0
$$191$$ −22.1467 −1.60248 −0.801239 0.598344i $$-0.795826\pi$$
−0.801239 + 0.598344i $$0.795826\pi$$
$$192$$ 0 0
$$193$$ 18.3653 1.32197 0.660983 0.750401i $$-0.270139\pi$$
0.660983 + 0.750401i $$0.270139\pi$$
$$194$$ 0 0
$$195$$ −33.5705 −2.40403
$$196$$ 0 0
$$197$$ −1.07831 −0.0768261 −0.0384131 0.999262i $$-0.512230\pi$$
−0.0384131 + 0.999262i $$0.512230\pi$$
$$198$$ 0 0
$$199$$ −5.55335 −0.393666 −0.196833 0.980437i $$-0.563066\pi$$
−0.196833 + 0.980437i $$0.563066\pi$$
$$200$$ 0 0
$$201$$ −28.6830 −2.02314
$$202$$ 0 0
$$203$$ 4.00190 0.280878
$$204$$ 0 0
$$205$$ 17.9688 1.25499
$$206$$ 0 0
$$207$$ −7.60729 −0.528743
$$208$$ 0 0
$$209$$ 17.7376 1.22693
$$210$$ 0 0
$$211$$ −19.3543 −1.33240 −0.666201 0.745772i $$-0.732081\pi$$
−0.666201 + 0.745772i $$0.732081\pi$$
$$212$$ 0 0
$$213$$ −47.3334 −3.24323
$$214$$ 0 0
$$215$$ 24.1024 1.64377
$$216$$ 0 0
$$217$$ −6.97185 −0.473280
$$218$$ 0 0
$$219$$ −43.0792 −2.91102
$$220$$ 0 0
$$221$$ −8.92509 −0.600367
$$222$$ 0 0
$$223$$ −2.69234 −0.180293 −0.0901464 0.995929i $$-0.528733\pi$$
−0.0901464 + 0.995929i $$0.528733\pi$$
$$224$$ 0 0
$$225$$ 32.6242 2.17495
$$226$$ 0 0
$$227$$ −8.26663 −0.548675 −0.274338 0.961633i $$-0.588459\pi$$
−0.274338 + 0.961633i $$0.588459\pi$$
$$228$$ 0 0
$$229$$ 26.2369 1.73378 0.866891 0.498499i $$-0.166115\pi$$
0.866891 + 0.498499i $$0.166115\pi$$
$$230$$ 0 0
$$231$$ −17.0180 −1.11970
$$232$$ 0 0
$$233$$ 1.17735 0.0771310 0.0385655 0.999256i $$-0.487721\pi$$
0.0385655 + 0.999256i $$0.487721\pi$$
$$234$$ 0 0
$$235$$ −29.1024 −1.89843
$$236$$ 0 0
$$237$$ 47.0288 3.05485
$$238$$ 0 0
$$239$$ 5.56911 0.360236 0.180118 0.983645i $$-0.442352\pi$$
0.180118 + 0.983645i $$0.442352\pi$$
$$240$$ 0 0
$$241$$ 8.12518 0.523389 0.261694 0.965151i $$-0.415719\pi$$
0.261694 + 0.965151i $$0.415719\pi$$
$$242$$ 0 0
$$243$$ 39.8223 2.55460
$$244$$ 0 0
$$245$$ −3.04771 −0.194711
$$246$$ 0 0
$$247$$ 11.4808 0.730505
$$248$$ 0 0
$$249$$ −2.25961 −0.143197
$$250$$ 0 0
$$251$$ 2.78542 0.175814 0.0879071 0.996129i $$-0.471982\pi$$
0.0879071 + 0.996129i $$0.471982\pi$$
$$252$$ 0 0
$$253$$ −5.22523 −0.328507
$$254$$ 0 0
$$255$$ 26.1943 1.64035
$$256$$ 0 0
$$257$$ −24.1150 −1.50425 −0.752126 0.659020i $$-0.770971\pi$$
−0.752126 + 0.659020i $$0.770971\pi$$
$$258$$ 0 0
$$259$$ 5.11916 0.318089
$$260$$ 0 0
$$261$$ −30.4436 −1.88441
$$262$$ 0 0
$$263$$ 3.40905 0.210211 0.105106 0.994461i $$-0.466482\pi$$
0.105106 + 0.994461i $$0.466482\pi$$
$$264$$ 0 0
$$265$$ 36.0239 2.21293
$$266$$ 0 0
$$267$$ 35.3407 2.16282
$$268$$ 0 0
$$269$$ 5.29044 0.322564 0.161282 0.986908i $$-0.448437\pi$$
0.161282 + 0.986908i $$0.448437\pi$$
$$270$$ 0 0
$$271$$ 16.2206 0.985331 0.492666 0.870219i $$-0.336023\pi$$
0.492666 + 0.870219i $$0.336023\pi$$
$$272$$ 0 0
$$273$$ −11.0150 −0.666658
$$274$$ 0 0
$$275$$ 22.4086 1.35129
$$276$$ 0 0
$$277$$ −22.2211 −1.33513 −0.667567 0.744550i $$-0.732664\pi$$
−0.667567 + 0.744550i $$0.732664\pi$$
$$278$$ 0 0
$$279$$ 53.0369 3.17524
$$280$$ 0 0
$$281$$ −5.74964 −0.342995 −0.171497 0.985185i $$-0.554860\pi$$
−0.171497 + 0.985185i $$0.554860\pi$$
$$282$$ 0 0
$$283$$ −14.0108 −0.832857 −0.416428 0.909169i $$-0.636718\pi$$
−0.416428 + 0.909169i $$0.636718\pi$$
$$284$$ 0 0
$$285$$ −33.6950 −1.99592
$$286$$ 0 0
$$287$$ 5.89583 0.348020
$$288$$ 0 0
$$289$$ −10.0360 −0.590350
$$290$$ 0 0
$$291$$ −10.5352 −0.617586
$$292$$ 0 0
$$293$$ 15.7616 0.920801 0.460400 0.887711i $$-0.347706\pi$$
0.460400 + 0.887711i $$0.347706\pi$$
$$294$$ 0 0
$$295$$ −4.63188 −0.269678
$$296$$ 0 0
$$297$$ 78.4066 4.54961
$$298$$ 0 0
$$299$$ −3.38206 −0.195590
$$300$$ 0 0
$$301$$ 7.90838 0.455831
$$302$$ 0 0
$$303$$ −27.1192 −1.55795
$$304$$ 0 0
$$305$$ −3.47237 −0.198827
$$306$$ 0 0
$$307$$ 11.4944 0.656018 0.328009 0.944675i $$-0.393622\pi$$
0.328009 + 0.944675i $$0.393622\pi$$
$$308$$ 0 0
$$309$$ 1.36250 0.0775100
$$310$$ 0 0
$$311$$ 4.41105 0.250128 0.125064 0.992149i $$-0.460086\pi$$
0.125064 + 0.992149i $$0.460086\pi$$
$$312$$ 0 0
$$313$$ −6.20154 −0.350532 −0.175266 0.984521i $$-0.556079\pi$$
−0.175266 + 0.984521i $$0.556079\pi$$
$$314$$ 0 0
$$315$$ 23.1848 1.30632
$$316$$ 0 0
$$317$$ −9.50769 −0.534005 −0.267003 0.963696i $$-0.586033\pi$$
−0.267003 + 0.963696i $$0.586033\pi$$
$$318$$ 0 0
$$319$$ −20.9108 −1.17078
$$320$$ 0 0
$$321$$ −38.7842 −2.16472
$$322$$ 0 0
$$323$$ −8.95819 −0.498447
$$324$$ 0 0
$$325$$ 14.5041 0.804544
$$326$$ 0 0
$$327$$ 58.3484 3.22667
$$328$$ 0 0
$$329$$ −9.54893 −0.526450
$$330$$ 0 0
$$331$$ 10.0087 0.550130 0.275065 0.961426i $$-0.411301\pi$$
0.275065 + 0.961426i $$0.411301\pi$$
$$332$$ 0 0
$$333$$ −38.9429 −2.13406
$$334$$ 0 0
$$335$$ 26.8408 1.46647
$$336$$ 0 0
$$337$$ 16.5389 0.900929 0.450464 0.892794i $$-0.351258\pi$$
0.450464 + 0.892794i $$0.351258\pi$$
$$338$$ 0 0
$$339$$ −41.1589 −2.23544
$$340$$ 0 0
$$341$$ 36.4295 1.97277
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 9.92604 0.534400
$$346$$ 0 0
$$347$$ 34.8917 1.87308 0.936541 0.350558i $$-0.114008\pi$$
0.936541 + 0.350558i $$0.114008\pi$$
$$348$$ 0 0
$$349$$ 6.92337 0.370599 0.185300 0.982682i $$-0.440674\pi$$
0.185300 + 0.982682i $$0.440674\pi$$
$$350$$ 0 0
$$351$$ 50.7493 2.70880
$$352$$ 0 0
$$353$$ 21.6050 1.14992 0.574959 0.818182i $$-0.305018\pi$$
0.574959 + 0.818182i $$0.305018\pi$$
$$354$$ 0 0
$$355$$ 44.2934 2.35085
$$356$$ 0 0
$$357$$ 8.59475 0.454882
$$358$$ 0 0
$$359$$ −26.2085 −1.38323 −0.691616 0.722265i $$-0.743101\pi$$
−0.691616 + 0.722265i $$0.743101\pi$$
$$360$$ 0 0
$$361$$ −7.47664 −0.393507
$$362$$ 0 0
$$363$$ 53.0969 2.78687
$$364$$ 0 0
$$365$$ 40.3124 2.11005
$$366$$ 0 0
$$367$$ 16.5650 0.864688 0.432344 0.901709i $$-0.357687\pi$$
0.432344 + 0.901709i $$0.357687\pi$$
$$368$$ 0 0
$$369$$ −44.8513 −2.33487
$$370$$ 0 0
$$371$$ 11.8200 0.613662
$$372$$ 0 0
$$373$$ 15.6116 0.808340 0.404170 0.914684i $$-0.367560\pi$$
0.404170 + 0.914684i $$0.367560\pi$$
$$374$$ 0 0
$$375$$ 7.06197 0.364678
$$376$$ 0 0
$$377$$ −13.5347 −0.697071
$$378$$ 0 0
$$379$$ −9.80933 −0.503871 −0.251936 0.967744i $$-0.581067\pi$$
−0.251936 + 0.967744i $$0.581067\pi$$
$$380$$ 0 0
$$381$$ −10.4980 −0.537829
$$382$$ 0 0
$$383$$ −18.4925 −0.944921 −0.472461 0.881352i $$-0.656634\pi$$
−0.472461 + 0.881352i $$0.656634\pi$$
$$384$$ 0 0
$$385$$ 15.9250 0.811612
$$386$$ 0 0
$$387$$ −60.1613 −3.05817
$$388$$ 0 0
$$389$$ 11.3030 0.573084 0.286542 0.958068i $$-0.407494\pi$$
0.286542 + 0.958068i $$0.407494\pi$$
$$390$$ 0 0
$$391$$ 2.63895 0.133457
$$392$$ 0 0
$$393$$ −45.8928 −2.31498
$$394$$ 0 0
$$395$$ −44.0084 −2.21430
$$396$$ 0 0
$$397$$ −1.20204 −0.0603285 −0.0301643 0.999545i $$-0.509603\pi$$
−0.0301643 + 0.999545i $$0.509603\pi$$
$$398$$ 0 0
$$399$$ −11.0558 −0.553484
$$400$$ 0 0
$$401$$ −3.01127 −0.150376 −0.0751878 0.997169i $$-0.523956\pi$$
−0.0751878 + 0.997169i $$0.523956\pi$$
$$402$$ 0 0
$$403$$ 23.5793 1.17457
$$404$$ 0 0
$$405$$ −79.3898 −3.94491
$$406$$ 0 0
$$407$$ −26.7488 −1.32589
$$408$$ 0 0
$$409$$ −33.4725 −1.65511 −0.827553 0.561387i $$-0.810268\pi$$
−0.827553 + 0.561387i $$0.810268\pi$$
$$410$$ 0 0
$$411$$ 8.70404 0.429338
$$412$$ 0 0
$$413$$ −1.51979 −0.0747839
$$414$$ 0 0
$$415$$ 2.11449 0.103796
$$416$$ 0 0
$$417$$ 20.3006 0.994126
$$418$$ 0 0
$$419$$ −6.40028 −0.312674 −0.156337 0.987704i $$-0.549969\pi$$
−0.156337 + 0.987704i $$0.549969\pi$$
$$420$$ 0 0
$$421$$ −8.01082 −0.390423 −0.195212 0.980761i $$-0.562539\pi$$
−0.195212 + 0.980761i $$0.562539\pi$$
$$422$$ 0 0
$$423$$ 72.6415 3.53195
$$424$$ 0 0
$$425$$ −11.3172 −0.548967
$$426$$ 0 0
$$427$$ −1.13934 −0.0551363
$$428$$ 0 0
$$429$$ 57.5558 2.77882
$$430$$ 0 0
$$431$$ −27.6084 −1.32985 −0.664925 0.746910i $$-0.731537\pi$$
−0.664925 + 0.746910i $$0.731537\pi$$
$$432$$ 0 0
$$433$$ −14.8797 −0.715074 −0.357537 0.933899i $$-0.616383\pi$$
−0.357537 + 0.933899i $$0.616383\pi$$
$$434$$ 0 0
$$435$$ 39.7230 1.90457
$$436$$ 0 0
$$437$$ −3.39461 −0.162386
$$438$$ 0 0
$$439$$ −36.1931 −1.72740 −0.863702 0.504004i $$-0.831860\pi$$
−0.863702 + 0.504004i $$0.831860\pi$$
$$440$$ 0 0
$$441$$ 7.60729 0.362252
$$442$$ 0 0
$$443$$ −10.5546 −0.501465 −0.250733 0.968056i $$-0.580671\pi$$
−0.250733 + 0.968056i $$0.580671\pi$$
$$444$$ 0 0
$$445$$ −33.0710 −1.56771
$$446$$ 0 0
$$447$$ −18.1788 −0.859828
$$448$$ 0 0
$$449$$ 39.9560 1.88564 0.942821 0.333301i $$-0.108162\pi$$
0.942821 + 0.333301i $$0.108162\pi$$
$$450$$ 0 0
$$451$$ −30.8071 −1.45065
$$452$$ 0 0
$$453$$ −38.5371 −1.81063
$$454$$ 0 0
$$455$$ 10.3076 0.483226
$$456$$ 0 0
$$457$$ 10.1870 0.476530 0.238265 0.971200i $$-0.423421\pi$$
0.238265 + 0.971200i $$0.423421\pi$$
$$458$$ 0 0
$$459$$ −39.5985 −1.84830
$$460$$ 0 0
$$461$$ −6.70956 −0.312495 −0.156248 0.987718i $$-0.549940\pi$$
−0.156248 + 0.987718i $$0.549940\pi$$
$$462$$ 0 0
$$463$$ 16.6512 0.773848 0.386924 0.922112i $$-0.373538\pi$$
0.386924 + 0.922112i $$0.373538\pi$$
$$464$$ 0 0
$$465$$ −69.2029 −3.20921
$$466$$ 0 0
$$467$$ 39.1863 1.81332 0.906662 0.421857i $$-0.138622\pi$$
0.906662 + 0.421857i $$0.138622\pi$$
$$468$$ 0 0
$$469$$ 8.80688 0.406664
$$470$$ 0 0
$$471$$ 56.3491 2.59643
$$472$$ 0 0
$$473$$ −41.3230 −1.90004
$$474$$ 0 0
$$475$$ 14.5579 0.667963
$$476$$ 0 0
$$477$$ −89.9180 −4.11706
$$478$$ 0 0
$$479$$ −36.2704 −1.65724 −0.828619 0.559813i $$-0.810873\pi$$
−0.828619 + 0.559813i $$0.810873\pi$$
$$480$$ 0 0
$$481$$ −17.3133 −0.789420
$$482$$ 0 0
$$483$$ 3.25688 0.148193
$$484$$ 0 0
$$485$$ 9.85861 0.447656
$$486$$ 0 0
$$487$$ 23.7517 1.07629 0.538146 0.842851i $$-0.319125\pi$$
0.538146 + 0.842851i $$0.319125\pi$$
$$488$$ 0 0
$$489$$ 46.6425 2.10925
$$490$$ 0 0
$$491$$ −22.3752 −1.00978 −0.504889 0.863185i $$-0.668466\pi$$
−0.504889 + 0.863185i $$0.668466\pi$$
$$492$$ 0 0
$$493$$ 10.5608 0.475635
$$494$$ 0 0
$$495$$ −121.146 −5.44510
$$496$$ 0 0
$$497$$ 14.5333 0.651909
$$498$$ 0 0
$$499$$ 16.2296 0.726535 0.363268 0.931685i $$-0.381661\pi$$
0.363268 + 0.931685i $$0.381661\pi$$
$$500$$ 0 0
$$501$$ 20.4649 0.914304
$$502$$ 0 0
$$503$$ −2.57708 −0.114906 −0.0574532 0.998348i $$-0.518298\pi$$
−0.0574532 + 0.998348i $$0.518298\pi$$
$$504$$ 0 0
$$505$$ 25.3774 1.12928
$$506$$ 0 0
$$507$$ −5.08608 −0.225881
$$508$$ 0 0
$$509$$ 33.4091 1.48083 0.740417 0.672148i $$-0.234628\pi$$
0.740417 + 0.672148i $$0.234628\pi$$
$$510$$ 0 0
$$511$$ 13.2271 0.585134
$$512$$ 0 0
$$513$$ 50.9375 2.24894
$$514$$ 0 0
$$515$$ −1.27500 −0.0561830
$$516$$ 0 0
$$517$$ 49.8953 2.19439
$$518$$ 0 0
$$519$$ −8.48546 −0.372470
$$520$$ 0 0
$$521$$ 12.4303 0.544580 0.272290 0.962215i $$-0.412219\pi$$
0.272290 + 0.962215i $$0.412219\pi$$
$$522$$ 0 0
$$523$$ 1.05262 0.0460279 0.0230140 0.999735i $$-0.492674\pi$$
0.0230140 + 0.999735i $$0.492674\pi$$
$$524$$ 0 0
$$525$$ −13.9673 −0.609582
$$526$$ 0 0
$$527$$ −18.3984 −0.801445
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 11.5615 0.501725
$$532$$ 0 0
$$533$$ −19.9401 −0.863701
$$534$$ 0 0
$$535$$ 36.2933 1.56910
$$536$$ 0 0
$$537$$ 82.1847 3.54653
$$538$$ 0 0
$$539$$ 5.22523 0.225066
$$540$$ 0 0
$$541$$ 29.4265 1.26514 0.632572 0.774502i $$-0.281999\pi$$
0.632572 + 0.774502i $$0.281999\pi$$
$$542$$ 0 0
$$543$$ −4.44515 −0.190760
$$544$$ 0 0
$$545$$ −54.6009 −2.33885
$$546$$ 0 0
$$547$$ −36.3851 −1.55571 −0.777857 0.628441i $$-0.783693\pi$$
−0.777857 + 0.628441i $$0.783693\pi$$
$$548$$ 0 0
$$549$$ 8.66726 0.369910
$$550$$ 0 0
$$551$$ −13.5849 −0.578735
$$552$$ 0 0
$$553$$ −14.4398 −0.614043
$$554$$ 0 0
$$555$$ 50.8130 2.15689
$$556$$ 0 0
$$557$$ −22.2666 −0.943467 −0.471734 0.881741i $$-0.656372\pi$$
−0.471734 + 0.881741i $$0.656372\pi$$
$$558$$ 0 0
$$559$$ −26.7466 −1.13126
$$560$$ 0 0
$$561$$ −44.9095 −1.89608
$$562$$ 0 0
$$563$$ 22.0300 0.928453 0.464227 0.885717i $$-0.346332\pi$$
0.464227 + 0.885717i $$0.346332\pi$$
$$564$$ 0 0
$$565$$ 38.5154 1.62036
$$566$$ 0 0
$$567$$ −26.0490 −1.09395
$$568$$ 0 0
$$569$$ 27.9918 1.17348 0.586738 0.809777i $$-0.300412\pi$$
0.586738 + 0.809777i $$0.300412\pi$$
$$570$$ 0 0
$$571$$ 19.7000 0.824421 0.412210 0.911089i $$-0.364757\pi$$
0.412210 + 0.911089i $$0.364757\pi$$
$$572$$ 0 0
$$573$$ −72.1292 −3.01324
$$574$$ 0 0
$$575$$ −4.28854 −0.178845
$$576$$ 0 0
$$577$$ 11.9624 0.498000 0.249000 0.968503i $$-0.419898\pi$$
0.249000 + 0.968503i $$0.419898\pi$$
$$578$$ 0 0
$$579$$ 59.8138 2.48578
$$580$$ 0 0
$$581$$ 0.693795 0.0287834
$$582$$ 0 0
$$583$$ −61.7620 −2.55792
$$584$$ 0 0
$$585$$ −78.4126 −3.24196
$$586$$ 0 0
$$587$$ −19.3278 −0.797743 −0.398872 0.917007i $$-0.630598\pi$$
−0.398872 + 0.917007i $$0.630598\pi$$
$$588$$ 0 0
$$589$$ 23.6667 0.975170
$$590$$ 0 0
$$591$$ −3.51192 −0.144461
$$592$$ 0 0
$$593$$ 8.45045 0.347018 0.173509 0.984832i $$-0.444489\pi$$
0.173509 + 0.984832i $$0.444489\pi$$
$$594$$ 0 0
$$595$$ −8.04275 −0.329721
$$596$$ 0 0
$$597$$ −18.0866 −0.740235
$$598$$ 0 0
$$599$$ 5.56581 0.227413 0.113706 0.993514i $$-0.463728\pi$$
0.113706 + 0.993514i $$0.463728\pi$$
$$600$$ 0 0
$$601$$ −18.2258 −0.743445 −0.371722 0.928344i $$-0.621233\pi$$
−0.371722 + 0.928344i $$0.621233\pi$$
$$602$$ 0 0
$$603$$ −66.9965 −2.72831
$$604$$ 0 0
$$605$$ −49.6868 −2.02005
$$606$$ 0 0
$$607$$ 28.3740 1.15166 0.575832 0.817568i $$-0.304678\pi$$
0.575832 + 0.817568i $$0.304678\pi$$
$$608$$ 0 0
$$609$$ 13.0337 0.528153
$$610$$ 0 0
$$611$$ 32.2951 1.30652
$$612$$ 0 0
$$613$$ 15.8620 0.640660 0.320330 0.947306i $$-0.396206\pi$$
0.320330 + 0.947306i $$0.396206\pi$$
$$614$$ 0 0
$$615$$ 58.5223 2.35985
$$616$$ 0 0
$$617$$ −14.9140 −0.600417 −0.300208 0.953874i $$-0.597056\pi$$
−0.300208 + 0.953874i $$0.597056\pi$$
$$618$$ 0 0
$$619$$ 32.3397 1.29984 0.649920 0.760002i $$-0.274802\pi$$
0.649920 + 0.760002i $$0.274802\pi$$
$$620$$ 0 0
$$621$$ −15.0054 −0.602146
$$622$$ 0 0
$$623$$ −10.8511 −0.434739
$$624$$ 0 0
$$625$$ −28.0511 −1.12204
$$626$$ 0 0
$$627$$ 57.7693 2.30708
$$628$$ 0 0
$$629$$ 13.5092 0.538647
$$630$$ 0 0
$$631$$ −15.3840 −0.612426 −0.306213 0.951963i $$-0.599062\pi$$
−0.306213 + 0.951963i $$0.599062\pi$$
$$632$$ 0 0
$$633$$ −63.0346 −2.50540
$$634$$ 0 0
$$635$$ 9.82377 0.389844
$$636$$ 0 0
$$637$$ 3.38206 0.134002
$$638$$ 0 0
$$639$$ −110.559 −4.37366
$$640$$ 0 0
$$641$$ 37.0193 1.46217 0.731087 0.682284i $$-0.239013\pi$$
0.731087 + 0.682284i $$0.239013\pi$$
$$642$$ 0 0
$$643$$ 24.7183 0.974796 0.487398 0.873180i $$-0.337946\pi$$
0.487398 + 0.873180i $$0.337946\pi$$
$$644$$ 0 0
$$645$$ 78.4988 3.09089
$$646$$ 0 0
$$647$$ −20.3207 −0.798887 −0.399444 0.916758i $$-0.630797\pi$$
−0.399444 + 0.916758i $$0.630797\pi$$
$$648$$ 0 0
$$649$$ 7.94124 0.311721
$$650$$ 0 0
$$651$$ −22.7065 −0.889938
$$652$$ 0 0
$$653$$ −13.7950 −0.539839 −0.269919 0.962883i $$-0.586997\pi$$
−0.269919 + 0.962883i $$0.586997\pi$$
$$654$$ 0 0
$$655$$ 42.9453 1.67801
$$656$$ 0 0
$$657$$ −100.623 −3.92566
$$658$$ 0 0
$$659$$ 39.4316 1.53604 0.768019 0.640427i $$-0.221243\pi$$
0.768019 + 0.640427i $$0.221243\pi$$
$$660$$ 0 0
$$661$$ 38.9211 1.51385 0.756927 0.653499i $$-0.226700\pi$$
0.756927 + 0.653499i $$0.226700\pi$$
$$662$$ 0 0
$$663$$ −29.0680 −1.12891
$$664$$ 0 0
$$665$$ 10.3458 0.401192
$$666$$ 0 0
$$667$$ 4.00190 0.154954
$$668$$ 0 0
$$669$$ −8.76865 −0.339016
$$670$$ 0 0
$$671$$ 5.95329 0.229824
$$672$$ 0 0
$$673$$ −43.7150 −1.68509 −0.842544 0.538627i $$-0.818943\pi$$
−0.842544 + 0.538627i $$0.818943\pi$$
$$674$$ 0 0
$$675$$ 64.3513 2.47688
$$676$$ 0 0
$$677$$ 16.6777 0.640977 0.320489 0.947252i $$-0.396153\pi$$
0.320489 + 0.947252i $$0.396153\pi$$
$$678$$ 0 0
$$679$$ 3.23476 0.124139
$$680$$ 0 0
$$681$$ −26.9234 −1.03171
$$682$$ 0 0
$$683$$ 48.7312 1.86465 0.932324 0.361625i $$-0.117778\pi$$
0.932324 + 0.361625i $$0.117778\pi$$
$$684$$ 0 0
$$685$$ −8.14502 −0.311205
$$686$$ 0 0
$$687$$ 85.4504 3.26014
$$688$$ 0 0
$$689$$ −39.9759 −1.52296
$$690$$ 0 0
$$691$$ −41.7882 −1.58970 −0.794849 0.606807i $$-0.792450\pi$$
−0.794849 + 0.606807i $$0.792450\pi$$
$$692$$ 0 0
$$693$$ −39.7498 −1.50997
$$694$$ 0 0
$$695$$ −18.9968 −0.720591
$$696$$ 0 0
$$697$$ 15.5588 0.589331
$$698$$ 0 0
$$699$$ 3.83450 0.145034
$$700$$ 0 0
$$701$$ −28.1499 −1.06321 −0.531604 0.846993i $$-0.678410\pi$$
−0.531604 + 0.846993i $$0.678410\pi$$
$$702$$ 0 0
$$703$$ −17.3775 −0.655406
$$704$$ 0 0
$$705$$ −94.7831 −3.56974
$$706$$ 0 0
$$707$$ 8.32672 0.313159
$$708$$ 0 0
$$709$$ 22.1680 0.832536 0.416268 0.909242i $$-0.363338\pi$$
0.416268 + 0.909242i $$0.363338\pi$$
$$710$$ 0 0
$$711$$ 109.848 4.11961
$$712$$ 0 0
$$713$$ −6.97185 −0.261098
$$714$$ 0 0
$$715$$ −53.8593 −2.01422
$$716$$ 0 0
$$717$$ 18.1379 0.677374
$$718$$ 0 0
$$719$$ −22.2623 −0.830243 −0.415122 0.909766i $$-0.636261\pi$$
−0.415122 + 0.909766i $$0.636261\pi$$
$$720$$ 0 0
$$721$$ −0.418345 −0.0155800
$$722$$ 0 0
$$723$$ 26.4628 0.984161
$$724$$ 0 0
$$725$$ −17.1623 −0.637392
$$726$$ 0 0
$$727$$ −21.6618 −0.803392 −0.401696 0.915773i $$-0.631579\pi$$
−0.401696 + 0.915773i $$0.631579\pi$$
$$728$$ 0 0
$$729$$ 51.5497 1.90925
$$730$$ 0 0
$$731$$ 20.8698 0.771897
$$732$$ 0 0
$$733$$ −4.50596 −0.166431 −0.0832156 0.996532i $$-0.526519\pi$$
−0.0832156 + 0.996532i $$0.526519\pi$$
$$734$$ 0 0
$$735$$ −9.92604 −0.366127
$$736$$ 0 0
$$737$$ −46.0179 −1.69509
$$738$$ 0 0
$$739$$ −15.7692 −0.580079 −0.290040 0.957015i $$-0.593669\pi$$
−0.290040 + 0.957015i $$0.593669\pi$$
$$740$$ 0 0
$$741$$ 37.3916 1.37361
$$742$$ 0 0
$$743$$ 30.8619 1.13222 0.566108 0.824331i $$-0.308449\pi$$
0.566108 + 0.824331i $$0.308449\pi$$
$$744$$ 0 0
$$745$$ 17.0113 0.623245
$$746$$ 0 0
$$747$$ −5.27790 −0.193108
$$748$$ 0 0
$$749$$ 11.9084 0.435123
$$750$$ 0 0
$$751$$ 38.1808 1.39324 0.696618 0.717442i $$-0.254687\pi$$
0.696618 + 0.717442i $$0.254687\pi$$
$$752$$ 0 0
$$753$$ 9.07179 0.330594
$$754$$ 0 0
$$755$$ 36.0621 1.31243
$$756$$ 0 0
$$757$$ −15.6733 −0.569655 −0.284828 0.958579i $$-0.591936\pi$$
−0.284828 + 0.958579i $$0.591936\pi$$
$$758$$ 0 0
$$759$$ −17.0180 −0.617713
$$760$$ 0 0
$$761$$ 41.8363 1.51657 0.758283 0.651926i $$-0.226039\pi$$
0.758283 + 0.651926i $$0.226039\pi$$
$$762$$ 0 0
$$763$$ −17.9154 −0.648581
$$764$$ 0 0
$$765$$ 61.1835 2.21210
$$766$$ 0 0
$$767$$ 5.14003 0.185596
$$768$$ 0 0
$$769$$ 5.39928 0.194703 0.0973515 0.995250i $$-0.468963\pi$$
0.0973515 + 0.995250i $$0.468963\pi$$
$$770$$ 0 0
$$771$$ −78.5397 −2.82854
$$772$$ 0 0
$$773$$ 37.1956 1.33783 0.668917 0.743337i $$-0.266758\pi$$
0.668917 + 0.743337i $$0.266758\pi$$
$$774$$ 0 0
$$775$$ 29.8991 1.07401
$$776$$ 0 0
$$777$$ 16.6725 0.598123
$$778$$ 0 0
$$779$$ −20.0140 −0.717077
$$780$$ 0 0
$$781$$ −75.9399 −2.71734
$$782$$ 0 0
$$783$$ −60.0501 −2.14602
$$784$$ 0 0
$$785$$ −52.7301 −1.88202
$$786$$ 0 0
$$787$$ −38.9942 −1.38999 −0.694997 0.719013i $$-0.744594\pi$$
−0.694997 + 0.719013i $$0.744594\pi$$
$$788$$ 0 0
$$789$$ 11.1029 0.395273
$$790$$ 0 0
$$791$$ 12.6375 0.449338
$$792$$ 0 0
$$793$$ 3.85331 0.136835
$$794$$ 0 0
$$795$$ 117.326 4.16111
$$796$$ 0 0
$$797$$ 25.8311 0.914986 0.457493 0.889213i $$-0.348747\pi$$
0.457493 + 0.889213i $$0.348747\pi$$
$$798$$ 0 0
$$799$$ −25.1991 −0.891482
$$800$$ 0 0
$$801$$ 82.5473 2.91667
$$802$$ 0 0
$$803$$ −69.1147 −2.43901
$$804$$ 0 0
$$805$$ −3.04771 −0.107418
$$806$$ 0 0
$$807$$ 17.2303 0.606537
$$808$$ 0 0
$$809$$ 26.7612 0.940875 0.470437 0.882433i $$-0.344096\pi$$
0.470437 + 0.882433i $$0.344096\pi$$
$$810$$ 0 0
$$811$$ 30.3856 1.06698 0.533492 0.845805i $$-0.320880\pi$$
0.533492 + 0.845805i $$0.320880\pi$$
$$812$$ 0 0
$$813$$ 52.8286 1.85278
$$814$$ 0 0
$$815$$ −43.6469 −1.52888
$$816$$ 0 0
$$817$$ −26.8458 −0.939217
$$818$$ 0 0
$$819$$ −25.7283 −0.899021
$$820$$ 0 0
$$821$$ −16.5069 −0.576095 −0.288048 0.957616i $$-0.593006\pi$$
−0.288048 + 0.957616i $$0.593006\pi$$
$$822$$ 0 0
$$823$$ −19.9562 −0.695631 −0.347816 0.937563i $$-0.613076\pi$$
−0.347816 + 0.937563i $$0.613076\pi$$
$$824$$ 0 0
$$825$$ 72.9822 2.54091
$$826$$ 0 0
$$827$$ −36.8443 −1.28120 −0.640601 0.767874i $$-0.721315\pi$$
−0.640601 + 0.767874i $$0.721315\pi$$
$$828$$ 0 0
$$829$$ 5.80916 0.201760 0.100880 0.994899i $$-0.467834\pi$$
0.100880 + 0.994899i $$0.467834\pi$$
$$830$$ 0 0
$$831$$ −72.3714 −2.51054
$$832$$ 0 0
$$833$$ −2.63895 −0.0914341
$$834$$ 0 0
$$835$$ −19.1505 −0.662732
$$836$$ 0 0
$$837$$ 104.615 3.61604
$$838$$ 0 0
$$839$$ −12.2733 −0.423722 −0.211861 0.977300i $$-0.567952\pi$$
−0.211861 + 0.977300i $$0.567952\pi$$
$$840$$ 0 0
$$841$$ −12.9848 −0.447752
$$842$$ 0 0
$$843$$ −18.7259 −0.644954
$$844$$ 0 0
$$845$$ 4.75942 0.163729
$$846$$ 0 0
$$847$$ −16.3030 −0.560177
$$848$$ 0 0
$$849$$ −45.6316 −1.56607
$$850$$ 0 0
$$851$$ 5.11916 0.175483
$$852$$ 0 0
$$853$$ −13.2560 −0.453878 −0.226939 0.973909i $$-0.572872\pi$$
−0.226939 + 0.973909i $$0.572872\pi$$
$$854$$ 0 0
$$855$$ −78.7034 −2.69160
$$856$$ 0 0
$$857$$ −19.1066 −0.652670 −0.326335 0.945254i $$-0.605814\pi$$
−0.326335 + 0.945254i $$0.605814\pi$$
$$858$$ 0 0
$$859$$ −46.1878 −1.57591 −0.787953 0.615735i $$-0.788859\pi$$
−0.787953 + 0.615735i $$0.788859\pi$$
$$860$$ 0 0
$$861$$ 19.2020 0.654404
$$862$$ 0 0
$$863$$ 11.5341 0.392625 0.196313 0.980541i $$-0.437103\pi$$
0.196313 + 0.980541i $$0.437103\pi$$
$$864$$ 0 0
$$865$$ 7.94048 0.269984
$$866$$ 0 0
$$867$$ −32.6859 −1.11007
$$868$$ 0 0
$$869$$ 75.4512 2.55951
$$870$$ 0 0
$$871$$ −29.7854 −1.00924
$$872$$ 0 0
$$873$$ −24.6077 −0.832846
$$874$$ 0 0
$$875$$ −2.16832 −0.0733026
$$876$$ 0 0
$$877$$ −16.9695 −0.573020 −0.286510 0.958077i $$-0.592495\pi$$
−0.286510 + 0.958077i $$0.592495\pi$$
$$878$$ 0 0
$$879$$ 51.3336 1.73144
$$880$$ 0 0
$$881$$ 11.4505 0.385775 0.192888 0.981221i $$-0.438215\pi$$
0.192888 + 0.981221i $$0.438215\pi$$
$$882$$ 0 0
$$883$$ 17.3634 0.584325 0.292162 0.956369i $$-0.405625\pi$$
0.292162 + 0.956369i $$0.405625\pi$$
$$884$$ 0 0
$$885$$ −15.0855 −0.507093
$$886$$ 0 0
$$887$$ 22.7378 0.763460 0.381730 0.924274i $$-0.375328\pi$$
0.381730 + 0.924274i $$0.375328\pi$$
$$888$$ 0 0
$$889$$ 3.22333 0.108107
$$890$$ 0 0
$$891$$ 136.112 4.55992
$$892$$ 0 0
$$893$$ 32.4149 1.08472
$$894$$ 0 0
$$895$$ −76.9064 −2.57070
$$896$$ 0 0
$$897$$ −11.0150 −0.367780
$$898$$ 0 0
$$899$$ −27.9006 −0.930539
$$900$$ 0 0
$$901$$ 31.1923 1.03917
$$902$$ 0 0
$$903$$ 25.7567 0.857128
$$904$$ 0 0
$$905$$ 4.15966 0.138272
$$906$$ 0 0
$$907$$ −41.5359 −1.37918 −0.689588 0.724202i $$-0.742208\pi$$
−0.689588 + 0.724202i $$0.742208\pi$$
$$908$$ 0 0
$$909$$ −63.3438 −2.10098
$$910$$ 0 0
$$911$$ 2.92749 0.0969921 0.0484961 0.998823i $$-0.484557\pi$$
0.0484961 + 0.998823i $$0.484557\pi$$
$$912$$ 0 0
$$913$$ −3.62523 −0.119978
$$914$$ 0 0
$$915$$ −11.3091 −0.373867
$$916$$ 0 0
$$917$$ 14.0910 0.465326
$$918$$ 0 0
$$919$$ 13.9648 0.460658 0.230329 0.973113i $$-0.426020\pi$$
0.230329 + 0.973113i $$0.426020\pi$$
$$920$$ 0 0
$$921$$ 37.4358 1.23355
$$922$$ 0 0
$$923$$ −49.1527 −1.61788
$$924$$ 0 0
$$925$$ −21.9537 −0.721834
$$926$$ 0 0
$$927$$ 3.18247 0.104526
$$928$$ 0 0
$$929$$ −23.3541 −0.766222 −0.383111 0.923702i $$-0.625147\pi$$
−0.383111 + 0.923702i $$0.625147\pi$$
$$930$$ 0 0
$$931$$ 3.39461 0.111254
$$932$$ 0 0
$$933$$ 14.3663 0.470331
$$934$$ 0 0
$$935$$ 42.0252 1.37437
$$936$$ 0 0
$$937$$ 11.1156 0.363130 0.181565 0.983379i $$-0.441884\pi$$
0.181565 + 0.983379i $$0.441884\pi$$
$$938$$ 0 0
$$939$$ −20.1977 −0.659127
$$940$$ 0 0
$$941$$ 5.13868 0.167516 0.0837580 0.996486i $$-0.473308\pi$$
0.0837580 + 0.996486i $$0.473308\pi$$
$$942$$ 0 0
$$943$$ 5.89583 0.191995
$$944$$ 0 0
$$945$$ 45.7321 1.48767
$$946$$ 0 0
$$947$$ 29.4147 0.955851 0.477925 0.878400i $$-0.341389\pi$$
0.477925 + 0.878400i $$0.341389\pi$$
$$948$$ 0 0
$$949$$ −44.7350 −1.45216
$$950$$ 0 0
$$951$$ −30.9655 −1.00412
$$952$$ 0 0
$$953$$ −5.67396 −0.183797 −0.0918987 0.995768i $$-0.529294\pi$$
−0.0918987 + 0.995768i $$0.529294\pi$$
$$954$$ 0 0
$$955$$ 67.4967 2.18414
$$956$$ 0 0
$$957$$ −68.1041 −2.20149
$$958$$ 0 0
$$959$$ −2.67250 −0.0862997
$$960$$ 0 0
$$961$$ 17.6067 0.567959
$$962$$ 0 0
$$963$$ −90.5905 −2.91924
$$964$$ 0 0
$$965$$ −55.9723 −1.80181
$$966$$ 0 0
$$967$$ −0.898731 −0.0289013 −0.0144506 0.999896i $$-0.504600\pi$$
−0.0144506 + 0.999896i $$0.504600\pi$$
$$968$$ 0 0
$$969$$ −29.1758 −0.937262
$$970$$ 0 0
$$971$$ 34.2084 1.09780 0.548899 0.835889i $$-0.315047\pi$$
0.548899 + 0.835889i $$0.315047\pi$$
$$972$$ 0 0
$$973$$ −6.23315 −0.199825
$$974$$ 0 0
$$975$$ 47.2382 1.51283
$$976$$ 0 0
$$977$$ 8.19697 0.262244 0.131122 0.991366i $$-0.458142\pi$$
0.131122 + 0.991366i $$0.458142\pi$$
$$978$$ 0 0
$$979$$ 56.6994 1.81212
$$980$$ 0 0
$$981$$ 136.288 4.35133
$$982$$ 0 0
$$983$$ −1.12997 −0.0360406 −0.0180203 0.999838i $$-0.505736\pi$$
−0.0180203 + 0.999838i $$0.505736\pi$$
$$984$$ 0 0
$$985$$ 3.28637 0.104712
$$986$$ 0 0
$$987$$ −31.0998 −0.989917
$$988$$ 0 0
$$989$$ 7.90838 0.251472
$$990$$ 0 0
$$991$$ 14.6766 0.466218 0.233109 0.972451i $$-0.425110\pi$$
0.233109 + 0.972451i $$0.425110\pi$$
$$992$$ 0 0
$$993$$ 32.5973 1.03444
$$994$$ 0 0
$$995$$ 16.9250 0.536558
$$996$$ 0 0
$$997$$ −1.64207 −0.0520049 −0.0260024 0.999662i $$-0.508278\pi$$
−0.0260024 + 0.999662i $$0.508278\pi$$
$$998$$ 0 0
$$999$$ −76.8151 −2.43032
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 644.2.a.d.1.5 5
3.2 odd 2 5796.2.a.t.1.4 5
4.3 odd 2 2576.2.a.bb.1.1 5
7.6 odd 2 4508.2.a.f.1.1 5

By twisted newform
Twist Min Dim Char Parity Ord Type
644.2.a.d.1.5 5 1.1 even 1 trivial
2576.2.a.bb.1.1 5 4.3 odd 2
4508.2.a.f.1.1 5 7.6 odd 2
5796.2.a.t.1.4 5 3.2 odd 2