# Properties

 Label 6240.2.a.bw.1.2 Level $6240$ Weight $2$ Character 6240.1 Self dual yes Analytic conductor $49.827$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6240.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$49.8266508613$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.11491$$ of defining polynomial Character $$\chi$$ $$=$$ 6240.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} -1.35793 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} -1.35793 q^{7} +1.00000 q^{9} -3.58774 q^{11} +1.00000 q^{13} +1.00000 q^{15} -7.81756 q^{17} -4.94567 q^{19} +1.35793 q^{21} -0.0737791 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.51396 q^{29} -4.00000 q^{31} +3.58774 q^{33} +1.35793 q^{35} -1.58774 q^{37} -1.00000 q^{39} -5.58774 q^{41} -7.17548 q^{43} -1.00000 q^{45} -2.56829 q^{47} -5.15604 q^{49} +7.81756 q^{51} +12.7632 q^{53} +3.58774 q^{55} +4.94567 q^{57} +8.45963 q^{59} -4.30359 q^{61} -1.35793 q^{63} -1.00000 q^{65} +0.0737791 q^{69} +0.871889 q^{71} +6.79811 q^{73} -1.00000 q^{75} +4.87189 q^{77} -12.8719 q^{79} +1.00000 q^{81} +5.43171 q^{83} +7.81756 q^{85} -5.51396 q^{87} -6.87189 q^{89} -1.35793 q^{91} +4.00000 q^{93} +4.94567 q^{95} +7.35793 q^{97} -3.58774 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 5 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 12 q^{31} - q^{33} + 5 q^{35} + 7 q^{37} - 3 q^{39} - 5 q^{41} + 2 q^{43} - 3 q^{45} - 4 q^{47} - q^{51} + 3 q^{53} - q^{55} + 4 q^{57} - 3 q^{61} - 5 q^{63} - 3 q^{65} + 3 q^{69} - 11 q^{71} + 4 q^{73} - 3 q^{75} + q^{77} - 25 q^{79} + 3 q^{81} + 20 q^{83} - q^{85} - 2 q^{87} - 7 q^{89} - 5 q^{91} + 12 q^{93} + 4 q^{95} + 23 q^{97} + q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 + q^11 + 3 * q^13 + 3 * q^15 + q^17 - 4 * q^19 + 5 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 12 * q^31 - q^33 + 5 * q^35 + 7 * q^37 - 3 * q^39 - 5 * q^41 + 2 * q^43 - 3 * q^45 - 4 * q^47 - q^51 + 3 * q^53 - q^55 + 4 * q^57 - 3 * q^61 - 5 * q^63 - 3 * q^65 + 3 * q^69 - 11 * q^71 + 4 * q^73 - 3 * q^75 + q^77 - 25 * q^79 + 3 * q^81 + 20 * q^83 - q^85 - 2 * q^87 - 7 * q^89 - 5 * q^91 + 12 * q^93 + 4 * q^95 + 23 * q^97 + q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −1.35793 −0.513248 −0.256624 0.966511i $$-0.582610\pi$$
−0.256624 + 0.966511i $$0.582610\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.58774 −1.08174 −0.540872 0.841105i $$-0.681906\pi$$
−0.540872 + 0.841105i $$0.681906\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −7.81756 −1.89604 −0.948018 0.318217i $$-0.896916\pi$$
−0.948018 + 0.318217i $$0.896916\pi$$
$$18$$ 0 0
$$19$$ −4.94567 −1.13461 −0.567307 0.823506i $$-0.692015\pi$$
−0.567307 + 0.823506i $$0.692015\pi$$
$$20$$ 0 0
$$21$$ 1.35793 0.296324
$$22$$ 0 0
$$23$$ −0.0737791 −0.0153840 −0.00769200 0.999970i $$-0.502448\pi$$
−0.00769200 + 0.999970i $$0.502448\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 5.51396 1.02392 0.511959 0.859010i $$-0.328920\pi$$
0.511959 + 0.859010i $$0.328920\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 3.58774 0.624546
$$34$$ 0 0
$$35$$ 1.35793 0.229531
$$36$$ 0 0
$$37$$ −1.58774 −0.261023 −0.130512 0.991447i $$-0.541662\pi$$
−0.130512 + 0.991447i $$0.541662\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −5.58774 −0.872659 −0.436329 0.899787i $$-0.643722\pi$$
−0.436329 + 0.899787i $$0.643722\pi$$
$$42$$ 0 0
$$43$$ −7.17548 −1.09425 −0.547125 0.837051i $$-0.684278\pi$$
−0.547125 + 0.837051i $$0.684278\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −2.56829 −0.374624 −0.187312 0.982300i $$-0.559978\pi$$
−0.187312 + 0.982300i $$0.559978\pi$$
$$48$$ 0 0
$$49$$ −5.15604 −0.736577
$$50$$ 0 0
$$51$$ 7.81756 1.09468
$$52$$ 0 0
$$53$$ 12.7632 1.75316 0.876582 0.481253i $$-0.159818\pi$$
0.876582 + 0.481253i $$0.159818\pi$$
$$54$$ 0 0
$$55$$ 3.58774 0.483771
$$56$$ 0 0
$$57$$ 4.94567 0.655070
$$58$$ 0 0
$$59$$ 8.45963 1.10135 0.550675 0.834720i $$-0.314370\pi$$
0.550675 + 0.834720i $$0.314370\pi$$
$$60$$ 0 0
$$61$$ −4.30359 −0.551019 −0.275509 0.961298i $$-0.588847\pi$$
−0.275509 + 0.961298i $$0.588847\pi$$
$$62$$ 0 0
$$63$$ −1.35793 −0.171083
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 0.0737791 0.00888196
$$70$$ 0 0
$$71$$ 0.871889 0.103474 0.0517371 0.998661i $$-0.483524\pi$$
0.0517371 + 0.998661i $$0.483524\pi$$
$$72$$ 0 0
$$73$$ 6.79811 0.795659 0.397829 0.917459i $$-0.369764\pi$$
0.397829 + 0.917459i $$0.369764\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ 4.87189 0.555203
$$78$$ 0 0
$$79$$ −12.8719 −1.44820 −0.724100 0.689695i $$-0.757745\pi$$
−0.724100 + 0.689695i $$0.757745\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 5.43171 0.596207 0.298104 0.954534i $$-0.403646\pi$$
0.298104 + 0.954534i $$0.403646\pi$$
$$84$$ 0 0
$$85$$ 7.81756 0.847933
$$86$$ 0 0
$$87$$ −5.51396 −0.591159
$$88$$ 0 0
$$89$$ −6.87189 −0.728419 −0.364209 0.931317i $$-0.618661\pi$$
−0.364209 + 0.931317i $$0.618661\pi$$
$$90$$ 0 0
$$91$$ −1.35793 −0.142349
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ 4.94567 0.507415
$$96$$ 0 0
$$97$$ 7.35793 0.747084 0.373542 0.927613i $$-0.378143\pi$$
0.373542 + 0.927613i $$0.378143\pi$$
$$98$$ 0 0
$$99$$ −3.58774 −0.360582
$$100$$ 0 0
$$101$$ −6.94567 −0.691120 −0.345560 0.938397i $$-0.612311\pi$$
−0.345560 + 0.938397i $$0.612311\pi$$
$$102$$ 0 0
$$103$$ −10.5683 −1.04133 −0.520663 0.853763i $$-0.674315\pi$$
−0.520663 + 0.853763i $$0.674315\pi$$
$$104$$ 0 0
$$105$$ −1.35793 −0.132520
$$106$$ 0 0
$$107$$ 1.69641 0.163998 0.0819989 0.996632i $$-0.473870\pi$$
0.0819989 + 0.996632i $$0.473870\pi$$
$$108$$ 0 0
$$109$$ −3.40530 −0.326168 −0.163084 0.986612i $$-0.552144\pi$$
−0.163084 + 0.986612i $$0.552144\pi$$
$$110$$ 0 0
$$111$$ 1.58774 0.150702
$$112$$ 0 0
$$113$$ 16.6894 1.57001 0.785005 0.619489i $$-0.212660\pi$$
0.785005 + 0.619489i $$0.212660\pi$$
$$114$$ 0 0
$$115$$ 0.0737791 0.00687994
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ 10.6157 0.973137
$$120$$ 0 0
$$121$$ 1.87189 0.170172
$$122$$ 0 0
$$123$$ 5.58774 0.503830
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −19.7827 −1.75543 −0.877714 0.479185i $$-0.840932\pi$$
−0.877714 + 0.479185i $$0.840932\pi$$
$$128$$ 0 0
$$129$$ 7.17548 0.631766
$$130$$ 0 0
$$131$$ 0.798110 0.0697312 0.0348656 0.999392i $$-0.488900\pi$$
0.0348656 + 0.999392i $$0.488900\pi$$
$$132$$ 0 0
$$133$$ 6.71585 0.582338
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 13.0279 1.11305 0.556525 0.830831i $$-0.312134\pi$$
0.556525 + 0.830831i $$0.312134\pi$$
$$138$$ 0 0
$$139$$ 8.19493 0.695085 0.347542 0.937664i $$-0.387016\pi$$
0.347542 + 0.937664i $$0.387016\pi$$
$$140$$ 0 0
$$141$$ 2.56829 0.216289
$$142$$ 0 0
$$143$$ −3.58774 −0.300022
$$144$$ 0 0
$$145$$ −5.51396 −0.457910
$$146$$ 0 0
$$147$$ 5.15604 0.425263
$$148$$ 0 0
$$149$$ 12.1560 0.995861 0.497931 0.867217i $$-0.334093\pi$$
0.497931 + 0.867217i $$0.334093\pi$$
$$150$$ 0 0
$$151$$ 12.9193 1.05135 0.525677 0.850684i $$-0.323812\pi$$
0.525677 + 0.850684i $$0.323812\pi$$
$$152$$ 0 0
$$153$$ −7.81756 −0.632012
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ 9.32304 0.744060 0.372030 0.928221i $$-0.378662\pi$$
0.372030 + 0.928221i $$0.378662\pi$$
$$158$$ 0 0
$$159$$ −12.7632 −1.01219
$$160$$ 0 0
$$161$$ 0.100187 0.00789581
$$162$$ 0 0
$$163$$ 14.7632 1.15634 0.578172 0.815915i $$-0.303766\pi$$
0.578172 + 0.815915i $$0.303766\pi$$
$$164$$ 0 0
$$165$$ −3.58774 −0.279305
$$166$$ 0 0
$$167$$ −4.45963 −0.345097 −0.172548 0.985001i $$-0.555200\pi$$
−0.172548 + 0.985001i $$0.555200\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −4.94567 −0.378205
$$172$$ 0 0
$$173$$ −6.45963 −0.491117 −0.245558 0.969382i $$-0.578971\pi$$
−0.245558 + 0.969382i $$0.578971\pi$$
$$174$$ 0 0
$$175$$ −1.35793 −0.102650
$$176$$ 0 0
$$177$$ −8.45963 −0.635865
$$178$$ 0 0
$$179$$ 8.94567 0.668631 0.334315 0.942461i $$-0.391495\pi$$
0.334315 + 0.942461i $$0.391495\pi$$
$$180$$ 0 0
$$181$$ −19.3315 −1.43690 −0.718450 0.695578i $$-0.755148\pi$$
−0.718450 + 0.695578i $$0.755148\pi$$
$$182$$ 0 0
$$183$$ 4.30359 0.318131
$$184$$ 0 0
$$185$$ 1.58774 0.116733
$$186$$ 0 0
$$187$$ 28.0474 2.05103
$$188$$ 0 0
$$189$$ 1.35793 0.0987746
$$190$$ 0 0
$$191$$ −9.74378 −0.705035 −0.352517 0.935805i $$-0.614674\pi$$
−0.352517 + 0.935805i $$0.614674\pi$$
$$192$$ 0 0
$$193$$ −4.27719 −0.307879 −0.153939 0.988080i $$-0.549196\pi$$
−0.153939 + 0.988080i $$0.549196\pi$$
$$194$$ 0 0
$$195$$ 1.00000 0.0716115
$$196$$ 0 0
$$197$$ 1.54037 0.109747 0.0548734 0.998493i $$-0.482524\pi$$
0.0548734 + 0.998493i $$0.482524\pi$$
$$198$$ 0 0
$$199$$ −16.9193 −1.19937 −0.599687 0.800234i $$-0.704708\pi$$
−0.599687 + 0.800234i $$0.704708\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −7.48755 −0.525523
$$204$$ 0 0
$$205$$ 5.58774 0.390265
$$206$$ 0 0
$$207$$ −0.0737791 −0.00512800
$$208$$ 0 0
$$209$$ 17.7438 1.22736
$$210$$ 0 0
$$211$$ −6.56829 −0.452180 −0.226090 0.974106i $$-0.572594\pi$$
−0.226090 + 0.974106i $$0.572594\pi$$
$$212$$ 0 0
$$213$$ −0.871889 −0.0597408
$$214$$ 0 0
$$215$$ 7.17548 0.489364
$$216$$ 0 0
$$217$$ 5.43171 0.368728
$$218$$ 0 0
$$219$$ −6.79811 −0.459374
$$220$$ 0 0
$$221$$ −7.81756 −0.525866
$$222$$ 0 0
$$223$$ −4.33848 −0.290526 −0.145263 0.989393i $$-0.546403\pi$$
−0.145263 + 0.989393i $$0.546403\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −20.0947 −1.33373 −0.666867 0.745176i $$-0.732365\pi$$
−0.666867 + 0.745176i $$0.732365\pi$$
$$228$$ 0 0
$$229$$ −0.837003 −0.0553107 −0.0276554 0.999618i $$-0.508804\pi$$
−0.0276554 + 0.999618i $$0.508804\pi$$
$$230$$ 0 0
$$231$$ −4.87189 −0.320547
$$232$$ 0 0
$$233$$ −3.21037 −0.210318 −0.105159 0.994455i $$-0.533535\pi$$
−0.105159 + 0.994455i $$0.533535\pi$$
$$234$$ 0 0
$$235$$ 2.56829 0.167537
$$236$$ 0 0
$$237$$ 12.8719 0.836119
$$238$$ 0 0
$$239$$ 10.3036 0.666484 0.333242 0.942841i $$-0.391857\pi$$
0.333242 + 0.942841i $$0.391857\pi$$
$$240$$ 0 0
$$241$$ 23.5264 1.51547 0.757736 0.652561i $$-0.226306\pi$$
0.757736 + 0.652561i $$0.226306\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 5.15604 0.329407
$$246$$ 0 0
$$247$$ −4.94567 −0.314685
$$248$$ 0 0
$$249$$ −5.43171 −0.344220
$$250$$ 0 0
$$251$$ 17.7702 1.12164 0.560822 0.827936i $$-0.310485\pi$$
0.560822 + 0.827936i $$0.310485\pi$$
$$252$$ 0 0
$$253$$ 0.264700 0.0166416
$$254$$ 0 0
$$255$$ −7.81756 −0.489554
$$256$$ 0 0
$$257$$ −17.4442 −1.08814 −0.544069 0.839040i $$-0.683117\pi$$
−0.544069 + 0.839040i $$0.683117\pi$$
$$258$$ 0 0
$$259$$ 2.15604 0.133970
$$260$$ 0 0
$$261$$ 5.51396 0.341306
$$262$$ 0 0
$$263$$ −11.1491 −0.687481 −0.343741 0.939065i $$-0.611694\pi$$
−0.343741 + 0.939065i $$0.611694\pi$$
$$264$$ 0 0
$$265$$ −12.7632 −0.784039
$$266$$ 0 0
$$267$$ 6.87189 0.420553
$$268$$ 0 0
$$269$$ −13.2966 −0.810710 −0.405355 0.914159i $$-0.632852\pi$$
−0.405355 + 0.914159i $$0.632852\pi$$
$$270$$ 0 0
$$271$$ 12.0947 0.734703 0.367352 0.930082i $$-0.380265\pi$$
0.367352 + 0.930082i $$0.380265\pi$$
$$272$$ 0 0
$$273$$ 1.35793 0.0821854
$$274$$ 0 0
$$275$$ −3.58774 −0.216349
$$276$$ 0 0
$$277$$ 27.0668 1.62629 0.813144 0.582063i $$-0.197754\pi$$
0.813144 + 0.582063i $$0.197754\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 17.7827 1.06083 0.530413 0.847740i $$-0.322037\pi$$
0.530413 + 0.847740i $$0.322037\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ −4.94567 −0.292956
$$286$$ 0 0
$$287$$ 7.58774 0.447890
$$288$$ 0 0
$$289$$ 44.1142 2.59495
$$290$$ 0 0
$$291$$ −7.35793 −0.431329
$$292$$ 0 0
$$293$$ 16.7159 0.976551 0.488275 0.872690i $$-0.337626\pi$$
0.488275 + 0.872690i $$0.337626\pi$$
$$294$$ 0 0
$$295$$ −8.45963 −0.492539
$$296$$ 0 0
$$297$$ 3.58774 0.208182
$$298$$ 0 0
$$299$$ −0.0737791 −0.00426676
$$300$$ 0 0
$$301$$ 9.74378 0.561622
$$302$$ 0 0
$$303$$ 6.94567 0.399018
$$304$$ 0 0
$$305$$ 4.30359 0.246423
$$306$$ 0 0
$$307$$ 22.0863 1.26053 0.630265 0.776380i $$-0.282946\pi$$
0.630265 + 0.776380i $$0.282946\pi$$
$$308$$ 0 0
$$309$$ 10.5683 0.601209
$$310$$ 0 0
$$311$$ 4.60719 0.261250 0.130625 0.991432i $$-0.458302\pi$$
0.130625 + 0.991432i $$0.458302\pi$$
$$312$$ 0 0
$$313$$ −16.8106 −0.950191 −0.475096 0.879934i $$-0.657587\pi$$
−0.475096 + 0.879934i $$0.657587\pi$$
$$314$$ 0 0
$$315$$ 1.35793 0.0765105
$$316$$ 0 0
$$317$$ 32.3510 1.81701 0.908506 0.417873i $$-0.137224\pi$$
0.908506 + 0.417873i $$0.137224\pi$$
$$318$$ 0 0
$$319$$ −19.7827 −1.10762
$$320$$ 0 0
$$321$$ −1.69641 −0.0946841
$$322$$ 0 0
$$323$$ 38.6630 2.15127
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 0 0
$$327$$ 3.40530 0.188313
$$328$$ 0 0
$$329$$ 3.48755 0.192275
$$330$$ 0 0
$$331$$ −28.2159 −1.55089 −0.775443 0.631418i $$-0.782473\pi$$
−0.775443 + 0.631418i $$0.782473\pi$$
$$332$$ 0 0
$$333$$ −1.58774 −0.0870077
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 19.5962 1.06747 0.533737 0.845651i $$-0.320787\pi$$
0.533737 + 0.845651i $$0.320787\pi$$
$$338$$ 0 0
$$339$$ −16.6894 −0.906446
$$340$$ 0 0
$$341$$ 14.3510 0.777148
$$342$$ 0 0
$$343$$ 16.5070 0.891294
$$344$$ 0 0
$$345$$ −0.0737791 −0.00397213
$$346$$ 0 0
$$347$$ 33.6436 1.80608 0.903041 0.429554i $$-0.141329\pi$$
0.903041 + 0.429554i $$0.141329\pi$$
$$348$$ 0 0
$$349$$ 23.5529 1.26076 0.630378 0.776289i $$-0.282900\pi$$
0.630378 + 0.776289i $$0.282900\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ 20.2034 1.07532 0.537659 0.843162i $$-0.319309\pi$$
0.537659 + 0.843162i $$0.319309\pi$$
$$354$$ 0 0
$$355$$ −0.871889 −0.0462750
$$356$$ 0 0
$$357$$ −10.6157 −0.561841
$$358$$ 0 0
$$359$$ 1.43171 0.0755625 0.0377813 0.999286i $$-0.487971\pi$$
0.0377813 + 0.999286i $$0.487971\pi$$
$$360$$ 0 0
$$361$$ 5.45963 0.287349
$$362$$ 0 0
$$363$$ −1.87189 −0.0982487
$$364$$ 0 0
$$365$$ −6.79811 −0.355829
$$366$$ 0 0
$$367$$ 9.74378 0.508621 0.254311 0.967123i $$-0.418151\pi$$
0.254311 + 0.967123i $$0.418151\pi$$
$$368$$ 0 0
$$369$$ −5.58774 −0.290886
$$370$$ 0 0
$$371$$ −17.3315 −0.899808
$$372$$ 0 0
$$373$$ 15.6740 0.811569 0.405785 0.913969i $$-0.366998\pi$$
0.405785 + 0.913969i $$0.366998\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 5.51396 0.283984
$$378$$ 0 0
$$379$$ −27.1491 −1.39455 −0.697277 0.716802i $$-0.745605\pi$$
−0.697277 + 0.716802i $$0.745605\pi$$
$$380$$ 0 0
$$381$$ 19.7827 1.01350
$$382$$ 0 0
$$383$$ 9.06682 0.463293 0.231646 0.972800i $$-0.425589\pi$$
0.231646 + 0.972800i $$0.425589\pi$$
$$384$$ 0 0
$$385$$ −4.87189 −0.248294
$$386$$ 0 0
$$387$$ −7.17548 −0.364750
$$388$$ 0 0
$$389$$ 15.7004 0.796043 0.398021 0.917376i $$-0.369697\pi$$
0.398021 + 0.917376i $$0.369697\pi$$
$$390$$ 0 0
$$391$$ 0.576772 0.0291686
$$392$$ 0 0
$$393$$ −0.798110 −0.0402593
$$394$$ 0 0
$$395$$ 12.8719 0.647655
$$396$$ 0 0
$$397$$ 11.5488 0.579620 0.289810 0.957084i $$-0.406408\pi$$
0.289810 + 0.957084i $$0.406408\pi$$
$$398$$ 0 0
$$399$$ −6.71585 −0.336213
$$400$$ 0 0
$$401$$ −16.8106 −0.839481 −0.419741 0.907644i $$-0.637879\pi$$
−0.419741 + 0.907644i $$0.637879\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 5.69641 0.282360
$$408$$ 0 0
$$409$$ 16.2034 0.801207 0.400603 0.916252i $$-0.368801\pi$$
0.400603 + 0.916252i $$0.368801\pi$$
$$410$$ 0 0
$$411$$ −13.0279 −0.642620
$$412$$ 0 0
$$413$$ −11.4876 −0.565266
$$414$$ 0 0
$$415$$ −5.43171 −0.266632
$$416$$ 0 0
$$417$$ −8.19493 −0.401307
$$418$$ 0 0
$$419$$ −34.1600 −1.66883 −0.834414 0.551139i $$-0.814194\pi$$
−0.834414 + 0.551139i $$0.814194\pi$$
$$420$$ 0 0
$$421$$ −1.80908 −0.0881691 −0.0440846 0.999028i $$-0.514037\pi$$
−0.0440846 + 0.999028i $$0.514037\pi$$
$$422$$ 0 0
$$423$$ −2.56829 −0.124875
$$424$$ 0 0
$$425$$ −7.81756 −0.379207
$$426$$ 0 0
$$427$$ 5.84396 0.282809
$$428$$ 0 0
$$429$$ 3.58774 0.173218
$$430$$ 0 0
$$431$$ 13.1366 0.632767 0.316384 0.948631i $$-0.397531\pi$$
0.316384 + 0.948631i $$0.397531\pi$$
$$432$$ 0 0
$$433$$ −18.7547 −0.901296 −0.450648 0.892702i $$-0.648807\pi$$
−0.450648 + 0.892702i $$0.648807\pi$$
$$434$$ 0 0
$$435$$ 5.51396 0.264374
$$436$$ 0 0
$$437$$ 0.364887 0.0174549
$$438$$ 0 0
$$439$$ 15.2229 0.726547 0.363274 0.931683i $$-0.381659\pi$$
0.363274 + 0.931683i $$0.381659\pi$$
$$440$$ 0 0
$$441$$ −5.15604 −0.245526
$$442$$ 0 0
$$443$$ −17.1142 −0.813120 −0.406560 0.913624i $$-0.633272\pi$$
−0.406560 + 0.913624i $$0.633272\pi$$
$$444$$ 0 0
$$445$$ 6.87189 0.325759
$$446$$ 0 0
$$447$$ −12.1560 −0.574961
$$448$$ 0 0
$$449$$ −22.9666 −1.08386 −0.541931 0.840423i $$-0.682307\pi$$
−0.541931 + 0.840423i $$0.682307\pi$$
$$450$$ 0 0
$$451$$ 20.0474 0.943994
$$452$$ 0 0
$$453$$ −12.9193 −0.607000
$$454$$ 0 0
$$455$$ 1.35793 0.0636606
$$456$$ 0 0
$$457$$ −11.2104 −0.524399 −0.262199 0.965014i $$-0.584448\pi$$
−0.262199 + 0.965014i $$0.584448\pi$$
$$458$$ 0 0
$$459$$ 7.81756 0.364892
$$460$$ 0 0
$$461$$ −20.6157 −0.960167 −0.480084 0.877223i $$-0.659394\pi$$
−0.480084 + 0.877223i $$0.659394\pi$$
$$462$$ 0 0
$$463$$ −5.81756 −0.270365 −0.135182 0.990821i $$-0.543162\pi$$
−0.135182 + 0.990821i $$0.543162\pi$$
$$464$$ 0 0
$$465$$ −4.00000 −0.185496
$$466$$ 0 0
$$467$$ 12.2647 0.567543 0.283771 0.958892i $$-0.408414\pi$$
0.283771 + 0.958892i $$0.408414\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −9.32304 −0.429583
$$472$$ 0 0
$$473$$ 25.7438 1.18370
$$474$$ 0 0
$$475$$ −4.94567 −0.226923
$$476$$ 0 0
$$477$$ 12.7632 0.584388
$$478$$ 0 0
$$479$$ 14.9108 0.681291 0.340646 0.940192i $$-0.389354\pi$$
0.340646 + 0.940192i $$0.389354\pi$$
$$480$$ 0 0
$$481$$ −1.58774 −0.0723948
$$482$$ 0 0
$$483$$ −0.100187 −0.00455865
$$484$$ 0 0
$$485$$ −7.35793 −0.334106
$$486$$ 0 0
$$487$$ −24.5334 −1.11171 −0.555857 0.831278i $$-0.687610\pi$$
−0.555857 + 0.831278i $$0.687610\pi$$
$$488$$ 0 0
$$489$$ −14.7632 −0.667616
$$490$$ 0 0
$$491$$ 30.3246 1.36853 0.684264 0.729234i $$-0.260124\pi$$
0.684264 + 0.729234i $$0.260124\pi$$
$$492$$ 0 0
$$493$$ −43.1057 −1.94138
$$494$$ 0 0
$$495$$ 3.58774 0.161257
$$496$$ 0 0
$$497$$ −1.18396 −0.0531079
$$498$$ 0 0
$$499$$ 10.0823 0.451344 0.225672 0.974203i $$-0.427542\pi$$
0.225672 + 0.974203i $$0.427542\pi$$
$$500$$ 0 0
$$501$$ 4.45963 0.199242
$$502$$ 0 0
$$503$$ −38.9317 −1.73588 −0.867940 0.496668i $$-0.834557\pi$$
−0.867940 + 0.496668i $$0.834557\pi$$
$$504$$ 0 0
$$505$$ 6.94567 0.309078
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ −14.3594 −0.636471 −0.318236 0.948012i $$-0.603090\pi$$
−0.318236 + 0.948012i $$0.603090\pi$$
$$510$$ 0 0
$$511$$ −9.23133 −0.408370
$$512$$ 0 0
$$513$$ 4.94567 0.218357
$$514$$ 0 0
$$515$$ 10.5683 0.465695
$$516$$ 0 0
$$517$$ 9.21438 0.405248
$$518$$ 0 0
$$519$$ 6.45963 0.283546
$$520$$ 0 0
$$521$$ 3.59622 0.157553 0.0787766 0.996892i $$-0.474899\pi$$
0.0787766 + 0.996892i $$0.474899\pi$$
$$522$$ 0 0
$$523$$ −7.68793 −0.336170 −0.168085 0.985773i $$-0.553758\pi$$
−0.168085 + 0.985773i $$0.553758\pi$$
$$524$$ 0 0
$$525$$ 1.35793 0.0592648
$$526$$ 0 0
$$527$$ 31.2702 1.36215
$$528$$ 0 0
$$529$$ −22.9946 −0.999763
$$530$$ 0 0
$$531$$ 8.45963 0.367117
$$532$$ 0 0
$$533$$ −5.58774 −0.242032
$$534$$ 0 0
$$535$$ −1.69641 −0.0733420
$$536$$ 0 0
$$537$$ −8.94567 −0.386034
$$538$$ 0 0
$$539$$ 18.4985 0.796788
$$540$$ 0 0
$$541$$ −1.41922 −0.0610170 −0.0305085 0.999535i $$-0.509713\pi$$
−0.0305085 + 0.999535i $$0.509713\pi$$
$$542$$ 0 0
$$543$$ 19.3315 0.829595
$$544$$ 0 0
$$545$$ 3.40530 0.145867
$$546$$ 0 0
$$547$$ 6.86341 0.293458 0.146729 0.989177i $$-0.453125\pi$$
0.146729 + 0.989177i $$0.453125\pi$$
$$548$$ 0 0
$$549$$ −4.30359 −0.183673
$$550$$ 0 0
$$551$$ −27.2702 −1.16175
$$552$$ 0 0
$$553$$ 17.4791 0.743286
$$554$$ 0 0
$$555$$ −1.58774 −0.0673959
$$556$$ 0 0
$$557$$ 35.7438 1.51451 0.757256 0.653118i $$-0.226539\pi$$
0.757256 + 0.653118i $$0.226539\pi$$
$$558$$ 0 0
$$559$$ −7.17548 −0.303491
$$560$$ 0 0
$$561$$ −28.0474 −1.18416
$$562$$ 0 0
$$563$$ 36.8021 1.55102 0.775512 0.631333i $$-0.217492\pi$$
0.775512 + 0.631333i $$0.217492\pi$$
$$564$$ 0 0
$$565$$ −16.6894 −0.702130
$$566$$ 0 0
$$567$$ −1.35793 −0.0570275
$$568$$ 0 0
$$569$$ 6.45963 0.270802 0.135401 0.990791i $$-0.456768\pi$$
0.135401 + 0.990791i $$0.456768\pi$$
$$570$$ 0 0
$$571$$ 31.3704 1.31281 0.656405 0.754408i $$-0.272076\pi$$
0.656405 + 0.754408i $$0.272076\pi$$
$$572$$ 0 0
$$573$$ 9.74378 0.407052
$$574$$ 0 0
$$575$$ −0.0737791 −0.00307680
$$576$$ 0 0
$$577$$ 40.9541 1.70494 0.852472 0.522773i $$-0.175103\pi$$
0.852472 + 0.522773i $$0.175103\pi$$
$$578$$ 0 0
$$579$$ 4.27719 0.177754
$$580$$ 0 0
$$581$$ −7.37586 −0.306002
$$582$$ 0 0
$$583$$ −45.7911 −1.89648
$$584$$ 0 0
$$585$$ −1.00000 −0.0413449
$$586$$ 0 0
$$587$$ 36.3899 1.50197 0.750985 0.660319i $$-0.229579\pi$$
0.750985 + 0.660319i $$0.229579\pi$$
$$588$$ 0 0
$$589$$ 19.7827 0.815131
$$590$$ 0 0
$$591$$ −1.54037 −0.0633623
$$592$$ 0 0
$$593$$ 8.03889 0.330118 0.165059 0.986284i $$-0.447219\pi$$
0.165059 + 0.986284i $$0.447219\pi$$
$$594$$ 0 0
$$595$$ −10.6157 −0.435200
$$596$$ 0 0
$$597$$ 16.9193 0.692459
$$598$$ 0 0
$$599$$ 44.7019 1.82647 0.913236 0.407432i $$-0.133576\pi$$
0.913236 + 0.407432i $$0.133576\pi$$
$$600$$ 0 0
$$601$$ −23.9387 −0.976480 −0.488240 0.872709i $$-0.662361\pi$$
−0.488240 + 0.872709i $$0.662361\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −1.87189 −0.0761031
$$606$$ 0 0
$$607$$ −6.25622 −0.253932 −0.126966 0.991907i $$-0.540524\pi$$
−0.126966 + 0.991907i $$0.540524\pi$$
$$608$$ 0 0
$$609$$ 7.48755 0.303411
$$610$$ 0 0
$$611$$ −2.56829 −0.103902
$$612$$ 0 0
$$613$$ −18.4123 −0.743664 −0.371832 0.928300i $$-0.621270\pi$$
−0.371832 + 0.928300i $$0.621270\pi$$
$$614$$ 0 0
$$615$$ −5.58774 −0.225319
$$616$$ 0 0
$$617$$ −11.5962 −0.466846 −0.233423 0.972375i $$-0.574993\pi$$
−0.233423 + 0.972375i $$0.574993\pi$$
$$618$$ 0 0
$$619$$ 19.8260 0.796876 0.398438 0.917195i $$-0.369552\pi$$
0.398438 + 0.917195i $$0.369552\pi$$
$$620$$ 0 0
$$621$$ 0.0737791 0.00296065
$$622$$ 0 0
$$623$$ 9.33152 0.373859
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −17.7438 −0.708618
$$628$$ 0 0
$$629$$ 12.4123 0.494909
$$630$$ 0 0
$$631$$ −33.0838 −1.31704 −0.658522 0.752561i $$-0.728818\pi$$
−0.658522 + 0.752561i $$0.728818\pi$$
$$632$$ 0 0
$$633$$ 6.56829 0.261066
$$634$$ 0 0
$$635$$ 19.7827 0.785051
$$636$$ 0 0
$$637$$ −5.15604 −0.204290
$$638$$ 0 0
$$639$$ 0.871889 0.0344914
$$640$$ 0 0
$$641$$ −11.4317 −0.451525 −0.225763 0.974182i $$-0.572487\pi$$
−0.225763 + 0.974182i $$0.572487\pi$$
$$642$$ 0 0
$$643$$ −8.26470 −0.325928 −0.162964 0.986632i $$-0.552105\pi$$
−0.162964 + 0.986632i $$0.552105\pi$$
$$644$$ 0 0
$$645$$ −7.17548 −0.282534
$$646$$ 0 0
$$647$$ −19.7089 −0.774837 −0.387418 0.921904i $$-0.626633\pi$$
−0.387418 + 0.921904i $$0.626633\pi$$
$$648$$ 0 0
$$649$$ −30.3510 −1.19138
$$650$$ 0 0
$$651$$ −5.43171 −0.212885
$$652$$ 0 0
$$653$$ −22.0778 −0.863971 −0.431985 0.901881i $$-0.642187\pi$$
−0.431985 + 0.901881i $$0.642187\pi$$
$$654$$ 0 0
$$655$$ −0.798110 −0.0311847
$$656$$ 0 0
$$657$$ 6.79811 0.265220
$$658$$ 0 0
$$659$$ −35.6087 −1.38712 −0.693559 0.720400i $$-0.743958\pi$$
−0.693559 + 0.720400i $$0.743958\pi$$
$$660$$ 0 0
$$661$$ −51.2577 −1.99370 −0.996848 0.0793414i $$-0.974718\pi$$
−0.996848 + 0.0793414i $$0.974718\pi$$
$$662$$ 0 0
$$663$$ 7.81756 0.303609
$$664$$ 0 0
$$665$$ −6.71585 −0.260430
$$666$$ 0 0
$$667$$ −0.406815 −0.0157519
$$668$$ 0 0
$$669$$ 4.33848 0.167735
$$670$$ 0 0
$$671$$ 15.4402 0.596062
$$672$$ 0 0
$$673$$ 25.1755 0.970444 0.485222 0.874391i $$-0.338739\pi$$
0.485222 + 0.874391i $$0.338739\pi$$
$$674$$ 0 0
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 13.7353 0.527890 0.263945 0.964538i $$-0.414976\pi$$
0.263945 + 0.964538i $$0.414976\pi$$
$$678$$ 0 0
$$679$$ −9.99152 −0.383439
$$680$$ 0 0
$$681$$ 20.0947 0.770032
$$682$$ 0 0
$$683$$ 40.3899 1.54548 0.772738 0.634726i $$-0.218887\pi$$
0.772738 + 0.634726i $$0.218887\pi$$
$$684$$ 0 0
$$685$$ −13.0279 −0.497771
$$686$$ 0 0
$$687$$ 0.837003 0.0319337
$$688$$ 0 0
$$689$$ 12.7632 0.486240
$$690$$ 0 0
$$691$$ −41.9427 −1.59558 −0.797788 0.602938i $$-0.793997\pi$$
−0.797788 + 0.602938i $$0.793997\pi$$
$$692$$ 0 0
$$693$$ 4.87189 0.185068
$$694$$ 0 0
$$695$$ −8.19493 −0.310851
$$696$$ 0 0
$$697$$ 43.6825 1.65459
$$698$$ 0 0
$$699$$ 3.21037 0.121427
$$700$$ 0 0
$$701$$ −42.7283 −1.61383 −0.806914 0.590670i $$-0.798864\pi$$
−0.806914 + 0.590670i $$0.798864\pi$$
$$702$$ 0 0
$$703$$ 7.85244 0.296160
$$704$$ 0 0
$$705$$ −2.56829 −0.0967276
$$706$$ 0 0
$$707$$ 9.43171 0.354716
$$708$$ 0 0
$$709$$ −11.1102 −0.417252 −0.208626 0.977996i $$-0.566899\pi$$
−0.208626 + 0.977996i $$0.566899\pi$$
$$710$$ 0 0
$$711$$ −12.8719 −0.482734
$$712$$ 0 0
$$713$$ 0.295116 0.0110522
$$714$$ 0 0
$$715$$ 3.58774 0.134174
$$716$$ 0 0
$$717$$ −10.3036 −0.384795
$$718$$ 0 0
$$719$$ 8.67696 0.323596 0.161798 0.986824i $$-0.448271\pi$$
0.161798 + 0.986824i $$0.448271\pi$$
$$720$$ 0 0
$$721$$ 14.3510 0.534458
$$722$$ 0 0
$$723$$ −23.5264 −0.874958
$$724$$ 0 0
$$725$$ 5.51396 0.204783
$$726$$ 0 0
$$727$$ 14.4985 0.537720 0.268860 0.963179i $$-0.413353\pi$$
0.268860 + 0.963179i $$0.413353\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 56.0947 2.07474
$$732$$ 0 0
$$733$$ −0.668481 −0.0246909 −0.0123455 0.999924i $$-0.503930\pi$$
−0.0123455 + 0.999924i $$0.503930\pi$$
$$734$$ 0 0
$$735$$ −5.15604 −0.190183
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −17.1102 −0.629408 −0.314704 0.949190i $$-0.601905\pi$$
−0.314704 + 0.949190i $$0.601905\pi$$
$$740$$ 0 0
$$741$$ 4.94567 0.181684
$$742$$ 0 0
$$743$$ −5.57926 −0.204683 −0.102342 0.994749i $$-0.532634\pi$$
−0.102342 + 0.994749i $$0.532634\pi$$
$$744$$ 0 0
$$745$$ −12.1560 −0.445363
$$746$$ 0 0
$$747$$ 5.43171 0.198736
$$748$$ 0 0
$$749$$ −2.30359 −0.0841715
$$750$$ 0 0
$$751$$ −5.47908 −0.199934 −0.0999672 0.994991i $$-0.531874\pi$$
−0.0999672 + 0.994991i $$0.531874\pi$$
$$752$$ 0 0
$$753$$ −17.7702 −0.647582
$$754$$ 0 0
$$755$$ −12.9193 −0.470180
$$756$$ 0 0
$$757$$ −48.7408 −1.77152 −0.885758 0.464148i $$-0.846361\pi$$
−0.885758 + 0.464148i $$0.846361\pi$$
$$758$$ 0 0
$$759$$ −0.264700 −0.00960801
$$760$$ 0 0
$$761$$ −29.9472 −1.08558 −0.542792 0.839867i $$-0.682633\pi$$
−0.542792 + 0.839867i $$0.682633\pi$$
$$762$$ 0 0
$$763$$ 4.62414 0.167405
$$764$$ 0 0
$$765$$ 7.81756 0.282644
$$766$$ 0 0
$$767$$ 8.45963 0.305460
$$768$$ 0 0
$$769$$ −24.1336 −0.870281 −0.435141 0.900363i $$-0.643301\pi$$
−0.435141 + 0.900363i $$0.643301\pi$$
$$770$$ 0 0
$$771$$ 17.4442 0.628237
$$772$$ 0 0
$$773$$ 42.5544 1.53057 0.765287 0.643689i $$-0.222597\pi$$
0.765287 + 0.643689i $$0.222597\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ −2.15604 −0.0773474
$$778$$ 0 0
$$779$$ 27.6351 0.990131
$$780$$ 0 0
$$781$$ −3.12811 −0.111933
$$782$$ 0 0
$$783$$ −5.51396 −0.197053
$$784$$ 0 0
$$785$$ −9.32304 −0.332754
$$786$$ 0 0
$$787$$ 0.676959 0.0241310 0.0120655 0.999927i $$-0.496159\pi$$
0.0120655 + 0.999927i $$0.496159\pi$$
$$788$$ 0 0
$$789$$ 11.1491 0.396918
$$790$$ 0 0
$$791$$ −22.6630 −0.805805
$$792$$ 0 0
$$793$$ −4.30359 −0.152825
$$794$$ 0 0
$$795$$ 12.7632 0.452665
$$796$$ 0 0
$$797$$ −7.84396 −0.277847 −0.138924 0.990303i $$-0.544364\pi$$
−0.138924 + 0.990303i $$0.544364\pi$$
$$798$$ 0 0
$$799$$ 20.0778 0.710301
$$800$$ 0 0
$$801$$ −6.87189 −0.242806
$$802$$ 0 0
$$803$$ −24.3899 −0.860699
$$804$$ 0 0
$$805$$ −0.100187 −0.00353111
$$806$$ 0 0
$$807$$ 13.2966 0.468064
$$808$$ 0 0
$$809$$ 7.13659 0.250909 0.125455 0.992099i $$-0.459961\pi$$
0.125455 + 0.992099i $$0.459961\pi$$
$$810$$ 0 0
$$811$$ 16.4860 0.578903 0.289452 0.957193i $$-0.406527\pi$$
0.289452 + 0.957193i $$0.406527\pi$$
$$812$$ 0 0
$$813$$ −12.0947 −0.424181
$$814$$ 0 0
$$815$$ −14.7632 −0.517133
$$816$$ 0 0
$$817$$ 35.4876 1.24155
$$818$$ 0 0
$$819$$ −1.35793 −0.0474498
$$820$$ 0 0
$$821$$ −9.80507 −0.342199 −0.171100 0.985254i $$-0.554732\pi$$
−0.171100 + 0.985254i $$0.554732\pi$$
$$822$$ 0 0
$$823$$ −26.5683 −0.926113 −0.463056 0.886329i $$-0.653247\pi$$
−0.463056 + 0.886329i $$0.653247\pi$$
$$824$$ 0 0
$$825$$ 3.58774 0.124909
$$826$$ 0 0
$$827$$ 17.6490 0.613717 0.306859 0.951755i $$-0.400722\pi$$
0.306859 + 0.951755i $$0.400722\pi$$
$$828$$ 0 0
$$829$$ 54.1895 1.88208 0.941039 0.338297i $$-0.109851\pi$$
0.941039 + 0.338297i $$0.109851\pi$$
$$830$$ 0 0
$$831$$ −27.0668 −0.938938
$$832$$ 0 0
$$833$$ 40.3076 1.39658
$$834$$ 0 0
$$835$$ 4.45963 0.154332
$$836$$ 0 0
$$837$$ 4.00000 0.138260
$$838$$ 0 0
$$839$$ −47.8300 −1.65128 −0.825638 0.564200i $$-0.809185\pi$$
−0.825638 + 0.564200i $$0.809185\pi$$
$$840$$ 0 0
$$841$$ 1.40378 0.0484062
$$842$$ 0 0
$$843$$ −17.7827 −0.612468
$$844$$ 0 0
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ −2.54189 −0.0873403
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0.117142 0.00401558
$$852$$ 0 0
$$853$$ 9.07530 0.310732 0.155366 0.987857i $$-0.450344\pi$$
0.155366 + 0.987857i $$0.450344\pi$$
$$854$$ 0 0
$$855$$ 4.94567 0.169138
$$856$$ 0 0
$$857$$ 1.54438 0.0527550 0.0263775 0.999652i $$-0.491603\pi$$
0.0263775 + 0.999652i $$0.491603\pi$$
$$858$$ 0 0
$$859$$ −9.69641 −0.330837 −0.165419 0.986223i $$-0.552897\pi$$
−0.165419 + 0.986223i $$0.552897\pi$$
$$860$$ 0 0
$$861$$ −7.58774 −0.258590
$$862$$ 0 0
$$863$$ 37.9083 1.29041 0.645207 0.764008i $$-0.276771\pi$$
0.645207 + 0.764008i $$0.276771\pi$$
$$864$$ 0 0
$$865$$ 6.45963 0.219634
$$866$$ 0 0
$$867$$ −44.1142 −1.49820
$$868$$ 0 0
$$869$$ 46.1810 1.56658
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 7.35793 0.249028
$$874$$ 0 0
$$875$$ 1.35793 0.0459063
$$876$$ 0 0
$$877$$ −22.4068 −0.756624 −0.378312 0.925678i $$-0.623495\pi$$
−0.378312 + 0.925678i $$0.623495\pi$$
$$878$$ 0 0
$$879$$ −16.7159 −0.563812
$$880$$ 0 0
$$881$$ −32.7159 −1.10223 −0.551113 0.834431i $$-0.685797\pi$$
−0.551113 + 0.834431i $$0.685797\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ 8.45963 0.284367
$$886$$ 0 0
$$887$$ 49.5305 1.66307 0.831535 0.555472i $$-0.187463\pi$$
0.831535 + 0.555472i $$0.187463\pi$$
$$888$$ 0 0
$$889$$ 26.8634 0.900970
$$890$$ 0 0
$$891$$ −3.58774 −0.120194
$$892$$ 0 0
$$893$$ 12.7019 0.425054
$$894$$ 0 0
$$895$$ −8.94567 −0.299021
$$896$$ 0 0
$$897$$ 0.0737791 0.00246341
$$898$$ 0 0
$$899$$ −22.0558 −0.735604
$$900$$ 0 0
$$901$$ −99.7772 −3.32406
$$902$$ 0 0
$$903$$ −9.74378 −0.324253
$$904$$ 0 0
$$905$$ 19.3315 0.642601
$$906$$ 0 0
$$907$$ 50.3679 1.67244 0.836220 0.548395i $$-0.184761\pi$$
0.836220 + 0.548395i $$0.184761\pi$$
$$908$$ 0 0
$$909$$ −6.94567 −0.230373
$$910$$ 0 0
$$911$$ −16.7717 −0.555671 −0.277836 0.960629i $$-0.589617\pi$$
−0.277836 + 0.960629i $$0.589617\pi$$
$$912$$ 0 0
$$913$$ −19.4876 −0.644944
$$914$$ 0 0
$$915$$ −4.30359 −0.142272
$$916$$ 0 0
$$917$$ −1.08377 −0.0357894
$$918$$ 0 0
$$919$$ 30.1032 0.993014 0.496507 0.868033i $$-0.334616\pi$$
0.496507 + 0.868033i $$0.334616\pi$$
$$920$$ 0 0
$$921$$ −22.0863 −0.727767
$$922$$ 0 0
$$923$$ 0.871889 0.0286986
$$924$$ 0 0
$$925$$ −1.58774 −0.0522046
$$926$$ 0 0
$$927$$ −10.5683 −0.347108
$$928$$ 0 0
$$929$$ 16.3983 0.538012 0.269006 0.963139i $$-0.413305\pi$$
0.269006 + 0.963139i $$0.413305\pi$$
$$930$$ 0 0
$$931$$ 25.5000 0.835730
$$932$$ 0 0
$$933$$ −4.60719 −0.150833
$$934$$ 0 0
$$935$$ −28.0474 −0.917247
$$936$$ 0 0
$$937$$ −23.3091 −0.761476 −0.380738 0.924683i $$-0.624330\pi$$
−0.380738 + 0.924683i $$0.624330\pi$$
$$938$$ 0 0
$$939$$ 16.8106 0.548593
$$940$$ 0 0
$$941$$ 33.1531 1.08076 0.540380 0.841421i $$-0.318281\pi$$
0.540380 + 0.841421i $$0.318281\pi$$
$$942$$ 0 0
$$943$$ 0.412259 0.0134250
$$944$$ 0 0
$$945$$ −1.35793 −0.0441733
$$946$$ 0 0
$$947$$ 43.7827 1.42275 0.711373 0.702815i $$-0.248074\pi$$
0.711373 + 0.702815i $$0.248074\pi$$
$$948$$ 0 0
$$949$$ 6.79811 0.220676
$$950$$ 0 0
$$951$$ −32.3510 −1.04905
$$952$$ 0 0
$$953$$ −29.2493 −0.947477 −0.473738 0.880666i $$-0.657096\pi$$
−0.473738 + 0.880666i $$0.657096\pi$$
$$954$$ 0 0
$$955$$ 9.74378 0.315301
$$956$$ 0 0
$$957$$ 19.7827 0.639483
$$958$$ 0 0
$$959$$ −17.6910 −0.571271
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 1.69641 0.0546659
$$964$$ 0 0
$$965$$ 4.27719 0.137688
$$966$$ 0 0
$$967$$ −28.5808 −0.919096 −0.459548 0.888153i $$-0.651989\pi$$
−0.459548 + 0.888153i $$0.651989\pi$$
$$968$$ 0 0
$$969$$ −38.6630 −1.24204
$$970$$ 0 0
$$971$$ −31.6785 −1.01661 −0.508305 0.861177i $$-0.669728\pi$$
−0.508305 + 0.861177i $$0.669728\pi$$
$$972$$ 0 0
$$973$$ −11.1281 −0.356751
$$974$$ 0 0
$$975$$ −1.00000 −0.0320256
$$976$$ 0 0
$$977$$ 13.2702 0.424552 0.212276 0.977210i $$-0.431912\pi$$
0.212276 + 0.977210i $$0.431912\pi$$
$$978$$ 0 0
$$979$$ 24.6546 0.787963
$$980$$ 0 0
$$981$$ −3.40530 −0.108723
$$982$$ 0 0
$$983$$ −2.18645 −0.0697370 −0.0348685 0.999392i $$-0.511101\pi$$
−0.0348685 + 0.999392i $$0.511101\pi$$
$$984$$ 0 0
$$985$$ −1.54037 −0.0490803
$$986$$ 0 0
$$987$$ −3.48755 −0.111010
$$988$$ 0 0
$$989$$ 0.529401 0.0168340
$$990$$ 0 0
$$991$$ 16.9497 0.538424 0.269212 0.963081i $$-0.413237\pi$$
0.269212 + 0.963081i $$0.413237\pi$$
$$992$$ 0 0
$$993$$ 28.2159 0.895404
$$994$$ 0 0
$$995$$ 16.9193 0.536377
$$996$$ 0 0
$$997$$ −20.7936 −0.658541 −0.329271 0.944236i $$-0.606803\pi$$
−0.329271 + 0.944236i $$0.606803\pi$$
$$998$$ 0 0
$$999$$ 1.58774 0.0502339
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 6240.2.a.bw.1.2 3
4.3 odd 2 6240.2.a.cb.1.2 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bw.1.2 3 1.1 even 1 trivial
6240.2.a.cb.1.2 yes 3 4.3 odd 2