# Properties

 Label 546.2.a.j.1.2 Level $546$ Weight $2$ Character 546.1 Self dual yes Analytic conductor $4.360$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 546.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.56155 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.56155 q^{10} -1.56155 q^{11} +1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +3.56155 q^{15} +1.00000 q^{16} -6.68466 q^{17} +1.00000 q^{18} -4.68466 q^{19} +3.56155 q^{20} -1.00000 q^{21} -1.56155 q^{22} -5.56155 q^{23} +1.00000 q^{24} +7.68466 q^{25} +1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.68466 q^{29} +3.56155 q^{30} +6.24621 q^{31} +1.00000 q^{32} -1.56155 q^{33} -6.68466 q^{34} -3.56155 q^{35} +1.00000 q^{36} -7.56155 q^{37} -4.68466 q^{38} +1.00000 q^{39} +3.56155 q^{40} -1.12311 q^{41} -1.00000 q^{42} -6.43845 q^{43} -1.56155 q^{44} +3.56155 q^{45} -5.56155 q^{46} +1.00000 q^{48} +1.00000 q^{49} +7.68466 q^{50} -6.68466 q^{51} +1.00000 q^{52} +12.2462 q^{53} +1.00000 q^{54} -5.56155 q^{55} -1.00000 q^{56} -4.68466 q^{57} +6.68466 q^{58} +2.24621 q^{59} +3.56155 q^{60} +6.68466 q^{61} +6.24621 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.56155 q^{65} -1.56155 q^{66} -7.12311 q^{67} -6.68466 q^{68} -5.56155 q^{69} -3.56155 q^{70} +8.00000 q^{71} +1.00000 q^{72} -3.56155 q^{73} -7.56155 q^{74} +7.68466 q^{75} -4.68466 q^{76} +1.56155 q^{77} +1.00000 q^{78} -11.1231 q^{79} +3.56155 q^{80} +1.00000 q^{81} -1.12311 q^{82} +8.87689 q^{83} -1.00000 q^{84} -23.8078 q^{85} -6.43845 q^{86} +6.68466 q^{87} -1.56155 q^{88} +10.0000 q^{89} +3.56155 q^{90} -1.00000 q^{91} -5.56155 q^{92} +6.24621 q^{93} -16.6847 q^{95} +1.00000 q^{96} +14.4924 q^{97} +1.00000 q^{98} -1.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{10} + q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 3 q^{15} + 2 q^{16} - q^{17} + 2 q^{18} + 3 q^{19} + 3 q^{20} - 2 q^{21} + q^{22} - 7 q^{23} + 2 q^{24} + 3 q^{25} + 2 q^{26} + 2 q^{27} - 2 q^{28} + q^{29} + 3 q^{30} - 4 q^{31} + 2 q^{32} + q^{33} - q^{34} - 3 q^{35} + 2 q^{36} - 11 q^{37} + 3 q^{38} + 2 q^{39} + 3 q^{40} + 6 q^{41} - 2 q^{42} - 17 q^{43} + q^{44} + 3 q^{45} - 7 q^{46} + 2 q^{48} + 2 q^{49} + 3 q^{50} - q^{51} + 2 q^{52} + 8 q^{53} + 2 q^{54} - 7 q^{55} - 2 q^{56} + 3 q^{57} + q^{58} - 12 q^{59} + 3 q^{60} + q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} + 3 q^{65} + q^{66} - 6 q^{67} - q^{68} - 7 q^{69} - 3 q^{70} + 16 q^{71} + 2 q^{72} - 3 q^{73} - 11 q^{74} + 3 q^{75} + 3 q^{76} - q^{77} + 2 q^{78} - 14 q^{79} + 3 q^{80} + 2 q^{81} + 6 q^{82} + 26 q^{83} - 2 q^{84} - 27 q^{85} - 17 q^{86} + q^{87} + q^{88} + 20 q^{89} + 3 q^{90} - 2 q^{91} - 7 q^{92} - 4 q^{93} - 21 q^{95} + 2 q^{96} - 4 q^{97} + 2 q^{98} + q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 3 * q^10 + q^11 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 3 * q^15 + 2 * q^16 - q^17 + 2 * q^18 + 3 * q^19 + 3 * q^20 - 2 * q^21 + q^22 - 7 * q^23 + 2 * q^24 + 3 * q^25 + 2 * q^26 + 2 * q^27 - 2 * q^28 + q^29 + 3 * q^30 - 4 * q^31 + 2 * q^32 + q^33 - q^34 - 3 * q^35 + 2 * q^36 - 11 * q^37 + 3 * q^38 + 2 * q^39 + 3 * q^40 + 6 * q^41 - 2 * q^42 - 17 * q^43 + q^44 + 3 * q^45 - 7 * q^46 + 2 * q^48 + 2 * q^49 + 3 * q^50 - q^51 + 2 * q^52 + 8 * q^53 + 2 * q^54 - 7 * q^55 - 2 * q^56 + 3 * q^57 + q^58 - 12 * q^59 + 3 * q^60 + q^61 - 4 * q^62 - 2 * q^63 + 2 * q^64 + 3 * q^65 + q^66 - 6 * q^67 - q^68 - 7 * q^69 - 3 * q^70 + 16 * q^71 + 2 * q^72 - 3 * q^73 - 11 * q^74 + 3 * q^75 + 3 * q^76 - q^77 + 2 * q^78 - 14 * q^79 + 3 * q^80 + 2 * q^81 + 6 * q^82 + 26 * q^83 - 2 * q^84 - 27 * q^85 - 17 * q^86 + q^87 + q^88 + 20 * q^89 + 3 * q^90 - 2 * q^91 - 7 * q^92 - 4 * q^93 - 21 * q^95 + 2 * q^96 - 4 * q^97 + 2 * q^98 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 3.56155 1.59277 0.796387 0.604787i $$-0.206742\pi$$
0.796387 + 0.604787i $$0.206742\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 3.56155 1.12626
$$11$$ −1.56155 −0.470826 −0.235413 0.971895i $$-0.575644\pi$$
−0.235413 + 0.971895i $$0.575644\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 1.00000 0.277350
$$14$$ −1.00000 −0.267261
$$15$$ 3.56155 0.919589
$$16$$ 1.00000 0.250000
$$17$$ −6.68466 −1.62127 −0.810634 0.585553i $$-0.800877\pi$$
−0.810634 + 0.585553i $$0.800877\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −4.68466 −1.07473 −0.537367 0.843348i $$-0.680581\pi$$
−0.537367 + 0.843348i $$0.680581\pi$$
$$20$$ 3.56155 0.796387
$$21$$ −1.00000 −0.218218
$$22$$ −1.56155 −0.332924
$$23$$ −5.56155 −1.15966 −0.579832 0.814736i $$-0.696882\pi$$
−0.579832 + 0.814736i $$0.696882\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 7.68466 1.53693
$$26$$ 1.00000 0.196116
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ 6.68466 1.24131 0.620655 0.784084i $$-0.286867\pi$$
0.620655 + 0.784084i $$0.286867\pi$$
$$30$$ 3.56155 0.650248
$$31$$ 6.24621 1.12185 0.560926 0.827866i $$-0.310445\pi$$
0.560926 + 0.827866i $$0.310445\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −1.56155 −0.271831
$$34$$ −6.68466 −1.14641
$$35$$ −3.56155 −0.602012
$$36$$ 1.00000 0.166667
$$37$$ −7.56155 −1.24311 −0.621556 0.783370i $$-0.713499\pi$$
−0.621556 + 0.783370i $$0.713499\pi$$
$$38$$ −4.68466 −0.759952
$$39$$ 1.00000 0.160128
$$40$$ 3.56155 0.563131
$$41$$ −1.12311 −0.175400 −0.0876998 0.996147i $$-0.527952\pi$$
−0.0876998 + 0.996147i $$0.527952\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −6.43845 −0.981854 −0.490927 0.871201i $$-0.663342\pi$$
−0.490927 + 0.871201i $$0.663342\pi$$
$$44$$ −1.56155 −0.235413
$$45$$ 3.56155 0.530925
$$46$$ −5.56155 −0.820006
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ 7.68466 1.08677
$$51$$ −6.68466 −0.936039
$$52$$ 1.00000 0.138675
$$53$$ 12.2462 1.68215 0.841073 0.540921i $$-0.181924\pi$$
0.841073 + 0.540921i $$0.181924\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −5.56155 −0.749920
$$56$$ −1.00000 −0.133631
$$57$$ −4.68466 −0.620498
$$58$$ 6.68466 0.877739
$$59$$ 2.24621 0.292432 0.146216 0.989253i $$-0.453291\pi$$
0.146216 + 0.989253i $$0.453291\pi$$
$$60$$ 3.56155 0.459794
$$61$$ 6.68466 0.855883 0.427941 0.903806i $$-0.359239\pi$$
0.427941 + 0.903806i $$0.359239\pi$$
$$62$$ 6.24621 0.793270
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 3.56155 0.441756
$$66$$ −1.56155 −0.192214
$$67$$ −7.12311 −0.870226 −0.435113 0.900376i $$-0.643292\pi$$
−0.435113 + 0.900376i $$0.643292\pi$$
$$68$$ −6.68466 −0.810634
$$69$$ −5.56155 −0.669532
$$70$$ −3.56155 −0.425687
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −3.56155 −0.416848 −0.208424 0.978039i $$-0.566833\pi$$
−0.208424 + 0.978039i $$0.566833\pi$$
$$74$$ −7.56155 −0.879013
$$75$$ 7.68466 0.887348
$$76$$ −4.68466 −0.537367
$$77$$ 1.56155 0.177955
$$78$$ 1.00000 0.113228
$$79$$ −11.1231 −1.25145 −0.625724 0.780045i $$-0.715196\pi$$
−0.625724 + 0.780045i $$0.715196\pi$$
$$80$$ 3.56155 0.398194
$$81$$ 1.00000 0.111111
$$82$$ −1.12311 −0.124026
$$83$$ 8.87689 0.974366 0.487183 0.873300i $$-0.338025\pi$$
0.487183 + 0.873300i $$0.338025\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ −23.8078 −2.58231
$$86$$ −6.43845 −0.694276
$$87$$ 6.68466 0.716671
$$88$$ −1.56155 −0.166462
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 3.56155 0.375421
$$91$$ −1.00000 −0.104828
$$92$$ −5.56155 −0.579832
$$93$$ 6.24621 0.647702
$$94$$ 0 0
$$95$$ −16.6847 −1.71181
$$96$$ 1.00000 0.102062
$$97$$ 14.4924 1.47148 0.735741 0.677263i $$-0.236834\pi$$
0.735741 + 0.677263i $$0.236834\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −1.56155 −0.156942
$$100$$ 7.68466 0.768466
$$101$$ −5.12311 −0.509768 −0.254884 0.966972i $$-0.582037\pi$$
−0.254884 + 0.966972i $$0.582037\pi$$
$$102$$ −6.68466 −0.661880
$$103$$ −0.684658 −0.0674614 −0.0337307 0.999431i $$-0.510739\pi$$
−0.0337307 + 0.999431i $$0.510739\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ −3.56155 −0.347572
$$106$$ 12.2462 1.18946
$$107$$ −8.87689 −0.858162 −0.429081 0.903266i $$-0.641162\pi$$
−0.429081 + 0.903266i $$0.641162\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −16.9309 −1.62168 −0.810842 0.585266i $$-0.800990\pi$$
−0.810842 + 0.585266i $$0.800990\pi$$
$$110$$ −5.56155 −0.530273
$$111$$ −7.56155 −0.717711
$$112$$ −1.00000 −0.0944911
$$113$$ −4.24621 −0.399450 −0.199725 0.979852i $$-0.564005\pi$$
−0.199725 + 0.979852i $$0.564005\pi$$
$$114$$ −4.68466 −0.438758
$$115$$ −19.8078 −1.84708
$$116$$ 6.68466 0.620655
$$117$$ 1.00000 0.0924500
$$118$$ 2.24621 0.206781
$$119$$ 6.68466 0.612782
$$120$$ 3.56155 0.325124
$$121$$ −8.56155 −0.778323
$$122$$ 6.68466 0.605201
$$123$$ −1.12311 −0.101267
$$124$$ 6.24621 0.560926
$$125$$ 9.56155 0.855211
$$126$$ −1.00000 −0.0890871
$$127$$ −4.87689 −0.432754 −0.216377 0.976310i $$-0.569424\pi$$
−0.216377 + 0.976310i $$0.569424\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −6.43845 −0.566874
$$130$$ 3.56155 0.312369
$$131$$ −9.56155 −0.835397 −0.417698 0.908586i $$-0.637163\pi$$
−0.417698 + 0.908586i $$0.637163\pi$$
$$132$$ −1.56155 −0.135916
$$133$$ 4.68466 0.406211
$$134$$ −7.12311 −0.615343
$$135$$ 3.56155 0.306530
$$136$$ −6.68466 −0.573205
$$137$$ −3.56155 −0.304284 −0.152142 0.988359i $$-0.548617\pi$$
−0.152142 + 0.988359i $$0.548617\pi$$
$$138$$ −5.56155 −0.473431
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ −3.56155 −0.301006
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ −1.56155 −0.130584
$$144$$ 1.00000 0.0833333
$$145$$ 23.8078 1.97713
$$146$$ −3.56155 −0.294756
$$147$$ 1.00000 0.0824786
$$148$$ −7.56155 −0.621556
$$149$$ −17.6155 −1.44312 −0.721560 0.692352i $$-0.756575\pi$$
−0.721560 + 0.692352i $$0.756575\pi$$
$$150$$ 7.68466 0.627450
$$151$$ 11.8078 0.960902 0.480451 0.877022i $$-0.340473\pi$$
0.480451 + 0.877022i $$0.340473\pi$$
$$152$$ −4.68466 −0.379976
$$153$$ −6.68466 −0.540423
$$154$$ 1.56155 0.125834
$$155$$ 22.2462 1.78686
$$156$$ 1.00000 0.0800641
$$157$$ −15.5616 −1.24195 −0.620974 0.783832i $$-0.713263\pi$$
−0.620974 + 0.783832i $$0.713263\pi$$
$$158$$ −11.1231 −0.884907
$$159$$ 12.2462 0.971188
$$160$$ 3.56155 0.281565
$$161$$ 5.56155 0.438312
$$162$$ 1.00000 0.0785674
$$163$$ −16.8769 −1.32190 −0.660950 0.750430i $$-0.729847\pi$$
−0.660950 + 0.750430i $$0.729847\pi$$
$$164$$ −1.12311 −0.0876998
$$165$$ −5.56155 −0.432966
$$166$$ 8.87689 0.688981
$$167$$ 22.9309 1.77444 0.887222 0.461343i $$-0.152632\pi$$
0.887222 + 0.461343i $$0.152632\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 1.00000 0.0769231
$$170$$ −23.8078 −1.82597
$$171$$ −4.68466 −0.358245
$$172$$ −6.43845 −0.490927
$$173$$ 20.2462 1.53929 0.769645 0.638471i $$-0.220433\pi$$
0.769645 + 0.638471i $$0.220433\pi$$
$$174$$ 6.68466 0.506763
$$175$$ −7.68466 −0.580906
$$176$$ −1.56155 −0.117706
$$177$$ 2.24621 0.168836
$$178$$ 10.0000 0.749532
$$179$$ 16.4924 1.23270 0.616351 0.787472i $$-0.288610\pi$$
0.616351 + 0.787472i $$0.288610\pi$$
$$180$$ 3.56155 0.265462
$$181$$ −0.246211 −0.0183007 −0.00915037 0.999958i $$-0.502913\pi$$
−0.00915037 + 0.999958i $$0.502913\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 6.68466 0.494144
$$184$$ −5.56155 −0.410003
$$185$$ −26.9309 −1.98000
$$186$$ 6.24621 0.457994
$$187$$ 10.4384 0.763335
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ −16.6847 −1.21043
$$191$$ 14.9309 1.08036 0.540180 0.841550i $$-0.318356\pi$$
0.540180 + 0.841550i $$0.318356\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 19.3693 1.39423 0.697117 0.716957i $$-0.254466\pi$$
0.697117 + 0.716957i $$0.254466\pi$$
$$194$$ 14.4924 1.04050
$$195$$ 3.56155 0.255048
$$196$$ 1.00000 0.0714286
$$197$$ 10.8769 0.774947 0.387473 0.921881i $$-0.373348\pi$$
0.387473 + 0.921881i $$0.373348\pi$$
$$198$$ −1.56155 −0.110975
$$199$$ 6.93087 0.491316 0.245658 0.969357i $$-0.420996\pi$$
0.245658 + 0.969357i $$0.420996\pi$$
$$200$$ 7.68466 0.543387
$$201$$ −7.12311 −0.502425
$$202$$ −5.12311 −0.360460
$$203$$ −6.68466 −0.469171
$$204$$ −6.68466 −0.468020
$$205$$ −4.00000 −0.279372
$$206$$ −0.684658 −0.0477024
$$207$$ −5.56155 −0.386555
$$208$$ 1.00000 0.0693375
$$209$$ 7.31534 0.506013
$$210$$ −3.56155 −0.245770
$$211$$ −15.8078 −1.08825 −0.544126 0.839004i $$-0.683139\pi$$
−0.544126 + 0.839004i $$0.683139\pi$$
$$212$$ 12.2462 0.841073
$$213$$ 8.00000 0.548151
$$214$$ −8.87689 −0.606812
$$215$$ −22.9309 −1.56387
$$216$$ 1.00000 0.0680414
$$217$$ −6.24621 −0.424020
$$218$$ −16.9309 −1.14670
$$219$$ −3.56155 −0.240667
$$220$$ −5.56155 −0.374960
$$221$$ −6.68466 −0.449659
$$222$$ −7.56155 −0.507498
$$223$$ −23.6155 −1.58141 −0.790706 0.612196i $$-0.790287\pi$$
−0.790706 + 0.612196i $$0.790287\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 7.68466 0.512311
$$226$$ −4.24621 −0.282454
$$227$$ −7.12311 −0.472777 −0.236389 0.971659i $$-0.575964\pi$$
−0.236389 + 0.971659i $$0.575964\pi$$
$$228$$ −4.68466 −0.310249
$$229$$ 28.2462 1.86656 0.933281 0.359147i $$-0.116932\pi$$
0.933281 + 0.359147i $$0.116932\pi$$
$$230$$ −19.8078 −1.30609
$$231$$ 1.56155 0.102743
$$232$$ 6.68466 0.438869
$$233$$ 27.3693 1.79302 0.896512 0.443020i $$-0.146093\pi$$
0.896512 + 0.443020i $$0.146093\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ 2.24621 0.146216
$$237$$ −11.1231 −0.722523
$$238$$ 6.68466 0.433302
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 3.56155 0.229897
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ −8.56155 −0.550357
$$243$$ 1.00000 0.0641500
$$244$$ 6.68466 0.427941
$$245$$ 3.56155 0.227539
$$246$$ −1.12311 −0.0716066
$$247$$ −4.68466 −0.298078
$$248$$ 6.24621 0.396635
$$249$$ 8.87689 0.562550
$$250$$ 9.56155 0.604726
$$251$$ −7.80776 −0.492822 −0.246411 0.969165i $$-0.579251\pi$$
−0.246411 + 0.969165i $$0.579251\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 8.68466 0.546000
$$254$$ −4.87689 −0.306004
$$255$$ −23.8078 −1.49090
$$256$$ 1.00000 0.0625000
$$257$$ −4.24621 −0.264871 −0.132436 0.991192i $$-0.542280\pi$$
−0.132436 + 0.991192i $$0.542280\pi$$
$$258$$ −6.43845 −0.400840
$$259$$ 7.56155 0.469852
$$260$$ 3.56155 0.220878
$$261$$ 6.68466 0.413770
$$262$$ −9.56155 −0.590715
$$263$$ −30.2462 −1.86506 −0.932531 0.361091i $$-0.882404\pi$$
−0.932531 + 0.361091i $$0.882404\pi$$
$$264$$ −1.56155 −0.0961069
$$265$$ 43.6155 2.67928
$$266$$ 4.68466 0.287235
$$267$$ 10.0000 0.611990
$$268$$ −7.12311 −0.435113
$$269$$ 20.2462 1.23443 0.617217 0.786793i $$-0.288260\pi$$
0.617217 + 0.786793i $$0.288260\pi$$
$$270$$ 3.56155 0.216749
$$271$$ 4.87689 0.296250 0.148125 0.988969i $$-0.452676\pi$$
0.148125 + 0.988969i $$0.452676\pi$$
$$272$$ −6.68466 −0.405317
$$273$$ −1.00000 −0.0605228
$$274$$ −3.56155 −0.215161
$$275$$ −12.0000 −0.723627
$$276$$ −5.56155 −0.334766
$$277$$ −22.4924 −1.35144 −0.675719 0.737159i $$-0.736167\pi$$
−0.675719 + 0.737159i $$0.736167\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 6.24621 0.373951
$$280$$ −3.56155 −0.212843
$$281$$ 16.2462 0.969168 0.484584 0.874745i $$-0.338971\pi$$
0.484584 + 0.874745i $$0.338971\pi$$
$$282$$ 0 0
$$283$$ 18.2462 1.08462 0.542312 0.840177i $$-0.317549\pi$$
0.542312 + 0.840177i $$0.317549\pi$$
$$284$$ 8.00000 0.474713
$$285$$ −16.6847 −0.988314
$$286$$ −1.56155 −0.0923366
$$287$$ 1.12311 0.0662948
$$288$$ 1.00000 0.0589256
$$289$$ 27.6847 1.62851
$$290$$ 23.8078 1.39804
$$291$$ 14.4924 0.849561
$$292$$ −3.56155 −0.208424
$$293$$ 24.7386 1.44525 0.722623 0.691242i $$-0.242936\pi$$
0.722623 + 0.691242i $$0.242936\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 8.00000 0.465778
$$296$$ −7.56155 −0.439506
$$297$$ −1.56155 −0.0906105
$$298$$ −17.6155 −1.02044
$$299$$ −5.56155 −0.321633
$$300$$ 7.68466 0.443674
$$301$$ 6.43845 0.371106
$$302$$ 11.8078 0.679460
$$303$$ −5.12311 −0.294315
$$304$$ −4.68466 −0.268684
$$305$$ 23.8078 1.36323
$$306$$ −6.68466 −0.382136
$$307$$ 26.2462 1.49795 0.748975 0.662598i $$-0.230546\pi$$
0.748975 + 0.662598i $$0.230546\pi$$
$$308$$ 1.56155 0.0889777
$$309$$ −0.684658 −0.0389489
$$310$$ 22.2462 1.26350
$$311$$ −12.8769 −0.730182 −0.365091 0.930972i $$-0.618962\pi$$
−0.365091 + 0.930972i $$0.618962\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ −13.6155 −0.769595 −0.384798 0.923001i $$-0.625729\pi$$
−0.384798 + 0.923001i $$0.625729\pi$$
$$314$$ −15.5616 −0.878189
$$315$$ −3.56155 −0.200671
$$316$$ −11.1231 −0.625724
$$317$$ 2.87689 0.161582 0.0807912 0.996731i $$-0.474255\pi$$
0.0807912 + 0.996731i $$0.474255\pi$$
$$318$$ 12.2462 0.686733
$$319$$ −10.4384 −0.584441
$$320$$ 3.56155 0.199097
$$321$$ −8.87689 −0.495460
$$322$$ 5.56155 0.309933
$$323$$ 31.3153 1.74243
$$324$$ 1.00000 0.0555556
$$325$$ 7.68466 0.426268
$$326$$ −16.8769 −0.934725
$$327$$ −16.9309 −0.936279
$$328$$ −1.12311 −0.0620131
$$329$$ 0 0
$$330$$ −5.56155 −0.306153
$$331$$ 13.3693 0.734844 0.367422 0.930054i $$-0.380240\pi$$
0.367422 + 0.930054i $$0.380240\pi$$
$$332$$ 8.87689 0.487183
$$333$$ −7.56155 −0.414371
$$334$$ 22.9309 1.25472
$$335$$ −25.3693 −1.38607
$$336$$ −1.00000 −0.0545545
$$337$$ 4.43845 0.241778 0.120889 0.992666i $$-0.461426\pi$$
0.120889 + 0.992666i $$0.461426\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ −4.24621 −0.230623
$$340$$ −23.8078 −1.29116
$$341$$ −9.75379 −0.528197
$$342$$ −4.68466 −0.253317
$$343$$ −1.00000 −0.0539949
$$344$$ −6.43845 −0.347138
$$345$$ −19.8078 −1.06641
$$346$$ 20.2462 1.08844
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 6.68466 0.358335
$$349$$ 7.75379 0.415051 0.207525 0.978230i $$-0.433459\pi$$
0.207525 + 0.978230i $$0.433459\pi$$
$$350$$ −7.68466 −0.410762
$$351$$ 1.00000 0.0533761
$$352$$ −1.56155 −0.0832310
$$353$$ 30.4924 1.62295 0.811474 0.584389i $$-0.198666\pi$$
0.811474 + 0.584389i $$0.198666\pi$$
$$354$$ 2.24621 0.119385
$$355$$ 28.4924 1.51222
$$356$$ 10.0000 0.529999
$$357$$ 6.68466 0.353790
$$358$$ 16.4924 0.871652
$$359$$ −26.7386 −1.41121 −0.705606 0.708605i $$-0.749325\pi$$
−0.705606 + 0.708605i $$0.749325\pi$$
$$360$$ 3.56155 0.187710
$$361$$ 2.94602 0.155054
$$362$$ −0.246211 −0.0129406
$$363$$ −8.56155 −0.449365
$$364$$ −1.00000 −0.0524142
$$365$$ −12.6847 −0.663945
$$366$$ 6.68466 0.349413
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ −5.56155 −0.289916
$$369$$ −1.12311 −0.0584665
$$370$$ −26.9309 −1.40007
$$371$$ −12.2462 −0.635792
$$372$$ 6.24621 0.323851
$$373$$ −17.6155 −0.912097 −0.456049 0.889955i $$-0.650736\pi$$
−0.456049 + 0.889955i $$0.650736\pi$$
$$374$$ 10.4384 0.539759
$$375$$ 9.56155 0.493756
$$376$$ 0 0
$$377$$ 6.68466 0.344277
$$378$$ −1.00000 −0.0514344
$$379$$ 34.2462 1.75911 0.879555 0.475797i $$-0.157840\pi$$
0.879555 + 0.475797i $$0.157840\pi$$
$$380$$ −16.6847 −0.855905
$$381$$ −4.87689 −0.249851
$$382$$ 14.9309 0.763930
$$383$$ 27.4233 1.40126 0.700632 0.713522i $$-0.252901\pi$$
0.700632 + 0.713522i $$0.252901\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 5.56155 0.283443
$$386$$ 19.3693 0.985872
$$387$$ −6.43845 −0.327285
$$388$$ 14.4924 0.735741
$$389$$ −28.7386 −1.45711 −0.728553 0.684989i $$-0.759807\pi$$
−0.728553 + 0.684989i $$0.759807\pi$$
$$390$$ 3.56155 0.180346
$$391$$ 37.1771 1.88013
$$392$$ 1.00000 0.0505076
$$393$$ −9.56155 −0.482317
$$394$$ 10.8769 0.547970
$$395$$ −39.6155 −1.99327
$$396$$ −1.56155 −0.0784710
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ 6.93087 0.347413
$$399$$ 4.68466 0.234526
$$400$$ 7.68466 0.384233
$$401$$ 8.24621 0.411796 0.205898 0.978573i $$-0.433988\pi$$
0.205898 + 0.978573i $$0.433988\pi$$
$$402$$ −7.12311 −0.355268
$$403$$ 6.24621 0.311146
$$404$$ −5.12311 −0.254884
$$405$$ 3.56155 0.176975
$$406$$ −6.68466 −0.331754
$$407$$ 11.8078 0.585289
$$408$$ −6.68466 −0.330940
$$409$$ −4.93087 −0.243816 −0.121908 0.992541i $$-0.538901\pi$$
−0.121908 + 0.992541i $$0.538901\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ −3.56155 −0.175678
$$412$$ −0.684658 −0.0337307
$$413$$ −2.24621 −0.110529
$$414$$ −5.56155 −0.273335
$$415$$ 31.6155 1.55195
$$416$$ 1.00000 0.0490290
$$417$$ 12.0000 0.587643
$$418$$ 7.31534 0.357805
$$419$$ −36.6847 −1.79216 −0.896081 0.443890i $$-0.853598\pi$$
−0.896081 + 0.443890i $$0.853598\pi$$
$$420$$ −3.56155 −0.173786
$$421$$ −3.75379 −0.182948 −0.0914742 0.995807i $$-0.529158\pi$$
−0.0914742 + 0.995807i $$0.529158\pi$$
$$422$$ −15.8078 −0.769510
$$423$$ 0 0
$$424$$ 12.2462 0.594729
$$425$$ −51.3693 −2.49178
$$426$$ 8.00000 0.387601
$$427$$ −6.68466 −0.323493
$$428$$ −8.87689 −0.429081
$$429$$ −1.56155 −0.0753925
$$430$$ −22.9309 −1.10582
$$431$$ −33.3693 −1.60734 −0.803672 0.595073i $$-0.797123\pi$$
−0.803672 + 0.595073i $$0.797123\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −31.3693 −1.50751 −0.753757 0.657154i $$-0.771760\pi$$
−0.753757 + 0.657154i $$0.771760\pi$$
$$434$$ −6.24621 −0.299828
$$435$$ 23.8078 1.14149
$$436$$ −16.9309 −0.810842
$$437$$ 26.0540 1.24633
$$438$$ −3.56155 −0.170178
$$439$$ −6.93087 −0.330792 −0.165396 0.986227i $$-0.552890\pi$$
−0.165396 + 0.986227i $$0.552890\pi$$
$$440$$ −5.56155 −0.265137
$$441$$ 1.00000 0.0476190
$$442$$ −6.68466 −0.317957
$$443$$ −5.36932 −0.255104 −0.127552 0.991832i $$-0.540712\pi$$
−0.127552 + 0.991832i $$0.540712\pi$$
$$444$$ −7.56155 −0.358855
$$445$$ 35.6155 1.68834
$$446$$ −23.6155 −1.11823
$$447$$ −17.6155 −0.833186
$$448$$ −1.00000 −0.0472456
$$449$$ 10.6847 0.504240 0.252120 0.967696i $$-0.418872\pi$$
0.252120 + 0.967696i $$0.418872\pi$$
$$450$$ 7.68466 0.362258
$$451$$ 1.75379 0.0825827
$$452$$ −4.24621 −0.199725
$$453$$ 11.8078 0.554777
$$454$$ −7.12311 −0.334304
$$455$$ −3.56155 −0.166968
$$456$$ −4.68466 −0.219379
$$457$$ 27.3693 1.28028 0.640141 0.768257i $$-0.278876\pi$$
0.640141 + 0.768257i $$0.278876\pi$$
$$458$$ 28.2462 1.31986
$$459$$ −6.68466 −0.312013
$$460$$ −19.8078 −0.923542
$$461$$ −28.0540 −1.30660 −0.653302 0.757097i $$-0.726617\pi$$
−0.653302 + 0.757097i $$0.726617\pi$$
$$462$$ 1.56155 0.0726500
$$463$$ −8.68466 −0.403610 −0.201805 0.979426i $$-0.564681\pi$$
−0.201805 + 0.979426i $$0.564681\pi$$
$$464$$ 6.68466 0.310327
$$465$$ 22.2462 1.03164
$$466$$ 27.3693 1.26786
$$467$$ −3.31534 −0.153416 −0.0767079 0.997054i $$-0.524441\pi$$
−0.0767079 + 0.997054i $$0.524441\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ 7.12311 0.328914
$$470$$ 0 0
$$471$$ −15.5616 −0.717039
$$472$$ 2.24621 0.103390
$$473$$ 10.0540 0.462282
$$474$$ −11.1231 −0.510901
$$475$$ −36.0000 −1.65179
$$476$$ 6.68466 0.306391
$$477$$ 12.2462 0.560715
$$478$$ 16.0000 0.731823
$$479$$ 32.3002 1.47583 0.737917 0.674892i $$-0.235810\pi$$
0.737917 + 0.674892i $$0.235810\pi$$
$$480$$ 3.56155 0.162562
$$481$$ −7.56155 −0.344777
$$482$$ −14.0000 −0.637683
$$483$$ 5.56155 0.253059
$$484$$ −8.56155 −0.389161
$$485$$ 51.6155 2.34374
$$486$$ 1.00000 0.0453609
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 6.68466 0.302600
$$489$$ −16.8769 −0.763200
$$490$$ 3.56155 0.160895
$$491$$ −8.87689 −0.400609 −0.200304 0.979734i $$-0.564193\pi$$
−0.200304 + 0.979734i $$0.564193\pi$$
$$492$$ −1.12311 −0.0506335
$$493$$ −44.6847 −2.01250
$$494$$ −4.68466 −0.210773
$$495$$ −5.56155 −0.249973
$$496$$ 6.24621 0.280463
$$497$$ −8.00000 −0.358849
$$498$$ 8.87689 0.397783
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ 9.56155 0.427606
$$501$$ 22.9309 1.02448
$$502$$ −7.80776 −0.348478
$$503$$ −3.12311 −0.139252 −0.0696262 0.997573i $$-0.522181\pi$$
−0.0696262 + 0.997573i $$0.522181\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ −18.2462 −0.811946
$$506$$ 8.68466 0.386080
$$507$$ 1.00000 0.0444116
$$508$$ −4.87689 −0.216377
$$509$$ −12.0540 −0.534283 −0.267142 0.963657i $$-0.586079\pi$$
−0.267142 + 0.963657i $$0.586079\pi$$
$$510$$ −23.8078 −1.05423
$$511$$ 3.56155 0.157554
$$512$$ 1.00000 0.0441942
$$513$$ −4.68466 −0.206833
$$514$$ −4.24621 −0.187292
$$515$$ −2.43845 −0.107451
$$516$$ −6.43845 −0.283437
$$517$$ 0 0
$$518$$ 7.56155 0.332236
$$519$$ 20.2462 0.888710
$$520$$ 3.56155 0.156184
$$521$$ 2.68466 0.117617 0.0588085 0.998269i $$-0.481270\pi$$
0.0588085 + 0.998269i $$0.481270\pi$$
$$522$$ 6.68466 0.292580
$$523$$ 21.7538 0.951227 0.475613 0.879654i $$-0.342226\pi$$
0.475613 + 0.879654i $$0.342226\pi$$
$$524$$ −9.56155 −0.417698
$$525$$ −7.68466 −0.335386
$$526$$ −30.2462 −1.31880
$$527$$ −41.7538 −1.81882
$$528$$ −1.56155 −0.0679579
$$529$$ 7.93087 0.344820
$$530$$ 43.6155 1.89454
$$531$$ 2.24621 0.0974773
$$532$$ 4.68466 0.203106
$$533$$ −1.12311 −0.0486471
$$534$$ 10.0000 0.432742
$$535$$ −31.6155 −1.36686
$$536$$ −7.12311 −0.307671
$$537$$ 16.4924 0.711701
$$538$$ 20.2462 0.872876
$$539$$ −1.56155 −0.0672608
$$540$$ 3.56155 0.153265
$$541$$ −32.9309 −1.41581 −0.707904 0.706308i $$-0.750359\pi$$
−0.707904 + 0.706308i $$0.750359\pi$$
$$542$$ 4.87689 0.209481
$$543$$ −0.246211 −0.0105659
$$544$$ −6.68466 −0.286602
$$545$$ −60.3002 −2.58298
$$546$$ −1.00000 −0.0427960
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −3.56155 −0.152142
$$549$$ 6.68466 0.285294
$$550$$ −12.0000 −0.511682
$$551$$ −31.3153 −1.33408
$$552$$ −5.56155 −0.236715
$$553$$ 11.1231 0.473003
$$554$$ −22.4924 −0.955611
$$555$$ −26.9309 −1.14315
$$556$$ 12.0000 0.508913
$$557$$ −3.36932 −0.142763 −0.0713813 0.997449i $$-0.522741\pi$$
−0.0713813 + 0.997449i $$0.522741\pi$$
$$558$$ 6.24621 0.264423
$$559$$ −6.43845 −0.272317
$$560$$ −3.56155 −0.150503
$$561$$ 10.4384 0.440712
$$562$$ 16.2462 0.685305
$$563$$ −6.05398 −0.255145 −0.127572 0.991829i $$-0.540718\pi$$
−0.127572 + 0.991829i $$0.540718\pi$$
$$564$$ 0 0
$$565$$ −15.1231 −0.636234
$$566$$ 18.2462 0.766945
$$567$$ −1.00000 −0.0419961
$$568$$ 8.00000 0.335673
$$569$$ 41.2311 1.72850 0.864248 0.503066i $$-0.167795\pi$$
0.864248 + 0.503066i $$0.167795\pi$$
$$570$$ −16.6847 −0.698843
$$571$$ 18.2462 0.763580 0.381790 0.924249i $$-0.375308\pi$$
0.381790 + 0.924249i $$0.375308\pi$$
$$572$$ −1.56155 −0.0652918
$$573$$ 14.9309 0.623746
$$574$$ 1.12311 0.0468775
$$575$$ −42.7386 −1.78232
$$576$$ 1.00000 0.0416667
$$577$$ −30.0000 −1.24892 −0.624458 0.781058i $$-0.714680\pi$$
−0.624458 + 0.781058i $$0.714680\pi$$
$$578$$ 27.6847 1.15153
$$579$$ 19.3693 0.804961
$$580$$ 23.8078 0.988564
$$581$$ −8.87689 −0.368276
$$582$$ 14.4924 0.600730
$$583$$ −19.1231 −0.791998
$$584$$ −3.56155 −0.147378
$$585$$ 3.56155 0.147252
$$586$$ 24.7386 1.02194
$$587$$ 13.3693 0.551811 0.275905 0.961185i $$-0.411022\pi$$
0.275905 + 0.961185i $$0.411022\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ −29.2614 −1.20569
$$590$$ 8.00000 0.329355
$$591$$ 10.8769 0.447416
$$592$$ −7.56155 −0.310778
$$593$$ −20.2462 −0.831412 −0.415706 0.909499i $$-0.636466\pi$$
−0.415706 + 0.909499i $$0.636466\pi$$
$$594$$ −1.56155 −0.0640713
$$595$$ 23.8078 0.976023
$$596$$ −17.6155 −0.721560
$$597$$ 6.93087 0.283662
$$598$$ −5.56155 −0.227429
$$599$$ 21.5616 0.880981 0.440491 0.897757i $$-0.354805\pi$$
0.440491 + 0.897757i $$0.354805\pi$$
$$600$$ 7.68466 0.313725
$$601$$ −10.8769 −0.443678 −0.221839 0.975083i $$-0.571206\pi$$
−0.221839 + 0.975083i $$0.571206\pi$$
$$602$$ 6.43845 0.262412
$$603$$ −7.12311 −0.290075
$$604$$ 11.8078 0.480451
$$605$$ −30.4924 −1.23969
$$606$$ −5.12311 −0.208112
$$607$$ −18.4384 −0.748393 −0.374197 0.927349i $$-0.622082\pi$$
−0.374197 + 0.927349i $$0.622082\pi$$
$$608$$ −4.68466 −0.189988
$$609$$ −6.68466 −0.270876
$$610$$ 23.8078 0.963948
$$611$$ 0 0
$$612$$ −6.68466 −0.270211
$$613$$ 44.5464 1.79921 0.899606 0.436702i $$-0.143854\pi$$
0.899606 + 0.436702i $$0.143854\pi$$
$$614$$ 26.2462 1.05921
$$615$$ −4.00000 −0.161296
$$616$$ 1.56155 0.0629168
$$617$$ −33.4233 −1.34557 −0.672786 0.739838i $$-0.734902\pi$$
−0.672786 + 0.739838i $$0.734902\pi$$
$$618$$ −0.684658 −0.0275410
$$619$$ 19.3153 0.776349 0.388175 0.921586i $$-0.373106\pi$$
0.388175 + 0.921586i $$0.373106\pi$$
$$620$$ 22.2462 0.893429
$$621$$ −5.56155 −0.223177
$$622$$ −12.8769 −0.516316
$$623$$ −10.0000 −0.400642
$$624$$ 1.00000 0.0400320
$$625$$ −4.36932 −0.174773
$$626$$ −13.6155 −0.544186
$$627$$ 7.31534 0.292147
$$628$$ −15.5616 −0.620974
$$629$$ 50.5464 2.01542
$$630$$ −3.56155 −0.141896
$$631$$ −22.9309 −0.912864 −0.456432 0.889758i $$-0.650873\pi$$
−0.456432 + 0.889758i $$0.650873\pi$$
$$632$$ −11.1231 −0.442453
$$633$$ −15.8078 −0.628302
$$634$$ 2.87689 0.114256
$$635$$ −17.3693 −0.689280
$$636$$ 12.2462 0.485594
$$637$$ 1.00000 0.0396214
$$638$$ −10.4384 −0.413262
$$639$$ 8.00000 0.316475
$$640$$ 3.56155 0.140783
$$641$$ −20.2462 −0.799677 −0.399839 0.916586i $$-0.630934\pi$$
−0.399839 + 0.916586i $$0.630934\pi$$
$$642$$ −8.87689 −0.350343
$$643$$ 16.1922 0.638559 0.319280 0.947661i $$-0.396559\pi$$
0.319280 + 0.947661i $$0.396559\pi$$
$$644$$ 5.56155 0.219156
$$645$$ −22.9309 −0.902902
$$646$$ 31.3153 1.23209
$$647$$ −14.2462 −0.560076 −0.280038 0.959989i $$-0.590347\pi$$
−0.280038 + 0.959989i $$0.590347\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −3.50758 −0.137684
$$650$$ 7.68466 0.301417
$$651$$ −6.24621 −0.244808
$$652$$ −16.8769 −0.660950
$$653$$ −16.9309 −0.662556 −0.331278 0.943533i $$-0.607480\pi$$
−0.331278 + 0.943533i $$0.607480\pi$$
$$654$$ −16.9309 −0.662049
$$655$$ −34.0540 −1.33060
$$656$$ −1.12311 −0.0438499
$$657$$ −3.56155 −0.138949
$$658$$ 0 0
$$659$$ 21.3693 0.832430 0.416215 0.909266i $$-0.363356\pi$$
0.416215 + 0.909266i $$0.363356\pi$$
$$660$$ −5.56155 −0.216483
$$661$$ −0.246211 −0.00957651 −0.00478825 0.999989i $$-0.501524\pi$$
−0.00478825 + 0.999989i $$0.501524\pi$$
$$662$$ 13.3693 0.519613
$$663$$ −6.68466 −0.259611
$$664$$ 8.87689 0.344490
$$665$$ 16.6847 0.647003
$$666$$ −7.56155 −0.293004
$$667$$ −37.1771 −1.43950
$$668$$ 22.9309 0.887222
$$669$$ −23.6155 −0.913029
$$670$$ −25.3693 −0.980102
$$671$$ −10.4384 −0.402972
$$672$$ −1.00000 −0.0385758
$$673$$ −33.8078 −1.30319 −0.651597 0.758566i $$-0.725901\pi$$
−0.651597 + 0.758566i $$0.725901\pi$$
$$674$$ 4.43845 0.170963
$$675$$ 7.68466 0.295783
$$676$$ 1.00000 0.0384615
$$677$$ 2.49242 0.0957916 0.0478958 0.998852i $$-0.484748\pi$$
0.0478958 + 0.998852i $$0.484748\pi$$
$$678$$ −4.24621 −0.163075
$$679$$ −14.4924 −0.556168
$$680$$ −23.8078 −0.912986
$$681$$ −7.12311 −0.272958
$$682$$ −9.75379 −0.373492
$$683$$ 35.3153 1.35130 0.675652 0.737221i $$-0.263862\pi$$
0.675652 + 0.737221i $$0.263862\pi$$
$$684$$ −4.68466 −0.179122
$$685$$ −12.6847 −0.484656
$$686$$ −1.00000 −0.0381802
$$687$$ 28.2462 1.07766
$$688$$ −6.43845 −0.245463
$$689$$ 12.2462 0.466543
$$690$$ −19.8078 −0.754069
$$691$$ 16.4924 0.627401 0.313701 0.949522i $$-0.398431\pi$$
0.313701 + 0.949522i $$0.398431\pi$$
$$692$$ 20.2462 0.769645
$$693$$ 1.56155 0.0593185
$$694$$ −20.0000 −0.759190
$$695$$ 42.7386 1.62117
$$696$$ 6.68466 0.253381
$$697$$ 7.50758 0.284370
$$698$$ 7.75379 0.293485
$$699$$ 27.3693 1.03520
$$700$$ −7.68466 −0.290453
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ 35.4233 1.33601
$$704$$ −1.56155 −0.0588532
$$705$$ 0 0
$$706$$ 30.4924 1.14760
$$707$$ 5.12311 0.192674
$$708$$ 2.24621 0.0844178
$$709$$ −16.2462 −0.610139 −0.305070 0.952330i $$-0.598680\pi$$
−0.305070 + 0.952330i $$0.598680\pi$$
$$710$$ 28.4924 1.06930
$$711$$ −11.1231 −0.417149
$$712$$ 10.0000 0.374766
$$713$$ −34.7386 −1.30097
$$714$$ 6.68466 0.250167
$$715$$ −5.56155 −0.207990
$$716$$ 16.4924 0.616351
$$717$$ 16.0000 0.597531
$$718$$ −26.7386 −0.997877
$$719$$ 43.1231 1.60822 0.804110 0.594480i $$-0.202642\pi$$
0.804110 + 0.594480i $$0.202642\pi$$
$$720$$ 3.56155 0.132731
$$721$$ 0.684658 0.0254980
$$722$$ 2.94602 0.109640
$$723$$ −14.0000 −0.520666
$$724$$ −0.246211 −0.00915037
$$725$$ 51.3693 1.90781
$$726$$ −8.56155 −0.317749
$$727$$ −6.93087 −0.257052 −0.128526 0.991706i $$-0.541025\pi$$
−0.128526 + 0.991706i $$0.541025\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ 1.00000 0.0370370
$$730$$ −12.6847 −0.469480
$$731$$ 43.0388 1.59185
$$732$$ 6.68466 0.247072
$$733$$ −41.6155 −1.53710 −0.768552 0.639787i $$-0.779023\pi$$
−0.768552 + 0.639787i $$0.779023\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 3.56155 0.131370
$$736$$ −5.56155 −0.205002
$$737$$ 11.1231 0.409725
$$738$$ −1.12311 −0.0413421
$$739$$ −32.1080 −1.18111 −0.590555 0.806997i $$-0.701091\pi$$
−0.590555 + 0.806997i $$0.701091\pi$$
$$740$$ −26.9309 −0.989998
$$741$$ −4.68466 −0.172095
$$742$$ −12.2462 −0.449573
$$743$$ 28.1080 1.03118 0.515590 0.856835i $$-0.327573\pi$$
0.515590 + 0.856835i $$0.327573\pi$$
$$744$$ 6.24621 0.228997
$$745$$ −62.7386 −2.29857
$$746$$ −17.6155 −0.644950
$$747$$ 8.87689 0.324789
$$748$$ 10.4384 0.381667
$$749$$ 8.87689 0.324355
$$750$$ 9.56155 0.349139
$$751$$ −6.24621 −0.227927 −0.113964 0.993485i $$-0.536355\pi$$
−0.113964 + 0.993485i $$0.536355\pi$$
$$752$$ 0 0
$$753$$ −7.80776 −0.284531
$$754$$ 6.68466 0.243441
$$755$$ 42.0540 1.53050
$$756$$ −1.00000 −0.0363696
$$757$$ 34.4924 1.25365 0.626824 0.779161i $$-0.284354\pi$$
0.626824 + 0.779161i $$0.284354\pi$$
$$758$$ 34.2462 1.24388
$$759$$ 8.68466 0.315233
$$760$$ −16.6847 −0.605216
$$761$$ 5.12311 0.185712 0.0928562 0.995680i $$-0.470400\pi$$
0.0928562 + 0.995680i $$0.470400\pi$$
$$762$$ −4.87689 −0.176671
$$763$$ 16.9309 0.612939
$$764$$ 14.9309 0.540180
$$765$$ −23.8078 −0.860772
$$766$$ 27.4233 0.990844
$$767$$ 2.24621 0.0811060
$$768$$ 1.00000 0.0360844
$$769$$ −16.4384 −0.592786 −0.296393 0.955066i $$-0.595784\pi$$
−0.296393 + 0.955066i $$0.595784\pi$$
$$770$$ 5.56155 0.200424
$$771$$ −4.24621 −0.152924
$$772$$ 19.3693 0.697117
$$773$$ 52.9309 1.90379 0.951896 0.306423i $$-0.0991321\pi$$
0.951896 + 0.306423i $$0.0991321\pi$$
$$774$$ −6.43845 −0.231425
$$775$$ 48.0000 1.72421
$$776$$ 14.4924 0.520248
$$777$$ 7.56155 0.271269
$$778$$ −28.7386 −1.03033
$$779$$ 5.26137 0.188508
$$780$$ 3.56155 0.127524
$$781$$ −12.4924 −0.447014
$$782$$ 37.1771 1.32945
$$783$$ 6.68466 0.238890
$$784$$ 1.00000 0.0357143
$$785$$ −55.4233 −1.97814
$$786$$ −9.56155 −0.341049
$$787$$ 20.3002 0.723624 0.361812 0.932251i $$-0.382158\pi$$
0.361812 + 0.932251i $$0.382158\pi$$
$$788$$ 10.8769 0.387473
$$789$$ −30.2462 −1.07679
$$790$$ −39.6155 −1.40946
$$791$$ 4.24621 0.150978
$$792$$ −1.56155 −0.0554874
$$793$$ 6.68466 0.237379
$$794$$ −18.0000 −0.638796
$$795$$ 43.6155 1.54688
$$796$$ 6.93087 0.245658
$$797$$ 31.3693 1.11116 0.555579 0.831464i $$-0.312497\pi$$
0.555579 + 0.831464i $$0.312497\pi$$
$$798$$ 4.68466 0.165835
$$799$$ 0 0
$$800$$ 7.68466 0.271694
$$801$$ 10.0000 0.353333
$$802$$ 8.24621 0.291184
$$803$$ 5.56155 0.196263
$$804$$ −7.12311 −0.251213
$$805$$ 19.8078 0.698132
$$806$$ 6.24621 0.220013
$$807$$ 20.2462 0.712700
$$808$$ −5.12311 −0.180230
$$809$$ −2.49242 −0.0876289 −0.0438145 0.999040i $$-0.513951\pi$$
−0.0438145 + 0.999040i $$0.513951\pi$$
$$810$$ 3.56155 0.125140
$$811$$ 41.5616 1.45942 0.729712 0.683755i $$-0.239654\pi$$
0.729712 + 0.683755i $$0.239654\pi$$
$$812$$ −6.68466 −0.234586
$$813$$ 4.87689 0.171040
$$814$$ 11.8078 0.413862
$$815$$ −60.1080 −2.10549
$$816$$ −6.68466 −0.234010
$$817$$ 30.1619 1.05523
$$818$$ −4.93087 −0.172404
$$819$$ −1.00000 −0.0349428
$$820$$ −4.00000 −0.139686
$$821$$ 14.3845 0.502022 0.251011 0.967984i $$-0.419237\pi$$
0.251011 + 0.967984i $$0.419237\pi$$
$$822$$ −3.56155 −0.124223
$$823$$ 4.49242 0.156596 0.0782980 0.996930i $$-0.475051\pi$$
0.0782980 + 0.996930i $$0.475051\pi$$
$$824$$ −0.684658 −0.0238512
$$825$$ −12.0000 −0.417786
$$826$$ −2.24621 −0.0781557
$$827$$ 26.9309 0.936478 0.468239 0.883602i $$-0.344889\pi$$
0.468239 + 0.883602i $$0.344889\pi$$
$$828$$ −5.56155 −0.193277
$$829$$ 38.6847 1.34357 0.671787 0.740744i $$-0.265527\pi$$
0.671787 + 0.740744i $$0.265527\pi$$
$$830$$ 31.6155 1.09739
$$831$$ −22.4924 −0.780253
$$832$$ 1.00000 0.0346688
$$833$$ −6.68466 −0.231610
$$834$$ 12.0000 0.415526
$$835$$ 81.6695 2.82629
$$836$$ 7.31534 0.253006
$$837$$ 6.24621 0.215901
$$838$$ −36.6847 −1.26725
$$839$$ −10.7386 −0.370739 −0.185369 0.982669i $$-0.559348\pi$$
−0.185369 + 0.982669i $$0.559348\pi$$
$$840$$ −3.56155 −0.122885
$$841$$ 15.6847 0.540850
$$842$$ −3.75379 −0.129364
$$843$$ 16.2462 0.559549
$$844$$ −15.8078 −0.544126
$$845$$ 3.56155 0.122521
$$846$$ 0 0
$$847$$ 8.56155 0.294178
$$848$$ 12.2462 0.420537
$$849$$ 18.2462 0.626208
$$850$$ −51.3693 −1.76195
$$851$$ 42.0540 1.44159
$$852$$ 8.00000 0.274075
$$853$$ −27.3693 −0.937108 −0.468554 0.883435i $$-0.655225\pi$$
−0.468554 + 0.883435i $$0.655225\pi$$
$$854$$ −6.68466 −0.228744
$$855$$ −16.6847 −0.570603
$$856$$ −8.87689 −0.303406
$$857$$ −34.4924 −1.17824 −0.589119 0.808046i $$-0.700525\pi$$
−0.589119 + 0.808046i $$0.700525\pi$$
$$858$$ −1.56155 −0.0533105
$$859$$ 5.75379 0.196317 0.0981584 0.995171i $$-0.468705\pi$$
0.0981584 + 0.995171i $$0.468705\pi$$
$$860$$ −22.9309 −0.781936
$$861$$ 1.12311 0.0382753
$$862$$ −33.3693 −1.13656
$$863$$ −17.3693 −0.591258 −0.295629 0.955303i $$-0.595529\pi$$
−0.295629 + 0.955303i $$0.595529\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 72.1080 2.45174
$$866$$ −31.3693 −1.06597
$$867$$ 27.6847 0.940220
$$868$$ −6.24621 −0.212010
$$869$$ 17.3693 0.589214
$$870$$ 23.8078 0.807159
$$871$$ −7.12311 −0.241357
$$872$$ −16.9309 −0.573352
$$873$$ 14.4924 0.490494
$$874$$ 26.0540 0.881289
$$875$$ −9.56155 −0.323239
$$876$$ −3.56155 −0.120334
$$877$$ −40.2462 −1.35902 −0.679509 0.733667i $$-0.737807\pi$$
−0.679509 + 0.733667i $$0.737807\pi$$
$$878$$ −6.93087 −0.233906
$$879$$ 24.7386 0.834413
$$880$$ −5.56155 −0.187480
$$881$$ −19.1771 −0.646092 −0.323046 0.946383i $$-0.604707\pi$$
−0.323046 + 0.946383i $$0.604707\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 1.56155 0.0525504 0.0262752 0.999655i $$-0.491635\pi$$
0.0262752 + 0.999655i $$0.491635\pi$$
$$884$$ −6.68466 −0.224829
$$885$$ 8.00000 0.268917
$$886$$ −5.36932 −0.180386
$$887$$ −52.4924 −1.76252 −0.881262 0.472629i $$-0.843305\pi$$
−0.881262 + 0.472629i $$0.843305\pi$$
$$888$$ −7.56155 −0.253749
$$889$$ 4.87689 0.163566
$$890$$ 35.6155 1.19384
$$891$$ −1.56155 −0.0523140
$$892$$ −23.6155 −0.790706
$$893$$ 0 0
$$894$$ −17.6155 −0.589151
$$895$$ 58.7386 1.96342
$$896$$ −1.00000 −0.0334077
$$897$$ −5.56155 −0.185695
$$898$$ 10.6847 0.356552
$$899$$ 41.7538 1.39257
$$900$$ 7.68466 0.256155
$$901$$ −81.8617 −2.72721
$$902$$ 1.75379 0.0583948
$$903$$ 6.43845 0.214258
$$904$$ −4.24621 −0.141227
$$905$$ −0.876894 −0.0291490
$$906$$ 11.8078 0.392287
$$907$$ −7.50758 −0.249285 −0.124643 0.992202i $$-0.539778\pi$$
−0.124643 + 0.992202i $$0.539778\pi$$
$$908$$ −7.12311 −0.236389
$$909$$ −5.12311 −0.169923
$$910$$ −3.56155 −0.118064
$$911$$ −11.4233 −0.378471 −0.189235 0.981932i $$-0.560601\pi$$
−0.189235 + 0.981932i $$0.560601\pi$$
$$912$$ −4.68466 −0.155125
$$913$$ −13.8617 −0.458757
$$914$$ 27.3693 0.905297
$$915$$ 23.8078 0.787060
$$916$$ 28.2462 0.933281
$$917$$ 9.56155 0.315750
$$918$$ −6.68466 −0.220627
$$919$$ 19.1231 0.630813 0.315407 0.948957i $$-0.397859\pi$$
0.315407 + 0.948957i $$0.397859\pi$$
$$920$$ −19.8078 −0.653043
$$921$$ 26.2462 0.864842
$$922$$ −28.0540 −0.923908
$$923$$ 8.00000 0.263323
$$924$$ 1.56155 0.0513713
$$925$$ −58.1080 −1.91058
$$926$$ −8.68466 −0.285396
$$927$$ −0.684658 −0.0224871
$$928$$ 6.68466 0.219435
$$929$$ 25.6155 0.840418 0.420209 0.907427i $$-0.361957\pi$$
0.420209 + 0.907427i $$0.361957\pi$$
$$930$$ 22.2462 0.729482
$$931$$ −4.68466 −0.153533
$$932$$ 27.3693 0.896512
$$933$$ −12.8769 −0.421571
$$934$$ −3.31534 −0.108481
$$935$$ 37.1771 1.21582
$$936$$ 1.00000 0.0326860
$$937$$ −46.9848 −1.53493 −0.767464 0.641092i $$-0.778482\pi$$
−0.767464 + 0.641092i $$0.778482\pi$$
$$938$$ 7.12311 0.232578
$$939$$ −13.6155 −0.444326
$$940$$ 0 0
$$941$$ −34.0000 −1.10837 −0.554184 0.832394i $$-0.686970\pi$$
−0.554184 + 0.832394i $$0.686970\pi$$
$$942$$ −15.5616 −0.507023
$$943$$ 6.24621 0.203405
$$944$$ 2.24621 0.0731079
$$945$$ −3.56155 −0.115857
$$946$$ 10.0540 0.326883
$$947$$ −40.7926 −1.32558 −0.662791 0.748805i $$-0.730628\pi$$
−0.662791 + 0.748805i $$0.730628\pi$$
$$948$$ −11.1231 −0.361262
$$949$$ −3.56155 −0.115613
$$950$$ −36.0000 −1.16799
$$951$$ 2.87689 0.0932897
$$952$$ 6.68466 0.216651
$$953$$ 11.3693 0.368288 0.184144 0.982899i $$-0.441049\pi$$
0.184144 + 0.982899i $$0.441049\pi$$
$$954$$ 12.2462 0.396486
$$955$$ 53.1771 1.72077
$$956$$ 16.0000 0.517477
$$957$$ −10.4384 −0.337427
$$958$$ 32.3002 1.04357
$$959$$ 3.56155 0.115009
$$960$$ 3.56155 0.114949
$$961$$ 8.01515 0.258553
$$962$$ −7.56155 −0.243794
$$963$$ −8.87689 −0.286054
$$964$$ −14.0000 −0.450910
$$965$$ 68.9848 2.22070
$$966$$ 5.56155 0.178940
$$967$$ −21.5616 −0.693373 −0.346686 0.937981i $$-0.612693\pi$$
−0.346686 + 0.937981i $$0.612693\pi$$
$$968$$ −8.56155 −0.275179
$$969$$ 31.3153 1.00599
$$970$$ 51.6155 1.65727
$$971$$ 2.24621 0.0720843 0.0360422 0.999350i $$-0.488525\pi$$
0.0360422 + 0.999350i $$0.488525\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −12.0000 −0.384702
$$974$$ −8.00000 −0.256337
$$975$$ 7.68466 0.246106
$$976$$ 6.68466 0.213971
$$977$$ −22.6847 −0.725747 −0.362873 0.931839i $$-0.618204\pi$$
−0.362873 + 0.931839i $$0.618204\pi$$
$$978$$ −16.8769 −0.539664
$$979$$ −15.6155 −0.499074
$$980$$ 3.56155 0.113770
$$981$$ −16.9309 −0.540561
$$982$$ −8.87689 −0.283273
$$983$$ −13.1771 −0.420284 −0.210142 0.977671i $$-0.567393\pi$$
−0.210142 + 0.977671i $$0.567393\pi$$
$$984$$ −1.12311 −0.0358033
$$985$$ 38.7386 1.23432
$$986$$ −44.6847 −1.42305
$$987$$ 0 0
$$988$$ −4.68466 −0.149039
$$989$$ 35.8078 1.13862
$$990$$ −5.56155 −0.176758
$$991$$ −7.61553 −0.241915 −0.120958 0.992658i $$-0.538597\pi$$
−0.120958 + 0.992658i $$0.538597\pi$$
$$992$$ 6.24621 0.198317
$$993$$ 13.3693 0.424262
$$994$$ −8.00000 −0.253745
$$995$$ 24.6847 0.782556
$$996$$ 8.87689 0.281275
$$997$$ 40.7386 1.29021 0.645103 0.764096i $$-0.276815\pi$$
0.645103 + 0.764096i $$0.276815\pi$$
$$998$$ 36.0000 1.13956
$$999$$ −7.56155 −0.239237
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 546.2.a.j.1.2 2
3.2 odd 2 1638.2.a.u.1.1 2
4.3 odd 2 4368.2.a.be.1.2 2
7.6 odd 2 3822.2.a.bo.1.1 2
13.12 even 2 7098.2.a.bl.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.j.1.2 2 1.1 even 1 trivial
1638.2.a.u.1.1 2 3.2 odd 2
3822.2.a.bo.1.1 2 7.6 odd 2
4368.2.a.be.1.2 2 4.3 odd 2
7098.2.a.bl.1.1 2 13.12 even 2