# Properties

 Label 1183.2.a.p.1.2 Level $1183$ Weight $2$ Character 1183.1 Self dual yes Analytic conductor $9.446$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.7674048.1 Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2$$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.82356$$ of defining polynomial Character $$\chi$$ $$=$$ 1183.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.823556 q^{2} +2.66029 q^{3} -1.32176 q^{4} +3.16209 q^{5} -2.19090 q^{6} -1.00000 q^{7} +2.73565 q^{8} +4.07715 q^{9} +O(q^{10})$$ $$q-0.823556 q^{2} +2.66029 q^{3} -1.32176 q^{4} +3.16209 q^{5} -2.19090 q^{6} -1.00000 q^{7} +2.73565 q^{8} +4.07715 q^{9} -2.60416 q^{10} +5.94270 q^{11} -3.51626 q^{12} +0.823556 q^{14} +8.41209 q^{15} +0.390549 q^{16} -2.69964 q^{17} -3.35776 q^{18} +1.95705 q^{19} -4.17951 q^{20} -2.66029 q^{21} -4.89414 q^{22} -2.72941 q^{23} +7.27763 q^{24} +4.99883 q^{25} +2.86554 q^{27} +1.32176 q^{28} -5.99845 q^{29} -6.92783 q^{30} +1.15155 q^{31} -5.79294 q^{32} +15.8093 q^{33} +2.22331 q^{34} -3.16209 q^{35} -5.38900 q^{36} -6.50454 q^{37} -1.61174 q^{38} +8.65038 q^{40} +3.73374 q^{41} +2.19090 q^{42} +6.99125 q^{43} -7.85479 q^{44} +12.8923 q^{45} +2.24783 q^{46} +0.456071 q^{47} +1.03897 q^{48} +1.00000 q^{49} -4.11682 q^{50} -7.18184 q^{51} +0.399286 q^{53} -2.35994 q^{54} +18.7914 q^{55} -2.73565 q^{56} +5.20632 q^{57} +4.94006 q^{58} -4.80586 q^{59} -11.1187 q^{60} -1.15703 q^{61} -0.948365 q^{62} -4.07715 q^{63} +3.98971 q^{64} -13.0199 q^{66} +6.27918 q^{67} +3.56827 q^{68} -7.26104 q^{69} +2.60416 q^{70} +4.50720 q^{71} +11.1537 q^{72} +8.30575 q^{73} +5.35685 q^{74} +13.2983 q^{75} -2.58674 q^{76} -5.94270 q^{77} -7.91410 q^{79} +1.23495 q^{80} -4.60828 q^{81} -3.07494 q^{82} -6.19795 q^{83} +3.51626 q^{84} -8.53652 q^{85} -5.75769 q^{86} -15.9576 q^{87} +16.2571 q^{88} +3.56136 q^{89} -10.6176 q^{90} +3.60762 q^{92} +3.06345 q^{93} -0.375600 q^{94} +6.18837 q^{95} -15.4109 q^{96} -3.42751 q^{97} -0.823556 q^{98} +24.2293 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 6 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10})$$ 6 * q + 4 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^6 - 6 * q^7 + 12 * q^8 + 4 * q^9 $$6 q + 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 6 q^{7} + 12 q^{8} + 4 q^{9} + 12 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{14} + 20 q^{15} + 8 q^{16} - 4 q^{17} - 16 q^{18} + 2 q^{19} + 26 q^{20} - 6 q^{22} - 12 q^{23} + 2 q^{24} + 10 q^{25} + 6 q^{27} - 4 q^{28} - 8 q^{29} + 8 q^{30} - 14 q^{31} + 8 q^{32} + 16 q^{33} - 2 q^{34} - 6 q^{35} - 10 q^{36} + 12 q^{37} - 2 q^{38} + 46 q^{40} + 28 q^{41} - 4 q^{42} + 2 q^{43} - 20 q^{44} + 16 q^{45} + 20 q^{46} + 14 q^{47} + 2 q^{48} + 6 q^{49} + 32 q^{50} - 26 q^{51} - 22 q^{53} + 14 q^{54} + 6 q^{55} - 12 q^{56} + 4 q^{58} - 2 q^{59} - 14 q^{61} - 4 q^{62} - 4 q^{63} + 26 q^{64} - 26 q^{66} + 24 q^{67} + 8 q^{68} + 4 q^{69} - 12 q^{70} + 4 q^{71} + 8 q^{72} + 36 q^{73} - 6 q^{74} + 46 q^{75} - 26 q^{76} - 4 q^{77} - 28 q^{79} + 36 q^{80} - 2 q^{81} + 14 q^{82} + 26 q^{83} - 2 q^{84} - 20 q^{85} - 24 q^{86} + 2 q^{87} - 14 q^{88} + 42 q^{89} - 12 q^{90} + 12 q^{92} - 4 q^{94} - 22 q^{95} - 42 q^{96} + 24 q^{97} + 4 q^{98} + 16 q^{99}+O(q^{100})$$ 6 * q + 4 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^6 - 6 * q^7 + 12 * q^8 + 4 * q^9 + 12 * q^10 + 4 * q^11 + 2 * q^12 - 4 * q^14 + 20 * q^15 + 8 * q^16 - 4 * q^17 - 16 * q^18 + 2 * q^19 + 26 * q^20 - 6 * q^22 - 12 * q^23 + 2 * q^24 + 10 * q^25 + 6 * q^27 - 4 * q^28 - 8 * q^29 + 8 * q^30 - 14 * q^31 + 8 * q^32 + 16 * q^33 - 2 * q^34 - 6 * q^35 - 10 * q^36 + 12 * q^37 - 2 * q^38 + 46 * q^40 + 28 * q^41 - 4 * q^42 + 2 * q^43 - 20 * q^44 + 16 * q^45 + 20 * q^46 + 14 * q^47 + 2 * q^48 + 6 * q^49 + 32 * q^50 - 26 * q^51 - 22 * q^53 + 14 * q^54 + 6 * q^55 - 12 * q^56 + 4 * q^58 - 2 * q^59 - 14 * q^61 - 4 * q^62 - 4 * q^63 + 26 * q^64 - 26 * q^66 + 24 * q^67 + 8 * q^68 + 4 * q^69 - 12 * q^70 + 4 * q^71 + 8 * q^72 + 36 * q^73 - 6 * q^74 + 46 * q^75 - 26 * q^76 - 4 * q^77 - 28 * q^79 + 36 * q^80 - 2 * q^81 + 14 * q^82 + 26 * q^83 - 2 * q^84 - 20 * q^85 - 24 * q^86 + 2 * q^87 - 14 * q^88 + 42 * q^89 - 12 * q^90 + 12 * q^92 - 4 * q^94 - 22 * q^95 - 42 * q^96 + 24 * q^97 + 4 * q^98 + 16 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.823556 −0.582342 −0.291171 0.956671i $$-0.594045\pi$$
−0.291171 + 0.956671i $$0.594045\pi$$
$$3$$ 2.66029 1.53592 0.767960 0.640498i $$-0.221272\pi$$
0.767960 + 0.640498i $$0.221272\pi$$
$$4$$ −1.32176 −0.660878
$$5$$ 3.16209 1.41413 0.707065 0.707148i $$-0.250019\pi$$
0.707065 + 0.707148i $$0.250019\pi$$
$$6$$ −2.19090 −0.894431
$$7$$ −1.00000 −0.377964
$$8$$ 2.73565 0.967199
$$9$$ 4.07715 1.35905
$$10$$ −2.60416 −0.823508
$$11$$ 5.94270 1.79179 0.895895 0.444265i $$-0.146535\pi$$
0.895895 + 0.444265i $$0.146535\pi$$
$$12$$ −3.51626 −1.01506
$$13$$ 0 0
$$14$$ 0.823556 0.220105
$$15$$ 8.41209 2.17199
$$16$$ 0.390549 0.0976372
$$17$$ −2.69964 −0.654760 −0.327380 0.944893i $$-0.606166\pi$$
−0.327380 + 0.944893i $$0.606166\pi$$
$$18$$ −3.35776 −0.791433
$$19$$ 1.95705 0.448978 0.224489 0.974477i $$-0.427929\pi$$
0.224489 + 0.974477i $$0.427929\pi$$
$$20$$ −4.17951 −0.934568
$$21$$ −2.66029 −0.580523
$$22$$ −4.89414 −1.04343
$$23$$ −2.72941 −0.569122 −0.284561 0.958658i $$-0.591848\pi$$
−0.284561 + 0.958658i $$0.591848\pi$$
$$24$$ 7.27763 1.48554
$$25$$ 4.99883 0.999766
$$26$$ 0 0
$$27$$ 2.86554 0.551474
$$28$$ 1.32176 0.249788
$$29$$ −5.99845 −1.11388 −0.556942 0.830551i $$-0.688026\pi$$
−0.556942 + 0.830551i $$0.688026\pi$$
$$30$$ −6.92783 −1.26484
$$31$$ 1.15155 0.206824 0.103412 0.994639i $$-0.467024\pi$$
0.103412 + 0.994639i $$0.467024\pi$$
$$32$$ −5.79294 −1.02406
$$33$$ 15.8093 2.75205
$$34$$ 2.22331 0.381294
$$35$$ −3.16209 −0.534491
$$36$$ −5.38900 −0.898167
$$37$$ −6.50454 −1.06934 −0.534670 0.845061i $$-0.679564\pi$$
−0.534670 + 0.845061i $$0.679564\pi$$
$$38$$ −1.61174 −0.261459
$$39$$ 0 0
$$40$$ 8.65038 1.36775
$$41$$ 3.73374 0.583112 0.291556 0.956554i $$-0.405827\pi$$
0.291556 + 0.956554i $$0.405827\pi$$
$$42$$ 2.19090 0.338063
$$43$$ 6.99125 1.06616 0.533078 0.846066i $$-0.321035\pi$$
0.533078 + 0.846066i $$0.321035\pi$$
$$44$$ −7.85479 −1.18415
$$45$$ 12.8923 1.92188
$$46$$ 2.24783 0.331424
$$47$$ 0.456071 0.0665248 0.0332624 0.999447i $$-0.489410\pi$$
0.0332624 + 0.999447i $$0.489410\pi$$
$$48$$ 1.03897 0.149963
$$49$$ 1.00000 0.142857
$$50$$ −4.11682 −0.582206
$$51$$ −7.18184 −1.00566
$$52$$ 0 0
$$53$$ 0.399286 0.0548462 0.0274231 0.999624i $$-0.491270\pi$$
0.0274231 + 0.999624i $$0.491270\pi$$
$$54$$ −2.35994 −0.321146
$$55$$ 18.7914 2.53383
$$56$$ −2.73565 −0.365567
$$57$$ 5.20632 0.689594
$$58$$ 4.94006 0.648662
$$59$$ −4.80586 −0.625670 −0.312835 0.949807i $$-0.601279\pi$$
−0.312835 + 0.949807i $$0.601279\pi$$
$$60$$ −11.1187 −1.43542
$$61$$ −1.15703 −0.148142 −0.0740711 0.997253i $$-0.523599\pi$$
−0.0740711 + 0.997253i $$0.523599\pi$$
$$62$$ −0.948365 −0.120442
$$63$$ −4.07715 −0.513673
$$64$$ 3.98971 0.498714
$$65$$ 0 0
$$66$$ −13.0199 −1.60263
$$67$$ 6.27918 0.767124 0.383562 0.923515i $$-0.374697\pi$$
0.383562 + 0.923515i $$0.374697\pi$$
$$68$$ 3.56827 0.432716
$$69$$ −7.26104 −0.874127
$$70$$ 2.60416 0.311257
$$71$$ 4.50720 0.534906 0.267453 0.963571i $$-0.413818\pi$$
0.267453 + 0.963571i $$0.413818\pi$$
$$72$$ 11.1537 1.31447
$$73$$ 8.30575 0.972115 0.486057 0.873927i $$-0.338435\pi$$
0.486057 + 0.873927i $$0.338435\pi$$
$$74$$ 5.35685 0.622721
$$75$$ 13.2983 1.53556
$$76$$ −2.58674 −0.296719
$$77$$ −5.94270 −0.677233
$$78$$ 0 0
$$79$$ −7.91410 −0.890405 −0.445203 0.895430i $$-0.646868\pi$$
−0.445203 + 0.895430i $$0.646868\pi$$
$$80$$ 1.23495 0.138072
$$81$$ −4.60828 −0.512031
$$82$$ −3.07494 −0.339571
$$83$$ −6.19795 −0.680313 −0.340156 0.940369i $$-0.610480\pi$$
−0.340156 + 0.940369i $$0.610480\pi$$
$$84$$ 3.51626 0.383655
$$85$$ −8.53652 −0.925916
$$86$$ −5.75769 −0.620867
$$87$$ −15.9576 −1.71084
$$88$$ 16.2571 1.73302
$$89$$ 3.56136 0.377504 0.188752 0.982025i $$-0.439556\pi$$
0.188752 + 0.982025i $$0.439556\pi$$
$$90$$ −10.6176 −1.11919
$$91$$ 0 0
$$92$$ 3.60762 0.376120
$$93$$ 3.06345 0.317665
$$94$$ −0.375600 −0.0387402
$$95$$ 6.18837 0.634913
$$96$$ −15.4109 −1.57287
$$97$$ −3.42751 −0.348011 −0.174005 0.984745i $$-0.555671\pi$$
−0.174005 + 0.984745i $$0.555671\pi$$
$$98$$ −0.823556 −0.0831917
$$99$$ 24.2293 2.43513
$$100$$ −6.60723 −0.660723
$$101$$ −13.3295 −1.32633 −0.663167 0.748472i $$-0.730788\pi$$
−0.663167 + 0.748472i $$0.730788\pi$$
$$102$$ 5.91465 0.585637
$$103$$ 11.6450 1.14741 0.573706 0.819061i $$-0.305505\pi$$
0.573706 + 0.819061i $$0.305505\pi$$
$$104$$ 0 0
$$105$$ −8.41209 −0.820936
$$106$$ −0.328834 −0.0319392
$$107$$ 3.92966 0.379894 0.189947 0.981794i $$-0.439168\pi$$
0.189947 + 0.981794i $$0.439168\pi$$
$$108$$ −3.78755 −0.364457
$$109$$ −11.2533 −1.07787 −0.538936 0.842346i $$-0.681174\pi$$
−0.538936 + 0.842346i $$0.681174\pi$$
$$110$$ −15.4757 −1.47555
$$111$$ −17.3040 −1.64242
$$112$$ −0.390549 −0.0369034
$$113$$ −5.77418 −0.543189 −0.271595 0.962412i $$-0.587551\pi$$
−0.271595 + 0.962412i $$0.587551\pi$$
$$114$$ −4.28770 −0.401579
$$115$$ −8.63066 −0.804813
$$116$$ 7.92849 0.736142
$$117$$ 0 0
$$118$$ 3.95790 0.364354
$$119$$ 2.69964 0.247476
$$120$$ 23.0125 2.10075
$$121$$ 24.3156 2.21051
$$122$$ 0.952877 0.0862694
$$123$$ 9.93284 0.895614
$$124$$ −1.52207 −0.136686
$$125$$ −0.00370455 −0.000331345 0
$$126$$ 3.35776 0.299133
$$127$$ 6.13117 0.544053 0.272027 0.962290i $$-0.412306\pi$$
0.272027 + 0.962290i $$0.412306\pi$$
$$128$$ 8.30013 0.733635
$$129$$ 18.5988 1.63753
$$130$$ 0 0
$$131$$ 10.2217 0.893073 0.446537 0.894765i $$-0.352657\pi$$
0.446537 + 0.894765i $$0.352657\pi$$
$$132$$ −20.8960 −1.81877
$$133$$ −1.95705 −0.169698
$$134$$ −5.17126 −0.446729
$$135$$ 9.06111 0.779856
$$136$$ −7.38528 −0.633283
$$137$$ 19.9475 1.70423 0.852116 0.523353i $$-0.175319\pi$$
0.852116 + 0.523353i $$0.175319\pi$$
$$138$$ 5.97987 0.509041
$$139$$ −20.3275 −1.72415 −0.862077 0.506777i $$-0.830837\pi$$
−0.862077 + 0.506777i $$0.830837\pi$$
$$140$$ 4.17951 0.353233
$$141$$ 1.21328 0.102177
$$142$$ −3.71193 −0.311498
$$143$$ 0 0
$$144$$ 1.59233 0.132694
$$145$$ −18.9677 −1.57518
$$146$$ −6.84025 −0.566103
$$147$$ 2.66029 0.219417
$$148$$ 8.59741 0.706703
$$149$$ −10.7162 −0.877901 −0.438951 0.898511i $$-0.644650\pi$$
−0.438951 + 0.898511i $$0.644650\pi$$
$$150$$ −10.9519 −0.894221
$$151$$ −8.74416 −0.711590 −0.355795 0.934564i $$-0.615790\pi$$
−0.355795 + 0.934564i $$0.615790\pi$$
$$152$$ 5.35380 0.434251
$$153$$ −11.0069 −0.889852
$$154$$ 4.89414 0.394381
$$155$$ 3.64130 0.292476
$$156$$ 0 0
$$157$$ 6.50734 0.519342 0.259671 0.965697i $$-0.416386\pi$$
0.259671 + 0.965697i $$0.416386\pi$$
$$158$$ 6.51770 0.518520
$$159$$ 1.06222 0.0842393
$$160$$ −18.3178 −1.44815
$$161$$ 2.72941 0.215108
$$162$$ 3.79518 0.298177
$$163$$ −2.61267 −0.204640 −0.102320 0.994752i $$-0.532627\pi$$
−0.102320 + 0.994752i $$0.532627\pi$$
$$164$$ −4.93509 −0.385366
$$165$$ 49.9905 3.89175
$$166$$ 5.10436 0.396175
$$167$$ −3.88624 −0.300726 −0.150363 0.988631i $$-0.548044\pi$$
−0.150363 + 0.988631i $$0.548044\pi$$
$$168$$ −7.27763 −0.561482
$$169$$ 0 0
$$170$$ 7.03030 0.539200
$$171$$ 7.97919 0.610184
$$172$$ −9.24072 −0.704599
$$173$$ 13.9768 1.06263 0.531317 0.847173i $$-0.321697\pi$$
0.531317 + 0.847173i $$0.321697\pi$$
$$174$$ 13.1420 0.996293
$$175$$ −4.99883 −0.377876
$$176$$ 2.32091 0.174945
$$177$$ −12.7850 −0.960979
$$178$$ −2.93298 −0.219836
$$179$$ −25.2843 −1.88984 −0.944919 0.327305i $$-0.893860\pi$$
−0.944919 + 0.327305i $$0.893860\pi$$
$$180$$ −17.0405 −1.27013
$$181$$ 0.864474 0.0642559 0.0321279 0.999484i $$-0.489772\pi$$
0.0321279 + 0.999484i $$0.489772\pi$$
$$182$$ 0 0
$$183$$ −3.07803 −0.227535
$$184$$ −7.46673 −0.550454
$$185$$ −20.5680 −1.51219
$$186$$ −2.52293 −0.184990
$$187$$ −16.0432 −1.17319
$$188$$ −0.602814 −0.0439647
$$189$$ −2.86554 −0.208438
$$190$$ −5.09647 −0.369737
$$191$$ 14.6676 1.06131 0.530657 0.847587i $$-0.321945\pi$$
0.530657 + 0.847587i $$0.321945\pi$$
$$192$$ 10.6138 0.765985
$$193$$ 16.4959 1.18740 0.593700 0.804686i $$-0.297667\pi$$
0.593700 + 0.804686i $$0.297667\pi$$
$$194$$ 2.82275 0.202661
$$195$$ 0 0
$$196$$ −1.32176 −0.0944111
$$197$$ −11.0102 −0.784443 −0.392222 0.919871i $$-0.628293\pi$$
−0.392222 + 0.919871i $$0.628293\pi$$
$$198$$ −19.9542 −1.41808
$$199$$ −21.2117 −1.50366 −0.751829 0.659358i $$-0.770828\pi$$
−0.751829 + 0.659358i $$0.770828\pi$$
$$200$$ 13.6751 0.966972
$$201$$ 16.7045 1.17824
$$202$$ 10.9776 0.772380
$$203$$ 5.99845 0.421009
$$204$$ 9.49264 0.664617
$$205$$ 11.8064 0.824597
$$206$$ −9.59027 −0.668186
$$207$$ −11.1282 −0.773466
$$208$$ 0 0
$$209$$ 11.6301 0.804474
$$210$$ 6.92783 0.478065
$$211$$ −17.9358 −1.23475 −0.617375 0.786669i $$-0.711804\pi$$
−0.617375 + 0.786669i $$0.711804\pi$$
$$212$$ −0.527759 −0.0362466
$$213$$ 11.9905 0.821572
$$214$$ −3.23629 −0.221228
$$215$$ 22.1070 1.50768
$$216$$ 7.83913 0.533385
$$217$$ −1.15155 −0.0781722
$$218$$ 9.26774 0.627691
$$219$$ 22.0957 1.49309
$$220$$ −24.8376 −1.67455
$$221$$ 0 0
$$222$$ 14.2508 0.956451
$$223$$ 16.0312 1.07353 0.536763 0.843733i $$-0.319647\pi$$
0.536763 + 0.843733i $$0.319647\pi$$
$$224$$ 5.79294 0.387057
$$225$$ 20.3810 1.35873
$$226$$ 4.75536 0.316322
$$227$$ −16.3750 −1.08685 −0.543424 0.839458i $$-0.682873\pi$$
−0.543424 + 0.839458i $$0.682873\pi$$
$$228$$ −6.88148 −0.455737
$$229$$ 27.0104 1.78490 0.892449 0.451148i $$-0.148985\pi$$
0.892449 + 0.451148i $$0.148985\pi$$
$$230$$ 7.10783 0.468677
$$231$$ −15.8093 −1.04018
$$232$$ −16.4097 −1.07735
$$233$$ 11.5681 0.757853 0.378926 0.925427i $$-0.376293\pi$$
0.378926 + 0.925427i $$0.376293\pi$$
$$234$$ 0 0
$$235$$ 1.44214 0.0940747
$$236$$ 6.35217 0.413491
$$237$$ −21.0538 −1.36759
$$238$$ −2.22331 −0.144116
$$239$$ 14.6731 0.949122 0.474561 0.880223i $$-0.342607\pi$$
0.474561 + 0.880223i $$0.342607\pi$$
$$240$$ 3.28533 0.212067
$$241$$ −14.3467 −0.924151 −0.462076 0.886841i $$-0.652895\pi$$
−0.462076 + 0.886841i $$0.652895\pi$$
$$242$$ −20.0253 −1.28727
$$243$$ −20.8560 −1.33791
$$244$$ 1.52931 0.0979039
$$245$$ 3.16209 0.202019
$$246$$ −8.18025 −0.521554
$$247$$ 0 0
$$248$$ 3.15024 0.200040
$$249$$ −16.4883 −1.04491
$$250$$ 0.00305091 0.000192956 0
$$251$$ 8.61452 0.543744 0.271872 0.962333i $$-0.412357\pi$$
0.271872 + 0.962333i $$0.412357\pi$$
$$252$$ 5.38900 0.339475
$$253$$ −16.2201 −1.01975
$$254$$ −5.04936 −0.316825
$$255$$ −22.7096 −1.42213
$$256$$ −14.8151 −0.925941
$$257$$ 10.3639 0.646485 0.323243 0.946316i $$-0.395227\pi$$
0.323243 + 0.946316i $$0.395227\pi$$
$$258$$ −15.3171 −0.953603
$$259$$ 6.50454 0.404172
$$260$$ 0 0
$$261$$ −24.4566 −1.51383
$$262$$ −8.41813 −0.520074
$$263$$ −22.0826 −1.36167 −0.680835 0.732436i $$-0.738383\pi$$
−0.680835 + 0.732436i $$0.738383\pi$$
$$264$$ 43.2488 2.66178
$$265$$ 1.26258 0.0775596
$$266$$ 1.61174 0.0988220
$$267$$ 9.47427 0.579816
$$268$$ −8.29954 −0.506975
$$269$$ 12.9399 0.788960 0.394480 0.918905i $$-0.370925\pi$$
0.394480 + 0.918905i $$0.370925\pi$$
$$270$$ −7.46233 −0.454143
$$271$$ −17.6749 −1.07367 −0.536837 0.843686i $$-0.680381\pi$$
−0.536837 + 0.843686i $$0.680381\pi$$
$$272$$ −1.05434 −0.0639289
$$273$$ 0 0
$$274$$ −16.4279 −0.992446
$$275$$ 29.7065 1.79137
$$276$$ 9.59732 0.577691
$$277$$ −18.0150 −1.08242 −0.541209 0.840888i $$-0.682033\pi$$
−0.541209 + 0.840888i $$0.682033\pi$$
$$278$$ 16.7408 1.00405
$$279$$ 4.69504 0.281085
$$280$$ −8.65038 −0.516959
$$281$$ 2.44178 0.145665 0.0728323 0.997344i $$-0.476796\pi$$
0.0728323 + 0.997344i $$0.476796\pi$$
$$282$$ −0.999205 −0.0595018
$$283$$ −28.7240 −1.70746 −0.853732 0.520713i $$-0.825666\pi$$
−0.853732 + 0.520713i $$0.825666\pi$$
$$284$$ −5.95741 −0.353507
$$285$$ 16.4629 0.975176
$$286$$ 0 0
$$287$$ −3.73374 −0.220396
$$288$$ −23.6187 −1.39175
$$289$$ −9.71193 −0.571290
$$290$$ 15.6209 0.917292
$$291$$ −9.11818 −0.534517
$$292$$ −10.9782 −0.642449
$$293$$ −29.3309 −1.71353 −0.856763 0.515710i $$-0.827528\pi$$
−0.856763 + 0.515710i $$0.827528\pi$$
$$294$$ −2.19090 −0.127776
$$295$$ −15.1966 −0.884779
$$296$$ −17.7942 −1.03426
$$297$$ 17.0291 0.988126
$$298$$ 8.82535 0.511239
$$299$$ 0 0
$$300$$ −17.5772 −1.01482
$$301$$ −6.99125 −0.402969
$$302$$ 7.20131 0.414389
$$303$$ −35.4603 −2.03714
$$304$$ 0.764323 0.0438369
$$305$$ −3.65863 −0.209492
$$306$$ 9.06476 0.518198
$$307$$ −7.06910 −0.403455 −0.201728 0.979442i $$-0.564656\pi$$
−0.201728 + 0.979442i $$0.564656\pi$$
$$308$$ 7.85479 0.447568
$$309$$ 30.9790 1.76233
$$310$$ −2.99882 −0.170321
$$311$$ −22.2686 −1.26274 −0.631368 0.775483i $$-0.717506\pi$$
−0.631368 + 0.775483i $$0.717506\pi$$
$$312$$ 0 0
$$313$$ −28.0840 −1.58740 −0.793700 0.608309i $$-0.791848\pi$$
−0.793700 + 0.608309i $$0.791848\pi$$
$$314$$ −5.35916 −0.302435
$$315$$ −12.8923 −0.726401
$$316$$ 10.4605 0.588449
$$317$$ 19.5155 1.09610 0.548049 0.836446i $$-0.315371\pi$$
0.548049 + 0.836446i $$0.315371\pi$$
$$318$$ −0.874796 −0.0490561
$$319$$ −35.6470 −1.99585
$$320$$ 12.6158 0.705247
$$321$$ 10.4540 0.583488
$$322$$ −2.24783 −0.125266
$$323$$ −5.28333 −0.293972
$$324$$ 6.09102 0.338390
$$325$$ 0 0
$$326$$ 2.15168 0.119171
$$327$$ −29.9371 −1.65553
$$328$$ 10.2142 0.563986
$$329$$ −0.456071 −0.0251440
$$330$$ −41.1700 −2.26633
$$331$$ 15.6308 0.859145 0.429573 0.903032i $$-0.358664\pi$$
0.429573 + 0.903032i $$0.358664\pi$$
$$332$$ 8.19217 0.449604
$$333$$ −26.5200 −1.45329
$$334$$ 3.20053 0.175125
$$335$$ 19.8553 1.08481
$$336$$ −1.03897 −0.0566807
$$337$$ −21.7501 −1.18480 −0.592401 0.805643i $$-0.701820\pi$$
−0.592401 + 0.805643i $$0.701820\pi$$
$$338$$ 0 0
$$339$$ −15.3610 −0.834295
$$340$$ 11.2832 0.611917
$$341$$ 6.84330 0.370586
$$342$$ −6.57131 −0.355336
$$343$$ −1.00000 −0.0539949
$$344$$ 19.1256 1.03118
$$345$$ −22.9601 −1.23613
$$346$$ −11.5106 −0.618816
$$347$$ −15.9590 −0.856726 −0.428363 0.903607i $$-0.640909\pi$$
−0.428363 + 0.903607i $$0.640909\pi$$
$$348$$ 21.0921 1.13065
$$349$$ 6.81706 0.364909 0.182455 0.983214i $$-0.441596\pi$$
0.182455 + 0.983214i $$0.441596\pi$$
$$350$$ 4.11682 0.220053
$$351$$ 0 0
$$352$$ −34.4257 −1.83490
$$353$$ −14.0033 −0.745318 −0.372659 0.927968i $$-0.621554\pi$$
−0.372659 + 0.927968i $$0.621554\pi$$
$$354$$ 10.5292 0.559619
$$355$$ 14.2522 0.756427
$$356$$ −4.70725 −0.249484
$$357$$ 7.18184 0.380103
$$358$$ 20.8230 1.10053
$$359$$ 5.41494 0.285789 0.142895 0.989738i $$-0.454359\pi$$
0.142895 + 0.989738i $$0.454359\pi$$
$$360$$ 35.2689 1.85884
$$361$$ −15.1700 −0.798419
$$362$$ −0.711943 −0.0374189
$$363$$ 64.6867 3.39517
$$364$$ 0 0
$$365$$ 26.2636 1.37470
$$366$$ 2.53493 0.132503
$$367$$ 30.0317 1.56764 0.783822 0.620985i $$-0.213267\pi$$
0.783822 + 0.620985i $$0.213267\pi$$
$$368$$ −1.06597 −0.0555675
$$369$$ 15.2230 0.792480
$$370$$ 16.9389 0.880610
$$371$$ −0.399286 −0.0207299
$$372$$ −4.04914 −0.209938
$$373$$ 21.4098 1.10856 0.554278 0.832332i $$-0.312995\pi$$
0.554278 + 0.832332i $$0.312995\pi$$
$$374$$ 13.2124 0.683199
$$375$$ −0.00985519 −0.000508920 0
$$376$$ 1.24765 0.0643427
$$377$$ 0 0
$$378$$ 2.35994 0.121382
$$379$$ 9.47655 0.486778 0.243389 0.969929i $$-0.421741\pi$$
0.243389 + 0.969929i $$0.421741\pi$$
$$380$$ −8.17951 −0.419600
$$381$$ 16.3107 0.835622
$$382$$ −12.0796 −0.618048
$$383$$ −5.43061 −0.277491 −0.138746 0.990328i $$-0.544307\pi$$
−0.138746 + 0.990328i $$0.544307\pi$$
$$384$$ 22.0808 1.12680
$$385$$ −18.7914 −0.957696
$$386$$ −13.5853 −0.691473
$$387$$ 28.5044 1.44896
$$388$$ 4.53033 0.229993
$$389$$ −10.6422 −0.539580 −0.269790 0.962919i $$-0.586954\pi$$
−0.269790 + 0.962919i $$0.586954\pi$$
$$390$$ 0 0
$$391$$ 7.36845 0.372638
$$392$$ 2.73565 0.138171
$$393$$ 27.1927 1.37169
$$394$$ 9.06750 0.456814
$$395$$ −25.0251 −1.25915
$$396$$ −32.0252 −1.60933
$$397$$ −37.1854 −1.86628 −0.933140 0.359512i $$-0.882943\pi$$
−0.933140 + 0.359512i $$0.882943\pi$$
$$398$$ 17.4690 0.875644
$$399$$ −5.20632 −0.260642
$$400$$ 1.95229 0.0976143
$$401$$ −0.896610 −0.0447746 −0.0223873 0.999749i $$-0.507127\pi$$
−0.0223873 + 0.999749i $$0.507127\pi$$
$$402$$ −13.7571 −0.686139
$$403$$ 0 0
$$404$$ 17.6183 0.876545
$$405$$ −14.5718 −0.724079
$$406$$ −4.94006 −0.245171
$$407$$ −38.6545 −1.91603
$$408$$ −19.6470 −0.972672
$$409$$ 24.5773 1.21527 0.607635 0.794217i $$-0.292119\pi$$
0.607635 + 0.794217i $$0.292119\pi$$
$$410$$ −9.72326 −0.480198
$$411$$ 53.0662 2.61756
$$412$$ −15.3918 −0.758299
$$413$$ 4.80586 0.236481
$$414$$ 9.16473 0.450422
$$415$$ −19.5985 −0.962052
$$416$$ 0 0
$$417$$ −54.0770 −2.64816
$$418$$ −9.57807 −0.468479
$$419$$ −7.64558 −0.373511 −0.186755 0.982406i $$-0.559797\pi$$
−0.186755 + 0.982406i $$0.559797\pi$$
$$420$$ 11.1187 0.542538
$$421$$ −25.0780 −1.22223 −0.611113 0.791544i $$-0.709278\pi$$
−0.611113 + 0.791544i $$0.709278\pi$$
$$422$$ 14.7711 0.719046
$$423$$ 1.85947 0.0904106
$$424$$ 1.09231 0.0530471
$$425$$ −13.4951 −0.654606
$$426$$ −9.87481 −0.478436
$$427$$ 1.15703 0.0559925
$$428$$ −5.19405 −0.251064
$$429$$ 0 0
$$430$$ −18.2063 −0.877987
$$431$$ −7.75404 −0.373499 −0.186750 0.982408i $$-0.559795\pi$$
−0.186750 + 0.982408i $$0.559795\pi$$
$$432$$ 1.11913 0.0538444
$$433$$ 35.9760 1.72890 0.864448 0.502722i $$-0.167668\pi$$
0.864448 + 0.502722i $$0.167668\pi$$
$$434$$ 0.948365 0.0455230
$$435$$ −50.4595 −2.41935
$$436$$ 14.8741 0.712342
$$437$$ −5.34160 −0.255523
$$438$$ −18.1971 −0.869489
$$439$$ −28.2350 −1.34758 −0.673792 0.738921i $$-0.735336\pi$$
−0.673792 + 0.738921i $$0.735336\pi$$
$$440$$ 51.4066 2.45071
$$441$$ 4.07715 0.194150
$$442$$ 0 0
$$443$$ 28.7918 1.36794 0.683970 0.729511i $$-0.260252\pi$$
0.683970 + 0.729511i $$0.260252\pi$$
$$444$$ 22.8716 1.08544
$$445$$ 11.2614 0.533840
$$446$$ −13.2026 −0.625159
$$447$$ −28.5081 −1.34839
$$448$$ −3.98971 −0.188496
$$449$$ 29.1902 1.37757 0.688785 0.724965i $$-0.258144\pi$$
0.688785 + 0.724965i $$0.258144\pi$$
$$450$$ −16.7849 −0.791247
$$451$$ 22.1885 1.04482
$$452$$ 7.63205 0.358982
$$453$$ −23.2620 −1.09295
$$454$$ 13.4858 0.632918
$$455$$ 0 0
$$456$$ 14.2427 0.666974
$$457$$ 31.6848 1.48215 0.741077 0.671420i $$-0.234316\pi$$
0.741077 + 0.671420i $$0.234316\pi$$
$$458$$ −22.2446 −1.03942
$$459$$ −7.73594 −0.361083
$$460$$ 11.4076 0.531883
$$461$$ −22.1018 −1.02938 −0.514691 0.857376i $$-0.672093\pi$$
−0.514691 + 0.857376i $$0.672093\pi$$
$$462$$ 13.0199 0.605738
$$463$$ −38.8811 −1.80696 −0.903479 0.428632i $$-0.858996\pi$$
−0.903479 + 0.428632i $$0.858996\pi$$
$$464$$ −2.34269 −0.108757
$$465$$ 9.68693 0.449220
$$466$$ −9.52699 −0.441329
$$467$$ 13.2823 0.614632 0.307316 0.951607i $$-0.400569\pi$$
0.307316 + 0.951607i $$0.400569\pi$$
$$468$$ 0 0
$$469$$ −6.27918 −0.289946
$$470$$ −1.18768 −0.0547837
$$471$$ 17.3114 0.797668
$$472$$ −13.1472 −0.605147
$$473$$ 41.5469 1.91033
$$474$$ 17.3390 0.796406
$$475$$ 9.78295 0.448872
$$476$$ −3.56827 −0.163551
$$477$$ 1.62795 0.0745387
$$478$$ −12.0841 −0.552714
$$479$$ 6.63512 0.303166 0.151583 0.988445i $$-0.451563\pi$$
0.151583 + 0.988445i $$0.451563\pi$$
$$480$$ −48.7307 −2.22424
$$481$$ 0 0
$$482$$ 11.8153 0.538172
$$483$$ 7.26104 0.330389
$$484$$ −32.1393 −1.46088
$$485$$ −10.8381 −0.492133
$$486$$ 17.1761 0.779123
$$487$$ 33.4701 1.51668 0.758338 0.651861i $$-0.226012\pi$$
0.758338 + 0.651861i $$0.226012\pi$$
$$488$$ −3.16522 −0.143283
$$489$$ −6.95047 −0.314311
$$490$$ −2.60416 −0.117644
$$491$$ −37.3287 −1.68462 −0.842310 0.538993i $$-0.818805\pi$$
−0.842310 + 0.538993i $$0.818805\pi$$
$$492$$ −13.1288 −0.591891
$$493$$ 16.1937 0.729327
$$494$$ 0 0
$$495$$ 76.6152 3.44360
$$496$$ 0.449736 0.0201937
$$497$$ −4.50720 −0.202175
$$498$$ 13.5791 0.608493
$$499$$ 34.1327 1.52799 0.763994 0.645223i $$-0.223236\pi$$
0.763994 + 0.645223i $$0.223236\pi$$
$$500$$ 0.00489651 0.000218979 0
$$501$$ −10.3385 −0.461891
$$502$$ −7.09454 −0.316645
$$503$$ 15.3089 0.682592 0.341296 0.939956i $$-0.389134\pi$$
0.341296 + 0.939956i $$0.389134\pi$$
$$504$$ −11.1537 −0.496824
$$505$$ −42.1491 −1.87561
$$506$$ 13.3581 0.593842
$$507$$ 0 0
$$508$$ −8.10390 −0.359553
$$509$$ 18.4970 0.819866 0.409933 0.912116i $$-0.365552\pi$$
0.409933 + 0.912116i $$0.365552\pi$$
$$510$$ 18.7027 0.828168
$$511$$ −8.30575 −0.367425
$$512$$ −4.39924 −0.194421
$$513$$ 5.60801 0.247599
$$514$$ −8.53529 −0.376476
$$515$$ 36.8224 1.62259
$$516$$ −24.5830 −1.08221
$$517$$ 2.71029 0.119198
$$518$$ −5.35685 −0.235367
$$519$$ 37.1823 1.63212
$$520$$ 0 0
$$521$$ 23.5865 1.03334 0.516671 0.856184i $$-0.327171\pi$$
0.516671 + 0.856184i $$0.327171\pi$$
$$522$$ 20.1414 0.881564
$$523$$ 12.3059 0.538099 0.269049 0.963126i $$-0.413291\pi$$
0.269049 + 0.963126i $$0.413291\pi$$
$$524$$ −13.5106 −0.590212
$$525$$ −13.2983 −0.580387
$$526$$ 18.1862 0.792958
$$527$$ −3.10877 −0.135420
$$528$$ 6.17431 0.268702
$$529$$ −15.5503 −0.676100
$$530$$ −1.03980 −0.0451662
$$531$$ −19.5942 −0.850317
$$532$$ 2.58674 0.112149
$$533$$ 0 0
$$534$$ −7.80259 −0.337651
$$535$$ 12.4259 0.537220
$$536$$ 17.1777 0.741962
$$537$$ −67.2636 −2.90264
$$538$$ −10.6567 −0.459444
$$539$$ 5.94270 0.255970
$$540$$ −11.9766 −0.515390
$$541$$ 19.4411 0.835838 0.417919 0.908484i $$-0.362760\pi$$
0.417919 + 0.908484i $$0.362760\pi$$
$$542$$ 14.5563 0.625245
$$543$$ 2.29975 0.0986919
$$544$$ 15.6389 0.670511
$$545$$ −35.5841 −1.52425
$$546$$ 0 0
$$547$$ 40.2163 1.71953 0.859763 0.510693i $$-0.170611\pi$$
0.859763 + 0.510693i $$0.170611\pi$$
$$548$$ −26.3658 −1.12629
$$549$$ −4.71738 −0.201333
$$550$$ −24.4650 −1.04319
$$551$$ −11.7393 −0.500109
$$552$$ −19.8637 −0.845454
$$553$$ 7.91410 0.336542
$$554$$ 14.8364 0.630337
$$555$$ −54.7168 −2.32260
$$556$$ 26.8680 1.13945
$$557$$ 7.96399 0.337445 0.168722 0.985664i $$-0.446036\pi$$
0.168722 + 0.985664i $$0.446036\pi$$
$$558$$ −3.86663 −0.163687
$$559$$ 0 0
$$560$$ −1.23495 −0.0521862
$$561$$ −42.6795 −1.80193
$$562$$ −2.01095 −0.0848266
$$563$$ −1.42396 −0.0600128 −0.0300064 0.999550i $$-0.509553\pi$$
−0.0300064 + 0.999550i $$0.509553\pi$$
$$564$$ −1.60366 −0.0675263
$$565$$ −18.2585 −0.768140
$$566$$ 23.6558 0.994327
$$567$$ 4.60828 0.193530
$$568$$ 12.3301 0.517360
$$569$$ −18.5189 −0.776353 −0.388177 0.921585i $$-0.626895\pi$$
−0.388177 + 0.921585i $$0.626895\pi$$
$$570$$ −13.5581 −0.567886
$$571$$ 4.35766 0.182362 0.0911812 0.995834i $$-0.470936\pi$$
0.0911812 + 0.995834i $$0.470936\pi$$
$$572$$ 0 0
$$573$$ 39.0202 1.63009
$$574$$ 3.07494 0.128346
$$575$$ −13.6439 −0.568989
$$576$$ 16.2667 0.677778
$$577$$ −9.56416 −0.398161 −0.199081 0.979983i $$-0.563796\pi$$
−0.199081 + 0.979983i $$0.563796\pi$$
$$578$$ 7.99832 0.332686
$$579$$ 43.8839 1.82375
$$580$$ 25.0706 1.04100
$$581$$ 6.19795 0.257134
$$582$$ 7.50933 0.311272
$$583$$ 2.37284 0.0982728
$$584$$ 22.7216 0.940228
$$585$$ 0 0
$$586$$ 24.1556 0.997859
$$587$$ 2.36053 0.0974296 0.0487148 0.998813i $$-0.484487\pi$$
0.0487148 + 0.998813i $$0.484487\pi$$
$$588$$ −3.51626 −0.145008
$$589$$ 2.25364 0.0928594
$$590$$ 12.5152 0.515244
$$591$$ −29.2903 −1.20484
$$592$$ −2.54034 −0.104407
$$593$$ 40.4292 1.66023 0.830114 0.557594i $$-0.188275\pi$$
0.830114 + 0.557594i $$0.188275\pi$$
$$594$$ −14.0244 −0.575427
$$595$$ 8.53652 0.349963
$$596$$ 14.1641 0.580185
$$597$$ −56.4294 −2.30950
$$598$$ 0 0
$$599$$ −38.5873 −1.57663 −0.788316 0.615270i $$-0.789047\pi$$
−0.788316 + 0.615270i $$0.789047\pi$$
$$600$$ 36.3796 1.48519
$$601$$ 8.16231 0.332948 0.166474 0.986046i $$-0.446762\pi$$
0.166474 + 0.986046i $$0.446762\pi$$
$$602$$ 5.75769 0.234666
$$603$$ 25.6012 1.04256
$$604$$ 11.5576 0.470274
$$605$$ 76.8883 3.12595
$$606$$ 29.2036 1.18631
$$607$$ 7.58525 0.307876 0.153938 0.988081i $$-0.450804\pi$$
0.153938 + 0.988081i $$0.450804\pi$$
$$608$$ −11.3371 −0.459779
$$609$$ 15.9576 0.646636
$$610$$ 3.01308 0.121996
$$611$$ 0 0
$$612$$ 14.5484 0.588083
$$613$$ 15.5778 0.629183 0.314592 0.949227i $$-0.398132\pi$$
0.314592 + 0.949227i $$0.398132\pi$$
$$614$$ 5.82180 0.234949
$$615$$ 31.4086 1.26652
$$616$$ −16.2571 −0.655019
$$617$$ 23.8687 0.960916 0.480458 0.877018i $$-0.340471\pi$$
0.480458 + 0.877018i $$0.340471\pi$$
$$618$$ −25.5129 −1.02628
$$619$$ −19.4963 −0.783622 −0.391811 0.920046i $$-0.628151\pi$$
−0.391811 + 0.920046i $$0.628151\pi$$
$$620$$ −4.81291 −0.193291
$$621$$ −7.82126 −0.313856
$$622$$ 18.3394 0.735344
$$623$$ −3.56136 −0.142683
$$624$$ 0 0
$$625$$ −25.0059 −1.00023
$$626$$ 23.1287 0.924410
$$627$$ 30.9396 1.23561
$$628$$ −8.60111 −0.343222
$$629$$ 17.5599 0.700161
$$630$$ 10.6176 0.423014
$$631$$ 25.6619 1.02158 0.510792 0.859704i $$-0.329352\pi$$
0.510792 + 0.859704i $$0.329352\pi$$
$$632$$ −21.6502 −0.861199
$$633$$ −47.7144 −1.89648
$$634$$ −16.0721 −0.638304
$$635$$ 19.3873 0.769362
$$636$$ −1.40399 −0.0556719
$$637$$ 0 0
$$638$$ 29.3573 1.16227
$$639$$ 18.3765 0.726964
$$640$$ 26.2458 1.03746
$$641$$ −1.10604 −0.0436860 −0.0218430 0.999761i $$-0.506953\pi$$
−0.0218430 + 0.999761i $$0.506953\pi$$
$$642$$ −8.60949 −0.339789
$$643$$ 12.6367 0.498341 0.249171 0.968460i $$-0.419842\pi$$
0.249171 + 0.968460i $$0.419842\pi$$
$$644$$ −3.60762 −0.142160
$$645$$ 58.8110 2.31568
$$646$$ 4.35112 0.171192
$$647$$ 25.7148 1.01095 0.505477 0.862840i $$-0.331317\pi$$
0.505477 + 0.862840i $$0.331317\pi$$
$$648$$ −12.6066 −0.495236
$$649$$ −28.5598 −1.12107
$$650$$ 0 0
$$651$$ −3.06345 −0.120066
$$652$$ 3.45331 0.135242
$$653$$ 25.2607 0.988527 0.494263 0.869312i $$-0.335438\pi$$
0.494263 + 0.869312i $$0.335438\pi$$
$$654$$ 24.6549 0.964083
$$655$$ 32.3219 1.26292
$$656$$ 1.45821 0.0569335
$$657$$ 33.8638 1.32115
$$658$$ 0.375600 0.0146424
$$659$$ 22.9764 0.895034 0.447517 0.894275i $$-0.352308\pi$$
0.447517 + 0.894275i $$0.352308\pi$$
$$660$$ −66.0752 −2.57197
$$661$$ 30.4326 1.18369 0.591845 0.806051i $$-0.298400\pi$$
0.591845 + 0.806051i $$0.298400\pi$$
$$662$$ −12.8728 −0.500316
$$663$$ 0 0
$$664$$ −16.9554 −0.657998
$$665$$ −6.18837 −0.239975
$$666$$ 21.8407 0.846310
$$667$$ 16.3723 0.633937
$$668$$ 5.13665 0.198743
$$669$$ 42.6475 1.64885
$$670$$ −16.3520 −0.631733
$$671$$ −6.87586 −0.265440
$$672$$ 15.4109 0.594489
$$673$$ 10.8387 0.417800 0.208900 0.977937i $$-0.433012\pi$$
0.208900 + 0.977937i $$0.433012\pi$$
$$674$$ 17.9124 0.689960
$$675$$ 14.3244 0.551345
$$676$$ 0 0
$$677$$ −18.1209 −0.696442 −0.348221 0.937412i $$-0.613214\pi$$
−0.348221 + 0.937412i $$0.613214\pi$$
$$678$$ 12.6506 0.485845
$$679$$ 3.42751 0.131536
$$680$$ −23.3529 −0.895545
$$681$$ −43.5624 −1.66931
$$682$$ −5.63584 −0.215808
$$683$$ −37.8352 −1.44772 −0.723861 0.689946i $$-0.757634\pi$$
−0.723861 + 0.689946i $$0.757634\pi$$
$$684$$ −10.5465 −0.403257
$$685$$ 63.0759 2.41001
$$686$$ 0.823556 0.0314435
$$687$$ 71.8556 2.74146
$$688$$ 2.73042 0.104096
$$689$$ 0 0
$$690$$ 18.9089 0.719850
$$691$$ −30.0261 −1.14225 −0.571124 0.820864i $$-0.693492\pi$$
−0.571124 + 0.820864i $$0.693492\pi$$
$$692$$ −18.4739 −0.702271
$$693$$ −24.2293 −0.920394
$$694$$ 13.1432 0.498907
$$695$$ −64.2774 −2.43818
$$696$$ −43.6545 −1.65472
$$697$$ −10.0798 −0.381798
$$698$$ −5.61423 −0.212502
$$699$$ 30.7746 1.16400
$$700$$ 6.60723 0.249730
$$701$$ −0.116177 −0.00438796 −0.00219398 0.999998i $$-0.500698\pi$$
−0.00219398 + 0.999998i $$0.500698\pi$$
$$702$$ 0 0
$$703$$ −12.7297 −0.480110
$$704$$ 23.7097 0.893591
$$705$$ 3.83651 0.144491
$$706$$ 11.5325 0.434030
$$707$$ 13.3295 0.501307
$$708$$ 16.8986 0.635090
$$709$$ −6.72993 −0.252748 −0.126374 0.991983i $$-0.540334\pi$$
−0.126374 + 0.991983i $$0.540334\pi$$
$$710$$ −11.7375 −0.440499
$$711$$ −32.2670 −1.21011
$$712$$ 9.74265 0.365121
$$713$$ −3.14305 −0.117708
$$714$$ −5.91465 −0.221350
$$715$$ 0 0
$$716$$ 33.4197 1.24895
$$717$$ 39.0347 1.45778
$$718$$ −4.45950 −0.166427
$$719$$ 46.8078 1.74564 0.872818 0.488046i $$-0.162290\pi$$
0.872818 + 0.488046i $$0.162290\pi$$
$$720$$ 5.03509 0.187647
$$721$$ −11.6450 −0.433681
$$722$$ 12.4933 0.464953
$$723$$ −38.1664 −1.41942
$$724$$ −1.14262 −0.0424653
$$725$$ −29.9852 −1.11362
$$726$$ −53.2731 −1.97715
$$727$$ −13.3362 −0.494611 −0.247305 0.968938i $$-0.579545\pi$$
−0.247305 + 0.968938i $$0.579545\pi$$
$$728$$ 0 0
$$729$$ −41.6582 −1.54290
$$730$$ −21.6295 −0.800544
$$731$$ −18.8739 −0.698076
$$732$$ 4.06840 0.150373
$$733$$ −29.4612 −1.08817 −0.544087 0.839029i $$-0.683124\pi$$
−0.544087 + 0.839029i $$0.683124\pi$$
$$734$$ −24.7328 −0.912905
$$735$$ 8.41209 0.310285
$$736$$ 15.8113 0.582814
$$737$$ 37.3153 1.37453
$$738$$ −12.5370 −0.461494
$$739$$ −12.0302 −0.442537 −0.221269 0.975213i $$-0.571020\pi$$
−0.221269 + 0.975213i $$0.571020\pi$$
$$740$$ 27.1858 0.999370
$$741$$ 0 0
$$742$$ 0.328834 0.0120719
$$743$$ 21.8826 0.802796 0.401398 0.915904i $$-0.368524\pi$$
0.401398 + 0.915904i $$0.368524\pi$$
$$744$$ 8.38055 0.307246
$$745$$ −33.8855 −1.24147
$$746$$ −17.6321 −0.645558
$$747$$ −25.2700 −0.924580
$$748$$ 21.2051 0.775337
$$749$$ −3.92966 −0.143587
$$750$$ 0.00811630 0.000296365 0
$$751$$ 34.7492 1.26802 0.634008 0.773327i $$-0.281409\pi$$
0.634008 + 0.773327i $$0.281409\pi$$
$$752$$ 0.178118 0.00649529
$$753$$ 22.9172 0.835147
$$754$$ 0 0
$$755$$ −27.6498 −1.00628
$$756$$ 3.78755 0.137752
$$757$$ 43.9263 1.59653 0.798265 0.602307i $$-0.205752\pi$$
0.798265 + 0.602307i $$0.205752\pi$$
$$758$$ −7.80447 −0.283471
$$759$$ −43.1502 −1.56625
$$760$$ 16.9292 0.614087
$$761$$ −0.141391 −0.00512543 −0.00256272 0.999997i $$-0.500816\pi$$
−0.00256272 + 0.999997i $$0.500816\pi$$
$$762$$ −13.4328 −0.486618
$$763$$ 11.2533 0.407398
$$764$$ −19.3870 −0.701399
$$765$$ −34.8047 −1.25837
$$766$$ 4.47241 0.161595
$$767$$ 0 0
$$768$$ −39.4124 −1.42217
$$769$$ −13.6486 −0.492180 −0.246090 0.969247i $$-0.579146\pi$$
−0.246090 + 0.969247i $$0.579146\pi$$
$$770$$ 15.4757 0.557707
$$771$$ 27.5711 0.992950
$$772$$ −21.8035 −0.784726
$$773$$ 17.5894 0.632646 0.316323 0.948652i $$-0.397552\pi$$
0.316323 + 0.948652i $$0.397552\pi$$
$$774$$ −23.4750 −0.843790
$$775$$ 5.75639 0.206776
$$776$$ −9.37647 −0.336596
$$777$$ 17.3040 0.620777
$$778$$ 8.76443 0.314220
$$779$$ 7.30711 0.261804
$$780$$ 0 0
$$781$$ 26.7849 0.958439
$$782$$ −6.06833 −0.217003
$$783$$ −17.1888 −0.614278
$$784$$ 0.390549 0.0139482
$$785$$ 20.5768 0.734418
$$786$$ −22.3947 −0.798792
$$787$$ −2.96845 −0.105814 −0.0529069 0.998599i $$-0.516849\pi$$
−0.0529069 + 0.998599i $$0.516849\pi$$
$$788$$ 14.5528 0.518421
$$789$$ −58.7461 −2.09142
$$790$$ 20.6096 0.733256
$$791$$ 5.77418 0.205306
$$792$$ 66.2829 2.35526
$$793$$ 0 0
$$794$$ 30.6242 1.08681
$$795$$ 3.35883 0.119125
$$796$$ 28.0367 0.993735
$$797$$ 9.45221 0.334815 0.167407 0.985888i $$-0.446461\pi$$
0.167407 + 0.985888i $$0.446461\pi$$
$$798$$ 4.28770 0.151783
$$799$$ −1.23123 −0.0435577
$$800$$ −28.9579 −1.02382
$$801$$ 14.5202 0.513047
$$802$$ 0.738409 0.0260741
$$803$$ 49.3586 1.74183
$$804$$ −22.0792 −0.778674
$$805$$ 8.63066 0.304191
$$806$$ 0 0
$$807$$ 34.4239 1.21178
$$808$$ −36.4648 −1.28283
$$809$$ −1.16255 −0.0408729 −0.0204365 0.999791i $$-0.506506\pi$$
−0.0204365 + 0.999791i $$0.506506\pi$$
$$810$$ 12.0007 0.421662
$$811$$ −19.5561 −0.686706 −0.343353 0.939206i $$-0.611563\pi$$
−0.343353 + 0.939206i $$0.611563\pi$$
$$812$$ −7.92849 −0.278235
$$813$$ −47.0204 −1.64908
$$814$$ 31.8342 1.11579
$$815$$ −8.26151 −0.289388
$$816$$ −2.80486 −0.0981897
$$817$$ 13.6822 0.478680
$$818$$ −20.2408 −0.707702
$$819$$ 0 0
$$820$$ −15.6052 −0.544958
$$821$$ −12.6189 −0.440403 −0.220201 0.975454i $$-0.570671\pi$$
−0.220201 + 0.975454i $$0.570671\pi$$
$$822$$ −43.7030 −1.52432
$$823$$ −6.56808 −0.228949 −0.114474 0.993426i $$-0.536518\pi$$
−0.114474 + 0.993426i $$0.536518\pi$$
$$824$$ 31.8565 1.10978
$$825$$ 79.0280 2.75140
$$826$$ −3.95790 −0.137713
$$827$$ −17.3050 −0.601754 −0.300877 0.953663i $$-0.597279\pi$$
−0.300877 + 0.953663i $$0.597279\pi$$
$$828$$ 14.7088 0.511167
$$829$$ −3.75674 −0.130477 −0.0652385 0.997870i $$-0.520781\pi$$
−0.0652385 + 0.997870i $$0.520781\pi$$
$$830$$ 16.1404 0.560243
$$831$$ −47.9252 −1.66251
$$832$$ 0 0
$$833$$ −2.69964 −0.0935371
$$834$$ 44.5355 1.54214
$$835$$ −12.2886 −0.425266
$$836$$ −15.3722 −0.531659
$$837$$ 3.29981 0.114058
$$838$$ 6.29656 0.217511
$$839$$ 46.3427 1.59993 0.799965 0.600047i $$-0.204852\pi$$
0.799965 + 0.600047i $$0.204852\pi$$
$$840$$ −23.0125 −0.794008
$$841$$ 6.98142 0.240739
$$842$$ 20.6531 0.711753
$$843$$ 6.49586 0.223729
$$844$$ 23.7067 0.816018
$$845$$ 0 0
$$846$$ −1.53138 −0.0526499
$$847$$ −24.3156 −0.835495
$$848$$ 0.155941 0.00535503
$$849$$ −76.4142 −2.62253
$$850$$ 11.1139 0.381205
$$851$$ 17.7536 0.608585
$$852$$ −15.8485 −0.542959
$$853$$ −15.3103 −0.524215 −0.262107 0.965039i $$-0.584417\pi$$
−0.262107 + 0.965039i $$0.584417\pi$$
$$854$$ −0.952877 −0.0326068
$$855$$ 25.2309 0.862879
$$856$$ 10.7502 0.367433
$$857$$ 2.59248 0.0885574 0.0442787 0.999019i $$-0.485901\pi$$
0.0442787 + 0.999019i $$0.485901\pi$$
$$858$$ 0 0
$$859$$ −13.7738 −0.469955 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$860$$ −29.2200 −0.996394
$$861$$ −9.93284 −0.338510
$$862$$ 6.38589 0.217504
$$863$$ −29.7592 −1.01302 −0.506508 0.862235i $$-0.669064\pi$$
−0.506508 + 0.862235i $$0.669064\pi$$
$$864$$ −16.5999 −0.564741
$$865$$ 44.1958 1.50270
$$866$$ −29.6282 −1.00681
$$867$$ −25.8366 −0.877456
$$868$$ 1.52207 0.0516623
$$869$$ −47.0311 −1.59542
$$870$$ 41.5562 1.40889
$$871$$ 0 0
$$872$$ −30.7852 −1.04252
$$873$$ −13.9745 −0.472965
$$874$$ 4.39910 0.148802
$$875$$ 0.00370455 0.000125237 0
$$876$$ −29.2051 −0.986750
$$877$$ −1.44332 −0.0487374 −0.0243687 0.999703i $$-0.507758\pi$$
−0.0243687 + 0.999703i $$0.507758\pi$$
$$878$$ 23.2531 0.784755
$$879$$ −78.0286 −2.63184
$$880$$ 7.33894 0.247396
$$881$$ −35.8804 −1.20884 −0.604420 0.796666i $$-0.706595\pi$$
−0.604420 + 0.796666i $$0.706595\pi$$
$$882$$ −3.35776 −0.113062
$$883$$ −10.5626 −0.355458 −0.177729 0.984079i $$-0.556875\pi$$
−0.177729 + 0.984079i $$0.556875\pi$$
$$884$$ 0 0
$$885$$ −40.4273 −1.35895
$$886$$ −23.7116 −0.796608
$$887$$ 12.2280 0.410577 0.205288 0.978702i $$-0.434187\pi$$
0.205288 + 0.978702i $$0.434187\pi$$
$$888$$ −47.3377 −1.58855
$$889$$ −6.13117 −0.205633
$$890$$ −9.27436 −0.310877
$$891$$ −27.3856 −0.917452
$$892$$ −21.1893 −0.709469
$$893$$ 0.892553 0.0298681
$$894$$ 23.4780 0.785222
$$895$$ −79.9513 −2.67248
$$896$$ −8.30013 −0.277288
$$897$$ 0 0
$$898$$ −24.0398 −0.802217
$$899$$ −6.90751 −0.230378
$$900$$ −26.9387 −0.897956
$$901$$ −1.07793 −0.0359110
$$902$$ −18.2735 −0.608440
$$903$$ −18.5988 −0.618928
$$904$$ −15.7961 −0.525372
$$905$$ 2.73355 0.0908662
$$906$$ 19.1576 0.636468
$$907$$ 4.52555 0.150269 0.0751343 0.997173i $$-0.476061\pi$$
0.0751343 + 0.997173i $$0.476061\pi$$
$$908$$ 21.6438 0.718274
$$909$$ −54.3464 −1.80256
$$910$$ 0 0
$$911$$ −57.2723 −1.89751 −0.948757 0.316006i $$-0.897658\pi$$
−0.948757 + 0.316006i $$0.897658\pi$$
$$912$$ 2.03332 0.0673300
$$913$$ −36.8325 −1.21898
$$914$$ −26.0942 −0.863120
$$915$$ −9.73302 −0.321764
$$916$$ −35.7012 −1.17960
$$917$$ −10.2217 −0.337550
$$918$$ 6.37098 0.210274
$$919$$ −40.7551 −1.34439 −0.672193 0.740376i $$-0.734648\pi$$
−0.672193 + 0.740376i $$0.734648\pi$$
$$920$$ −23.6105 −0.778415
$$921$$ −18.8059 −0.619675
$$922$$ 18.2021 0.599453
$$923$$ 0 0
$$924$$ 20.8960 0.687429
$$925$$ −32.5151 −1.06909
$$926$$ 32.0208 1.05227
$$927$$ 47.4783 1.55939
$$928$$ 34.7487 1.14068
$$929$$ 52.2791 1.71522 0.857611 0.514298i $$-0.171948\pi$$
0.857611 + 0.514298i $$0.171948\pi$$
$$930$$ −7.97773 −0.261600
$$931$$ 1.95705 0.0641397
$$932$$ −15.2902 −0.500848
$$933$$ −59.2410 −1.93946
$$934$$ −10.9387 −0.357926
$$935$$ −50.7300 −1.65905
$$936$$ 0 0
$$937$$ −6.38634 −0.208633 −0.104316 0.994544i $$-0.533265\pi$$
−0.104316 + 0.994544i $$0.533265\pi$$
$$938$$ 5.17126 0.168848
$$939$$ −74.7116 −2.43812
$$940$$ −1.90615 −0.0621719
$$941$$ 25.3711 0.827073 0.413536 0.910488i $$-0.364293\pi$$
0.413536 + 0.910488i $$0.364293\pi$$
$$942$$ −14.2569 −0.464516
$$943$$ −10.1909 −0.331862
$$944$$ −1.87692 −0.0610887
$$945$$ −9.06111 −0.294758
$$946$$ −34.2162 −1.11246
$$947$$ 27.2061 0.884080 0.442040 0.896995i $$-0.354255\pi$$
0.442040 + 0.896995i $$0.354255\pi$$
$$948$$ 27.8280 0.903811
$$949$$ 0 0
$$950$$ −8.05681 −0.261397
$$951$$ 51.9169 1.68352
$$952$$ 7.38528 0.239358
$$953$$ 26.5879 0.861265 0.430633 0.902527i $$-0.358290\pi$$
0.430633 + 0.902527i $$0.358290\pi$$
$$954$$ −1.34071 −0.0434070
$$955$$ 46.3805 1.50084
$$956$$ −19.3942 −0.627254
$$957$$ −94.8314 −3.06546
$$958$$ −5.46439 −0.176546
$$959$$ −19.9475 −0.644139
$$960$$ 33.5618 1.08320
$$961$$ −29.6739 −0.957224
$$962$$ 0 0
$$963$$ 16.0218 0.516296
$$964$$ 18.9628 0.610751
$$965$$ 52.1615 1.67914
$$966$$ −5.97987 −0.192399
$$967$$ −35.2467 −1.13346 −0.566729 0.823904i $$-0.691791\pi$$
−0.566729 + 0.823904i $$0.691791\pi$$
$$968$$ 66.5191 2.13801
$$969$$ −14.0552 −0.451518
$$970$$ 8.92578 0.286590
$$971$$ 36.9783 1.18669 0.593344 0.804949i $$-0.297807\pi$$
0.593344 + 0.804949i $$0.297807\pi$$
$$972$$ 27.5665 0.884197
$$973$$ 20.3275 0.651669
$$974$$ −27.5645 −0.883224
$$975$$ 0 0
$$976$$ −0.451876 −0.0144642
$$977$$ 24.7525 0.791902 0.395951 0.918272i $$-0.370415\pi$$
0.395951 + 0.918272i $$0.370415\pi$$
$$978$$ 5.72410 0.183036
$$979$$ 21.1641 0.676408
$$980$$ −4.17951 −0.133510
$$981$$ −45.8815 −1.46488
$$982$$ 30.7423 0.981025
$$983$$ 4.55736 0.145357 0.0726786 0.997355i $$-0.476845\pi$$
0.0726786 + 0.997355i $$0.476845\pi$$
$$984$$ 27.1728 0.866237
$$985$$ −34.8152 −1.10930
$$986$$ −13.3364 −0.424718
$$987$$ −1.21328 −0.0386192
$$988$$ 0 0
$$989$$ −19.0820 −0.606773
$$990$$ −63.0969 −2.00535
$$991$$ −27.1963 −0.863919 −0.431960 0.901893i $$-0.642178\pi$$
−0.431960 + 0.901893i $$0.642178\pi$$
$$992$$ −6.67085 −0.211800
$$993$$ 41.5824 1.31958
$$994$$ 3.71193 0.117735
$$995$$ −67.0734 −2.12637
$$996$$ 21.7936 0.690556
$$997$$ −30.2274 −0.957312 −0.478656 0.878002i $$-0.658876\pi$$
−0.478656 + 0.878002i $$0.658876\pi$$
$$998$$ −28.1102 −0.889812
$$999$$ −18.6390 −0.589713
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 1183.2.a.p.1.2 6
7.6 odd 2 8281.2.a.ch.1.2 6
13.5 odd 4 1183.2.c.i.337.8 12
13.6 odd 12 91.2.q.a.36.4 12
13.8 odd 4 1183.2.c.i.337.5 12
13.11 odd 12 91.2.q.a.43.4 yes 12
13.12 even 2 1183.2.a.m.1.5 6
39.11 even 12 819.2.ct.a.316.3 12
39.32 even 12 819.2.ct.a.127.3 12
52.11 even 12 1456.2.cc.c.225.6 12
52.19 even 12 1456.2.cc.c.673.6 12
91.6 even 12 637.2.q.h.491.4 12
91.11 odd 12 637.2.u.h.30.3 12
91.19 even 12 637.2.u.i.361.3 12
91.24 even 12 637.2.u.i.30.3 12
91.32 odd 12 637.2.k.h.569.4 12
91.37 odd 12 637.2.k.h.459.3 12
91.45 even 12 637.2.k.g.569.4 12
91.58 odd 12 637.2.u.h.361.3 12
91.76 even 12 637.2.q.h.589.4 12
91.89 even 12 637.2.k.g.459.3 12
91.90 odd 2 8281.2.a.by.1.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.q.a.36.4 12 13.6 odd 12
91.2.q.a.43.4 yes 12 13.11 odd 12
637.2.k.g.459.3 12 91.89 even 12
637.2.k.g.569.4 12 91.45 even 12
637.2.k.h.459.3 12 91.37 odd 12
637.2.k.h.569.4 12 91.32 odd 12
637.2.q.h.491.4 12 91.6 even 12
637.2.q.h.589.4 12 91.76 even 12
637.2.u.h.30.3 12 91.11 odd 12
637.2.u.h.361.3 12 91.58 odd 12
637.2.u.i.30.3 12 91.24 even 12
637.2.u.i.361.3 12 91.19 even 12
819.2.ct.a.127.3 12 39.32 even 12
819.2.ct.a.316.3 12 39.11 even 12
1183.2.a.m.1.5 6 13.12 even 2
1183.2.a.p.1.2 6 1.1 even 1 trivial
1183.2.c.i.337.5 12 13.8 odd 4
1183.2.c.i.337.8 12 13.5 odd 4
1456.2.cc.c.225.6 12 52.11 even 12
1456.2.cc.c.673.6 12 52.19 even 12
8281.2.a.by.1.5 6 91.90 odd 2
8281.2.a.ch.1.2 6 7.6 odd 2