# Properties

 Label 8550.2.a.cr.1.3 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +4.42864 q^{7} +1.00000 q^{8} +5.80642 q^{11} -6.42864 q^{13} +4.42864 q^{14} +1.00000 q^{16} +3.37778 q^{17} -1.00000 q^{19} +5.80642 q^{22} +6.42864 q^{23} -6.42864 q^{26} +4.42864 q^{28} -7.80642 q^{29} +9.05086 q^{31} +1.00000 q^{32} +3.37778 q^{34} +3.67307 q^{37} -1.00000 q^{38} -4.42864 q^{41} -1.05086 q^{43} +5.80642 q^{44} +6.42864 q^{46} -5.18421 q^{47} +12.6128 q^{49} -6.42864 q^{52} +4.75557 q^{53} +4.42864 q^{56} -7.80642 q^{58} -4.62222 q^{59} +2.00000 q^{61} +9.05086 q^{62} +1.00000 q^{64} +2.75557 q^{67} +3.37778 q^{68} -7.61285 q^{71} +11.6128 q^{73} +3.67307 q^{74} -1.00000 q^{76} +25.7146 q^{77} +2.94914 q^{79} -4.42864 q^{82} +0.133353 q^{83} -1.05086 q^{86} +5.80642 q^{88} +3.18421 q^{89} -28.4701 q^{91} +6.42864 q^{92} -5.18421 q^{94} -11.4193 q^{97} +12.6128 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 $$3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 4 q^{11} - 6 q^{13} + 3 q^{16} + 10 q^{17} - 3 q^{19} + 4 q^{22} + 6 q^{23} - 6 q^{26} - 10 q^{29} + 14 q^{31} + 3 q^{32} + 10 q^{34} - 2 q^{37} - 3 q^{38} + 10 q^{43} + 4 q^{44} + 6 q^{46} - 2 q^{47} + 11 q^{49} - 6 q^{52} + 14 q^{53} - 10 q^{58} - 14 q^{59} + 6 q^{61} + 14 q^{62} + 3 q^{64} + 8 q^{67} + 10 q^{68} + 4 q^{71} + 8 q^{73} - 2 q^{74} - 3 q^{76} + 24 q^{77} + 22 q^{79} + 10 q^{86} + 4 q^{88} - 4 q^{89} - 32 q^{91} + 6 q^{92} - 2 q^{94} + 6 q^{97} + 11 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 4 * q^11 - 6 * q^13 + 3 * q^16 + 10 * q^17 - 3 * q^19 + 4 * q^22 + 6 * q^23 - 6 * q^26 - 10 * q^29 + 14 * q^31 + 3 * q^32 + 10 * q^34 - 2 * q^37 - 3 * q^38 + 10 * q^43 + 4 * q^44 + 6 * q^46 - 2 * q^47 + 11 * q^49 - 6 * q^52 + 14 * q^53 - 10 * q^58 - 14 * q^59 + 6 * q^61 + 14 * q^62 + 3 * q^64 + 8 * q^67 + 10 * q^68 + 4 * q^71 + 8 * q^73 - 2 * q^74 - 3 * q^76 + 24 * q^77 + 22 * q^79 + 10 * q^86 + 4 * q^88 - 4 * q^89 - 32 * q^91 + 6 * q^92 - 2 * q^94 + 6 * q^97 + 11 * q^98

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.42864 1.67387 0.836934 0.547304i $$-0.184346\pi$$
0.836934 + 0.547304i $$0.184346\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.80642 1.75070 0.875351 0.483487i $$-0.160630\pi$$
0.875351 + 0.483487i $$0.160630\pi$$
$$12$$ 0 0
$$13$$ −6.42864 −1.78298 −0.891492 0.453037i $$-0.850341\pi$$
−0.891492 + 0.453037i $$0.850341\pi$$
$$14$$ 4.42864 1.18360
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.37778 0.819233 0.409617 0.912258i $$-0.365663\pi$$
0.409617 + 0.912258i $$0.365663\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 5.80642 1.23793
$$23$$ 6.42864 1.34046 0.670232 0.742152i $$-0.266195\pi$$
0.670232 + 0.742152i $$0.266195\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −6.42864 −1.26076
$$27$$ 0 0
$$28$$ 4.42864 0.836934
$$29$$ −7.80642 −1.44962 −0.724808 0.688951i $$-0.758072\pi$$
−0.724808 + 0.688951i $$0.758072\pi$$
$$30$$ 0 0
$$31$$ 9.05086 1.62558 0.812791 0.582556i $$-0.197947\pi$$
0.812791 + 0.582556i $$0.197947\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 3.37778 0.579285
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.67307 0.603849 0.301925 0.953332i $$-0.402371\pi$$
0.301925 + 0.953332i $$0.402371\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.42864 −0.691637 −0.345819 0.938301i $$-0.612399\pi$$
−0.345819 + 0.938301i $$0.612399\pi$$
$$42$$ 0 0
$$43$$ −1.05086 −0.160254 −0.0801270 0.996785i $$-0.525533\pi$$
−0.0801270 + 0.996785i $$0.525533\pi$$
$$44$$ 5.80642 0.875351
$$45$$ 0 0
$$46$$ 6.42864 0.947851
$$47$$ −5.18421 −0.756194 −0.378097 0.925766i $$-0.623422\pi$$
−0.378097 + 0.925766i $$0.623422\pi$$
$$48$$ 0 0
$$49$$ 12.6128 1.80184
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −6.42864 −0.891492
$$53$$ 4.75557 0.653228 0.326614 0.945158i $$-0.394092\pi$$
0.326614 + 0.945158i $$0.394092\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.42864 0.591802
$$57$$ 0 0
$$58$$ −7.80642 −1.02503
$$59$$ −4.62222 −0.601761 −0.300881 0.953662i $$-0.597281\pi$$
−0.300881 + 0.953662i $$0.597281\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 9.05086 1.14946
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.75557 0.336646 0.168323 0.985732i $$-0.446165\pi$$
0.168323 + 0.985732i $$0.446165\pi$$
$$68$$ 3.37778 0.409617
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −7.61285 −0.903479 −0.451739 0.892150i $$-0.649196\pi$$
−0.451739 + 0.892150i $$0.649196\pi$$
$$72$$ 0 0
$$73$$ 11.6128 1.35918 0.679591 0.733592i $$-0.262157\pi$$
0.679591 + 0.733592i $$0.262157\pi$$
$$74$$ 3.67307 0.426986
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 25.7146 2.93045
$$78$$ 0 0
$$79$$ 2.94914 0.331805 0.165902 0.986142i $$-0.446946\pi$$
0.165902 + 0.986142i $$0.446946\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −4.42864 −0.489061
$$83$$ 0.133353 0.0146374 0.00731870 0.999973i $$-0.497670\pi$$
0.00731870 + 0.999973i $$0.497670\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1.05086 −0.113317
$$87$$ 0 0
$$88$$ 5.80642 0.618967
$$89$$ 3.18421 0.337525 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$90$$ 0 0
$$91$$ −28.4701 −2.98448
$$92$$ 6.42864 0.670232
$$93$$ 0 0
$$94$$ −5.18421 −0.534710
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −11.4193 −1.15945 −0.579726 0.814812i $$-0.696840\pi$$
−0.579726 + 0.814812i $$0.696840\pi$$
$$98$$ 12.6128 1.27409
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1.86665 0.185738 0.0928692 0.995678i $$-0.470396\pi$$
0.0928692 + 0.995678i $$0.470396\pi$$
$$102$$ 0 0
$$103$$ 10.6222 1.04664 0.523319 0.852137i $$-0.324694\pi$$
0.523319 + 0.852137i $$0.324694\pi$$
$$104$$ −6.42864 −0.630380
$$105$$ 0 0
$$106$$ 4.75557 0.461902
$$107$$ −7.61285 −0.735962 −0.367981 0.929833i $$-0.619951\pi$$
−0.367981 + 0.929833i $$0.619951\pi$$
$$108$$ 0 0
$$109$$ 5.53972 0.530609 0.265304 0.964165i $$-0.414528\pi$$
0.265304 + 0.964165i $$0.414528\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.42864 0.418467
$$113$$ −12.3684 −1.16352 −0.581761 0.813360i $$-0.697636\pi$$
−0.581761 + 0.813360i $$0.697636\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −7.80642 −0.724808
$$117$$ 0 0
$$118$$ −4.62222 −0.425509
$$119$$ 14.9590 1.37129
$$120$$ 0 0
$$121$$ 22.7146 2.06496
$$122$$ 2.00000 0.181071
$$123$$ 0 0
$$124$$ 9.05086 0.812791
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1.76494 0.156613 0.0783064 0.996929i $$-0.475049\pi$$
0.0783064 + 0.996929i $$0.475049\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −9.80642 −0.856791 −0.428396 0.903591i $$-0.640921\pi$$
−0.428396 + 0.903591i $$0.640921\pi$$
$$132$$ 0 0
$$133$$ −4.42864 −0.384012
$$134$$ 2.75557 0.238045
$$135$$ 0 0
$$136$$ 3.37778 0.289643
$$137$$ −1.47949 −0.126402 −0.0632009 0.998001i $$-0.520131\pi$$
−0.0632009 + 0.998001i $$0.520131\pi$$
$$138$$ 0 0
$$139$$ −4.85728 −0.411989 −0.205995 0.978553i $$-0.566043\pi$$
−0.205995 + 0.978553i $$0.566043\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −7.61285 −0.638856
$$143$$ −37.3274 −3.12147
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 11.6128 0.961086
$$147$$ 0 0
$$148$$ 3.67307 0.301925
$$149$$ −4.62222 −0.378667 −0.189333 0.981913i $$-0.560633\pi$$
−0.189333 + 0.981913i $$0.560633\pi$$
$$150$$ 0 0
$$151$$ −11.4193 −0.929287 −0.464644 0.885498i $$-0.653818\pi$$
−0.464644 + 0.885498i $$0.653818\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ 25.7146 2.07214
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 7.37778 0.588811 0.294406 0.955681i $$-0.404878\pi$$
0.294406 + 0.955681i $$0.404878\pi$$
$$158$$ 2.94914 0.234621
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 28.4701 2.24376
$$162$$ 0 0
$$163$$ −5.90813 −0.462761 −0.231380 0.972863i $$-0.574324\pi$$
−0.231380 + 0.972863i $$0.574324\pi$$
$$164$$ −4.42864 −0.345819
$$165$$ 0 0
$$166$$ 0.133353 0.0103502
$$167$$ −2.75557 −0.213232 −0.106616 0.994300i $$-0.534002\pi$$
−0.106616 + 0.994300i $$0.534002\pi$$
$$168$$ 0 0
$$169$$ 28.3274 2.17903
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1.05086 −0.0801270
$$173$$ 8.10171 0.615962 0.307981 0.951393i $$-0.400347\pi$$
0.307981 + 0.951393i $$0.400347\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 5.80642 0.437676
$$177$$ 0 0
$$178$$ 3.18421 0.238666
$$179$$ 0.235063 0.0175695 0.00878473 0.999961i $$-0.497204\pi$$
0.00878473 + 0.999961i $$0.497204\pi$$
$$180$$ 0 0
$$181$$ 11.3176 0.841228 0.420614 0.907240i $$-0.361815\pi$$
0.420614 + 0.907240i $$0.361815\pi$$
$$182$$ −28.4701 −2.11035
$$183$$ 0 0
$$184$$ 6.42864 0.473926
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 19.6128 1.43423
$$188$$ −5.18421 −0.378097
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.32693 0.457801 0.228900 0.973450i $$-0.426487\pi$$
0.228900 + 0.973450i $$0.426487\pi$$
$$192$$ 0 0
$$193$$ 11.5397 0.830647 0.415324 0.909674i $$-0.363668\pi$$
0.415324 + 0.909674i $$0.363668\pi$$
$$194$$ −11.4193 −0.819856
$$195$$ 0 0
$$196$$ 12.6128 0.900918
$$197$$ −20.0415 −1.42790 −0.713948 0.700198i $$-0.753095\pi$$
−0.713948 + 0.700198i $$0.753095\pi$$
$$198$$ 0 0
$$199$$ −20.4701 −1.45109 −0.725544 0.688175i $$-0.758412\pi$$
−0.725544 + 0.688175i $$0.758412\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 1.86665 0.131337
$$203$$ −34.5718 −2.42647
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 10.6222 0.740085
$$207$$ 0 0
$$208$$ −6.42864 −0.445746
$$209$$ −5.80642 −0.401639
$$210$$ 0 0
$$211$$ 2.75557 0.189701 0.0948506 0.995492i $$-0.469763\pi$$
0.0948506 + 0.995492i $$0.469763\pi$$
$$212$$ 4.75557 0.326614
$$213$$ 0 0
$$214$$ −7.61285 −0.520404
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 40.0830 2.72101
$$218$$ 5.53972 0.375197
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −21.7146 −1.46068
$$222$$ 0 0
$$223$$ 10.6222 0.711316 0.355658 0.934616i $$-0.384257\pi$$
0.355658 + 0.934616i $$0.384257\pi$$
$$224$$ 4.42864 0.295901
$$225$$ 0 0
$$226$$ −12.3684 −0.822735
$$227$$ −15.3461 −1.01856 −0.509280 0.860601i $$-0.670088\pi$$
−0.509280 + 0.860601i $$0.670088\pi$$
$$228$$ 0 0
$$229$$ −19.7146 −1.30277 −0.651387 0.758745i $$-0.725813\pi$$
−0.651387 + 0.758745i $$0.725813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −7.80642 −0.512517
$$233$$ 29.5625 1.93670 0.968351 0.249593i $$-0.0802968\pi$$
0.968351 + 0.249593i $$0.0802968\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.62222 −0.300881
$$237$$ 0 0
$$238$$ 14.9590 0.969647
$$239$$ −1.67307 −0.108222 −0.0541110 0.998535i $$-0.517232\pi$$
−0.0541110 + 0.998535i $$0.517232\pi$$
$$240$$ 0 0
$$241$$ −16.9590 −1.09242 −0.546212 0.837647i $$-0.683931\pi$$
−0.546212 + 0.837647i $$0.683931\pi$$
$$242$$ 22.7146 1.46015
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.42864 0.409045
$$248$$ 9.05086 0.574730
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4.94914 −0.312387 −0.156194 0.987726i $$-0.549922\pi$$
−0.156194 + 0.987726i $$0.549922\pi$$
$$252$$ 0 0
$$253$$ 37.3274 2.34675
$$254$$ 1.76494 0.110742
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −5.34614 −0.333483 −0.166742 0.986001i $$-0.553325\pi$$
−0.166742 + 0.986001i $$0.553325\pi$$
$$258$$ 0 0
$$259$$ 16.2667 1.01076
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −9.80642 −0.605843
$$263$$ 9.45091 0.582768 0.291384 0.956606i $$-0.405884\pi$$
0.291384 + 0.956606i $$0.405884\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.42864 −0.271537
$$267$$ 0 0
$$268$$ 2.75557 0.168323
$$269$$ −2.94914 −0.179813 −0.0899063 0.995950i $$-0.528657\pi$$
−0.0899063 + 0.995950i $$0.528657\pi$$
$$270$$ 0 0
$$271$$ 30.9590 1.88062 0.940312 0.340313i $$-0.110533\pi$$
0.940312 + 0.340313i $$0.110533\pi$$
$$272$$ 3.37778 0.204808
$$273$$ 0 0
$$274$$ −1.47949 −0.0893795
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −13.0923 −0.786643 −0.393321 0.919401i $$-0.628674\pi$$
−0.393321 + 0.919401i $$0.628674\pi$$
$$278$$ −4.85728 −0.291320
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −18.5303 −1.10543 −0.552714 0.833371i $$-0.686408\pi$$
−0.552714 + 0.833371i $$0.686408\pi$$
$$282$$ 0 0
$$283$$ −26.3783 −1.56802 −0.784012 0.620745i $$-0.786830\pi$$
−0.784012 + 0.620745i $$0.786830\pi$$
$$284$$ −7.61285 −0.451739
$$285$$ 0 0
$$286$$ −37.3274 −2.20722
$$287$$ −19.6128 −1.15771
$$288$$ 0 0
$$289$$ −5.59057 −0.328857
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 11.6128 0.679591
$$293$$ 28.5718 1.66918 0.834592 0.550868i $$-0.185703\pi$$
0.834592 + 0.550868i $$0.185703\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 3.67307 0.213493
$$297$$ 0 0
$$298$$ −4.62222 −0.267758
$$299$$ −41.3274 −2.39003
$$300$$ 0 0
$$301$$ −4.65386 −0.268244
$$302$$ −11.4193 −0.657105
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 22.7556 1.29873 0.649364 0.760477i $$-0.275035\pi$$
0.649364 + 0.760477i $$0.275035\pi$$
$$308$$ 25.7146 1.46522
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 6.32693 0.358767 0.179384 0.983779i $$-0.442590\pi$$
0.179384 + 0.983779i $$0.442590\pi$$
$$312$$ 0 0
$$313$$ −13.7146 −0.775193 −0.387596 0.921829i $$-0.626695\pi$$
−0.387596 + 0.921829i $$0.626695\pi$$
$$314$$ 7.37778 0.416352
$$315$$ 0 0
$$316$$ 2.94914 0.165902
$$317$$ 32.5718 1.82942 0.914708 0.404115i $$-0.132420\pi$$
0.914708 + 0.404115i $$0.132420\pi$$
$$318$$ 0 0
$$319$$ −45.3274 −2.53785
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 28.4701 1.58658
$$323$$ −3.37778 −0.187945
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −5.90813 −0.327221
$$327$$ 0 0
$$328$$ −4.42864 −0.244531
$$329$$ −22.9590 −1.26577
$$330$$ 0 0
$$331$$ 5.63158 0.309540 0.154770 0.987951i $$-0.450536\pi$$
0.154770 + 0.987951i $$0.450536\pi$$
$$332$$ 0.133353 0.00731870
$$333$$ 0 0
$$334$$ −2.75557 −0.150778
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 5.70471 0.310756 0.155378 0.987855i $$-0.450341\pi$$
0.155378 + 0.987855i $$0.450341\pi$$
$$338$$ 28.3274 1.54081
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 52.5531 2.84591
$$342$$ 0 0
$$343$$ 24.8573 1.34217
$$344$$ −1.05086 −0.0566583
$$345$$ 0 0
$$346$$ 8.10171 0.435551
$$347$$ 2.62222 0.140768 0.0703840 0.997520i $$-0.477578\pi$$
0.0703840 + 0.997520i $$0.477578\pi$$
$$348$$ 0 0
$$349$$ −24.1017 −1.29013 −0.645067 0.764126i $$-0.723171\pi$$
−0.645067 + 0.764126i $$0.723171\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5.80642 0.309483
$$353$$ −3.64449 −0.193977 −0.0969883 0.995286i $$-0.530921\pi$$
−0.0969883 + 0.995286i $$0.530921\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 3.18421 0.168763
$$357$$ 0 0
$$358$$ 0.235063 0.0124235
$$359$$ −9.08250 −0.479356 −0.239678 0.970852i $$-0.577042\pi$$
−0.239678 + 0.970852i $$0.577042\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 11.3176 0.594838
$$363$$ 0 0
$$364$$ −28.4701 −1.49224
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3.95851 0.206633 0.103316 0.994649i $$-0.467055\pi$$
0.103316 + 0.994649i $$0.467055\pi$$
$$368$$ 6.42864 0.335116
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 21.0607 1.09342
$$372$$ 0 0
$$373$$ 12.5303 0.648797 0.324398 0.945921i $$-0.394838\pi$$
0.324398 + 0.945921i $$0.394838\pi$$
$$374$$ 19.6128 1.01416
$$375$$ 0 0
$$376$$ −5.18421 −0.267355
$$377$$ 50.1847 2.58464
$$378$$ 0 0
$$379$$ 26.8385 1.37860 0.689302 0.724474i $$-0.257917\pi$$
0.689302 + 0.724474i $$0.257917\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 6.32693 0.323714
$$383$$ −17.5111 −0.894777 −0.447389 0.894340i $$-0.647646\pi$$
−0.447389 + 0.894340i $$0.647646\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 11.5397 0.587356
$$387$$ 0 0
$$388$$ −11.4193 −0.579726
$$389$$ 29.4795 1.49467 0.747335 0.664448i $$-0.231333\pi$$
0.747335 + 0.664448i $$0.231333\pi$$
$$390$$ 0 0
$$391$$ 21.7146 1.09815
$$392$$ 12.6128 0.637045
$$393$$ 0 0
$$394$$ −20.0415 −1.00968
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 9.21279 0.462377 0.231188 0.972909i $$-0.425739\pi$$
0.231188 + 0.972909i $$0.425739\pi$$
$$398$$ −20.4701 −1.02607
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −29.2859 −1.46247 −0.731234 0.682126i $$-0.761055\pi$$
−0.731234 + 0.682126i $$0.761055\pi$$
$$402$$ 0 0
$$403$$ −58.1847 −2.89839
$$404$$ 1.86665 0.0928692
$$405$$ 0 0
$$406$$ −34.5718 −1.71577
$$407$$ 21.3274 1.05716
$$408$$ 0 0
$$409$$ −17.8796 −0.884087 −0.442044 0.896994i $$-0.645746\pi$$
−0.442044 + 0.896994i $$0.645746\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 10.6222 0.523319
$$413$$ −20.4701 −1.00727
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −6.42864 −0.315190
$$417$$ 0 0
$$418$$ −5.80642 −0.284001
$$419$$ −30.8671 −1.50796 −0.753979 0.656899i $$-0.771868\pi$$
−0.753979 + 0.656899i $$0.771868\pi$$
$$420$$ 0 0
$$421$$ −18.3970 −0.896615 −0.448307 0.893879i $$-0.647973\pi$$
−0.448307 + 0.893879i $$0.647973\pi$$
$$422$$ 2.75557 0.134139
$$423$$ 0 0
$$424$$ 4.75557 0.230951
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.85728 0.428634
$$428$$ −7.61285 −0.367981
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −29.5941 −1.42550 −0.712749 0.701419i $$-0.752550\pi$$
−0.712749 + 0.701419i $$0.752550\pi$$
$$432$$ 0 0
$$433$$ 27.2988 1.31190 0.655949 0.754805i $$-0.272269\pi$$
0.655949 + 0.754805i $$0.272269\pi$$
$$434$$ 40.0830 1.92404
$$435$$ 0 0
$$436$$ 5.53972 0.265304
$$437$$ −6.42864 −0.307524
$$438$$ 0 0
$$439$$ 9.64143 0.460160 0.230080 0.973172i $$-0.426101\pi$$
0.230080 + 0.973172i $$0.426101\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −21.7146 −1.03286
$$443$$ −6.70519 −0.318573 −0.159287 0.987232i $$-0.550919\pi$$
−0.159287 + 0.987232i $$0.550919\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 10.6222 0.502976
$$447$$ 0 0
$$448$$ 4.42864 0.209234
$$449$$ 13.9398 0.657859 0.328929 0.944355i $$-0.393312\pi$$
0.328929 + 0.944355i $$0.393312\pi$$
$$450$$ 0 0
$$451$$ −25.7146 −1.21085
$$452$$ −12.3684 −0.581761
$$453$$ 0 0
$$454$$ −15.3461 −0.720230
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 23.2257 1.08645 0.543226 0.839586i $$-0.317203\pi$$
0.543226 + 0.839586i $$0.317203\pi$$
$$458$$ −19.7146 −0.921201
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −19.9684 −0.930019 −0.465010 0.885306i $$-0.653949\pi$$
−0.465010 + 0.885306i $$0.653949\pi$$
$$462$$ 0 0
$$463$$ −2.79706 −0.129990 −0.0649951 0.997886i $$-0.520703\pi$$
−0.0649951 + 0.997886i $$0.520703\pi$$
$$464$$ −7.80642 −0.362404
$$465$$ 0 0
$$466$$ 29.5625 1.36945
$$467$$ 30.9719 1.43321 0.716604 0.697480i $$-0.245695\pi$$
0.716604 + 0.697480i $$0.245695\pi$$
$$468$$ 0 0
$$469$$ 12.2034 0.563502
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −4.62222 −0.212755
$$473$$ −6.10171 −0.280557
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 14.9590 0.685644
$$477$$ 0 0
$$478$$ −1.67307 −0.0765245
$$479$$ −29.8765 −1.36509 −0.682546 0.730843i $$-0.739127\pi$$
−0.682546 + 0.730843i $$0.739127\pi$$
$$480$$ 0 0
$$481$$ −23.6128 −1.07665
$$482$$ −16.9590 −0.772461
$$483$$ 0 0
$$484$$ 22.7146 1.03248
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 17.8479 0.808766 0.404383 0.914590i $$-0.367486\pi$$
0.404383 + 0.914590i $$0.367486\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 8.68244 0.391833 0.195916 0.980621i $$-0.437232\pi$$
0.195916 + 0.980621i $$0.437232\pi$$
$$492$$ 0 0
$$493$$ −26.3684 −1.18757
$$494$$ 6.42864 0.289238
$$495$$ 0 0
$$496$$ 9.05086 0.406395
$$497$$ −33.7146 −1.51230
$$498$$ 0 0
$$499$$ 11.2257 0.502531 0.251266 0.967918i $$-0.419153\pi$$
0.251266 + 0.967918i $$0.419153\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −4.94914 −0.220891
$$503$$ 17.6543 0.787168 0.393584 0.919289i $$-0.371235\pi$$
0.393584 + 0.919289i $$0.371235\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 37.3274 1.65941
$$507$$ 0 0
$$508$$ 1.76494 0.0783064
$$509$$ −14.1748 −0.628289 −0.314144 0.949375i $$-0.601718\pi$$
−0.314144 + 0.949375i $$0.601718\pi$$
$$510$$ 0 0
$$511$$ 51.4291 2.27509
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −5.34614 −0.235808
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −30.1017 −1.32387
$$518$$ 16.2667 0.714718
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 15.0005 0.657183 0.328591 0.944472i $$-0.393426\pi$$
0.328591 + 0.944472i $$0.393426\pi$$
$$522$$ 0 0
$$523$$ 40.2864 1.76160 0.880801 0.473488i $$-0.157005\pi$$
0.880801 + 0.473488i $$0.157005\pi$$
$$524$$ −9.80642 −0.428396
$$525$$ 0 0
$$526$$ 9.45091 0.412079
$$527$$ 30.5718 1.33173
$$528$$ 0 0
$$529$$ 18.3274 0.796844
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −4.42864 −0.192006
$$533$$ 28.4701 1.23318
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 2.75557 0.119022
$$537$$ 0 0
$$538$$ −2.94914 −0.127147
$$539$$ 73.2355 3.15448
$$540$$ 0 0
$$541$$ −22.3872 −0.962499 −0.481249 0.876584i $$-0.659817\pi$$
−0.481249 + 0.876584i $$0.659817\pi$$
$$542$$ 30.9590 1.32980
$$543$$ 0 0
$$544$$ 3.37778 0.144821
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 19.7333 0.843735 0.421867 0.906658i $$-0.361375\pi$$
0.421867 + 0.906658i $$0.361375\pi$$
$$548$$ −1.47949 −0.0632009
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 7.80642 0.332565
$$552$$ 0 0
$$553$$ 13.0607 0.555397
$$554$$ −13.0923 −0.556240
$$555$$ 0 0
$$556$$ −4.85728 −0.205995
$$557$$ −15.6543 −0.663295 −0.331648 0.943403i $$-0.607605\pi$$
−0.331648 + 0.943403i $$0.607605\pi$$
$$558$$ 0 0
$$559$$ 6.75557 0.285730
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −18.5303 −0.781656
$$563$$ −36.9403 −1.55685 −0.778423 0.627740i $$-0.783980\pi$$
−0.778423 + 0.627740i $$0.783980\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −26.3783 −1.10876
$$567$$ 0 0
$$568$$ −7.61285 −0.319428
$$569$$ 1.20294 0.0504300 0.0252150 0.999682i $$-0.491973\pi$$
0.0252150 + 0.999682i $$0.491973\pi$$
$$570$$ 0 0
$$571$$ 37.7975 1.58178 0.790889 0.611960i $$-0.209619\pi$$
0.790889 + 0.611960i $$0.209619\pi$$
$$572$$ −37.3274 −1.56074
$$573$$ 0 0
$$574$$ −19.6128 −0.818624
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 23.2257 0.966898 0.483449 0.875372i $$-0.339384\pi$$
0.483449 + 0.875372i $$0.339384\pi$$
$$578$$ −5.59057 −0.232537
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0.590573 0.0245011
$$582$$ 0 0
$$583$$ 27.6128 1.14361
$$584$$ 11.6128 0.480543
$$585$$ 0 0
$$586$$ 28.5718 1.18029
$$587$$ −16.8069 −0.693695 −0.346848 0.937922i $$-0.612748\pi$$
−0.346848 + 0.937922i $$0.612748\pi$$
$$588$$ 0 0
$$589$$ −9.05086 −0.372934
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 3.67307 0.150962
$$593$$ 16.3555 0.671640 0.335820 0.941926i $$-0.390987\pi$$
0.335820 + 0.941926i $$0.390987\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −4.62222 −0.189333
$$597$$ 0 0
$$598$$ −41.3274 −1.69000
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 7.89829 0.322178 0.161089 0.986940i $$-0.448499\pi$$
0.161089 + 0.986940i $$0.448499\pi$$
$$602$$ −4.65386 −0.189677
$$603$$ 0 0
$$604$$ −11.4193 −0.464644
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 17.8479 0.724424 0.362212 0.932096i $$-0.382022\pi$$
0.362212 + 0.932096i $$0.382022\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 33.3274 1.34828
$$612$$ 0 0
$$613$$ 0.622216 0.0251311 0.0125655 0.999921i $$-0.496000\pi$$
0.0125655 + 0.999921i $$0.496000\pi$$
$$614$$ 22.7556 0.918340
$$615$$ 0 0
$$616$$ 25.7146 1.03607
$$617$$ 37.4795 1.50887 0.754434 0.656376i $$-0.227912\pi$$
0.754434 + 0.656376i $$0.227912\pi$$
$$618$$ 0 0
$$619$$ 28.6735 1.15249 0.576244 0.817278i $$-0.304518\pi$$
0.576244 + 0.817278i $$0.304518\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 6.32693 0.253687
$$623$$ 14.1017 0.564973
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −13.7146 −0.548144
$$627$$ 0 0
$$628$$ 7.37778 0.294406
$$629$$ 12.4068 0.494693
$$630$$ 0 0
$$631$$ 1.24443 0.0495400 0.0247700 0.999693i $$-0.492115\pi$$
0.0247700 + 0.999693i $$0.492115\pi$$
$$632$$ 2.94914 0.117311
$$633$$ 0 0
$$634$$ 32.5718 1.29359
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −81.0835 −3.21264
$$638$$ −45.3274 −1.79453
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 17.4064 0.687510 0.343755 0.939059i $$-0.388301\pi$$
0.343755 + 0.939059i $$0.388301\pi$$
$$642$$ 0 0
$$643$$ −35.1526 −1.38628 −0.693141 0.720802i $$-0.743774\pi$$
−0.693141 + 0.720802i $$0.743774\pi$$
$$644$$ 28.4701 1.12188
$$645$$ 0 0
$$646$$ −3.37778 −0.132897
$$647$$ −34.4286 −1.35353 −0.676765 0.736199i $$-0.736619\pi$$
−0.676765 + 0.736199i $$0.736619\pi$$
$$648$$ 0 0
$$649$$ −26.8385 −1.05350
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −5.90813 −0.231380
$$653$$ 4.30819 0.168593 0.0842963 0.996441i $$-0.473136\pi$$
0.0842963 + 0.996441i $$0.473136\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −4.42864 −0.172909
$$657$$ 0 0
$$658$$ −22.9590 −0.895035
$$659$$ 27.0736 1.05464 0.527319 0.849667i $$-0.323197\pi$$
0.527319 + 0.849667i $$0.323197\pi$$
$$660$$ 0 0
$$661$$ −43.2543 −1.68240 −0.841198 0.540727i $$-0.818149\pi$$
−0.841198 + 0.540727i $$0.818149\pi$$
$$662$$ 5.63158 0.218878
$$663$$ 0 0
$$664$$ 0.133353 0.00517510
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −50.1847 −1.94316
$$668$$ −2.75557 −0.106616
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 11.6128 0.448309
$$672$$ 0 0
$$673$$ 15.0321 0.579446 0.289723 0.957111i $$-0.406437\pi$$
0.289723 + 0.957111i $$0.406437\pi$$
$$674$$ 5.70471 0.219737
$$675$$ 0 0
$$676$$ 28.3274 1.08952
$$677$$ −22.9403 −0.881666 −0.440833 0.897589i $$-0.645317\pi$$
−0.440833 + 0.897589i $$0.645317\pi$$
$$678$$ 0 0
$$679$$ −50.5718 −1.94077
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 52.5531 2.01236
$$683$$ −9.77784 −0.374139 −0.187069 0.982347i $$-0.559899\pi$$
−0.187069 + 0.982347i $$0.559899\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 24.8573 0.949055
$$687$$ 0 0
$$688$$ −1.05086 −0.0400635
$$689$$ −30.5718 −1.16469
$$690$$ 0 0
$$691$$ −44.0830 −1.67700 −0.838498 0.544905i $$-0.816566\pi$$
−0.838498 + 0.544905i $$0.816566\pi$$
$$692$$ 8.10171 0.307981
$$693$$ 0 0
$$694$$ 2.62222 0.0995379
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −14.9590 −0.566612
$$698$$ −24.1017 −0.912263
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 12.7685 0.482259 0.241129 0.970493i $$-0.422482\pi$$
0.241129 + 0.970493i $$0.422482\pi$$
$$702$$ 0 0
$$703$$ −3.67307 −0.138532
$$704$$ 5.80642 0.218838
$$705$$ 0 0
$$706$$ −3.64449 −0.137162
$$707$$ 8.26671 0.310901
$$708$$ 0 0
$$709$$ −47.5941 −1.78743 −0.893717 0.448631i $$-0.851912\pi$$
−0.893717 + 0.448631i $$0.851912\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 3.18421 0.119333
$$713$$ 58.1847 2.17903
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0.235063 0.00878473
$$717$$ 0 0
$$718$$ −9.08250 −0.338956
$$719$$ 47.0005 1.75282 0.876411 0.481564i $$-0.159931\pi$$
0.876411 + 0.481564i $$0.159931\pi$$
$$720$$ 0 0
$$721$$ 47.0420 1.75193
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ 11.3176 0.420614
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 52.6321 1.95202 0.976008 0.217737i $$-0.0698674\pi$$
0.976008 + 0.217737i $$0.0698674\pi$$
$$728$$ −28.4701 −1.05517
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −3.54956 −0.131285
$$732$$ 0 0
$$733$$ 27.3145 1.00888 0.504442 0.863446i $$-0.331698\pi$$
0.504442 + 0.863446i $$0.331698\pi$$
$$734$$ 3.95851 0.146111
$$735$$ 0 0
$$736$$ 6.42864 0.236963
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ 25.3274 0.931684 0.465842 0.884868i $$-0.345752\pi$$
0.465842 + 0.884868i $$0.345752\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 21.0607 0.773163
$$743$$ 11.8796 0.435819 0.217909 0.975969i $$-0.430076\pi$$
0.217909 + 0.975969i $$0.430076\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 12.5303 0.458769
$$747$$ 0 0
$$748$$ 19.6128 0.717117
$$749$$ −33.7146 −1.23190
$$750$$ 0 0
$$751$$ −45.5210 −1.66108 −0.830542 0.556956i $$-0.811969\pi$$
−0.830542 + 0.556956i $$0.811969\pi$$
$$752$$ −5.18421 −0.189049
$$753$$ 0 0
$$754$$ 50.1847 1.82762
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −51.7275 −1.88007 −0.940033 0.341083i $$-0.889206\pi$$
−0.940033 + 0.341083i $$0.889206\pi$$
$$758$$ 26.8385 0.974820
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −28.3684 −1.02835 −0.514177 0.857684i $$-0.671903\pi$$
−0.514177 + 0.857684i $$0.671903\pi$$
$$762$$ 0 0
$$763$$ 24.5334 0.888169
$$764$$ 6.32693 0.228900
$$765$$ 0 0
$$766$$ −17.5111 −0.632703
$$767$$ 29.7146 1.07293
$$768$$ 0 0
$$769$$ −0.285442 −0.0102933 −0.00514665 0.999987i $$-0.501638\pi$$
−0.00514665 + 0.999987i $$0.501638\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 11.5397 0.415324
$$773$$ −49.8163 −1.79177 −0.895883 0.444289i $$-0.853456\pi$$
−0.895883 + 0.444289i $$0.853456\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −11.4193 −0.409928
$$777$$ 0 0
$$778$$ 29.4795 1.05689
$$779$$ 4.42864 0.158672
$$780$$ 0 0
$$781$$ −44.2034 −1.58172
$$782$$ 21.7146 0.776511
$$783$$ 0 0
$$784$$ 12.6128 0.450459
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −20.7368 −0.739188 −0.369594 0.929193i $$-0.620503\pi$$
−0.369594 + 0.929193i $$0.620503\pi$$
$$788$$ −20.0415 −0.713948
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −54.7753 −1.94758
$$792$$ 0 0
$$793$$ −12.8573 −0.456575
$$794$$ 9.21279 0.326950
$$795$$ 0 0
$$796$$ −20.4701 −0.725544
$$797$$ 21.3461 0.756119 0.378060 0.925781i $$-0.376591\pi$$
0.378060 + 0.925781i $$0.376591\pi$$
$$798$$ 0 0
$$799$$ −17.5111 −0.619500
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −29.2859 −1.03412
$$803$$ 67.4291 2.37952
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −58.1847 −2.04947
$$807$$ 0 0
$$808$$ 1.86665 0.0656684
$$809$$ 18.6735 0.656527 0.328263 0.944586i $$-0.393537\pi$$
0.328263 + 0.944586i $$0.393537\pi$$
$$810$$ 0 0
$$811$$ −14.8385 −0.521052 −0.260526 0.965467i $$-0.583896\pi$$
−0.260526 + 0.965467i $$0.583896\pi$$
$$812$$ −34.5718 −1.21323
$$813$$ 0 0
$$814$$ 21.3274 0.747525
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.05086 0.0367648
$$818$$ −17.8796 −0.625144
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16.1521 −0.563712 −0.281856 0.959457i $$-0.590950\pi$$
−0.281856 + 0.959457i $$0.590950\pi$$
$$822$$ 0 0
$$823$$ −47.6543 −1.66113 −0.830563 0.556925i $$-0.811981\pi$$
−0.830563 + 0.556925i $$0.811981\pi$$
$$824$$ 10.6222 0.370042
$$825$$ 0 0
$$826$$ −20.4701 −0.712247
$$827$$ −26.1017 −0.907645 −0.453823 0.891092i $$-0.649940\pi$$
−0.453823 + 0.891092i $$0.649940\pi$$
$$828$$ 0 0
$$829$$ −13.8894 −0.482399 −0.241199 0.970476i $$-0.577541\pi$$
−0.241199 + 0.970476i $$0.577541\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −6.42864 −0.222873
$$833$$ 42.6035 1.47612
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −5.80642 −0.200819
$$837$$ 0 0
$$838$$ −30.8671 −1.06629
$$839$$ −31.6958 −1.09426 −0.547131 0.837047i $$-0.684280\pi$$
−0.547131 + 0.837047i $$0.684280\pi$$
$$840$$ 0 0
$$841$$ 31.9403 1.10139
$$842$$ −18.3970 −0.634002
$$843$$ 0 0
$$844$$ 2.75557 0.0948506
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 100.595 3.45647
$$848$$ 4.75557 0.163307
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 23.6128 0.809438
$$852$$ 0 0
$$853$$ 34.8702 1.19393 0.596966 0.802266i $$-0.296373\pi$$
0.596966 + 0.802266i $$0.296373\pi$$
$$854$$ 8.85728 0.303090
$$855$$ 0 0
$$856$$ −7.61285 −0.260202
$$857$$ 33.9367 1.15926 0.579628 0.814881i $$-0.303198\pi$$
0.579628 + 0.814881i $$0.303198\pi$$
$$858$$ 0 0
$$859$$ −37.4479 −1.27770 −0.638852 0.769330i $$-0.720590\pi$$
−0.638852 + 0.769330i $$0.720590\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −29.5941 −1.00798
$$863$$ −15.5299 −0.528643 −0.264322 0.964435i $$-0.585148\pi$$
−0.264322 + 0.964435i $$0.585148\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 27.2988 0.927652
$$867$$ 0 0
$$868$$ 40.0830 1.36050
$$869$$ 17.1240 0.580891
$$870$$ 0 0
$$871$$ −17.7146 −0.600235
$$872$$ 5.53972 0.187599
$$873$$ 0 0
$$874$$ −6.42864 −0.217452
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −7.87649 −0.265970 −0.132985 0.991118i $$-0.542456\pi$$
−0.132985 + 0.991118i $$0.542456\pi$$
$$878$$ 9.64143 0.325382
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −33.5496 −1.13031 −0.565157 0.824984i $$-0.691184\pi$$
−0.565157 + 0.824984i $$0.691184\pi$$
$$882$$ 0 0
$$883$$ 50.9688 1.71524 0.857619 0.514286i $$-0.171943\pi$$
0.857619 + 0.514286i $$0.171943\pi$$
$$884$$ −21.7146 −0.730340
$$885$$ 0 0
$$886$$ −6.70519 −0.225265
$$887$$ −17.0035 −0.570923 −0.285461 0.958390i $$-0.592147\pi$$
−0.285461 + 0.958390i $$0.592147\pi$$
$$888$$ 0 0
$$889$$ 7.81627 0.262149
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 10.6222 0.355658
$$893$$ 5.18421 0.173483
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 4.42864 0.147950
$$897$$ 0 0
$$898$$ 13.9398 0.465176
$$899$$ −70.6548 −2.35647
$$900$$ 0 0
$$901$$ 16.0633 0.535146
$$902$$ −25.7146 −0.856201
$$903$$ 0 0
$$904$$ −12.3684 −0.411367
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −21.7778 −0.723121 −0.361561 0.932349i $$-0.617756\pi$$
−0.361561 + 0.932349i $$0.617756\pi$$
$$908$$ −15.3461 −0.509280
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −10.6351 −0.352357 −0.176179 0.984358i $$-0.556374\pi$$
−0.176179 + 0.984358i $$0.556374\pi$$
$$912$$ 0 0
$$913$$ 0.774305 0.0256257
$$914$$ 23.2257 0.768238
$$915$$ 0 0
$$916$$ −19.7146 −0.651387
$$917$$ −43.4291 −1.43416
$$918$$ 0 0
$$919$$ 12.2667 0.404641 0.202321 0.979319i $$-0.435152\pi$$
0.202321 + 0.979319i $$0.435152\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −19.9684 −0.657623
$$923$$ 48.9403 1.61089
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −2.79706 −0.0919170
$$927$$ 0 0
$$928$$ −7.80642 −0.256258
$$929$$ 27.9180 0.915959 0.457980 0.888963i $$-0.348573\pi$$
0.457980 + 0.888963i $$0.348573\pi$$
$$930$$ 0 0
$$931$$ −12.6128 −0.413369
$$932$$ 29.5625 0.968351
$$933$$ 0 0
$$934$$ 30.9719 1.01343
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −3.14272 −0.102668 −0.0513341 0.998682i $$-0.516347\pi$$
−0.0513341 + 0.998682i $$0.516347\pi$$
$$938$$ 12.2034 0.398456
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 19.6227 0.639681 0.319841 0.947471i $$-0.396371\pi$$
0.319841 + 0.947471i $$0.396371\pi$$
$$942$$ 0 0
$$943$$ −28.4701 −0.927115
$$944$$ −4.62222 −0.150440
$$945$$ 0 0
$$946$$ −6.10171 −0.198384
$$947$$ −34.5018 −1.12116 −0.560578 0.828101i $$-0.689421\pi$$
−0.560578 + 0.828101i $$0.689421\pi$$
$$948$$ 0 0
$$949$$ −74.6548 −2.42340
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 14.9590 0.484824
$$953$$ 19.0035 0.615585 0.307793 0.951454i $$-0.400410\pi$$
0.307793 + 0.951454i $$0.400410\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −1.67307 −0.0541110
$$957$$ 0 0
$$958$$ −29.8765 −0.965266
$$959$$ −6.55215 −0.211580
$$960$$ 0 0
$$961$$ 50.9180 1.64252
$$962$$ −23.6128 −0.761309
$$963$$ 0 0
$$964$$ −16.9590 −0.546212
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −17.8765 −0.574869 −0.287435 0.957800i $$-0.592802\pi$$
−0.287435 + 0.957800i $$0.592802\pi$$
$$968$$ 22.7146 0.730074
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −34.0701 −1.09336 −0.546680 0.837341i $$-0.684109\pi$$
−0.546680 + 0.837341i $$0.684109\pi$$
$$972$$ 0 0
$$973$$ −21.5111 −0.689615
$$974$$ 17.8479 0.571884
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 54.0830 1.73027 0.865134 0.501541i $$-0.167233\pi$$
0.865134 + 0.501541i $$0.167233\pi$$
$$978$$ 0 0
$$979$$ 18.4889 0.590907
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 8.68244 0.277068
$$983$$ −56.9403 −1.81611 −0.908056 0.418849i $$-0.862434\pi$$
−0.908056 + 0.418849i $$0.862434\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −26.3684 −0.839741
$$987$$ 0 0
$$988$$ 6.42864 0.204522
$$989$$ −6.75557 −0.214815
$$990$$ 0 0
$$991$$ −47.0321 −1.49402 −0.747012 0.664810i $$-0.768512\pi$$
−0.747012 + 0.664810i $$0.768512\pi$$
$$992$$ 9.05086 0.287365
$$993$$ 0 0
$$994$$ −33.7146 −1.06936
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −20.8256 −0.659555 −0.329777 0.944059i $$-0.606974\pi$$
−0.329777 + 0.944059i $$0.606974\pi$$
$$998$$ 11.2257 0.355343
$$999$$ 0 0
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 8550.2.a.cr.1.3 3
3.2 odd 2 2850.2.a.bk.1.3 3
5.2 odd 4 1710.2.d.e.1369.5 6
5.3 odd 4 1710.2.d.e.1369.2 6
5.4 even 2 8550.2.a.cf.1.1 3
15.2 even 4 570.2.d.d.229.2 6
15.8 even 4 570.2.d.d.229.5 yes 6
15.14 odd 2 2850.2.a.bn.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.d.229.2 6 15.2 even 4
570.2.d.d.229.5 yes 6 15.8 even 4
1710.2.d.e.1369.2 6 5.3 odd 4
1710.2.d.e.1369.5 6 5.2 odd 4
2850.2.a.bk.1.3 3 3.2 odd 2
2850.2.a.bn.1.1 3 15.14 odd 2
8550.2.a.cf.1.1 3 5.4 even 2
8550.2.a.cr.1.3 3 1.1 even 1 trivial