Properties

 Label 7728.2.a.w.1.1 Level $7728$ Weight $2$ Character 7728.1 Self dual yes Analytic conductor $61.708$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7728.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$61.7083906820$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1932) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 7728.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.09017 q^{13} +3.61803 q^{15} -5.00000 q^{17} +3.47214 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.09017 q^{25} -1.00000 q^{27} -6.23607 q^{29} +8.70820 q^{31} -1.00000 q^{33} +3.61803 q^{35} +1.47214 q^{37} -5.09017 q^{39} -5.76393 q^{41} -6.32624 q^{43} -3.61803 q^{45} -3.70820 q^{47} +1.00000 q^{49} +5.00000 q^{51} +0.381966 q^{53} -3.61803 q^{55} -3.47214 q^{57} +3.61803 q^{59} -7.56231 q^{61} -1.00000 q^{63} -18.4164 q^{65} +7.32624 q^{67} +1.00000 q^{69} +4.32624 q^{71} -0.527864 q^{73} -8.09017 q^{75} -1.00000 q^{77} +10.7082 q^{79} +1.00000 q^{81} -9.18034 q^{83} +18.0902 q^{85} +6.23607 q^{87} -16.6180 q^{89} -5.09017 q^{91} -8.70820 q^{93} -12.5623 q^{95} -12.2361 q^{97} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - q^{13} + 5 q^{15} - 10 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + q^{39} - 16 q^{41} + 3 q^{43} - 5 q^{45} + 6 q^{47} + 2 q^{49} + 10 q^{51} + 3 q^{53} - 5 q^{55} + 2 q^{57} + 5 q^{59} + 5 q^{61} - 2 q^{63} - 10 q^{65} - q^{67} + 2 q^{69} - 7 q^{71} - 10 q^{73} - 5 q^{75} - 2 q^{77} + 8 q^{79} + 2 q^{81} + 4 q^{83} + 25 q^{85} + 8 q^{87} - 31 q^{89} + q^{91} - 4 q^{93} - 5 q^{95} - 20 q^{97} + 2 q^{99} + O(q^{100})$$

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −3.61803 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ 5.09017 1.41176 0.705880 0.708332i $$-0.250552\pi$$
0.705880 + 0.708332i $$0.250552\pi$$
$$14$$ 0 0
$$15$$ 3.61803 0.934172
$$16$$ 0 0
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 0 0
$$19$$ 3.47214 0.796563 0.398281 0.917263i $$-0.369607\pi$$
0.398281 + 0.917263i $$0.369607\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 8.09017 1.61803
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −6.23607 −1.15801 −0.579004 0.815324i $$-0.696559\pi$$
−0.579004 + 0.815324i $$0.696559\pi$$
$$30$$ 0 0
$$31$$ 8.70820 1.56404 0.782020 0.623254i $$-0.214190\pi$$
0.782020 + 0.623254i $$0.214190\pi$$
$$32$$ 0 0
$$33$$ −1.00000 −0.174078
$$34$$ 0 0
$$35$$ 3.61803 0.611559
$$36$$ 0 0
$$37$$ 1.47214 0.242018 0.121009 0.992651i $$-0.461387\pi$$
0.121009 + 0.992651i $$0.461387\pi$$
$$38$$ 0 0
$$39$$ −5.09017 −0.815080
$$40$$ 0 0
$$41$$ −5.76393 −0.900175 −0.450087 0.892984i $$-0.648607\pi$$
−0.450087 + 0.892984i $$0.648607\pi$$
$$42$$ 0 0
$$43$$ −6.32624 −0.964742 −0.482371 0.875967i $$-0.660224\pi$$
−0.482371 + 0.875967i $$0.660224\pi$$
$$44$$ 0 0
$$45$$ −3.61803 −0.539345
$$46$$ 0 0
$$47$$ −3.70820 −0.540897 −0.270449 0.962734i $$-0.587172\pi$$
−0.270449 + 0.962734i $$0.587172\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 5.00000 0.700140
$$52$$ 0 0
$$53$$ 0.381966 0.0524671 0.0262335 0.999656i $$-0.491649\pi$$
0.0262335 + 0.999656i $$0.491649\pi$$
$$54$$ 0 0
$$55$$ −3.61803 −0.487856
$$56$$ 0 0
$$57$$ −3.47214 −0.459896
$$58$$ 0 0
$$59$$ 3.61803 0.471028 0.235514 0.971871i $$-0.424323\pi$$
0.235514 + 0.971871i $$0.424323\pi$$
$$60$$ 0 0
$$61$$ −7.56231 −0.968254 −0.484127 0.874998i $$-0.660863\pi$$
−0.484127 + 0.874998i $$0.660863\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ −18.4164 −2.28427
$$66$$ 0 0
$$67$$ 7.32624 0.895042 0.447521 0.894273i $$-0.352307\pi$$
0.447521 + 0.894273i $$0.352307\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 4.32624 0.513430 0.256715 0.966487i $$-0.417360\pi$$
0.256715 + 0.966487i $$0.417360\pi$$
$$72$$ 0 0
$$73$$ −0.527864 −0.0617818 −0.0308909 0.999523i $$-0.509834\pi$$
−0.0308909 + 0.999523i $$0.509834\pi$$
$$74$$ 0 0
$$75$$ −8.09017 −0.934172
$$76$$ 0 0
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ 10.7082 1.20477 0.602384 0.798207i $$-0.294218\pi$$
0.602384 + 0.798207i $$0.294218\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −9.18034 −1.00767 −0.503837 0.863799i $$-0.668079\pi$$
−0.503837 + 0.863799i $$0.668079\pi$$
$$84$$ 0 0
$$85$$ 18.0902 1.96215
$$86$$ 0 0
$$87$$ 6.23607 0.668577
$$88$$ 0 0
$$89$$ −16.6180 −1.76151 −0.880754 0.473574i $$-0.842964\pi$$
−0.880754 + 0.473574i $$0.842964\pi$$
$$90$$ 0 0
$$91$$ −5.09017 −0.533595
$$92$$ 0 0
$$93$$ −8.70820 −0.902999
$$94$$ 0 0
$$95$$ −12.5623 −1.28887
$$96$$ 0 0
$$97$$ −12.2361 −1.24238 −0.621192 0.783658i $$-0.713351\pi$$
−0.621192 + 0.783658i $$0.713351\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −5.61803 −0.559015 −0.279508 0.960143i $$-0.590171\pi$$
−0.279508 + 0.960143i $$0.590171\pi$$
$$102$$ 0 0
$$103$$ −2.47214 −0.243587 −0.121793 0.992555i $$-0.538865\pi$$
−0.121793 + 0.992555i $$0.538865\pi$$
$$104$$ 0 0
$$105$$ −3.61803 −0.353084
$$106$$ 0 0
$$107$$ 14.6180 1.41318 0.706589 0.707624i $$-0.250233\pi$$
0.706589 + 0.707624i $$0.250233\pi$$
$$108$$ 0 0
$$109$$ −17.5066 −1.67683 −0.838413 0.545035i $$-0.816516\pi$$
−0.838413 + 0.545035i $$0.816516\pi$$
$$110$$ 0 0
$$111$$ −1.47214 −0.139729
$$112$$ 0 0
$$113$$ 3.14590 0.295941 0.147971 0.988992i $$-0.452726\pi$$
0.147971 + 0.988992i $$0.452726\pi$$
$$114$$ 0 0
$$115$$ 3.61803 0.337383
$$116$$ 0 0
$$117$$ 5.09017 0.470586
$$118$$ 0 0
$$119$$ 5.00000 0.458349
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ 5.76393 0.519716
$$124$$ 0 0
$$125$$ −11.1803 −1.00000
$$126$$ 0 0
$$127$$ −4.61803 −0.409784 −0.204892 0.978785i $$-0.565684\pi$$
−0.204892 + 0.978785i $$0.565684\pi$$
$$128$$ 0 0
$$129$$ 6.32624 0.556994
$$130$$ 0 0
$$131$$ 9.18034 0.802090 0.401045 0.916058i $$-0.368647\pi$$
0.401045 + 0.916058i $$0.368647\pi$$
$$132$$ 0 0
$$133$$ −3.47214 −0.301072
$$134$$ 0 0
$$135$$ 3.61803 0.311391
$$136$$ 0 0
$$137$$ −4.52786 −0.386842 −0.193421 0.981116i $$-0.561958\pi$$
−0.193421 + 0.981116i $$0.561958\pi$$
$$138$$ 0 0
$$139$$ −0.673762 −0.0571478 −0.0285739 0.999592i $$-0.509097\pi$$
−0.0285739 + 0.999592i $$0.509097\pi$$
$$140$$ 0 0
$$141$$ 3.70820 0.312287
$$142$$ 0 0
$$143$$ 5.09017 0.425661
$$144$$ 0 0
$$145$$ 22.5623 1.87370
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 19.1246 1.56675 0.783375 0.621550i $$-0.213497\pi$$
0.783375 + 0.621550i $$0.213497\pi$$
$$150$$ 0 0
$$151$$ −5.23607 −0.426105 −0.213053 0.977041i $$-0.568341\pi$$
−0.213053 + 0.977041i $$0.568341\pi$$
$$152$$ 0 0
$$153$$ −5.00000 −0.404226
$$154$$ 0 0
$$155$$ −31.5066 −2.53067
$$156$$ 0 0
$$157$$ 14.6525 1.16939 0.584697 0.811251i $$-0.301213\pi$$
0.584697 + 0.811251i $$0.301213\pi$$
$$158$$ 0 0
$$159$$ −0.381966 −0.0302919
$$160$$ 0 0
$$161$$ 1.00000 0.0788110
$$162$$ 0 0
$$163$$ 14.0902 1.10363 0.551814 0.833967i $$-0.313936\pi$$
0.551814 + 0.833967i $$0.313936\pi$$
$$164$$ 0 0
$$165$$ 3.61803 0.281664
$$166$$ 0 0
$$167$$ 3.18034 0.246102 0.123051 0.992400i $$-0.460732\pi$$
0.123051 + 0.992400i $$0.460732\pi$$
$$168$$ 0 0
$$169$$ 12.9098 0.993064
$$170$$ 0 0
$$171$$ 3.47214 0.265521
$$172$$ 0 0
$$173$$ 9.18034 0.697968 0.348984 0.937129i $$-0.386527\pi$$
0.348984 + 0.937129i $$0.386527\pi$$
$$174$$ 0 0
$$175$$ −8.09017 −0.611559
$$176$$ 0 0
$$177$$ −3.61803 −0.271948
$$178$$ 0 0
$$179$$ −10.2705 −0.767654 −0.383827 0.923405i $$-0.625394\pi$$
−0.383827 + 0.923405i $$0.625394\pi$$
$$180$$ 0 0
$$181$$ −0.236068 −0.0175468 −0.00877340 0.999962i $$-0.502793\pi$$
−0.00877340 + 0.999962i $$0.502793\pi$$
$$182$$ 0 0
$$183$$ 7.56231 0.559022
$$184$$ 0 0
$$185$$ −5.32624 −0.391593
$$186$$ 0 0
$$187$$ −5.00000 −0.365636
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −16.6525 −1.20493 −0.602465 0.798145i $$-0.705815\pi$$
−0.602465 + 0.798145i $$0.705815\pi$$
$$192$$ 0 0
$$193$$ 21.7082 1.56259 0.781295 0.624161i $$-0.214559\pi$$
0.781295 + 0.624161i $$0.214559\pi$$
$$194$$ 0 0
$$195$$ 18.4164 1.31883
$$196$$ 0 0
$$197$$ 22.6180 1.61147 0.805734 0.592277i $$-0.201771\pi$$
0.805734 + 0.592277i $$0.201771\pi$$
$$198$$ 0 0
$$199$$ −3.79837 −0.269260 −0.134630 0.990896i $$-0.542985\pi$$
−0.134630 + 0.990896i $$0.542985\pi$$
$$200$$ 0 0
$$201$$ −7.32624 −0.516753
$$202$$ 0 0
$$203$$ 6.23607 0.437686
$$204$$ 0 0
$$205$$ 20.8541 1.45651
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ 3.47214 0.240173
$$210$$ 0 0
$$211$$ −18.2361 −1.25542 −0.627711 0.778446i $$-0.716008\pi$$
−0.627711 + 0.778446i $$0.716008\pi$$
$$212$$ 0 0
$$213$$ −4.32624 −0.296429
$$214$$ 0 0
$$215$$ 22.8885 1.56099
$$216$$ 0 0
$$217$$ −8.70820 −0.591151
$$218$$ 0 0
$$219$$ 0.527864 0.0356697
$$220$$ 0 0
$$221$$ −25.4508 −1.71201
$$222$$ 0 0
$$223$$ −10.5623 −0.707304 −0.353652 0.935377i $$-0.615060\pi$$
−0.353652 + 0.935377i $$0.615060\pi$$
$$224$$ 0 0
$$225$$ 8.09017 0.539345
$$226$$ 0 0
$$227$$ 16.8541 1.11865 0.559323 0.828950i $$-0.311061\pi$$
0.559323 + 0.828950i $$0.311061\pi$$
$$228$$ 0 0
$$229$$ −7.79837 −0.515331 −0.257666 0.966234i $$-0.582953\pi$$
−0.257666 + 0.966234i $$0.582953\pi$$
$$230$$ 0 0
$$231$$ 1.00000 0.0657952
$$232$$ 0 0
$$233$$ 18.5623 1.21606 0.608029 0.793915i $$-0.291961\pi$$
0.608029 + 0.793915i $$0.291961\pi$$
$$234$$ 0 0
$$235$$ 13.4164 0.875190
$$236$$ 0 0
$$237$$ −10.7082 −0.695573
$$238$$ 0 0
$$239$$ −12.6738 −0.819798 −0.409899 0.912131i $$-0.634436\pi$$
−0.409899 + 0.912131i $$0.634436\pi$$
$$240$$ 0 0
$$241$$ −25.6525 −1.65242 −0.826211 0.563361i $$-0.809508\pi$$
−0.826211 + 0.563361i $$0.809508\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −3.61803 −0.231148
$$246$$ 0 0
$$247$$ 17.6738 1.12455
$$248$$ 0 0
$$249$$ 9.18034 0.581780
$$250$$ 0 0
$$251$$ −3.81966 −0.241095 −0.120547 0.992708i $$-0.538465\pi$$
−0.120547 + 0.992708i $$0.538465\pi$$
$$252$$ 0 0
$$253$$ −1.00000 −0.0628695
$$254$$ 0 0
$$255$$ −18.0902 −1.13285
$$256$$ 0 0
$$257$$ −0.291796 −0.0182017 −0.00910087 0.999959i $$-0.502897\pi$$
−0.00910087 + 0.999959i $$0.502897\pi$$
$$258$$ 0 0
$$259$$ −1.47214 −0.0914741
$$260$$ 0 0
$$261$$ −6.23607 −0.386003
$$262$$ 0 0
$$263$$ 27.1803 1.67601 0.838006 0.545661i $$-0.183721\pi$$
0.838006 + 0.545661i $$0.183721\pi$$
$$264$$ 0 0
$$265$$ −1.38197 −0.0848935
$$266$$ 0 0
$$267$$ 16.6180 1.01701
$$268$$ 0 0
$$269$$ 19.8541 1.21053 0.605263 0.796026i $$-0.293068\pi$$
0.605263 + 0.796026i $$0.293068\pi$$
$$270$$ 0 0
$$271$$ 6.52786 0.396540 0.198270 0.980147i $$-0.436468\pi$$
0.198270 + 0.980147i $$0.436468\pi$$
$$272$$ 0 0
$$273$$ 5.09017 0.308071
$$274$$ 0 0
$$275$$ 8.09017 0.487856
$$276$$ 0 0
$$277$$ 31.0902 1.86803 0.934014 0.357237i $$-0.116281\pi$$
0.934014 + 0.357237i $$0.116281\pi$$
$$278$$ 0 0
$$279$$ 8.70820 0.521347
$$280$$ 0 0
$$281$$ 8.18034 0.487998 0.243999 0.969775i $$-0.421541\pi$$
0.243999 + 0.969775i $$0.421541\pi$$
$$282$$ 0 0
$$283$$ −2.96556 −0.176284 −0.0881421 0.996108i $$-0.528093\pi$$
−0.0881421 + 0.996108i $$0.528093\pi$$
$$284$$ 0 0
$$285$$ 12.5623 0.744127
$$286$$ 0 0
$$287$$ 5.76393 0.340234
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 12.2361 0.717291
$$292$$ 0 0
$$293$$ 2.47214 0.144424 0.0722119 0.997389i $$-0.476994\pi$$
0.0722119 + 0.997389i $$0.476994\pi$$
$$294$$ 0 0
$$295$$ −13.0902 −0.762139
$$296$$ 0 0
$$297$$ −1.00000 −0.0580259
$$298$$ 0 0
$$299$$ −5.09017 −0.294372
$$300$$ 0 0
$$301$$ 6.32624 0.364638
$$302$$ 0 0
$$303$$ 5.61803 0.322748
$$304$$ 0 0
$$305$$ 27.3607 1.56667
$$306$$ 0 0
$$307$$ −29.0689 −1.65905 −0.829524 0.558470i $$-0.811388\pi$$
−0.829524 + 0.558470i $$0.811388\pi$$
$$308$$ 0 0
$$309$$ 2.47214 0.140635
$$310$$ 0 0
$$311$$ 15.0902 0.855685 0.427843 0.903853i $$-0.359274\pi$$
0.427843 + 0.903853i $$0.359274\pi$$
$$312$$ 0 0
$$313$$ −0.583592 −0.0329866 −0.0164933 0.999864i $$-0.505250\pi$$
−0.0164933 + 0.999864i $$0.505250\pi$$
$$314$$ 0 0
$$315$$ 3.61803 0.203853
$$316$$ 0 0
$$317$$ 26.3820 1.48176 0.740879 0.671638i $$-0.234409\pi$$
0.740879 + 0.671638i $$0.234409\pi$$
$$318$$ 0 0
$$319$$ −6.23607 −0.349153
$$320$$ 0 0
$$321$$ −14.6180 −0.815899
$$322$$ 0 0
$$323$$ −17.3607 −0.965974
$$324$$ 0 0
$$325$$ 41.1803 2.28427
$$326$$ 0 0
$$327$$ 17.5066 0.968116
$$328$$ 0 0
$$329$$ 3.70820 0.204440
$$330$$ 0 0
$$331$$ 33.4164 1.83673 0.918366 0.395732i $$-0.129509\pi$$
0.918366 + 0.395732i $$0.129509\pi$$
$$332$$ 0 0
$$333$$ 1.47214 0.0806726
$$334$$ 0 0
$$335$$ −26.5066 −1.44821
$$336$$ 0 0
$$337$$ −12.9787 −0.706996 −0.353498 0.935435i $$-0.615008\pi$$
−0.353498 + 0.935435i $$0.615008\pi$$
$$338$$ 0 0
$$339$$ −3.14590 −0.170862
$$340$$ 0 0
$$341$$ 8.70820 0.471576
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −3.61803 −0.194788
$$346$$ 0 0
$$347$$ 34.3050 1.84159 0.920793 0.390051i $$-0.127543\pi$$
0.920793 + 0.390051i $$0.127543\pi$$
$$348$$ 0 0
$$349$$ 2.96556 0.158743 0.0793713 0.996845i $$-0.474709\pi$$
0.0793713 + 0.996845i $$0.474709\pi$$
$$350$$ 0 0
$$351$$ −5.09017 −0.271693
$$352$$ 0 0
$$353$$ −15.6525 −0.833097 −0.416549 0.909113i $$-0.636760\pi$$
−0.416549 + 0.909113i $$0.636760\pi$$
$$354$$ 0 0
$$355$$ −15.6525 −0.830747
$$356$$ 0 0
$$357$$ −5.00000 −0.264628
$$358$$ 0 0
$$359$$ 16.5623 0.874125 0.437063 0.899431i $$-0.356019\pi$$
0.437063 + 0.899431i $$0.356019\pi$$
$$360$$ 0 0
$$361$$ −6.94427 −0.365488
$$362$$ 0 0
$$363$$ 10.0000 0.524864
$$364$$ 0 0
$$365$$ 1.90983 0.0999651
$$366$$ 0 0
$$367$$ 27.2705 1.42351 0.711755 0.702428i $$-0.247901\pi$$
0.711755 + 0.702428i $$0.247901\pi$$
$$368$$ 0 0
$$369$$ −5.76393 −0.300058
$$370$$ 0 0
$$371$$ −0.381966 −0.0198307
$$372$$ 0 0
$$373$$ 32.8885 1.70290 0.851452 0.524432i $$-0.175722\pi$$
0.851452 + 0.524432i $$0.175722\pi$$
$$374$$ 0 0
$$375$$ 11.1803 0.577350
$$376$$ 0 0
$$377$$ −31.7426 −1.63483
$$378$$ 0 0
$$379$$ 17.8885 0.918873 0.459436 0.888211i $$-0.348051\pi$$
0.459436 + 0.888211i $$0.348051\pi$$
$$380$$ 0 0
$$381$$ 4.61803 0.236589
$$382$$ 0 0
$$383$$ 16.2361 0.829624 0.414812 0.909907i $$-0.363847\pi$$
0.414812 + 0.909907i $$0.363847\pi$$
$$384$$ 0 0
$$385$$ 3.61803 0.184392
$$386$$ 0 0
$$387$$ −6.32624 −0.321581
$$388$$ 0 0
$$389$$ 20.1246 1.02036 0.510179 0.860068i $$-0.329579\pi$$
0.510179 + 0.860068i $$0.329579\pi$$
$$390$$ 0 0
$$391$$ 5.00000 0.252861
$$392$$ 0 0
$$393$$ −9.18034 −0.463087
$$394$$ 0 0
$$395$$ −38.7426 −1.94935
$$396$$ 0 0
$$397$$ −2.18034 −0.109428 −0.0547141 0.998502i $$-0.517425\pi$$
−0.0547141 + 0.998502i $$0.517425\pi$$
$$398$$ 0 0
$$399$$ 3.47214 0.173824
$$400$$ 0 0
$$401$$ −0.708204 −0.0353660 −0.0176830 0.999844i $$-0.505629\pi$$
−0.0176830 + 0.999844i $$0.505629\pi$$
$$402$$ 0 0
$$403$$ 44.3262 2.20805
$$404$$ 0 0
$$405$$ −3.61803 −0.179782
$$406$$ 0 0
$$407$$ 1.47214 0.0729711
$$408$$ 0 0
$$409$$ 25.1803 1.24509 0.622544 0.782585i $$-0.286099\pi$$
0.622544 + 0.782585i $$0.286099\pi$$
$$410$$ 0 0
$$411$$ 4.52786 0.223343
$$412$$ 0 0
$$413$$ −3.61803 −0.178032
$$414$$ 0 0
$$415$$ 33.2148 1.63045
$$416$$ 0 0
$$417$$ 0.673762 0.0329943
$$418$$ 0 0
$$419$$ 3.14590 0.153687 0.0768436 0.997043i $$-0.475516\pi$$
0.0768436 + 0.997043i $$0.475516\pi$$
$$420$$ 0 0
$$421$$ 38.0344 1.85369 0.926843 0.375450i $$-0.122512\pi$$
0.926843 + 0.375450i $$0.122512\pi$$
$$422$$ 0 0
$$423$$ −3.70820 −0.180299
$$424$$ 0 0
$$425$$ −40.4508 −1.96215
$$426$$ 0 0
$$427$$ 7.56231 0.365966
$$428$$ 0 0
$$429$$ −5.09017 −0.245756
$$430$$ 0 0
$$431$$ −18.0344 −0.868688 −0.434344 0.900747i $$-0.643020\pi$$
−0.434344 + 0.900747i $$0.643020\pi$$
$$432$$ 0 0
$$433$$ −5.70820 −0.274319 −0.137159 0.990549i $$-0.543797\pi$$
−0.137159 + 0.990549i $$0.543797\pi$$
$$434$$ 0 0
$$435$$ −22.5623 −1.08178
$$436$$ 0 0
$$437$$ −3.47214 −0.166095
$$438$$ 0 0
$$439$$ 28.7082 1.37017 0.685084 0.728464i $$-0.259766\pi$$
0.685084 + 0.728464i $$0.259766\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 8.94427 0.424955 0.212478 0.977166i $$-0.431847\pi$$
0.212478 + 0.977166i $$0.431847\pi$$
$$444$$ 0 0
$$445$$ 60.1246 2.85018
$$446$$ 0 0
$$447$$ −19.1246 −0.904563
$$448$$ 0 0
$$449$$ −19.1459 −0.903551 −0.451775 0.892132i $$-0.649209\pi$$
−0.451775 + 0.892132i $$0.649209\pi$$
$$450$$ 0 0
$$451$$ −5.76393 −0.271413
$$452$$ 0 0
$$453$$ 5.23607 0.246012
$$454$$ 0 0
$$455$$ 18.4164 0.863375
$$456$$ 0 0
$$457$$ −6.27051 −0.293322 −0.146661 0.989187i $$-0.546853\pi$$
−0.146661 + 0.989187i $$0.546853\pi$$
$$458$$ 0 0
$$459$$ 5.00000 0.233380
$$460$$ 0 0
$$461$$ −3.72949 −0.173700 −0.0868498 0.996221i $$-0.527680\pi$$
−0.0868498 + 0.996221i $$0.527680\pi$$
$$462$$ 0 0
$$463$$ −22.5279 −1.04696 −0.523479 0.852038i $$-0.675366\pi$$
−0.523479 + 0.852038i $$0.675366\pi$$
$$464$$ 0 0
$$465$$ 31.5066 1.46108
$$466$$ 0 0
$$467$$ −34.5967 −1.60095 −0.800473 0.599368i $$-0.795418\pi$$
−0.800473 + 0.599368i $$0.795418\pi$$
$$468$$ 0 0
$$469$$ −7.32624 −0.338294
$$470$$ 0 0
$$471$$ −14.6525 −0.675150
$$472$$ 0 0
$$473$$ −6.32624 −0.290881
$$474$$ 0 0
$$475$$ 28.0902 1.28887
$$476$$ 0 0
$$477$$ 0.381966 0.0174890
$$478$$ 0 0
$$479$$ 24.8885 1.13719 0.568593 0.822619i $$-0.307488\pi$$
0.568593 + 0.822619i $$0.307488\pi$$
$$480$$ 0 0
$$481$$ 7.49342 0.341671
$$482$$ 0 0
$$483$$ −1.00000 −0.0455016
$$484$$ 0 0
$$485$$ 44.2705 2.01022
$$486$$ 0 0
$$487$$ −42.7771 −1.93841 −0.969207 0.246246i $$-0.920803\pi$$
−0.969207 + 0.246246i $$0.920803\pi$$
$$488$$ 0 0
$$489$$ −14.0902 −0.637180
$$490$$ 0 0
$$491$$ −5.09017 −0.229716 −0.114858 0.993382i $$-0.536641\pi$$
−0.114858 + 0.993382i $$0.536641\pi$$
$$492$$ 0 0
$$493$$ 31.1803 1.40429
$$494$$ 0 0
$$495$$ −3.61803 −0.162619
$$496$$ 0 0
$$497$$ −4.32624 −0.194058
$$498$$ 0 0
$$499$$ 21.0344 0.941631 0.470815 0.882232i $$-0.343960\pi$$
0.470815 + 0.882232i $$0.343960\pi$$
$$500$$ 0 0
$$501$$ −3.18034 −0.142087
$$502$$ 0 0
$$503$$ 19.3262 0.861714 0.430857 0.902420i $$-0.358211\pi$$
0.430857 + 0.902420i $$0.358211\pi$$
$$504$$ 0 0
$$505$$ 20.3262 0.904506
$$506$$ 0 0
$$507$$ −12.9098 −0.573346
$$508$$ 0 0
$$509$$ 6.76393 0.299806 0.149903 0.988701i $$-0.452104\pi$$
0.149903 + 0.988701i $$0.452104\pi$$
$$510$$ 0 0
$$511$$ 0.527864 0.0233513
$$512$$ 0 0
$$513$$ −3.47214 −0.153299
$$514$$ 0 0
$$515$$ 8.94427 0.394132
$$516$$ 0 0
$$517$$ −3.70820 −0.163087
$$518$$ 0 0
$$519$$ −9.18034 −0.402972
$$520$$ 0 0
$$521$$ 31.3050 1.37149 0.685747 0.727840i $$-0.259475\pi$$
0.685747 + 0.727840i $$0.259475\pi$$
$$522$$ 0 0
$$523$$ 8.47214 0.370461 0.185230 0.982695i $$-0.440697\pi$$
0.185230 + 0.982695i $$0.440697\pi$$
$$524$$ 0 0
$$525$$ 8.09017 0.353084
$$526$$ 0 0
$$527$$ −43.5410 −1.89668
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 3.61803 0.157009
$$532$$ 0 0
$$533$$ −29.3394 −1.27083
$$534$$ 0 0
$$535$$ −52.8885 −2.28657
$$536$$ 0 0
$$537$$ 10.2705 0.443205
$$538$$ 0 0
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ −5.23607 −0.225116 −0.112558 0.993645i $$-0.535904\pi$$
−0.112558 + 0.993645i $$0.535904\pi$$
$$542$$ 0 0
$$543$$ 0.236068 0.0101306
$$544$$ 0 0
$$545$$ 63.3394 2.71316
$$546$$ 0 0
$$547$$ 32.4508 1.38750 0.693749 0.720217i $$-0.255958\pi$$
0.693749 + 0.720217i $$0.255958\pi$$
$$548$$ 0 0
$$549$$ −7.56231 −0.322751
$$550$$ 0 0
$$551$$ −21.6525 −0.922426
$$552$$ 0 0
$$553$$ −10.7082 −0.455359
$$554$$ 0 0
$$555$$ 5.32624 0.226086
$$556$$ 0 0
$$557$$ −24.1803 −1.02455 −0.512277 0.858820i $$-0.671198\pi$$
−0.512277 + 0.858820i $$0.671198\pi$$
$$558$$ 0 0
$$559$$ −32.2016 −1.36198
$$560$$ 0 0
$$561$$ 5.00000 0.211100
$$562$$ 0 0
$$563$$ −37.5623 −1.58306 −0.791531 0.611129i $$-0.790716\pi$$
−0.791531 + 0.611129i $$0.790716\pi$$
$$564$$ 0 0
$$565$$ −11.3820 −0.478843
$$566$$ 0 0
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −41.8885 −1.75606 −0.878030 0.478606i $$-0.841142\pi$$
−0.878030 + 0.478606i $$0.841142\pi$$
$$570$$ 0 0
$$571$$ −21.1803 −0.886370 −0.443185 0.896430i $$-0.646151\pi$$
−0.443185 + 0.896430i $$0.646151\pi$$
$$572$$ 0 0
$$573$$ 16.6525 0.695667
$$574$$ 0 0
$$575$$ −8.09017 −0.337383
$$576$$ 0 0
$$577$$ −5.41641 −0.225488 −0.112744 0.993624i $$-0.535964\pi$$
−0.112744 + 0.993624i $$0.535964\pi$$
$$578$$ 0 0
$$579$$ −21.7082 −0.902162
$$580$$ 0 0
$$581$$ 9.18034 0.380865
$$582$$ 0 0
$$583$$ 0.381966 0.0158194
$$584$$ 0 0
$$585$$ −18.4164 −0.761425
$$586$$ 0 0
$$587$$ 42.6180 1.75903 0.879517 0.475867i $$-0.157866\pi$$
0.879517 + 0.475867i $$0.157866\pi$$
$$588$$ 0 0
$$589$$ 30.2361 1.24586
$$590$$ 0 0
$$591$$ −22.6180 −0.930382
$$592$$ 0 0
$$593$$ −35.0132 −1.43782 −0.718909 0.695104i $$-0.755358\pi$$
−0.718909 + 0.695104i $$0.755358\pi$$
$$594$$ 0 0
$$595$$ −18.0902 −0.741625
$$596$$ 0 0
$$597$$ 3.79837 0.155457
$$598$$ 0 0
$$599$$ 4.79837 0.196056 0.0980281 0.995184i $$-0.468746\pi$$
0.0980281 + 0.995184i $$0.468746\pi$$
$$600$$ 0 0
$$601$$ −29.9098 −1.22005 −0.610024 0.792383i $$-0.708840\pi$$
−0.610024 + 0.792383i $$0.708840\pi$$
$$602$$ 0 0
$$603$$ 7.32624 0.298347
$$604$$ 0 0
$$605$$ 36.1803 1.47094
$$606$$ 0 0
$$607$$ −45.3394 −1.84027 −0.920135 0.391602i $$-0.871921\pi$$
−0.920135 + 0.391602i $$0.871921\pi$$
$$608$$ 0 0
$$609$$ −6.23607 −0.252698
$$610$$ 0 0
$$611$$ −18.8754 −0.763616
$$612$$ 0 0
$$613$$ 37.0689 1.49720 0.748599 0.663023i $$-0.230727\pi$$
0.748599 + 0.663023i $$0.230727\pi$$
$$614$$ 0 0
$$615$$ −20.8541 −0.840919
$$616$$ 0 0
$$617$$ −30.0344 −1.20914 −0.604571 0.796552i $$-0.706655\pi$$
−0.604571 + 0.796552i $$0.706655\pi$$
$$618$$ 0 0
$$619$$ −19.9230 −0.800772 −0.400386 0.916346i $$-0.631124\pi$$
−0.400386 + 0.916346i $$0.631124\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 16.6180 0.665787
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −3.47214 −0.138664
$$628$$ 0 0
$$629$$ −7.36068 −0.293490
$$630$$ 0 0
$$631$$ −0.527864 −0.0210139 −0.0105070 0.999945i $$-0.503345\pi$$
−0.0105070 + 0.999945i $$0.503345\pi$$
$$632$$ 0 0
$$633$$ 18.2361 0.724819
$$634$$ 0 0
$$635$$ 16.7082 0.663045
$$636$$ 0 0
$$637$$ 5.09017 0.201680
$$638$$ 0 0
$$639$$ 4.32624 0.171143
$$640$$ 0 0
$$641$$ −16.6180 −0.656373 −0.328186 0.944613i $$-0.606437\pi$$
−0.328186 + 0.944613i $$0.606437\pi$$
$$642$$ 0 0
$$643$$ 10.9098 0.430242 0.215121 0.976587i $$-0.430985\pi$$
0.215121 + 0.976587i $$0.430985\pi$$
$$644$$ 0 0
$$645$$ −22.8885 −0.901236
$$646$$ 0 0
$$647$$ −15.9098 −0.625480 −0.312740 0.949839i $$-0.601247\pi$$
−0.312740 + 0.949839i $$0.601247\pi$$
$$648$$ 0 0
$$649$$ 3.61803 0.142020
$$650$$ 0 0
$$651$$ 8.70820 0.341301
$$652$$ 0 0
$$653$$ 34.5623 1.35253 0.676264 0.736660i $$-0.263598\pi$$
0.676264 + 0.736660i $$0.263598\pi$$
$$654$$ 0 0
$$655$$ −33.2148 −1.29781
$$656$$ 0 0
$$657$$ −0.527864 −0.0205939
$$658$$ 0 0
$$659$$ 1.00000 0.0389545 0.0194772 0.999810i $$-0.493800\pi$$
0.0194772 + 0.999810i $$0.493800\pi$$
$$660$$ 0 0
$$661$$ −38.7082 −1.50557 −0.752787 0.658264i $$-0.771291\pi$$
−0.752787 + 0.658264i $$0.771291\pi$$
$$662$$ 0 0
$$663$$ 25.4508 0.988429
$$664$$ 0 0
$$665$$ 12.5623 0.487145
$$666$$ 0 0
$$667$$ 6.23607 0.241462
$$668$$ 0 0
$$669$$ 10.5623 0.408362
$$670$$ 0 0
$$671$$ −7.56231 −0.291940
$$672$$ 0 0
$$673$$ 29.9443 1.15427 0.577133 0.816650i $$-0.304171\pi$$
0.577133 + 0.816650i $$0.304171\pi$$
$$674$$ 0 0
$$675$$ −8.09017 −0.311391
$$676$$ 0 0
$$677$$ −0.437694 −0.0168220 −0.00841098 0.999965i $$-0.502677\pi$$
−0.00841098 + 0.999965i $$0.502677\pi$$
$$678$$ 0 0
$$679$$ 12.2361 0.469577
$$680$$ 0 0
$$681$$ −16.8541 −0.645851
$$682$$ 0 0
$$683$$ 15.9443 0.610091 0.305045 0.952338i $$-0.401328\pi$$
0.305045 + 0.952338i $$0.401328\pi$$
$$684$$ 0 0
$$685$$ 16.3820 0.625923
$$686$$ 0 0
$$687$$ 7.79837 0.297527
$$688$$ 0 0
$$689$$ 1.94427 0.0740709
$$690$$ 0 0
$$691$$ −14.2148 −0.540756 −0.270378 0.962754i $$-0.587149\pi$$
−0.270378 + 0.962754i $$0.587149\pi$$
$$692$$ 0 0
$$693$$ −1.00000 −0.0379869
$$694$$ 0 0
$$695$$ 2.43769 0.0924670
$$696$$ 0 0
$$697$$ 28.8197 1.09162
$$698$$ 0 0
$$699$$ −18.5623 −0.702091
$$700$$ 0 0
$$701$$ −25.9098 −0.978601 −0.489300 0.872115i $$-0.662748\pi$$
−0.489300 + 0.872115i $$0.662748\pi$$
$$702$$ 0 0
$$703$$ 5.11146 0.192782
$$704$$ 0 0
$$705$$ −13.4164 −0.505291
$$706$$ 0 0
$$707$$ 5.61803 0.211288
$$708$$ 0 0
$$709$$ 27.1591 1.01998 0.509990 0.860180i $$-0.329649\pi$$
0.509990 + 0.860180i $$0.329649\pi$$
$$710$$ 0 0
$$711$$ 10.7082 0.401589
$$712$$ 0 0
$$713$$ −8.70820 −0.326125
$$714$$ 0 0
$$715$$ −18.4164 −0.688735
$$716$$ 0 0
$$717$$ 12.6738 0.473310
$$718$$ 0 0
$$719$$ 20.7771 0.774855 0.387427 0.921900i $$-0.373364\pi$$
0.387427 + 0.921900i $$0.373364\pi$$
$$720$$ 0 0
$$721$$ 2.47214 0.0920672
$$722$$ 0 0
$$723$$ 25.6525 0.954026
$$724$$ 0 0
$$725$$ −50.4508 −1.87370
$$726$$ 0 0
$$727$$ 37.2492 1.38150 0.690749 0.723095i $$-0.257281\pi$$
0.690749 + 0.723095i $$0.257281\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 31.6312 1.16992
$$732$$ 0 0
$$733$$ 2.29180 0.0846494 0.0423247 0.999104i $$-0.486524\pi$$
0.0423247 + 0.999104i $$0.486524\pi$$
$$734$$ 0 0
$$735$$ 3.61803 0.133453
$$736$$ 0 0
$$737$$ 7.32624 0.269865
$$738$$ 0 0
$$739$$ 20.2361 0.744396 0.372198 0.928153i $$-0.378604\pi$$
0.372198 + 0.928153i $$0.378604\pi$$
$$740$$ 0 0
$$741$$ −17.6738 −0.649262
$$742$$ 0 0
$$743$$ −6.85410 −0.251453 −0.125726 0.992065i $$-0.540126\pi$$
−0.125726 + 0.992065i $$0.540126\pi$$
$$744$$ 0 0
$$745$$ −69.1935 −2.53505
$$746$$ 0 0
$$747$$ −9.18034 −0.335891
$$748$$ 0 0
$$749$$ −14.6180 −0.534131
$$750$$ 0 0
$$751$$ 22.6738 0.827377 0.413689 0.910418i $$-0.364240\pi$$
0.413689 + 0.910418i $$0.364240\pi$$
$$752$$ 0 0
$$753$$ 3.81966 0.139196
$$754$$ 0 0
$$755$$ 18.9443 0.689453
$$756$$ 0 0
$$757$$ 16.4164 0.596664 0.298332 0.954462i $$-0.403570\pi$$
0.298332 + 0.954462i $$0.403570\pi$$
$$758$$ 0 0
$$759$$ 1.00000 0.0362977
$$760$$ 0 0
$$761$$ −7.05573 −0.255770 −0.127885 0.991789i $$-0.540819\pi$$
−0.127885 + 0.991789i $$0.540819\pi$$
$$762$$ 0 0
$$763$$ 17.5066 0.633781
$$764$$ 0 0
$$765$$ 18.0902 0.654051
$$766$$ 0 0
$$767$$ 18.4164 0.664978
$$768$$ 0 0
$$769$$ −13.3475 −0.481324 −0.240662 0.970609i $$-0.577365\pi$$
−0.240662 + 0.970609i $$0.577365\pi$$
$$770$$ 0 0
$$771$$ 0.291796 0.0105088
$$772$$ 0 0
$$773$$ −1.29180 −0.0464627 −0.0232313 0.999730i $$-0.507395\pi$$
−0.0232313 + 0.999730i $$0.507395\pi$$
$$774$$ 0 0
$$775$$ 70.4508 2.53067
$$776$$ 0 0
$$777$$ 1.47214 0.0528126
$$778$$ 0 0
$$779$$ −20.0132 −0.717046
$$780$$ 0 0
$$781$$ 4.32624 0.154805
$$782$$ 0 0
$$783$$ 6.23607 0.222859
$$784$$ 0 0
$$785$$ −53.0132 −1.89212
$$786$$ 0 0
$$787$$ −24.7426 −0.881980 −0.440990 0.897512i $$-0.645373\pi$$
−0.440990 + 0.897512i $$0.645373\pi$$
$$788$$ 0 0
$$789$$ −27.1803 −0.967646
$$790$$ 0 0
$$791$$ −3.14590 −0.111855
$$792$$ 0 0
$$793$$ −38.4934 −1.36694
$$794$$ 0 0
$$795$$ 1.38197 0.0490133
$$796$$ 0 0
$$797$$ 40.0689 1.41931 0.709656 0.704548i $$-0.248850\pi$$
0.709656 + 0.704548i $$0.248850\pi$$
$$798$$ 0 0
$$799$$ 18.5410 0.655934
$$800$$ 0 0
$$801$$ −16.6180 −0.587169
$$802$$ 0 0
$$803$$ −0.527864 −0.0186279
$$804$$ 0 0
$$805$$ −3.61803 −0.127519
$$806$$ 0 0
$$807$$ −19.8541 −0.698897
$$808$$ 0 0
$$809$$ 47.3820 1.66586 0.832931 0.553377i $$-0.186661\pi$$
0.832931 + 0.553377i $$0.186661\pi$$
$$810$$ 0 0
$$811$$ 47.3607 1.66306 0.831529 0.555481i $$-0.187466\pi$$
0.831529 + 0.555481i $$0.187466\pi$$
$$812$$ 0 0
$$813$$ −6.52786 −0.228942
$$814$$ 0 0
$$815$$ −50.9787 −1.78571
$$816$$ 0 0
$$817$$ −21.9656 −0.768478
$$818$$ 0 0
$$819$$ −5.09017 −0.177865
$$820$$ 0 0
$$821$$ 7.34752 0.256430 0.128215 0.991746i $$-0.459075\pi$$
0.128215 + 0.991746i $$0.459075\pi$$
$$822$$ 0 0
$$823$$ 13.7984 0.480981 0.240491 0.970651i $$-0.422692\pi$$
0.240491 + 0.970651i $$0.422692\pi$$
$$824$$ 0 0
$$825$$ −8.09017 −0.281664
$$826$$ 0 0
$$827$$ −53.7984 −1.87075 −0.935376 0.353654i $$-0.884939\pi$$
−0.935376 + 0.353654i $$0.884939\pi$$
$$828$$ 0 0
$$829$$ −4.23607 −0.147125 −0.0735624 0.997291i $$-0.523437\pi$$
−0.0735624 + 0.997291i $$0.523437\pi$$
$$830$$ 0 0
$$831$$ −31.0902 −1.07851
$$832$$ 0 0
$$833$$ −5.00000 −0.173240
$$834$$ 0 0
$$835$$ −11.5066 −0.398202
$$836$$ 0 0
$$837$$ −8.70820 −0.301000
$$838$$ 0 0
$$839$$ 7.27051 0.251006 0.125503 0.992093i $$-0.459946\pi$$
0.125503 + 0.992093i $$0.459946\pi$$
$$840$$ 0 0
$$841$$ 9.88854 0.340984
$$842$$ 0 0
$$843$$ −8.18034 −0.281746
$$844$$ 0 0
$$845$$ −46.7082 −1.60681
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 0 0
$$849$$ 2.96556 0.101778
$$850$$ 0 0
$$851$$ −1.47214 −0.0504642
$$852$$ 0 0
$$853$$ −44.5410 −1.52506 −0.762528 0.646956i $$-0.776042\pi$$
−0.762528 + 0.646956i $$0.776042\pi$$
$$854$$ 0 0
$$855$$ −12.5623 −0.429622
$$856$$ 0 0
$$857$$ 22.6525 0.773794 0.386897 0.922123i $$-0.373547\pi$$
0.386897 + 0.922123i $$0.373547\pi$$
$$858$$ 0 0
$$859$$ −23.8885 −0.815067 −0.407533 0.913190i $$-0.633611\pi$$
−0.407533 + 0.913190i $$0.633611\pi$$
$$860$$ 0 0
$$861$$ −5.76393 −0.196434
$$862$$ 0 0
$$863$$ 53.4853 1.82066 0.910330 0.413883i $$-0.135828\pi$$
0.910330 + 0.413883i $$0.135828\pi$$
$$864$$ 0 0
$$865$$ −33.2148 −1.12934
$$866$$ 0 0
$$867$$ −8.00000 −0.271694
$$868$$ 0 0
$$869$$ 10.7082 0.363251
$$870$$ 0 0
$$871$$ 37.2918 1.26358
$$872$$ 0 0
$$873$$ −12.2361 −0.414128
$$874$$ 0 0
$$875$$ 11.1803 0.377964
$$876$$ 0 0
$$877$$ −57.3050 −1.93505 −0.967525 0.252774i $$-0.918657\pi$$
−0.967525 + 0.252774i $$0.918657\pi$$
$$878$$ 0 0
$$879$$ −2.47214 −0.0833831
$$880$$ 0 0
$$881$$ −10.8754 −0.366401 −0.183201 0.983076i $$-0.558646\pi$$
−0.183201 + 0.983076i $$0.558646\pi$$
$$882$$ 0 0
$$883$$ −26.3262 −0.885948 −0.442974 0.896534i $$-0.646077\pi$$
−0.442974 + 0.896534i $$0.646077\pi$$
$$884$$ 0 0
$$885$$ 13.0902 0.440021
$$886$$ 0 0
$$887$$ 14.7295 0.494568 0.247284 0.968943i $$-0.420462\pi$$
0.247284 + 0.968943i $$0.420462\pi$$
$$888$$ 0 0
$$889$$ 4.61803 0.154884
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 0 0
$$893$$ −12.8754 −0.430858
$$894$$ 0 0
$$895$$ 37.1591 1.24209
$$896$$ 0 0
$$897$$ 5.09017 0.169956
$$898$$ 0 0
$$899$$ −54.3050 −1.81117
$$900$$ 0 0
$$901$$ −1.90983 −0.0636257
$$902$$ 0 0
$$903$$ −6.32624 −0.210524
$$904$$ 0 0
$$905$$ 0.854102 0.0283913
$$906$$ 0 0
$$907$$ 48.9230 1.62446 0.812231 0.583337i $$-0.198253\pi$$
0.812231 + 0.583337i $$0.198253\pi$$
$$908$$ 0 0
$$909$$ −5.61803 −0.186338
$$910$$ 0 0
$$911$$ −42.5410 −1.40945 −0.704723 0.709482i $$-0.748929\pi$$
−0.704723 + 0.709482i $$0.748929\pi$$
$$912$$ 0 0
$$913$$ −9.18034 −0.303825
$$914$$ 0 0
$$915$$ −27.3607 −0.904516
$$916$$ 0 0
$$917$$ −9.18034 −0.303162
$$918$$ 0 0
$$919$$ −19.0000 −0.626752 −0.313376 0.949629i $$-0.601460\pi$$
−0.313376 + 0.949629i $$0.601460\pi$$
$$920$$ 0 0
$$921$$ 29.0689 0.957852
$$922$$ 0 0
$$923$$ 22.0213 0.724839
$$924$$ 0 0
$$925$$ 11.9098 0.391593
$$926$$ 0 0
$$927$$ −2.47214 −0.0811956
$$928$$ 0 0
$$929$$ 39.0344 1.28068 0.640339 0.768092i $$-0.278794\pi$$
0.640339 + 0.768092i $$0.278794\pi$$
$$930$$ 0 0
$$931$$ 3.47214 0.113795
$$932$$ 0 0
$$933$$ −15.0902 −0.494030
$$934$$ 0 0
$$935$$ 18.0902 0.591612
$$936$$ 0 0
$$937$$ −13.0000 −0.424691 −0.212346 0.977195i $$-0.568110\pi$$
−0.212346 + 0.977195i $$0.568110\pi$$
$$938$$ 0 0
$$939$$ 0.583592 0.0190448
$$940$$ 0 0
$$941$$ −10.8328 −0.353140 −0.176570 0.984288i $$-0.556500\pi$$
−0.176570 + 0.984288i $$0.556500\pi$$
$$942$$ 0 0
$$943$$ 5.76393 0.187699
$$944$$ 0 0
$$945$$ −3.61803 −0.117695
$$946$$ 0 0
$$947$$ −2.36068 −0.0767118 −0.0383559 0.999264i $$-0.512212\pi$$
−0.0383559 + 0.999264i $$0.512212\pi$$
$$948$$ 0 0
$$949$$ −2.68692 −0.0872210
$$950$$ 0 0
$$951$$ −26.3820 −0.855494
$$952$$ 0 0
$$953$$ 23.6312 0.765489 0.382745 0.923854i $$-0.374979\pi$$
0.382745 + 0.923854i $$0.374979\pi$$
$$954$$ 0 0
$$955$$ 60.2492 1.94962
$$956$$ 0 0
$$957$$ 6.23607 0.201583
$$958$$ 0 0
$$959$$ 4.52786 0.146212
$$960$$ 0 0
$$961$$ 44.8328 1.44622
$$962$$ 0 0
$$963$$ 14.6180 0.471060
$$964$$ 0 0
$$965$$ −78.5410 −2.52832
$$966$$ 0 0
$$967$$ 46.9443 1.50963 0.754813 0.655940i $$-0.227728\pi$$
0.754813 + 0.655940i $$0.227728\pi$$
$$968$$ 0 0
$$969$$ 17.3607 0.557705
$$970$$ 0 0
$$971$$ −15.7426 −0.505206 −0.252603 0.967570i $$-0.581287\pi$$
−0.252603 + 0.967570i $$0.581287\pi$$
$$972$$ 0 0
$$973$$ 0.673762 0.0215998
$$974$$ 0 0
$$975$$ −41.1803 −1.31883
$$976$$ 0 0
$$977$$ 59.7426 1.91134 0.955668 0.294445i $$-0.0951349\pi$$
0.955668 + 0.294445i $$0.0951349\pi$$
$$978$$ 0 0
$$979$$ −16.6180 −0.531115
$$980$$ 0 0
$$981$$ −17.5066 −0.558942
$$982$$ 0 0
$$983$$ −8.70820 −0.277749 −0.138874 0.990310i $$-0.544348\pi$$
−0.138874 + 0.990310i $$0.544348\pi$$
$$984$$ 0 0
$$985$$ −81.8328 −2.60741
$$986$$ 0 0
$$987$$ −3.70820 −0.118033
$$988$$ 0 0
$$989$$ 6.32624 0.201163
$$990$$ 0 0
$$991$$ 10.7295 0.340833 0.170417 0.985372i $$-0.445489\pi$$
0.170417 + 0.985372i $$0.445489\pi$$
$$992$$ 0 0
$$993$$ −33.4164 −1.06044
$$994$$ 0 0
$$995$$ 13.7426 0.435671
$$996$$ 0 0
$$997$$ −17.7639 −0.562589 −0.281295 0.959621i $$-0.590764\pi$$
−0.281295 + 0.959621i $$0.590764\pi$$
$$998$$ 0 0
$$999$$ −1.47214 −0.0465763
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 7728.2.a.w.1.1 2
4.3 odd 2 1932.2.a.f.1.1 2
12.11 even 2 5796.2.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.f.1.1 2 4.3 odd 2
5796.2.a.n.1.2 2 12.11 even 2
7728.2.a.w.1.1 2 1.1 even 1 trivial