# Properties

 Label 1728.3.b.i.1567.3 Level $1728$ Weight $3$ Character 1728.1567 Analytic conductor $47.085$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.116304318664704.2 Defining polynomial: $$x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64$$ x^12 - 2*x^11 + x^10 + 6*x^9 - 9*x^8 - 2*x^7 + 18*x^6 - 4*x^5 - 36*x^4 + 48*x^3 + 16*x^2 - 64*x + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{22}\cdot 3^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1567.3 Root $$1.33544 - 0.465413i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1567 Dual form 1728.3.b.i.1567.10

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.99718i q^{5} -7.74742i q^{7} +O(q^{10})$$ $$q-3.99718i q^{5} -7.74742i q^{7} +7.68970 q^{11} +1.42738i q^{13} -22.3189 q^{17} -21.9464 q^{19} -18.1656i q^{23} +9.02254 q^{25} -15.2838i q^{29} +10.2197i q^{31} -30.9678 q^{35} -59.1765i q^{37} -41.3743 q^{41} -11.0016 q^{43} +63.6933i q^{47} -11.0225 q^{49} -19.2810i q^{53} -30.7371i q^{55} +16.1799 q^{59} +27.7128i q^{61} +5.70551 q^{65} +60.6039 q^{67} +134.803i q^{71} +90.0019 q^{73} -59.5753i q^{77} +12.0122i q^{79} -164.757 q^{83} +89.2129i q^{85} -35.6688 q^{89} +11.0585 q^{91} +87.7236i q^{95} -147.070 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q - 48 q^{17} - 72 q^{25} - 336 q^{41} + 48 q^{49} - 912 q^{65} + 60 q^{73} - 1248 q^{89} - 204 q^{97}+O(q^{100})$$ 12 * q - 48 * q^17 - 72 * q^25 - 336 * q^41 + 48 * q^49 - 912 * q^65 + 60 * q^73 - 1248 * q^89 - 204 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ − 3.99718i − 0.799436i −0.916638 0.399718i $$-0.869108\pi$$
0.916638 0.399718i $$-0.130892\pi$$
$$6$$ 0 0
$$7$$ − 7.74742i − 1.10677i −0.832924 0.553387i $$-0.813335\pi$$
0.832924 0.553387i $$-0.186665\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 7.68970 0.699063 0.349532 0.936925i $$-0.386341\pi$$
0.349532 + 0.936925i $$0.386341\pi$$
$$12$$ 0 0
$$13$$ 1.42738i 0.109799i 0.998492 + 0.0548994i $$0.0174838\pi$$
−0.998492 + 0.0548994i $$0.982516\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −22.3189 −1.31288 −0.656440 0.754379i $$-0.727938\pi$$
−0.656440 + 0.754379i $$0.727938\pi$$
$$18$$ 0 0
$$19$$ −21.9464 −1.15507 −0.577536 0.816365i $$-0.695986\pi$$
−0.577536 + 0.816365i $$0.695986\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 18.1656i − 0.789808i −0.918722 0.394904i $$-0.870778\pi$$
0.918722 0.394904i $$-0.129222\pi$$
$$24$$ 0 0
$$25$$ 9.02254 0.360902
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 15.2838i − 0.527027i −0.964656 0.263514i $$-0.915119\pi$$
0.964656 0.263514i $$-0.0848814\pi$$
$$30$$ 0 0
$$31$$ 10.2197i 0.329668i 0.986321 + 0.164834i $$0.0527089\pi$$
−0.986321 + 0.164834i $$0.947291\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −30.9678 −0.884796
$$36$$ 0 0
$$37$$ − 59.1765i − 1.59937i −0.600423 0.799683i $$-0.705001\pi$$
0.600423 0.799683i $$-0.294999\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −41.3743 −1.00913 −0.504565 0.863374i $$-0.668347\pi$$
−0.504565 + 0.863374i $$0.668347\pi$$
$$42$$ 0 0
$$43$$ −11.0016 −0.255852 −0.127926 0.991784i $$-0.540832\pi$$
−0.127926 + 0.991784i $$0.540832\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 63.6933i 1.35518i 0.735442 + 0.677588i $$0.236975\pi$$
−0.735442 + 0.677588i $$0.763025\pi$$
$$48$$ 0 0
$$49$$ −11.0225 −0.224950
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 19.2810i − 0.363792i −0.983318 0.181896i $$-0.941777\pi$$
0.983318 0.181896i $$-0.0582234\pi$$
$$54$$ 0 0
$$55$$ − 30.7371i − 0.558857i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 16.1799 0.274236 0.137118 0.990555i $$-0.456216\pi$$
0.137118 + 0.990555i $$0.456216\pi$$
$$60$$ 0 0
$$61$$ 27.7128i 0.454308i 0.973859 + 0.227154i $$0.0729421\pi$$
−0.973859 + 0.227154i $$0.927058\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 5.70551 0.0877771
$$66$$ 0 0
$$67$$ 60.6039 0.904536 0.452268 0.891882i $$-0.350615\pi$$
0.452268 + 0.891882i $$0.350615\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 134.803i 1.89864i 0.314311 + 0.949320i $$0.398227\pi$$
−0.314311 + 0.949320i $$0.601773\pi$$
$$72$$ 0 0
$$73$$ 90.0019 1.23290 0.616451 0.787393i $$-0.288570\pi$$
0.616451 + 0.787393i $$0.288570\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 59.5753i − 0.773705i
$$78$$ 0 0
$$79$$ 12.0122i 0.152054i 0.997106 + 0.0760268i $$0.0242234\pi$$
−0.997106 + 0.0760268i $$0.975777\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −164.757 −1.98503 −0.992513 0.122137i $$-0.961025\pi$$
−0.992513 + 0.122137i $$0.961025\pi$$
$$84$$ 0 0
$$85$$ 89.2129i 1.04956i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −35.6688 −0.400773 −0.200387 0.979717i $$-0.564220\pi$$
−0.200387 + 0.979717i $$0.564220\pi$$
$$90$$ 0 0
$$91$$ 11.0585 0.121522
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 87.7236i 0.923407i
$$96$$ 0 0
$$97$$ −147.070 −1.51618 −0.758090 0.652149i $$-0.773867\pi$$
−0.758090 + 0.652149i $$0.773867\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 74.2140i 0.734793i 0.930065 + 0.367396i $$0.119751\pi$$
−0.930065 + 0.367396i $$0.880249\pi$$
$$102$$ 0 0
$$103$$ 55.0676i 0.534637i 0.963608 + 0.267319i $$0.0861376\pi$$
−0.963608 + 0.267319i $$0.913862\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −32.6046 −0.304716 −0.152358 0.988325i $$-0.548687\pi$$
−0.152358 + 0.988325i $$0.548687\pi$$
$$108$$ 0 0
$$109$$ − 29.6714i − 0.272215i −0.990694 0.136108i $$-0.956541\pi$$
0.990694 0.136108i $$-0.0434593\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −213.399 −1.88848 −0.944242 0.329251i $$-0.893204\pi$$
−0.944242 + 0.329251i $$0.893204\pi$$
$$114$$ 0 0
$$115$$ −72.6111 −0.631401
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 172.914i 1.45306i
$$120$$ 0 0
$$121$$ −61.8686 −0.511311
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 135.994i − 1.08795i
$$126$$ 0 0
$$127$$ 64.2526i 0.505926i 0.967476 + 0.252963i $$0.0814051\pi$$
−0.967476 + 0.252963i $$0.918595\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −92.1030 −0.703076 −0.351538 0.936174i $$-0.614341\pi$$
−0.351538 + 0.936174i $$0.614341\pi$$
$$132$$ 0 0
$$133$$ 170.028i 1.27840i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −169.582 −1.23783 −0.618914 0.785459i $$-0.712427\pi$$
−0.618914 + 0.785459i $$0.712427\pi$$
$$138$$ 0 0
$$139$$ 71.6056 0.515148 0.257574 0.966259i $$-0.417077\pi$$
0.257574 + 0.966259i $$0.417077\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 10.9761i 0.0767563i
$$144$$ 0 0
$$145$$ −61.0921 −0.421325
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 198.277i − 1.33072i −0.746525 0.665358i $$-0.768279\pi$$
0.746525 0.665358i $$-0.231721\pi$$
$$150$$ 0 0
$$151$$ − 72.3305i − 0.479010i −0.970895 0.239505i $$-0.923015\pi$$
0.970895 0.239505i $$-0.0769852\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 40.8501 0.263549
$$156$$ 0 0
$$157$$ − 124.063i − 0.790208i −0.918636 0.395104i $$-0.870709\pi$$
0.918636 0.395104i $$-0.129291\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −140.736 −0.874139
$$162$$ 0 0
$$163$$ 232.533 1.42659 0.713293 0.700866i $$-0.247203\pi$$
0.713293 + 0.700866i $$0.247203\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 43.9755i 0.263327i 0.991294 + 0.131663i $$0.0420318\pi$$
−0.991294 + 0.131663i $$0.957968\pi$$
$$168$$ 0 0
$$169$$ 166.963 0.987944
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 305.908i − 1.76826i −0.467245 0.884128i $$-0.654753\pi$$
0.467245 0.884128i $$-0.345247\pi$$
$$174$$ 0 0
$$175$$ − 69.9014i − 0.399437i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2.19254 0.0122488 0.00612442 0.999981i $$-0.498051\pi$$
0.00612442 + 0.999981i $$0.498051\pi$$
$$180$$ 0 0
$$181$$ − 132.855i − 0.734003i −0.930220 0.367001i $$-0.880384\pi$$
0.930220 0.367001i $$-0.119616\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −236.539 −1.27859
$$186$$ 0 0
$$187$$ −171.626 −0.917785
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 151.417i 0.792759i 0.918087 + 0.396379i $$0.129733\pi$$
−0.918087 + 0.396379i $$0.870267\pi$$
$$192$$ 0 0
$$193$$ 229.850 1.19093 0.595466 0.803381i $$-0.296967\pi$$
0.595466 + 0.803381i $$0.296967\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 189.195i 0.960380i 0.877164 + 0.480190i $$0.159432\pi$$
−0.877164 + 0.480190i $$0.840568\pi$$
$$198$$ 0 0
$$199$$ 217.686i 1.09390i 0.837166 + 0.546949i $$0.184211\pi$$
−0.837166 + 0.546949i $$0.815789\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −118.410 −0.583300
$$204$$ 0 0
$$205$$ 165.381i 0.806735i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −168.761 −0.807468
$$210$$ 0 0
$$211$$ −247.120 −1.17118 −0.585591 0.810606i $$-0.699138\pi$$
−0.585591 + 0.810606i $$0.699138\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 43.9755i 0.204537i
$$216$$ 0 0
$$217$$ 79.1765 0.364869
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 31.8577i − 0.144153i
$$222$$ 0 0
$$223$$ − 361.006i − 1.61886i −0.587216 0.809430i $$-0.699776\pi$$
0.587216 0.809430i $$-0.300224\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −21.0546 −0.0927515 −0.0463758 0.998924i $$-0.514767\pi$$
−0.0463758 + 0.998924i $$0.514767\pi$$
$$228$$ 0 0
$$229$$ − 319.313i − 1.39438i −0.716886 0.697191i $$-0.754433\pi$$
0.716886 0.697191i $$-0.245567\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −197.607 −0.848098 −0.424049 0.905639i $$-0.639392\pi$$
−0.424049 + 0.905639i $$0.639392\pi$$
$$234$$ 0 0
$$235$$ 254.594 1.08338
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 256.325i 1.07249i 0.844062 + 0.536245i $$0.180158\pi$$
−0.844062 + 0.536245i $$0.819842\pi$$
$$240$$ 0 0
$$241$$ −375.717 −1.55899 −0.779495 0.626409i $$-0.784524\pi$$
−0.779495 + 0.626409i $$0.784524\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 44.0591i 0.179833i
$$246$$ 0 0
$$247$$ − 31.3259i − 0.126825i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −423.810 −1.68849 −0.844243 0.535961i $$-0.819950\pi$$
−0.844243 + 0.535961i $$0.819950\pi$$
$$252$$ 0 0
$$253$$ − 139.688i − 0.552126i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 205.754 0.800601 0.400300 0.916384i $$-0.368906\pi$$
0.400300 + 0.916384i $$0.368906\pi$$
$$258$$ 0 0
$$259$$ −458.466 −1.77014
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 75.9941i − 0.288951i −0.989508 0.144476i $$-0.953851\pi$$
0.989508 0.144476i $$-0.0461495\pi$$
$$264$$ 0 0
$$265$$ −77.0695 −0.290828
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 327.627i − 1.21794i −0.793192 0.608971i $$-0.791582\pi$$
0.793192 0.608971i $$-0.208418\pi$$
$$270$$ 0 0
$$271$$ 277.780i 1.02502i 0.858681 + 0.512510i $$0.171284\pi$$
−0.858681 + 0.512510i $$0.828716\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 69.3806 0.252293
$$276$$ 0 0
$$277$$ 92.9929i 0.335714i 0.985811 + 0.167857i $$0.0536847\pi$$
−0.985811 + 0.167857i $$0.946315\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 190.784 0.678947 0.339474 0.940616i $$-0.389751\pi$$
0.339474 + 0.940616i $$0.389751\pi$$
$$282$$ 0 0
$$283$$ −153.511 −0.542441 −0.271220 0.962517i $$-0.587427\pi$$
−0.271220 + 0.962517i $$0.587427\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 320.544i 1.11688i
$$288$$ 0 0
$$289$$ 209.135 0.723651
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 550.059i 1.87733i 0.344825 + 0.938667i $$0.387938\pi$$
−0.344825 + 0.938667i $$0.612062\pi$$
$$294$$ 0 0
$$295$$ − 64.6741i − 0.219234i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 25.9293 0.0867200
$$300$$ 0 0
$$301$$ 85.2343i 0.283171i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 110.773 0.363191
$$306$$ 0 0
$$307$$ 403.571 1.31456 0.657282 0.753645i $$-0.271706\pi$$
0.657282 + 0.753645i $$0.271706\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 328.988i 1.05784i 0.848672 + 0.528919i $$0.177402\pi$$
−0.848672 + 0.528919i $$0.822598\pi$$
$$312$$ 0 0
$$313$$ −168.133 −0.537167 −0.268584 0.963256i $$-0.586556\pi$$
−0.268584 + 0.963256i $$0.586556\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 171.437i − 0.540809i −0.962747 0.270405i $$-0.912843\pi$$
0.962747 0.270405i $$-0.0871575\pi$$
$$318$$ 0 0
$$319$$ − 117.528i − 0.368425i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 489.820 1.51647
$$324$$ 0 0
$$325$$ 12.8786i 0.0396266i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 493.459 1.49987
$$330$$ 0 0
$$331$$ −520.075 −1.57122 −0.785612 0.618720i $$-0.787652\pi$$
−0.785612 + 0.618720i $$0.787652\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 242.245i − 0.723119i
$$336$$ 0 0
$$337$$ 411.987 1.22251 0.611257 0.791432i $$-0.290664\pi$$
0.611257 + 0.791432i $$0.290664\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 78.5866i 0.230459i
$$342$$ 0 0
$$343$$ − 294.227i − 0.857806i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −309.229 −0.891151 −0.445575 0.895244i $$-0.647001\pi$$
−0.445575 + 0.895244i $$0.647001\pi$$
$$348$$ 0 0
$$349$$ 609.516i 1.74646i 0.487306 + 0.873231i $$0.337980\pi$$
−0.487306 + 0.873231i $$0.662020\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −376.662 −1.06703 −0.533515 0.845791i $$-0.679129\pi$$
−0.533515 + 0.845791i $$0.679129\pi$$
$$354$$ 0 0
$$355$$ 538.834 1.51784
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 186.196i − 0.518652i −0.965790 0.259326i $$-0.916500\pi$$
0.965790 0.259326i $$-0.0835003\pi$$
$$360$$ 0 0
$$361$$ 120.643 0.334192
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 359.754i − 0.985627i
$$366$$ 0 0
$$367$$ 89.1514i 0.242919i 0.992596 + 0.121460i $$0.0387575\pi$$
−0.992596 + 0.121460i $$0.961243\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −149.378 −0.402636
$$372$$ 0 0
$$373$$ − 103.487i − 0.277444i −0.990331 0.138722i $$-0.955701\pi$$
0.990331 0.138722i $$-0.0442995\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 21.8158 0.0578669
$$378$$ 0 0
$$379$$ 636.105 1.67838 0.839188 0.543841i $$-0.183031\pi$$
0.839188 + 0.543841i $$0.183031\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 617.514i − 1.61231i −0.591707 0.806153i $$-0.701546\pi$$
0.591707 0.806153i $$-0.298454\pi$$
$$384$$ 0 0
$$385$$ −238.133 −0.618528
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 674.534i 1.73402i 0.498290 + 0.867010i $$0.333961\pi$$
−0.498290 + 0.867010i $$0.666039\pi$$
$$390$$ 0 0
$$391$$ 405.437i 1.03692i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 48.0151 0.121557
$$396$$ 0 0
$$397$$ − 758.512i − 1.91061i −0.295622 0.955305i $$-0.595527\pi$$
0.295622 0.955305i $$-0.404473\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 314.381 0.783993 0.391997 0.919967i $$-0.371784\pi$$
0.391997 + 0.919967i $$0.371784\pi$$
$$402$$ 0 0
$$403$$ −14.5875 −0.0361972
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 455.050i − 1.11806i
$$408$$ 0 0
$$409$$ −591.015 −1.44503 −0.722513 0.691358i $$-0.757013\pi$$
−0.722513 + 0.691358i $$0.757013\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 125.353i − 0.303517i
$$414$$ 0 0
$$415$$ 658.564i 1.58690i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 371.686 0.887078 0.443539 0.896255i $$-0.353723\pi$$
0.443539 + 0.896255i $$0.353723\pi$$
$$420$$ 0 0
$$421$$ 660.963i 1.56998i 0.619507 + 0.784991i $$0.287333\pi$$
−0.619507 + 0.784991i $$0.712667\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −201.374 −0.473820
$$426$$ 0 0
$$427$$ 214.703 0.502817
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 376.295i − 0.873074i −0.899686 0.436537i $$-0.856205\pi$$
0.899686 0.436537i $$-0.143795\pi$$
$$432$$ 0 0
$$433$$ 301.156 0.695510 0.347755 0.937585i $$-0.386944\pi$$
0.347755 + 0.937585i $$0.386944\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 398.669i 0.912285i
$$438$$ 0 0
$$439$$ − 463.627i − 1.05610i −0.849214 0.528049i $$-0.822924\pi$$
0.849214 0.528049i $$-0.177076\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 655.755 1.48026 0.740130 0.672464i $$-0.234764\pi$$
0.740130 + 0.672464i $$0.234764\pi$$
$$444$$ 0 0
$$445$$ 142.575i 0.320393i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 167.436 0.372909 0.186455 0.982464i $$-0.440300\pi$$
0.186455 + 0.982464i $$0.440300\pi$$
$$450$$ 0 0
$$451$$ −318.156 −0.705446
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 44.2030i − 0.0971495i
$$456$$ 0 0
$$457$$ −527.115 −1.15342 −0.576712 0.816948i $$-0.695664\pi$$
−0.576712 + 0.816948i $$0.695664\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 395.196i − 0.857259i −0.903480 0.428629i $$-0.858997\pi$$
0.903480 0.428629i $$-0.141003\pi$$
$$462$$ 0 0
$$463$$ 159.767i 0.345070i 0.985003 + 0.172535i $$0.0551958\pi$$
−0.985003 + 0.172535i $$0.944804\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 275.233 0.589363 0.294682 0.955595i $$-0.404786\pi$$
0.294682 + 0.955595i $$0.404786\pi$$
$$468$$ 0 0
$$469$$ − 469.524i − 1.00112i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −84.5993 −0.178857
$$474$$ 0 0
$$475$$ −198.012 −0.416867
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 551.742i − 1.15186i −0.817498 0.575931i $$-0.804640\pi$$
0.817498 0.575931i $$-0.195360\pi$$
$$480$$ 0 0
$$481$$ 84.4676 0.175608
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 587.864i 1.21209i
$$486$$ 0 0
$$487$$ 219.363i 0.450437i 0.974308 + 0.225218i $$0.0723095\pi$$
−0.974308 + 0.225218i $$0.927690\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 346.740 0.706191 0.353095 0.935587i $$-0.385129\pi$$
0.353095 + 0.935587i $$0.385129\pi$$
$$492$$ 0 0
$$493$$ 341.118i 0.691923i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1044.38 2.10137
$$498$$ 0 0
$$499$$ 101.861 0.204130 0.102065 0.994778i $$-0.467455\pi$$
0.102065 + 0.994778i $$0.467455\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 752.544i − 1.49611i −0.663635 0.748056i $$-0.730987\pi$$
0.663635 0.748056i $$-0.269013\pi$$
$$504$$ 0 0
$$505$$ 296.647 0.587420
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 339.596i − 0.667182i −0.942718 0.333591i $$-0.891740\pi$$
0.942718 0.333591i $$-0.108260\pi$$
$$510$$ 0 0
$$511$$ − 697.283i − 1.36455i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 220.115 0.427408
$$516$$ 0 0
$$517$$ 489.782i 0.947354i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 683.717 1.31232 0.656158 0.754623i $$-0.272180\pi$$
0.656158 + 0.754623i $$0.272180\pi$$
$$522$$ 0 0
$$523$$ −112.671 −0.215433 −0.107716 0.994182i $$-0.534354\pi$$
−0.107716 + 0.994182i $$0.534354\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 228.093i − 0.432815i
$$528$$ 0 0
$$529$$ 199.011 0.376203
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 59.0571i − 0.110801i
$$534$$ 0 0
$$535$$ 130.327i 0.243601i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −84.7600 −0.157254
$$540$$ 0 0
$$541$$ − 752.788i − 1.39147i −0.718296 0.695737i $$-0.755078\pi$$
0.718296 0.695737i $$-0.244922\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −118.602 −0.217619
$$546$$ 0 0
$$547$$ −892.328 −1.63131 −0.815657 0.578536i $$-0.803624\pi$$
−0.815657 + 0.578536i $$0.803624\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 335.424i 0.608754i
$$552$$ 0 0
$$553$$ 93.0638 0.168289
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 105.929i 0.190177i 0.995469 + 0.0950887i $$0.0303135\pi$$
−0.995469 + 0.0950887i $$0.969687\pi$$
$$558$$ 0 0
$$559$$ − 15.7036i − 0.0280922i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −221.818 −0.393993 −0.196997 0.980404i $$-0.563119\pi$$
−0.196997 + 0.980404i $$0.563119\pi$$
$$564$$ 0 0
$$565$$ 852.994i 1.50972i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 604.024 1.06155 0.530777 0.847511i $$-0.321900\pi$$
0.530777 + 0.847511i $$0.321900\pi$$
$$570$$ 0 0
$$571$$ 470.074 0.823248 0.411624 0.911354i $$-0.364962\pi$$
0.411624 + 0.911354i $$0.364962\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 163.900i − 0.285043i
$$576$$ 0 0
$$577$$ 484.209 0.839183 0.419592 0.907713i $$-0.362173\pi$$
0.419592 + 0.907713i $$0.362173\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 1276.44i 2.19698i
$$582$$ 0 0
$$583$$ − 148.265i − 0.254314i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 361.426 0.615717 0.307858 0.951432i $$-0.400388\pi$$
0.307858 + 0.951432i $$0.400388\pi$$
$$588$$ 0 0
$$589$$ − 224.286i − 0.380791i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 277.115 0.467310 0.233655 0.972320i $$-0.424931\pi$$
0.233655 + 0.972320i $$0.424931\pi$$
$$594$$ 0 0
$$595$$ 691.170 1.16163
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 654.345i − 1.09240i −0.837656 0.546198i $$-0.816075\pi$$
0.837656 0.546198i $$-0.183925\pi$$
$$600$$ 0 0
$$601$$ −138.002 −0.229620 −0.114810 0.993387i $$-0.536626\pi$$
−0.114810 + 0.993387i $$0.536626\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 247.300i 0.408760i
$$606$$ 0 0
$$607$$ − 569.498i − 0.938217i −0.883140 0.469109i $$-0.844575\pi$$
0.883140 0.469109i $$-0.155425\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −90.9148 −0.148797
$$612$$ 0 0
$$613$$ − 1132.55i − 1.84756i −0.382925 0.923779i $$-0.625083\pi$$
0.382925 0.923779i $$-0.374917\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 651.926 1.05661 0.528303 0.849056i $$-0.322829\pi$$
0.528303 + 0.849056i $$0.322829\pi$$
$$618$$ 0 0
$$619$$ −691.283 −1.11677 −0.558387 0.829580i $$-0.688580\pi$$
−0.558387 + 0.829580i $$0.688580\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 276.341i 0.443566i
$$624$$ 0 0
$$625$$ −318.030 −0.508848
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1320.76i 2.09977i
$$630$$ 0 0
$$631$$ − 1008.39i − 1.59808i −0.601280 0.799038i $$-0.705342\pi$$
0.601280 0.799038i $$-0.294658\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 256.829 0.404455
$$636$$ 0 0
$$637$$ − 15.7334i − 0.0246992i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 609.937 0.951540 0.475770 0.879570i $$-0.342169\pi$$
0.475770 + 0.879570i $$0.342169\pi$$
$$642$$ 0 0
$$643$$ −453.534 −0.705341 −0.352670 0.935748i $$-0.614726\pi$$
−0.352670 + 0.935748i $$0.614726\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 422.050i − 0.652319i −0.945315 0.326159i $$-0.894245\pi$$
0.945315 0.326159i $$-0.105755\pi$$
$$648$$ 0 0
$$649$$ 124.419 0.191708
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 912.070i − 1.39674i −0.715738 0.698369i $$-0.753909\pi$$
0.715738 0.698369i $$-0.246091\pi$$
$$654$$ 0 0
$$655$$ 368.152i 0.562065i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −1130.79 −1.71591 −0.857957 0.513722i $$-0.828266\pi$$
−0.857957 + 0.513722i $$0.828266\pi$$
$$660$$ 0 0
$$661$$ − 669.361i − 1.01265i −0.862343 0.506324i $$-0.831004\pi$$
0.862343 0.506324i $$-0.168996\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 679.632 1.02200
$$666$$ 0 0
$$667$$ −277.639 −0.416250
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 213.103i 0.317590i
$$672$$ 0 0
$$673$$ −504.561 −0.749719 −0.374859 0.927082i $$-0.622309\pi$$
−0.374859 + 0.927082i $$0.622309\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 863.692i 1.27576i 0.770134 + 0.637882i $$0.220189\pi$$
−0.770134 + 0.637882i $$0.779811\pi$$
$$678$$ 0 0
$$679$$ 1139.41i 1.67807i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 845.591 1.23805 0.619027 0.785369i $$-0.287527\pi$$
0.619027 + 0.785369i $$0.287527\pi$$
$$684$$ 0 0
$$685$$ 677.852i 0.989565i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 27.5213 0.0399439
$$690$$ 0 0
$$691$$ −165.670 −0.239753 −0.119877 0.992789i $$-0.538250\pi$$
−0.119877 + 0.992789i $$0.538250\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 286.220i − 0.411828i
$$696$$ 0 0
$$697$$ 923.432 1.32487
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 947.047i − 1.35099i −0.737362 0.675497i $$-0.763929\pi$$
0.737362 0.675497i $$-0.236071\pi$$
$$702$$ 0 0
$$703$$ 1298.71i 1.84738i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 574.967 0.813250
$$708$$ 0 0
$$709$$ − 835.746i − 1.17877i −0.807853 0.589383i $$-0.799371\pi$$
0.807853 0.589383i $$-0.200629\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 185.647 0.260375
$$714$$ 0 0
$$715$$ 43.8737 0.0613618
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 20.1727i − 0.0280566i −0.999902 0.0140283i $$-0.995535\pi$$
0.999902 0.0140283i $$-0.00446549\pi$$
$$720$$ 0 0
$$721$$ 426.632 0.591723
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 137.899i − 0.190205i
$$726$$ 0 0
$$727$$ 276.178i 0.379887i 0.981795 + 0.189943i $$0.0608305\pi$$
−0.981795 + 0.189943i $$0.939170\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 245.545 0.335903
$$732$$ 0 0
$$733$$ − 294.820i − 0.402210i −0.979570 0.201105i $$-0.935547\pi$$
0.979570 0.201105i $$-0.0644533\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 466.026 0.632328
$$738$$ 0 0
$$739$$ 914.711 1.23777 0.618885 0.785482i $$-0.287585\pi$$
0.618885 + 0.785482i $$0.287585\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 1068.78i − 1.43846i −0.694772 0.719230i $$-0.744495\pi$$
0.694772 0.719230i $$-0.255505\pi$$
$$744$$ 0 0
$$745$$ −792.548 −1.06382
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 252.602i 0.337252i
$$750$$ 0 0
$$751$$ − 1147.59i − 1.52809i −0.645164 0.764044i $$-0.723211\pi$$
0.645164 0.764044i $$-0.276789\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −289.118 −0.382938
$$756$$ 0 0
$$757$$ 91.4460i 0.120801i 0.998174 + 0.0604003i $$0.0192377\pi$$
−0.998174 + 0.0604003i $$0.980762\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −762.676 −1.00220 −0.501101 0.865389i $$-0.667072\pi$$
−0.501101 + 0.865389i $$0.667072\pi$$
$$762$$ 0 0
$$763$$ −229.877 −0.301281
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 23.0950i 0.0301108i
$$768$$ 0 0
$$769$$ −214.567 −0.279021 −0.139511 0.990221i $$-0.544553\pi$$
−0.139511 + 0.990221i $$0.544553\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 459.360i − 0.594257i −0.954838 0.297128i $$-0.903971\pi$$
0.954838 0.297128i $$-0.0960289\pi$$
$$774$$ 0 0
$$775$$ 92.2079i 0.118978i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 908.017 1.16562
$$780$$ 0 0
$$781$$ 1036.60i 1.32727i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −495.901 −0.631721
$$786$$ 0 0
$$787$$ 76.5940 0.0973241 0.0486620 0.998815i $$-0.484504\pi$$
0.0486620 + 0.998815i $$0.484504\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1653.29i 2.09013i
$$792$$ 0 0
$$793$$ −39.5568 −0.0498825
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 273.548i − 0.343223i −0.985165 0.171611i $$-0.945103\pi$$
0.985165 0.171611i $$-0.0548973\pi$$
$$798$$ 0 0
$$799$$ − 1421.57i − 1.77918i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 692.087 0.861877
$$804$$ 0 0
$$805$$ 562.549i 0.698819i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −999.127 −1.23502 −0.617508 0.786565i $$-0.711858\pi$$
−0.617508 + 0.786565i $$0.711858\pi$$
$$810$$ 0 0
$$811$$ 1189.75 1.46702 0.733509 0.679679i $$-0.237881\pi$$
0.733509 + 0.679679i $$0.237881\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 929.478i − 1.14046i
$$816$$ 0 0
$$817$$ 241.446 0.295528
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 985.174i 1.19997i 0.800012 + 0.599984i $$0.204826\pi$$
−0.800012 + 0.599984i $$0.795174\pi$$
$$822$$ 0 0
$$823$$ 186.578i 0.226705i 0.993555 + 0.113352i $$0.0361589\pi$$
−0.993555 + 0.113352i $$0.963841\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −189.836 −0.229548 −0.114774 0.993392i $$-0.536614\pi$$
−0.114774 + 0.993392i $$0.536614\pi$$
$$828$$ 0 0
$$829$$ 920.478i 1.11035i 0.831735 + 0.555174i $$0.187348\pi$$
−0.831735 + 0.555174i $$0.812652\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 246.011 0.295332
$$834$$ 0 0
$$835$$ 175.778 0.210513
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 584.287i − 0.696408i −0.937419 0.348204i $$-0.886792\pi$$
0.937419 0.348204i $$-0.113208\pi$$
$$840$$ 0 0
$$841$$ 607.406 0.722242
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 667.380i − 0.789798i
$$846$$ 0 0
$$847$$ 479.322i 0.565906i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1074.98 −1.26319
$$852$$ 0 0
$$853$$ 767.535i 0.899806i 0.893077 + 0.449903i $$0.148542\pi$$
−0.893077 + 0.449903i $$0.851458\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −30.7621 −0.0358951 −0.0179476 0.999839i $$-0.505713\pi$$
−0.0179476 + 0.999839i $$0.505713\pi$$
$$858$$ 0 0
$$859$$ −795.268 −0.925807 −0.462903 0.886409i $$-0.653192\pi$$
−0.462903 + 0.886409i $$0.653192\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 1107.46i − 1.28327i −0.767012 0.641633i $$-0.778257\pi$$
0.767012 0.641633i $$-0.221743\pi$$
$$864$$ 0 0
$$865$$ −1222.77 −1.41361
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 92.3704i 0.106295i
$$870$$ 0 0
$$871$$ 86.5051i 0.0993169i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1053.60 −1.20412
$$876$$ 0 0
$$877$$ − 649.012i − 0.740037i −0.929024 0.370018i $$-0.879351\pi$$
0.929024 0.370018i $$-0.120649\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −951.863 −1.08043 −0.540217 0.841526i $$-0.681658\pi$$
−0.540217 + 0.841526i $$0.681658\pi$$
$$882$$ 0 0
$$883$$ −431.588 −0.488774 −0.244387 0.969678i $$-0.578587\pi$$
−0.244387 + 0.969678i $$0.578587\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 101.304i − 0.114209i −0.998368 0.0571047i $$-0.981813\pi$$
0.998368 0.0571047i $$-0.0181869\pi$$
$$888$$ 0 0
$$889$$ 497.792 0.559946
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 1397.84i − 1.56533i
$$894$$ 0 0
$$895$$ − 8.76399i − 0.00979216i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 156.196 0.173744
$$900$$ 0 0
$$901$$ 430.331i 0.477615i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −531.044 −0.586789
$$906$$ 0 0
$$907$$ 489.763 0.539981 0.269991 0.962863i $$-0.412979\pi$$
0.269991 + 0.962863i $$0.412979\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 253.105i 0.277832i 0.990304 + 0.138916i $$0.0443617\pi$$
−0.990304 + 0.138916i $$0.955638\pi$$
$$912$$ 0 0
$$913$$ −1266.93 −1.38766
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 713.561i 0.778147i
$$918$$ 0 0
$$919$$ − 68.3565i − 0.0743813i −0.999308 0.0371907i $$-0.988159\pi$$
0.999308 0.0371907i $$-0.0118409\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −192.416 −0.208468
$$924$$ 0 0
$$925$$ − 533.923i − 0.577214i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −319.311 −0.343715 −0.171857 0.985122i $$-0.554977\pi$$
−0.171857 + 0.985122i $$0.554977\pi$$
$$930$$ 0 0
$$931$$ 241.905 0.259833
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 686.020i 0.733711i
$$936$$ 0 0
$$937$$ −680.412 −0.726161 −0.363080 0.931758i $$-0.618275\pi$$
−0.363080 + 0.931758i $$0.618275\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 265.255i − 0.281886i −0.990018 0.140943i $$-0.954987\pi$$
0.990018 0.140943i $$-0.0450134\pi$$
$$942$$ 0 0
$$943$$ 751.589i 0.797019i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 412.117 0.435182 0.217591 0.976040i $$-0.430180\pi$$
0.217591 + 0.976040i $$0.430180\pi$$
$$948$$ 0 0
$$949$$ 128.467i 0.135371i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −149.202 −0.156561 −0.0782804 0.996931i $$-0.524943\pi$$
−0.0782804 + 0.996931i $$0.524943\pi$$
$$954$$ 0 0
$$955$$ 605.241 0.633760
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 1313.83i 1.37000i
$$960$$ 0 0
$$961$$ 856.557 0.891319
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 918.752i − 0.952074i
$$966$$ 0 0
$$967$$ 91.2392i 0.0943528i 0.998887 + 0.0471764i $$0.0150223\pi$$
−0.998887 + 0.0471764i $$0.984978\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −117.427 −0.120934 −0.0604669 0.998170i $$-0.519259\pi$$
−0.0604669 + 0.998170i $$0.519259\pi$$
$$972$$ 0 0
$$973$$ − 554.758i − 0.570153i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −576.207 −0.589772 −0.294886 0.955532i $$-0.595282\pi$$
−0.294886 + 0.955532i $$0.595282\pi$$
$$978$$ 0 0
$$979$$ −274.282 −0.280166
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 1338.61i 1.36176i 0.732395 + 0.680880i $$0.238403\pi$$
−0.732395 + 0.680880i $$0.761597\pi$$
$$984$$ 0 0
$$985$$ 756.246 0.767763
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 199.851i 0.202074i
$$990$$ 0 0
$$991$$ 1519.83i 1.53363i 0.641867 + 0.766816i $$0.278160\pi$$
−0.641867 + 0.766816i $$0.721840\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 870.129 0.874501
$$996$$ 0 0
$$997$$ 492.073i 0.493554i 0.969072 + 0.246777i $$0.0793715\pi$$
−0.969072 + 0.246777i $$0.920628\pi$$
$$998$$ 0 0
$$999$$ 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 1728.3.b.i.1567.3 12
3.2 odd 2 1728.3.b.j.1567.9 yes 12
4.3 odd 2 inner 1728.3.b.i.1567.4 yes 12
8.3 odd 2 inner 1728.3.b.i.1567.10 yes 12
8.5 even 2 inner 1728.3.b.i.1567.9 yes 12
12.11 even 2 1728.3.b.j.1567.10 yes 12
24.5 odd 2 1728.3.b.j.1567.3 yes 12
24.11 even 2 1728.3.b.j.1567.4 yes 12

By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.i.1567.3 12 1.1 even 1 trivial
1728.3.b.i.1567.4 yes 12 4.3 odd 2 inner
1728.3.b.i.1567.9 yes 12 8.5 even 2 inner
1728.3.b.i.1567.10 yes 12 8.3 odd 2 inner
1728.3.b.j.1567.3 yes 12 24.5 odd 2
1728.3.b.j.1567.4 yes 12 24.11 even 2
1728.3.b.j.1567.9 yes 12 3.2 odd 2
1728.3.b.j.1567.10 yes 12 12.11 even 2