# Properties

 Label 1600.2.f.i.1249.3 Level $1600$ Weight $2$ Character 1600.1249 Analytic conductor $12.776$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 320) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.3 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1600.1249 Dual form 1600.2.f.i.1249.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.73205 q^{3} -4.73205i q^{7} +4.46410 q^{9} +O(q^{10})$$ $$q+2.73205 q^{3} -4.73205i q^{7} +4.46410 q^{9} +3.46410i q^{11} +3.46410 q^{13} -3.46410i q^{17} +2.00000i q^{19} -12.9282i q^{21} -2.19615i q^{23} +4.00000 q^{27} +2.53590 q^{31} +9.46410i q^{33} -6.00000 q^{37} +9.46410 q^{39} +9.46410 q^{41} -0.196152 q^{43} -2.19615i q^{47} -15.3923 q^{49} -9.46410i q^{51} +10.3923 q^{53} +5.46410i q^{57} -6.00000i q^{59} -0.928203i q^{61} -21.1244i q^{63} +0.196152 q^{67} -6.00000i q^{69} -16.3923 q^{71} -6.39230i q^{73} +16.3923 q^{77} -12.0000 q^{79} -2.46410 q^{81} +1.26795 q^{83} +12.9282 q^{89} -16.3923i q^{91} +6.92820 q^{93} +14.3923i q^{97} +15.4641i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{3} + 4 q^{9} + 16 q^{27} + 24 q^{31} - 24 q^{37} + 24 q^{39} + 24 q^{41} + 20 q^{43} - 20 q^{49} - 20 q^{67} - 24 q^{71} + 24 q^{77} - 48 q^{79} + 4 q^{81} + 12 q^{83} + 24 q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ 2.73205 1.57735 0.788675 0.614810i $$-0.210767\pi$$
0.788675 + 0.614810i $$0.210767\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 4.73205i − 1.78855i −0.447521 0.894274i $$-0.647693\pi$$
0.447521 0.894274i $$-0.352307\pi$$
$$8$$ 0 0
$$9$$ 4.46410 1.48803
$$10$$ 0 0
$$11$$ 3.46410i 1.04447i 0.852803 + 0.522233i $$0.174901\pi$$
−0.852803 + 0.522233i $$0.825099\pi$$
$$12$$ 0 0
$$13$$ 3.46410 0.960769 0.480384 0.877058i $$-0.340497\pi$$
0.480384 + 0.877058i $$0.340497\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 3.46410i − 0.840168i −0.907485 0.420084i $$-0.862001\pi$$
0.907485 0.420084i $$-0.137999\pi$$
$$18$$ 0 0
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ 0 0
$$21$$ − 12.9282i − 2.82117i
$$22$$ 0 0
$$23$$ − 2.19615i − 0.457929i −0.973435 0.228965i $$-0.926466\pi$$
0.973435 0.228965i $$-0.0735340\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 2.53590 0.455461 0.227730 0.973724i $$-0.426870\pi$$
0.227730 + 0.973724i $$0.426870\pi$$
$$32$$ 0 0
$$33$$ 9.46410i 1.64749i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 9.46410 1.51547
$$40$$ 0 0
$$41$$ 9.46410 1.47804 0.739022 0.673681i $$-0.235288\pi$$
0.739022 + 0.673681i $$0.235288\pi$$
$$42$$ 0 0
$$43$$ −0.196152 −0.0299130 −0.0149565 0.999888i $$-0.504761\pi$$
−0.0149565 + 0.999888i $$0.504761\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 2.19615i − 0.320342i −0.987089 0.160171i $$-0.948795\pi$$
0.987089 0.160171i $$-0.0512045\pi$$
$$48$$ 0 0
$$49$$ −15.3923 −2.19890
$$50$$ 0 0
$$51$$ − 9.46410i − 1.32524i
$$52$$ 0 0
$$53$$ 10.3923 1.42749 0.713746 0.700404i $$-0.246997\pi$$
0.713746 + 0.700404i $$0.246997\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 5.46410i 0.723738i
$$58$$ 0 0
$$59$$ − 6.00000i − 0.781133i −0.920575 0.390567i $$-0.872279\pi$$
0.920575 0.390567i $$-0.127721\pi$$
$$60$$ 0 0
$$61$$ − 0.928203i − 0.118844i −0.998233 0.0594221i $$-0.981074\pi$$
0.998233 0.0594221i $$-0.0189258\pi$$
$$62$$ 0 0
$$63$$ − 21.1244i − 2.66142i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.196152 0.0239638 0.0119819 0.999928i $$-0.496186\pi$$
0.0119819 + 0.999928i $$0.496186\pi$$
$$68$$ 0 0
$$69$$ − 6.00000i − 0.722315i
$$70$$ 0 0
$$71$$ −16.3923 −1.94541 −0.972704 0.232048i $$-0.925457\pi$$
−0.972704 + 0.232048i $$0.925457\pi$$
$$72$$ 0 0
$$73$$ − 6.39230i − 0.748163i −0.927396 0.374081i $$-0.877958\pi$$
0.927396 0.374081i $$-0.122042\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 16.3923 1.86808
$$78$$ 0 0
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ 0 0
$$83$$ 1.26795 0.139176 0.0695878 0.997576i $$-0.477832\pi$$
0.0695878 + 0.997576i $$0.477832\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 12.9282 1.37039 0.685193 0.728361i $$-0.259718\pi$$
0.685193 + 0.728361i $$0.259718\pi$$
$$90$$ 0 0
$$91$$ − 16.3923i − 1.71838i
$$92$$ 0 0
$$93$$ 6.92820 0.718421
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.3923i 1.46132i 0.682743 + 0.730659i $$0.260787\pi$$
−0.682743 + 0.730659i $$0.739213\pi$$
$$98$$ 0 0
$$99$$ 15.4641i 1.55420i
$$100$$ 0 0
$$101$$ 12.0000i 1.19404i 0.802225 + 0.597022i $$0.203650\pi$$
−0.802225 + 0.597022i $$0.796350\pi$$
$$102$$ 0 0
$$103$$ − 2.19615i − 0.216393i −0.994130 0.108197i $$-0.965492\pi$$
0.994130 0.108197i $$-0.0345076\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 13.2679 1.28266 0.641331 0.767265i $$-0.278383\pi$$
0.641331 + 0.767265i $$0.278383\pi$$
$$108$$ 0 0
$$109$$ 12.9282i 1.23830i 0.785274 + 0.619149i $$0.212522\pi$$
−0.785274 + 0.619149i $$0.787478\pi$$
$$110$$ 0 0
$$111$$ −16.3923 −1.55589
$$112$$ 0 0
$$113$$ 12.9282i 1.21618i 0.793867 + 0.608092i $$0.208065\pi$$
−0.793867 + 0.608092i $$0.791935\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 15.4641 1.42966
$$118$$ 0 0
$$119$$ −16.3923 −1.50268
$$120$$ 0 0
$$121$$ −1.00000 −0.0909091
$$122$$ 0 0
$$123$$ 25.8564 2.33139
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 14.1962i 1.25970i 0.776715 + 0.629852i $$0.216885\pi$$
−0.776715 + 0.629852i $$0.783115\pi$$
$$128$$ 0 0
$$129$$ −0.535898 −0.0471832
$$130$$ 0 0
$$131$$ 10.3923i 0.907980i 0.891007 + 0.453990i $$0.150000\pi$$
−0.891007 + 0.453990i $$0.850000\pi$$
$$132$$ 0 0
$$133$$ 9.46410 0.820642
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 0.928203i − 0.0793018i −0.999214 0.0396509i $$-0.987375\pi$$
0.999214 0.0396509i $$-0.0126246\pi$$
$$138$$ 0 0
$$139$$ − 10.0000i − 0.848189i −0.905618 0.424094i $$-0.860592\pi$$
0.905618 0.424094i $$-0.139408\pi$$
$$140$$ 0 0
$$141$$ − 6.00000i − 0.505291i
$$142$$ 0 0
$$143$$ 12.0000i 1.00349i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −42.0526 −3.46844
$$148$$ 0 0
$$149$$ 18.0000i 1.47462i 0.675556 + 0.737309i $$0.263904\pi$$
−0.675556 + 0.737309i $$0.736096\pi$$
$$150$$ 0 0
$$151$$ −9.46410 −0.770178 −0.385089 0.922880i $$-0.625829\pi$$
−0.385089 + 0.922880i $$0.625829\pi$$
$$152$$ 0 0
$$153$$ − 15.4641i − 1.25020i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −12.9282 −1.03178 −0.515891 0.856654i $$-0.672539\pi$$
−0.515891 + 0.856654i $$0.672539\pi$$
$$158$$ 0 0
$$159$$ 28.3923 2.25166
$$160$$ 0 0
$$161$$ −10.3923 −0.819028
$$162$$ 0 0
$$163$$ −16.1962 −1.26858 −0.634290 0.773095i $$-0.718708\pi$$
−0.634290 + 0.773095i $$0.718708\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 2.19615i − 0.169943i −0.996383 0.0849717i $$-0.972920\pi$$
0.996383 0.0849717i $$-0.0270800\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 8.92820i 0.682757i
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 16.3923i − 1.23212i
$$178$$ 0 0
$$179$$ − 7.85641i − 0.587215i −0.955926 0.293608i $$-0.905144\pi$$
0.955926 0.293608i $$-0.0948559\pi$$
$$180$$ 0 0
$$181$$ 6.92820i 0.514969i 0.966282 + 0.257485i $$0.0828937\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ 0 0
$$183$$ − 2.53590i − 0.187459i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ 0 0
$$189$$ − 18.9282i − 1.37682i
$$190$$ 0 0
$$191$$ 4.39230 0.317816 0.158908 0.987293i $$-0.449203\pi$$
0.158908 + 0.987293i $$0.449203\pi$$
$$192$$ 0 0
$$193$$ 14.3923i 1.03598i 0.855386 + 0.517990i $$0.173320\pi$$
−0.855386 + 0.517990i $$0.826680\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 10.3923 0.740421 0.370211 0.928948i $$-0.379286\pi$$
0.370211 + 0.928948i $$0.379286\pi$$
$$198$$ 0 0
$$199$$ 6.92820 0.491127 0.245564 0.969380i $$-0.421027\pi$$
0.245564 + 0.969380i $$0.421027\pi$$
$$200$$ 0 0
$$201$$ 0.535898 0.0377994
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 9.80385i − 0.681415i
$$208$$ 0 0
$$209$$ −6.92820 −0.479234
$$210$$ 0 0
$$211$$ − 14.3923i − 0.990807i −0.868663 0.495404i $$-0.835020\pi$$
0.868663 0.495404i $$-0.164980\pi$$
$$212$$ 0 0
$$213$$ −44.7846 −3.06859
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 12.0000i − 0.814613i
$$218$$ 0 0
$$219$$ − 17.4641i − 1.18011i
$$220$$ 0 0
$$221$$ − 12.0000i − 0.807207i
$$222$$ 0 0
$$223$$ 9.12436i 0.611012i 0.952190 + 0.305506i $$0.0988256\pi$$
−0.952190 + 0.305506i $$0.901174\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.5885 0.835525 0.417763 0.908556i $$-0.362814\pi$$
0.417763 + 0.908556i $$0.362814\pi$$
$$228$$ 0 0
$$229$$ − 5.07180i − 0.335154i −0.985859 0.167577i $$-0.946406\pi$$
0.985859 0.167577i $$-0.0535942\pi$$
$$230$$ 0 0
$$231$$ 44.7846 2.94661
$$232$$ 0 0
$$233$$ 22.3923i 1.46697i 0.679706 + 0.733484i $$0.262107\pi$$
−0.679706 + 0.733484i $$0.737893\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −32.7846 −2.12959
$$238$$ 0 0
$$239$$ −20.7846 −1.34444 −0.672222 0.740349i $$-0.734660\pi$$
−0.672222 + 0.740349i $$0.734660\pi$$
$$240$$ 0 0
$$241$$ 0.392305 0.0252706 0.0126353 0.999920i $$-0.495978\pi$$
0.0126353 + 0.999920i $$0.495978\pi$$
$$242$$ 0 0
$$243$$ −18.7321 −1.20166
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.92820i 0.440831i
$$248$$ 0 0
$$249$$ 3.46410 0.219529
$$250$$ 0 0
$$251$$ 8.53590i 0.538781i 0.963031 + 0.269391i $$0.0868223\pi$$
−0.963031 + 0.269391i $$0.913178\pi$$
$$252$$ 0 0
$$253$$ 7.60770 0.478292
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ 0 0
$$259$$ 28.3923i 1.76421i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 9.80385i − 0.604531i −0.953224 0.302266i $$-0.902257\pi$$
0.953224 0.302266i $$-0.0977429\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 35.3205 2.16158
$$268$$ 0 0
$$269$$ − 26.7846i − 1.63309i −0.577284 0.816543i $$-0.695888\pi$$
0.577284 0.816543i $$-0.304112\pi$$
$$270$$ 0 0
$$271$$ 16.3923 0.995762 0.497881 0.867245i $$-0.334112\pi$$
0.497881 + 0.867245i $$0.334112\pi$$
$$272$$ 0 0
$$273$$ − 44.7846i − 2.71049i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −7.85641 −0.472046 −0.236023 0.971748i $$-0.575844\pi$$
−0.236023 + 0.971748i $$0.575844\pi$$
$$278$$ 0 0
$$279$$ 11.3205 0.677741
$$280$$ 0 0
$$281$$ −7.60770 −0.453837 −0.226919 0.973914i $$-0.572865\pi$$
−0.226919 + 0.973914i $$0.572865\pi$$
$$282$$ 0 0
$$283$$ −20.5885 −1.22386 −0.611928 0.790913i $$-0.709606\pi$$
−0.611928 + 0.790913i $$0.709606\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 44.7846i − 2.64355i
$$288$$ 0 0
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ 39.3205i 2.30501i
$$292$$ 0 0
$$293$$ 30.0000 1.75262 0.876309 0.481749i $$-0.159998\pi$$
0.876309 + 0.481749i $$0.159998\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 13.8564i 0.804030i
$$298$$ 0 0
$$299$$ − 7.60770i − 0.439964i
$$300$$ 0 0
$$301$$ 0.928203i 0.0535007i
$$302$$ 0 0
$$303$$ 32.7846i 1.88343i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −23.8038 −1.35856 −0.679279 0.733880i $$-0.737707\pi$$
−0.679279 + 0.733880i $$0.737707\pi$$
$$308$$ 0 0
$$309$$ − 6.00000i − 0.341328i
$$310$$ 0 0
$$311$$ 28.3923 1.60998 0.804990 0.593288i $$-0.202171\pi$$
0.804990 + 0.593288i $$0.202171\pi$$
$$312$$ 0 0
$$313$$ − 22.0000i − 1.24351i −0.783210 0.621757i $$-0.786419\pi$$
0.783210 0.621757i $$-0.213581\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1.60770 −0.0902972 −0.0451486 0.998980i $$-0.514376\pi$$
−0.0451486 + 0.998980i $$0.514376\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 36.2487 2.02321
$$322$$ 0 0
$$323$$ 6.92820 0.385496
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 35.3205i 1.95323i
$$328$$ 0 0
$$329$$ −10.3923 −0.572946
$$330$$ 0 0
$$331$$ 26.3923i 1.45065i 0.688405 + 0.725326i $$0.258311\pi$$
−0.688405 + 0.725326i $$0.741689\pi$$
$$332$$ 0 0
$$333$$ −26.7846 −1.46779
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ 0 0
$$339$$ 35.3205i 1.91835i
$$340$$ 0 0
$$341$$ 8.78461i 0.475713i
$$342$$ 0 0
$$343$$ 39.7128i 2.14429i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −10.0526 −0.539650 −0.269825 0.962909i $$-0.586966\pi$$
−0.269825 + 0.962909i $$0.586966\pi$$
$$348$$ 0 0
$$349$$ 32.7846i 1.75492i 0.479650 + 0.877460i $$0.340764\pi$$
−0.479650 + 0.877460i $$0.659236\pi$$
$$350$$ 0 0
$$351$$ 13.8564 0.739600
$$352$$ 0 0
$$353$$ − 26.7846i − 1.42560i −0.701367 0.712800i $$-0.747427\pi$$
0.701367 0.712800i $$-0.252573\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −44.7846 −2.37025
$$358$$ 0 0
$$359$$ −32.7846 −1.73031 −0.865153 0.501508i $$-0.832779\pi$$
−0.865153 + 0.501508i $$0.832779\pi$$
$$360$$ 0 0
$$361$$ 15.0000 0.789474
$$362$$ 0 0
$$363$$ −2.73205 −0.143395
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 28.0526i 1.46433i 0.681126 + 0.732166i $$0.261490\pi$$
−0.681126 + 0.732166i $$0.738510\pi$$
$$368$$ 0 0
$$369$$ 42.2487 2.19938
$$370$$ 0 0
$$371$$ − 49.1769i − 2.55314i
$$372$$ 0 0
$$373$$ −19.8564 −1.02813 −0.514063 0.857753i $$-0.671860\pi$$
−0.514063 + 0.857753i $$0.671860\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 2.00000i − 0.102733i −0.998680 0.0513665i $$-0.983642\pi$$
0.998680 0.0513665i $$-0.0163577\pi$$
$$380$$ 0 0
$$381$$ 38.7846i 1.98700i
$$382$$ 0 0
$$383$$ 14.1962i 0.725390i 0.931908 + 0.362695i $$0.118143\pi$$
−0.931908 + 0.362695i $$0.881857\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −0.875644 −0.0445115
$$388$$ 0 0
$$389$$ − 14.7846i − 0.749609i −0.927104 0.374805i $$-0.877710\pi$$
0.927104 0.374805i $$-0.122290\pi$$
$$390$$ 0 0
$$391$$ −7.60770 −0.384738
$$392$$ 0 0
$$393$$ 28.3923i 1.43220i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 20.5359 1.03067 0.515334 0.856990i $$-0.327668\pi$$
0.515334 + 0.856990i $$0.327668\pi$$
$$398$$ 0 0
$$399$$ 25.8564 1.29444
$$400$$ 0 0
$$401$$ −31.8564 −1.59083 −0.795417 0.606063i $$-0.792748\pi$$
−0.795417 + 0.606063i $$0.792748\pi$$
$$402$$ 0 0
$$403$$ 8.78461 0.437593
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 20.7846i − 1.03025i
$$408$$ 0 0
$$409$$ −24.3923 −1.20612 −0.603061 0.797695i $$-0.706052\pi$$
−0.603061 + 0.797695i $$0.706052\pi$$
$$410$$ 0 0
$$411$$ − 2.53590i − 0.125087i
$$412$$ 0 0
$$413$$ −28.3923 −1.39709
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 27.3205i − 1.33789i
$$418$$ 0 0
$$419$$ − 12.9282i − 0.631584i −0.948828 0.315792i $$-0.897730\pi$$
0.948828 0.315792i $$-0.102270\pi$$
$$420$$ 0 0
$$421$$ 6.00000i 0.292422i 0.989253 + 0.146211i $$0.0467079\pi$$
−0.989253 + 0.146211i $$0.953292\pi$$
$$422$$ 0 0
$$423$$ − 9.80385i − 0.476679i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −4.39230 −0.212559
$$428$$ 0 0
$$429$$ 32.7846i 1.58286i
$$430$$ 0 0
$$431$$ −7.60770 −0.366450 −0.183225 0.983071i $$-0.558654\pi$$
−0.183225 + 0.983071i $$0.558654\pi$$
$$432$$ 0 0
$$433$$ − 5.60770i − 0.269489i −0.990880 0.134744i $$-0.956979\pi$$
0.990880 0.134744i $$-0.0430213\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.39230 0.210112
$$438$$ 0 0
$$439$$ 5.07180 0.242064 0.121032 0.992649i $$-0.461380\pi$$
0.121032 + 0.992649i $$0.461380\pi$$
$$440$$ 0 0
$$441$$ −68.7128 −3.27204
$$442$$ 0 0
$$443$$ −17.6603 −0.839064 −0.419532 0.907741i $$-0.637806\pi$$
−0.419532 + 0.907741i $$0.637806\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 49.1769i 2.32599i
$$448$$ 0 0
$$449$$ 9.46410 0.446639 0.223319 0.974745i $$-0.428311\pi$$
0.223319 + 0.974745i $$0.428311\pi$$
$$450$$ 0 0
$$451$$ 32.7846i 1.54377i
$$452$$ 0 0
$$453$$ −25.8564 −1.21484
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 18.7846i − 0.878707i −0.898314 0.439353i $$-0.855208\pi$$
0.898314 0.439353i $$-0.144792\pi$$
$$458$$ 0 0
$$459$$ − 13.8564i − 0.646762i
$$460$$ 0 0
$$461$$ 12.0000i 0.558896i 0.960161 + 0.279448i $$0.0901514\pi$$
−0.960161 + 0.279448i $$0.909849\pi$$
$$462$$ 0 0
$$463$$ − 26.1962i − 1.21744i −0.793386 0.608719i $$-0.791684\pi$$
0.793386 0.608719i $$-0.208316\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −28.9808 −1.34107 −0.670535 0.741878i $$-0.733935\pi$$
−0.670535 + 0.741878i $$0.733935\pi$$
$$468$$ 0 0
$$469$$ − 0.928203i − 0.0428604i
$$470$$ 0 0
$$471$$ −35.3205 −1.62748
$$472$$ 0 0
$$473$$ − 0.679492i − 0.0312431i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 46.3923 2.12416
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −20.7846 −0.947697
$$482$$ 0 0
$$483$$ −28.3923 −1.29189
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 14.8756i − 0.674080i −0.941490 0.337040i $$-0.890574\pi$$
0.941490 0.337040i $$-0.109426\pi$$
$$488$$ 0 0
$$489$$ −44.2487 −2.00100
$$490$$ 0 0
$$491$$ 1.60770i 0.0725543i 0.999342 + 0.0362771i $$0.0115499\pi$$
−0.999342 + 0.0362771i $$0.988450\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 77.5692i 3.47946i
$$498$$ 0 0
$$499$$ − 43.5692i − 1.95043i −0.221268 0.975213i $$-0.571020\pi$$
0.221268 0.975213i $$-0.428980\pi$$
$$500$$ 0 0
$$501$$ − 6.00000i − 0.268060i
$$502$$ 0 0
$$503$$ − 2.19615i − 0.0979216i −0.998801 0.0489608i $$-0.984409\pi$$
0.998801 0.0489608i $$-0.0155909\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −2.73205 −0.121335
$$508$$ 0 0
$$509$$ − 32.7846i − 1.45315i −0.687086 0.726576i $$-0.741110\pi$$
0.687086 0.726576i $$-0.258890\pi$$
$$510$$ 0 0
$$511$$ −30.2487 −1.33812
$$512$$ 0 0
$$513$$ 8.00000i 0.353209i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 7.60770 0.334586
$$518$$ 0 0
$$519$$ −16.3923 −0.719542
$$520$$ 0 0
$$521$$ 4.14359 0.181534 0.0907671 0.995872i $$-0.471068\pi$$
0.0907671 + 0.995872i $$0.471068\pi$$
$$522$$ 0 0
$$523$$ 36.9808 1.61706 0.808528 0.588458i $$-0.200265\pi$$
0.808528 + 0.588458i $$0.200265\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 8.78461i − 0.382664i
$$528$$ 0 0
$$529$$ 18.1769 0.790301
$$530$$ 0 0
$$531$$ − 26.7846i − 1.16235i
$$532$$ 0 0
$$533$$ 32.7846 1.42006
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 21.4641i − 0.926244i
$$538$$ 0 0
$$539$$ − 53.3205i − 2.29668i
$$540$$ 0 0
$$541$$ 39.7128i 1.70739i 0.520776 + 0.853694i $$0.325643\pi$$
−0.520776 + 0.853694i $$0.674357\pi$$
$$542$$ 0 0
$$543$$ 18.9282i 0.812287i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −12.1962 −0.521470 −0.260735 0.965410i $$-0.583965\pi$$
−0.260735 + 0.965410i $$0.583965\pi$$
$$548$$ 0 0
$$549$$ − 4.14359i − 0.176844i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 56.7846i 2.41473i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 26.7846 1.13490 0.567450 0.823408i $$-0.307930\pi$$
0.567450 + 0.823408i $$0.307930\pi$$
$$558$$ 0 0
$$559$$ −0.679492 −0.0287394
$$560$$ 0 0
$$561$$ 32.7846 1.38417
$$562$$ 0 0
$$563$$ 15.8038 0.666053 0.333026 0.942918i $$-0.391930\pi$$
0.333026 + 0.942918i $$0.391930\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 11.6603i 0.489685i
$$568$$ 0 0
$$569$$ −6.24871 −0.261960 −0.130980 0.991385i $$-0.541812\pi$$
−0.130980 + 0.991385i $$0.541812\pi$$
$$570$$ 0 0
$$571$$ − 9.60770i − 0.402070i −0.979584 0.201035i $$-0.935570\pi$$
0.979584 0.201035i $$-0.0644304\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 0 0
$$579$$ 39.3205i 1.63410i
$$580$$ 0 0
$$581$$ − 6.00000i − 0.248922i
$$582$$ 0 0
$$583$$ 36.0000i 1.49097i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 31.5167 1.30083 0.650416 0.759578i $$-0.274595\pi$$
0.650416 + 0.759578i $$0.274595\pi$$
$$588$$ 0 0
$$589$$ 5.07180i 0.208980i
$$590$$ 0 0
$$591$$ 28.3923 1.16790
$$592$$ 0 0
$$593$$ − 12.9282i − 0.530898i −0.964125 0.265449i $$-0.914480\pi$$
0.964125 0.265449i $$-0.0855201\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 18.9282 0.774680
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 0.392305 0.0160024 0.00800122 0.999968i $$-0.497453\pi$$
0.00800122 + 0.999968i $$0.497453\pi$$
$$602$$ 0 0
$$603$$ 0.875644 0.0356590
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 2.19615i − 0.0891391i −0.999006 0.0445695i $$-0.985808\pi$$
0.999006 0.0445695i $$-0.0141916\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 7.60770i − 0.307774i
$$612$$ 0 0
$$613$$ −34.3923 −1.38909 −0.694546 0.719448i $$-0.744395\pi$$
−0.694546 + 0.719448i $$0.744395\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 34.3923i 1.38458i 0.721618 + 0.692291i $$0.243399\pi$$
−0.721618 + 0.692291i $$0.756601\pi$$
$$618$$ 0 0
$$619$$ 34.7846i 1.39811i 0.715067 + 0.699056i $$0.246396\pi$$
−0.715067 + 0.699056i $$0.753604\pi$$
$$620$$ 0 0
$$621$$ − 8.78461i − 0.352514i
$$622$$ 0 0
$$623$$ − 61.1769i − 2.45100i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −18.9282 −0.755920
$$628$$ 0 0
$$629$$ 20.7846i 0.828737i
$$630$$ 0 0
$$631$$ −14.5359 −0.578665 −0.289332 0.957229i $$-0.593433\pi$$
−0.289332 + 0.957229i $$0.593433\pi$$
$$632$$ 0 0
$$633$$ − 39.3205i − 1.56285i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −53.3205 −2.11264
$$638$$ 0 0
$$639$$ −73.1769 −2.89483
$$640$$ 0 0
$$641$$ 16.3923 0.647457 0.323729 0.946150i $$-0.395064\pi$$
0.323729 + 0.946150i $$0.395064\pi$$
$$642$$ 0 0
$$643$$ −20.5885 −0.811929 −0.405965 0.913889i $$-0.633064\pi$$
−0.405965 + 0.913889i $$0.633064\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 5.41154i 0.212750i 0.994326 + 0.106375i $$0.0339244\pi$$
−0.994326 + 0.106375i $$0.966076\pi$$
$$648$$ 0 0
$$649$$ 20.7846 0.815867
$$650$$ 0 0
$$651$$ − 32.7846i − 1.28493i
$$652$$ 0 0
$$653$$ 43.1769 1.68964 0.844822 0.535048i $$-0.179707\pi$$
0.844822 + 0.535048i $$0.179707\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 28.5359i − 1.11329i
$$658$$ 0 0
$$659$$ 28.6410i 1.11570i 0.829943 + 0.557848i $$0.188373\pi$$
−0.829943 + 0.557848i $$0.811627\pi$$
$$660$$ 0 0
$$661$$ − 47.5692i − 1.85023i −0.379689 0.925114i $$-0.623969\pi$$
0.379689 0.925114i $$-0.376031\pi$$
$$662$$ 0 0
$$663$$ − 32.7846i − 1.27325i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 24.9282i 0.963780i
$$670$$ 0 0
$$671$$ 3.21539 0.124129
$$672$$ 0 0
$$673$$ − 23.1769i − 0.893404i −0.894683 0.446702i $$-0.852598\pi$$
0.894683 0.446702i $$-0.147402\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 19.1769 0.737029 0.368514 0.929622i $$-0.379867\pi$$
0.368514 + 0.929622i $$0.379867\pi$$
$$678$$ 0 0
$$679$$ 68.1051 2.61363
$$680$$ 0 0
$$681$$ 34.3923 1.31792
$$682$$ 0 0
$$683$$ 5.66025 0.216584 0.108292 0.994119i $$-0.465462\pi$$
0.108292 + 0.994119i $$0.465462\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 13.8564i − 0.528655i
$$688$$ 0 0
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ 26.3923i 1.00401i 0.864865 + 0.502005i $$0.167404\pi$$
−0.864865 + 0.502005i $$0.832596\pi$$
$$692$$ 0 0
$$693$$ 73.1769 2.77976
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 32.7846i − 1.24181i
$$698$$ 0 0
$$699$$ 61.1769i 2.31392i
$$700$$ 0 0
$$701$$ − 26.7846i − 1.01164i −0.862639 0.505820i $$-0.831190\pi$$
0.862639 0.505820i $$-0.168810\pi$$
$$702$$ 0 0
$$703$$ − 12.0000i − 0.452589i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 56.7846 2.13561
$$708$$ 0 0
$$709$$ 37.8564i 1.42173i 0.703330 + 0.710864i $$0.251696\pi$$
−0.703330 + 0.710864i $$0.748304\pi$$
$$710$$ 0 0
$$711$$ −53.5692 −2.00900
$$712$$ 0 0
$$713$$ − 5.56922i − 0.208569i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −56.7846 −2.12066
$$718$$ 0 0
$$719$$ 3.21539 0.119914 0.0599569 0.998201i $$-0.480904\pi$$
0.0599569 + 0.998201i $$0.480904\pi$$
$$720$$ 0 0
$$721$$ −10.3923 −0.387030
$$722$$ 0 0
$$723$$ 1.07180 0.0398606
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 4.05256i 0.150301i 0.997172 + 0.0751505i $$0.0239437\pi$$
−0.997172 + 0.0751505i $$0.976056\pi$$
$$728$$ 0 0
$$729$$ −43.7846 −1.62165
$$730$$ 0 0
$$731$$ 0.679492i 0.0251319i
$$732$$ 0 0
$$733$$ 2.78461 0.102852 0.0514260 0.998677i $$-0.483623\pi$$
0.0514260 + 0.998677i $$0.483623\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0.679492i 0.0250294i
$$738$$ 0 0
$$739$$ − 2.00000i − 0.0735712i −0.999323 0.0367856i $$-0.988288\pi$$
0.999323 0.0367856i $$-0.0117119\pi$$
$$740$$ 0 0
$$741$$ 18.9282i 0.695345i
$$742$$ 0 0
$$743$$ 5.41154i 0.198530i 0.995061 + 0.0992651i $$0.0316492\pi$$
−0.995061 + 0.0992651i $$0.968351\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 5.66025 0.207098
$$748$$ 0 0
$$749$$ − 62.7846i − 2.29410i
$$750$$ 0 0
$$751$$ 19.6077 0.715495 0.357747 0.933818i $$-0.383545\pi$$
0.357747 + 0.933818i $$0.383545\pi$$
$$752$$ 0 0
$$753$$ 23.3205i 0.849847i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −36.9282 −1.34218 −0.671089 0.741377i $$-0.734173\pi$$
−0.671089 + 0.741377i $$0.734173\pi$$
$$758$$ 0 0
$$759$$ 20.7846 0.754434
$$760$$ 0 0
$$761$$ 33.7128 1.22209 0.611044 0.791596i $$-0.290750\pi$$
0.611044 + 0.791596i $$0.290750\pi$$
$$762$$ 0 0
$$763$$ 61.1769 2.21475
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 20.7846i − 0.750489i
$$768$$ 0 0
$$769$$ −34.7846 −1.25437 −0.627183 0.778872i $$-0.715792\pi$$
−0.627183 + 0.778872i $$0.715792\pi$$
$$770$$ 0 0
$$771$$ 16.3923i 0.590354i
$$772$$ 0 0
$$773$$ −25.6077 −0.921045 −0.460522 0.887648i $$-0.652338\pi$$
−0.460522 + 0.887648i $$0.652338\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 77.5692i 2.78278i
$$778$$ 0 0
$$779$$ 18.9282i 0.678173i
$$780$$ 0 0
$$781$$ − 56.7846i − 2.03191i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 16.1962 0.577330 0.288665 0.957430i $$-0.406789\pi$$
0.288665 + 0.957430i $$0.406789\pi$$
$$788$$ 0 0
$$789$$ − 26.7846i − 0.953557i
$$790$$ 0 0
$$791$$ 61.1769 2.17520
$$792$$ 0 0
$$793$$ − 3.21539i − 0.114182i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 22.3923 0.793176 0.396588 0.917997i $$-0.370194\pi$$
0.396588 + 0.917997i $$0.370194\pi$$
$$798$$ 0 0
$$799$$ −7.60770 −0.269141
$$800$$ 0 0
$$801$$ 57.7128 2.03918
$$802$$ 0 0
$$803$$ 22.1436 0.781430
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 73.1769i − 2.57595i
$$808$$ 0 0
$$809$$ 47.5692 1.67244 0.836222 0.548391i $$-0.184759\pi$$
0.836222 + 0.548391i $$0.184759\pi$$
$$810$$ 0 0
$$811$$ − 17.6077i − 0.618290i −0.951015 0.309145i $$-0.899957\pi$$
0.951015 0.309145i $$-0.100043\pi$$
$$812$$ 0 0
$$813$$ 44.7846 1.57066
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 0.392305i − 0.0137250i
$$818$$ 0 0
$$819$$ − 73.1769i − 2.55701i
$$820$$ 0 0
$$821$$ 9.21539i 0.321619i 0.986985 + 0.160810i $$0.0514105\pi$$
−0.986985 + 0.160810i $$0.948589\pi$$
$$822$$ 0 0
$$823$$ − 48.8372i − 1.70236i −0.524877 0.851178i $$-0.675889\pi$$
0.524877 0.851178i $$-0.324111\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1.26795 −0.0440909 −0.0220455 0.999757i $$-0.507018\pi$$
−0.0220455 + 0.999757i $$0.507018\pi$$
$$828$$ 0 0
$$829$$ − 9.21539i − 0.320064i −0.987112 0.160032i $$-0.948840\pi$$
0.987112 0.160032i $$-0.0511597\pi$$
$$830$$ 0 0
$$831$$ −21.4641 −0.744581
$$832$$ 0 0
$$833$$ 53.3205i 1.84745i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 10.1436 0.350614
$$838$$ 0 0
$$839$$ −32.7846 −1.13185 −0.565925 0.824457i $$-0.691481\pi$$
−0.565925 + 0.824457i $$0.691481\pi$$
$$840$$ 0 0
$$841$$ 29.0000 1.00000
$$842$$ 0 0
$$843$$ −20.7846 −0.715860
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4.73205i 0.162595i
$$848$$ 0 0
$$849$$ −56.2487 −1.93045
$$850$$ 0 0
$$851$$ 13.1769i 0.451699i
$$852$$ 0 0
$$853$$ 3.46410 0.118609 0.0593043 0.998240i $$-0.481112\pi$$
0.0593043 + 0.998240i $$0.481112\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 43.8564i − 1.49811i −0.662510 0.749053i $$-0.730509\pi$$
0.662510 0.749053i $$-0.269491\pi$$
$$858$$ 0 0
$$859$$ − 31.5692i − 1.07713i −0.842585 0.538564i $$-0.818967\pi$$
0.842585 0.538564i $$-0.181033\pi$$
$$860$$ 0 0
$$861$$ − 122.354i − 4.16981i
$$862$$ 0 0
$$863$$ − 10.9808i − 0.373789i −0.982380 0.186895i $$-0.940158\pi$$
0.982380 0.186895i $$-0.0598423\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.6603 0.463927
$$868$$ 0 0
$$869$$ − 41.5692i − 1.41014i
$$870$$ 0 0
$$871$$ 0.679492 0.0230237
$$872$$ 0 0
$$873$$ 64.2487i 2.17449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 19.8564 0.670503 0.335252 0.942129i $$-0.391179\pi$$
0.335252 + 0.942129i $$0.391179\pi$$
$$878$$ 0 0
$$879$$ 81.9615 2.76449
$$880$$ 0 0
$$881$$ −28.3923 −0.956561 −0.478281 0.878207i $$-0.658740\pi$$
−0.478281 + 0.878207i $$0.658740\pi$$
$$882$$ 0 0
$$883$$ −16.1962 −0.545044 −0.272522 0.962150i $$-0.587858\pi$$
−0.272522 + 0.962150i $$0.587858\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 51.3731i − 1.72494i −0.506109 0.862469i $$-0.668917\pi$$
0.506109 0.862469i $$-0.331083\pi$$
$$888$$ 0 0
$$889$$ 67.1769 2.25304
$$890$$ 0 0
$$891$$ − 8.53590i − 0.285963i
$$892$$ 0 0
$$893$$ 4.39230 0.146983
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 20.7846i − 0.693978i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ − 36.0000i − 1.19933i
$$902$$ 0 0
$$903$$ 2.53590i 0.0843894i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −32.5885 −1.08208 −0.541041 0.840996i $$-0.681970\pi$$
−0.541041 + 0.840996i $$0.681970\pi$$
$$908$$ 0 0
$$909$$ 53.5692i 1.77678i
$$910$$ 0 0
$$911$$ −25.1769 −0.834148 −0.417074 0.908872i $$-0.636944\pi$$
−0.417074 + 0.908872i $$0.636944\pi$$
$$912$$ 0 0
$$913$$ 4.39230i 0.145364i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 49.1769 1.62396
$$918$$ 0 0
$$919$$ −15.7128 −0.518318 −0.259159 0.965835i $$-0.583445\pi$$
−0.259159 + 0.965835i $$0.583445\pi$$
$$920$$ 0 0
$$921$$ −65.0333 −2.14292
$$922$$ 0 0
$$923$$ −56.7846 −1.86909
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 9.80385i − 0.322001i
$$928$$ 0 0
$$929$$ −28.3923 −0.931521 −0.465761 0.884911i $$-0.654219\pi$$
−0.465761 + 0.884911i $$0.654219\pi$$
$$930$$ 0 0
$$931$$ − 30.7846i − 1.00892i
$$932$$ 0 0
$$933$$ 77.5692 2.53950
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 30.3923i − 0.992873i −0.868073 0.496437i $$-0.834641\pi$$
0.868073 0.496437i $$-0.165359\pi$$
$$938$$ 0 0
$$939$$ − 60.1051i − 1.96146i
$$940$$ 0 0
$$941$$ 20.7846i 0.677559i 0.940866 + 0.338779i $$0.110014\pi$$
−0.940866 + 0.338779i $$0.889986\pi$$
$$942$$ 0 0
$$943$$ − 20.7846i − 0.676840i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 56.1962 1.82613 0.913065 0.407815i $$-0.133709\pi$$
0.913065 + 0.407815i $$0.133709\pi$$
$$948$$ 0 0
$$949$$ − 22.1436i − 0.718811i
$$950$$ 0 0
$$951$$ −4.39230 −0.142430
$$952$$ 0 0
$$953$$ − 11.0718i − 0.358651i −0.983790 0.179325i $$-0.942609\pi$$
0.983790 0.179325i $$-0.0573915\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −4.39230 −0.141835
$$960$$ 0 0
$$961$$ −24.5692 −0.792555
$$962$$ 0 0
$$963$$ 59.2295 1.90864
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 2.19615i − 0.0706235i −0.999376 0.0353118i $$-0.988758\pi$$
0.999376 0.0353118i $$-0.0112424\pi$$
$$968$$ 0 0
$$969$$ 18.9282 0.608061
$$970$$ 0 0
$$971$$ 57.0333i 1.83029i 0.403129 + 0.915143i $$0.367923\pi$$
−0.403129 + 0.915143i $$0.632077\pi$$
$$972$$ 0 0
$$973$$ −47.3205 −1.51703
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 29.3205i − 0.938046i −0.883186 0.469023i $$-0.844606\pi$$
0.883186 0.469023i $$-0.155394\pi$$
$$978$$ 0 0
$$979$$ 44.7846i 1.43132i
$$980$$ 0 0
$$981$$ 57.7128i 1.84263i
$$982$$ 0 0
$$983$$ − 42.5885i − 1.35836i −0.733971 0.679180i $$-0.762335\pi$$
0.733971 0.679180i $$-0.237665\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −28.3923 −0.903737
$$988$$ 0 0
$$989$$ 0.430781i 0.0136980i
$$990$$ 0 0
$$991$$ 32.1051 1.01985 0.509926 0.860218i $$-0.329673\pi$$
0.509926 + 0.860218i $$0.329673\pi$$
$$992$$ 0 0
$$993$$ 72.1051i 2.28819i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 46.3923 1.46926 0.734630 0.678468i $$-0.237356\pi$$
0.734630 + 0.678468i $$0.237356\pi$$
$$998$$ 0 0
$$999$$ −24.0000 −0.759326
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 1600.2.f.i.1249.3 4
4.3 odd 2 1600.2.f.d.1249.2 4
5.2 odd 4 1600.2.d.h.801.1 4
5.3 odd 4 320.2.d.a.161.4 yes 4
5.4 even 2 1600.2.f.e.1249.2 4
8.3 odd 2 1600.2.f.h.1249.4 4
8.5 even 2 1600.2.f.e.1249.1 4
15.8 even 4 2880.2.k.e.1441.3 4
20.3 even 4 320.2.d.b.161.1 yes 4
20.7 even 4 1600.2.d.b.801.4 4
20.19 odd 2 1600.2.f.h.1249.3 4
40.3 even 4 320.2.d.b.161.4 yes 4
40.13 odd 4 320.2.d.a.161.1 4
40.19 odd 2 1600.2.f.d.1249.1 4
40.27 even 4 1600.2.d.b.801.1 4
40.29 even 2 inner 1600.2.f.i.1249.4 4
40.37 odd 4 1600.2.d.h.801.4 4
60.23 odd 4 2880.2.k.l.1441.4 4
80.3 even 4 1280.2.a.m.1.2 2
80.13 odd 4 1280.2.a.b.1.1 2
80.27 even 4 6400.2.a.ck.1.2 2
80.37 odd 4 6400.2.a.y.1.1 2
80.43 even 4 1280.2.a.c.1.1 2
80.53 odd 4 1280.2.a.p.1.2 2
80.67 even 4 6400.2.a.bf.1.1 2
80.77 odd 4 6400.2.a.cd.1.2 2
120.53 even 4 2880.2.k.e.1441.1 4
120.83 odd 4 2880.2.k.l.1441.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.1 4 40.13 odd 4
320.2.d.a.161.4 yes 4 5.3 odd 4
320.2.d.b.161.1 yes 4 20.3 even 4
320.2.d.b.161.4 yes 4 40.3 even 4
1280.2.a.b.1.1 2 80.13 odd 4
1280.2.a.c.1.1 2 80.43 even 4
1280.2.a.m.1.2 2 80.3 even 4
1280.2.a.p.1.2 2 80.53 odd 4
1600.2.d.b.801.1 4 40.27 even 4
1600.2.d.b.801.4 4 20.7 even 4
1600.2.d.h.801.1 4 5.2 odd 4
1600.2.d.h.801.4 4 40.37 odd 4
1600.2.f.d.1249.1 4 40.19 odd 2
1600.2.f.d.1249.2 4 4.3 odd 2
1600.2.f.e.1249.1 4 8.5 even 2
1600.2.f.e.1249.2 4 5.4 even 2
1600.2.f.h.1249.3 4 20.19 odd 2
1600.2.f.h.1249.4 4 8.3 odd 2
1600.2.f.i.1249.3 4 1.1 even 1 trivial
1600.2.f.i.1249.4 4 40.29 even 2 inner
2880.2.k.e.1441.1 4 120.53 even 4
2880.2.k.e.1441.3 4 15.8 even 4
2880.2.k.l.1441.2 4 120.83 odd 4
2880.2.k.l.1441.4 4 60.23 odd 4
6400.2.a.y.1.1 2 80.37 odd 4
6400.2.a.bf.1.1 2 80.67 even 4
6400.2.a.cd.1.2 2 80.77 odd 4
6400.2.a.ck.1.2 2 80.27 even 4