Properties

 Label 7728.2.a.cg.1.4 Level $7728$ Weight $2$ Character 7728.1 Self dual yes Analytic conductor $61.708$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} - 13 x^{4} + 7 x^{3} + 31 x^{2} - 17 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 3864) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$3.52361$$ of defining polynomial Character $$\chi$$ $$=$$ 7728.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.16465 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.16465 q^{5} +1.00000 q^{7} +1.00000 q^{9} -6.56356 q^{11} -1.77879 q^{13} -1.16465 q^{15} +4.94344 q^{17} -1.64398 q^{19} -1.00000 q^{21} -1.00000 q^{23} -3.64359 q^{25} -1.00000 q^{27} -0.671372 q^{29} -2.94344 q^{31} +6.56356 q^{33} +1.16465 q^{35} -0.671372 q^{37} +1.77879 q^{39} +3.91605 q^{41} +10.3156 q^{43} +1.16465 q^{45} -8.81177 q^{47} +1.00000 q^{49} -4.94344 q^{51} +5.05684 q^{53} -7.64426 q^{55} +1.64398 q^{57} +3.05684 q^{59} -12.2078 q^{61} +1.00000 q^{63} -2.07167 q^{65} -6.98566 q^{67} +1.00000 q^{69} +9.62041 q^{71} +9.47894 q^{73} +3.64359 q^{75} -6.56356 q^{77} +9.48029 q^{79} +1.00000 q^{81} +0.614137 q^{83} +5.75738 q^{85} +0.671372 q^{87} -1.08356 q^{89} -1.77879 q^{91} +2.94344 q^{93} -1.91466 q^{95} +0.463829 q^{97} -6.56356 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} + O(q^{10})$$ $$6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99} + O(q^{100})$$

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
$$5$$ 1.16465 0.520848 0.260424 0.965494i $$-0.416138\pi$$
0.260424 + 0.965494i $$0.416138\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −6.56356 −1.97899 −0.989494 0.144571i $$-0.953820\pi$$
−0.989494 + 0.144571i $$0.953820\pi$$
$$12$$ 0 0
$$13$$ −1.77879 −0.493347 −0.246674 0.969099i $$-0.579338\pi$$
−0.246674 + 0.969099i $$0.579338\pi$$
$$14$$ 0 0
$$15$$ −1.16465 −0.300712
$$16$$ 0 0
$$17$$ 4.94344 1.19896 0.599480 0.800390i $$-0.295374\pi$$
0.599480 + 0.800390i $$0.295374\pi$$
$$18$$ 0 0
$$19$$ −1.64398 −0.377155 −0.188577 0.982058i $$-0.560388\pi$$
−0.188577 + 0.982058i $$0.560388\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ −3.64359 −0.728717
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −0.671372 −0.124671 −0.0623353 0.998055i $$-0.519855\pi$$
−0.0623353 + 0.998055i $$0.519855\pi$$
$$30$$ 0 0
$$31$$ −2.94344 −0.528657 −0.264329 0.964433i $$-0.585150\pi$$
−0.264329 + 0.964433i $$0.585150\pi$$
$$32$$ 0 0
$$33$$ 6.56356 1.14257
$$34$$ 0 0
$$35$$ 1.16465 0.196862
$$36$$ 0 0
$$37$$ −0.671372 −0.110373 −0.0551865 0.998476i $$-0.517575\pi$$
−0.0551865 + 0.998476i $$0.517575\pi$$
$$38$$ 0 0
$$39$$ 1.77879 0.284834
$$40$$ 0 0
$$41$$ 3.91605 0.611584 0.305792 0.952098i $$-0.401079\pi$$
0.305792 + 0.952098i $$0.401079\pi$$
$$42$$ 0 0
$$43$$ 10.3156 1.57312 0.786560 0.617514i $$-0.211860\pi$$
0.786560 + 0.617514i $$0.211860\pi$$
$$44$$ 0 0
$$45$$ 1.16465 0.173616
$$46$$ 0 0
$$47$$ −8.81177 −1.28533 −0.642665 0.766148i $$-0.722171\pi$$
−0.642665 + 0.766148i $$0.722171\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −4.94344 −0.692220
$$52$$ 0 0
$$53$$ 5.05684 0.694611 0.347305 0.937752i $$-0.387097\pi$$
0.347305 + 0.937752i $$0.387097\pi$$
$$54$$ 0 0
$$55$$ −7.64426 −1.03075
$$56$$ 0 0
$$57$$ 1.64398 0.217751
$$58$$ 0 0
$$59$$ 3.05684 0.397967 0.198983 0.980003i $$-0.436236\pi$$
0.198983 + 0.980003i $$0.436236\pi$$
$$60$$ 0 0
$$61$$ −12.2078 −1.56305 −0.781526 0.623873i $$-0.785558\pi$$
−0.781526 + 0.623873i $$0.785558\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ −2.07167 −0.256959
$$66$$ 0 0
$$67$$ −6.98566 −0.853434 −0.426717 0.904385i $$-0.640330\pi$$
−0.426717 + 0.904385i $$0.640330\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 9.62041 1.14173 0.570866 0.821043i $$-0.306608\pi$$
0.570866 + 0.821043i $$0.306608\pi$$
$$72$$ 0 0
$$73$$ 9.47894 1.10943 0.554713 0.832042i $$-0.312828\pi$$
0.554713 + 0.832042i $$0.312828\pi$$
$$74$$ 0 0
$$75$$ 3.64359 0.420725
$$76$$ 0 0
$$77$$ −6.56356 −0.747987
$$78$$ 0 0
$$79$$ 9.48029 1.06662 0.533308 0.845921i $$-0.320949\pi$$
0.533308 + 0.845921i $$0.320949\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0.614137 0.0674103 0.0337051 0.999432i $$-0.489269\pi$$
0.0337051 + 0.999432i $$0.489269\pi$$
$$84$$ 0 0
$$85$$ 5.75738 0.624476
$$86$$ 0 0
$$87$$ 0.671372 0.0719786
$$88$$ 0 0
$$89$$ −1.08356 −0.114857 −0.0574285 0.998350i $$-0.518290\pi$$
−0.0574285 + 0.998350i $$0.518290\pi$$
$$90$$ 0 0
$$91$$ −1.77879 −0.186468
$$92$$ 0 0
$$93$$ 2.94344 0.305220
$$94$$ 0 0
$$95$$ −1.91466 −0.196440
$$96$$ 0 0
$$97$$ 0.463829 0.0470947 0.0235473 0.999723i $$-0.492504\pi$$
0.0235473 + 0.999723i $$0.492504\pi$$
$$98$$ 0 0
$$99$$ −6.56356 −0.659663
$$100$$ 0 0
$$101$$ 11.3669 1.13105 0.565524 0.824732i $$-0.308674\pi$$
0.565524 + 0.824732i $$0.308674\pi$$
$$102$$ 0 0
$$103$$ 15.3486 1.51234 0.756172 0.654373i $$-0.227067\pi$$
0.756172 + 0.654373i $$0.227067\pi$$
$$104$$ 0 0
$$105$$ −1.16465 −0.113658
$$106$$ 0 0
$$107$$ −15.2911 −1.47825 −0.739123 0.673570i $$-0.764760\pi$$
−0.739123 + 0.673570i $$0.764760\pi$$
$$108$$ 0 0
$$109$$ 17.5797 1.68383 0.841917 0.539607i $$-0.181427\pi$$
0.841917 + 0.539607i $$0.181427\pi$$
$$110$$ 0 0
$$111$$ 0.671372 0.0637238
$$112$$ 0 0
$$113$$ 8.29043 0.779898 0.389949 0.920836i $$-0.372493\pi$$
0.389949 + 0.920836i $$0.372493\pi$$
$$114$$ 0 0
$$115$$ −1.16465 −0.108604
$$116$$ 0 0
$$117$$ −1.77879 −0.164449
$$118$$ 0 0
$$119$$ 4.94344 0.453164
$$120$$ 0 0
$$121$$ 32.0804 2.91640
$$122$$ 0 0
$$123$$ −3.91605 −0.353098
$$124$$ 0 0
$$125$$ −10.0668 −0.900399
$$126$$ 0 0
$$127$$ 5.34413 0.474215 0.237107 0.971483i $$-0.423801\pi$$
0.237107 + 0.971483i $$0.423801\pi$$
$$128$$ 0 0
$$129$$ −10.3156 −0.908241
$$130$$ 0 0
$$131$$ −8.46107 −0.739247 −0.369623 0.929182i $$-0.620513\pi$$
−0.369623 + 0.929182i $$0.620513\pi$$
$$132$$ 0 0
$$133$$ −1.64398 −0.142551
$$134$$ 0 0
$$135$$ −1.16465 −0.100237
$$136$$ 0 0
$$137$$ −5.25879 −0.449289 −0.224644 0.974441i $$-0.572122\pi$$
−0.224644 + 0.974441i $$0.572122\pi$$
$$138$$ 0 0
$$139$$ 5.96248 0.505731 0.252865 0.967502i $$-0.418627\pi$$
0.252865 + 0.967502i $$0.418627\pi$$
$$140$$ 0 0
$$141$$ 8.81177 0.742085
$$142$$ 0 0
$$143$$ 11.6752 0.976328
$$144$$ 0 0
$$145$$ −0.781914 −0.0649345
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ 0 0
$$149$$ 10.5397 0.863446 0.431723 0.902006i $$-0.357906\pi$$
0.431723 + 0.902006i $$0.357906\pi$$
$$150$$ 0 0
$$151$$ 8.55080 0.695854 0.347927 0.937522i $$-0.386886\pi$$
0.347927 + 0.937522i $$0.386886\pi$$
$$152$$ 0 0
$$153$$ 4.94344 0.399653
$$154$$ 0 0
$$155$$ −3.42808 −0.275350
$$156$$ 0 0
$$157$$ 5.41258 0.431971 0.215985 0.976397i $$-0.430704\pi$$
0.215985 + 0.976397i $$0.430704\pi$$
$$158$$ 0 0
$$159$$ −5.05684 −0.401034
$$160$$ 0 0
$$161$$ −1.00000 −0.0788110
$$162$$ 0 0
$$163$$ −21.1426 −1.65602 −0.828009 0.560715i $$-0.810527\pi$$
−0.828009 + 0.560715i $$0.810527\pi$$
$$164$$ 0 0
$$165$$ 7.64426 0.595105
$$166$$ 0 0
$$167$$ 2.35602 0.182314 0.0911571 0.995837i $$-0.470943\pi$$
0.0911571 + 0.995837i $$0.470943\pi$$
$$168$$ 0 0
$$169$$ −9.83591 −0.756609
$$170$$ 0 0
$$171$$ −1.64398 −0.125718
$$172$$ 0 0
$$173$$ 1.85438 0.140986 0.0704931 0.997512i $$-0.477543\pi$$
0.0704931 + 0.997512i $$0.477543\pi$$
$$174$$ 0 0
$$175$$ −3.64359 −0.275429
$$176$$ 0 0
$$177$$ −3.05684 −0.229766
$$178$$ 0 0
$$179$$ 6.40490 0.478725 0.239362 0.970930i $$-0.423062\pi$$
0.239362 + 0.970930i $$0.423062\pi$$
$$180$$ 0 0
$$181$$ −12.9354 −0.961478 −0.480739 0.876864i $$-0.659632\pi$$
−0.480739 + 0.876864i $$0.659632\pi$$
$$182$$ 0 0
$$183$$ 12.2078 0.902428
$$184$$ 0 0
$$185$$ −0.781914 −0.0574875
$$186$$ 0 0
$$187$$ −32.4466 −2.37273
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −1.87403 −0.135600 −0.0678001 0.997699i $$-0.521598\pi$$
−0.0678001 + 0.997699i $$0.521598\pi$$
$$192$$ 0 0
$$193$$ 24.9062 1.79279 0.896394 0.443259i $$-0.146178\pi$$
0.896394 + 0.443259i $$0.146178\pi$$
$$194$$ 0 0
$$195$$ 2.07167 0.148355
$$196$$ 0 0
$$197$$ −1.41396 −0.100741 −0.0503704 0.998731i $$-0.516040\pi$$
−0.0503704 + 0.998731i $$0.516040\pi$$
$$198$$ 0 0
$$199$$ 2.92556 0.207388 0.103694 0.994609i $$-0.466934\pi$$
0.103694 + 0.994609i $$0.466934\pi$$
$$200$$ 0 0
$$201$$ 6.98566 0.492730
$$202$$ 0 0
$$203$$ −0.671372 −0.0471211
$$204$$ 0 0
$$205$$ 4.56083 0.318542
$$206$$ 0 0
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ 10.7904 0.746385
$$210$$ 0 0
$$211$$ 6.56356 0.451854 0.225927 0.974144i $$-0.427459\pi$$
0.225927 + 0.974144i $$0.427459\pi$$
$$212$$ 0 0
$$213$$ −9.62041 −0.659179
$$214$$ 0 0
$$215$$ 12.0141 0.819356
$$216$$ 0 0
$$217$$ −2.94344 −0.199814
$$218$$ 0 0
$$219$$ −9.47894 −0.640527
$$220$$ 0 0
$$221$$ −8.79333 −0.591504
$$222$$ 0 0
$$223$$ −14.5652 −0.975358 −0.487679 0.873023i $$-0.662156\pi$$
−0.487679 + 0.873023i $$0.662156\pi$$
$$224$$ 0 0
$$225$$ −3.64359 −0.242906
$$226$$ 0 0
$$227$$ 18.8153 1.24882 0.624408 0.781099i $$-0.285340\pi$$
0.624408 + 0.781099i $$0.285340\pi$$
$$228$$ 0 0
$$229$$ 9.18899 0.607226 0.303613 0.952796i $$-0.401807\pi$$
0.303613 + 0.952796i $$0.401807\pi$$
$$230$$ 0 0
$$231$$ 6.56356 0.431851
$$232$$ 0 0
$$233$$ −29.8388 −1.95481 −0.977403 0.211383i $$-0.932203\pi$$
−0.977403 + 0.211383i $$0.932203\pi$$
$$234$$ 0 0
$$235$$ −10.2626 −0.669461
$$236$$ 0 0
$$237$$ −9.48029 −0.615811
$$238$$ 0 0
$$239$$ −8.87254 −0.573917 −0.286958 0.957943i $$-0.592644\pi$$
−0.286958 + 0.957943i $$0.592644\pi$$
$$240$$ 0 0
$$241$$ 29.5203 1.90157 0.950786 0.309849i $$-0.100278\pi$$
0.950786 + 0.309849i $$0.100278\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 1.16465 0.0744068
$$246$$ 0 0
$$247$$ 2.92429 0.186068
$$248$$ 0 0
$$249$$ −0.614137 −0.0389194
$$250$$ 0 0
$$251$$ −13.7833 −0.869993 −0.434996 0.900432i $$-0.643250\pi$$
−0.434996 + 0.900432i $$0.643250\pi$$
$$252$$ 0 0
$$253$$ 6.56356 0.412648
$$254$$ 0 0
$$255$$ −5.75738 −0.360541
$$256$$ 0 0
$$257$$ −17.2531 −1.07622 −0.538109 0.842875i $$-0.680861\pi$$
−0.538109 + 0.842875i $$0.680861\pi$$
$$258$$ 0 0
$$259$$ −0.671372 −0.0417170
$$260$$ 0 0
$$261$$ −0.671372 −0.0415569
$$262$$ 0 0
$$263$$ 22.7629 1.40362 0.701811 0.712363i $$-0.252375\pi$$
0.701811 + 0.712363i $$0.252375\pi$$
$$264$$ 0 0
$$265$$ 5.88946 0.361787
$$266$$ 0 0
$$267$$ 1.08356 0.0663128
$$268$$ 0 0
$$269$$ −4.80332 −0.292864 −0.146432 0.989221i $$-0.546779\pi$$
−0.146432 + 0.989221i $$0.546779\pi$$
$$270$$ 0 0
$$271$$ 11.8544 0.720103 0.360051 0.932933i $$-0.382759\pi$$
0.360051 + 0.932933i $$0.382759\pi$$
$$272$$ 0 0
$$273$$ 1.77879 0.107657
$$274$$ 0 0
$$275$$ 23.9149 1.44212
$$276$$ 0 0
$$277$$ 23.7654 1.42792 0.713962 0.700184i $$-0.246899\pi$$
0.713962 + 0.700184i $$0.246899\pi$$
$$278$$ 0 0
$$279$$ −2.94344 −0.176219
$$280$$ 0 0
$$281$$ 18.5534 1.10680 0.553402 0.832914i $$-0.313329\pi$$
0.553402 + 0.832914i $$0.313329\pi$$
$$282$$ 0 0
$$283$$ 29.0913 1.72930 0.864648 0.502378i $$-0.167541\pi$$
0.864648 + 0.502378i $$0.167541\pi$$
$$284$$ 0 0
$$285$$ 1.91466 0.113415
$$286$$ 0 0
$$287$$ 3.91605 0.231157
$$288$$ 0 0
$$289$$ 7.43759 0.437506
$$290$$ 0 0
$$291$$ −0.463829 −0.0271901
$$292$$ 0 0
$$293$$ −23.5422 −1.37535 −0.687675 0.726018i $$-0.741369\pi$$
−0.687675 + 0.726018i $$0.741369\pi$$
$$294$$ 0 0
$$295$$ 3.56016 0.207280
$$296$$ 0 0
$$297$$ 6.56356 0.380857
$$298$$ 0 0
$$299$$ 1.77879 0.102870
$$300$$ 0 0
$$301$$ 10.3156 0.594583
$$302$$ 0 0
$$303$$ −11.3669 −0.653010
$$304$$ 0 0
$$305$$ −14.2179 −0.814112
$$306$$ 0 0
$$307$$ 9.08511 0.518515 0.259257 0.965808i $$-0.416522\pi$$
0.259257 + 0.965808i $$0.416522\pi$$
$$308$$ 0 0
$$309$$ −15.3486 −0.873152
$$310$$ 0 0
$$311$$ 6.98628 0.396155 0.198078 0.980186i $$-0.436530\pi$$
0.198078 + 0.980186i $$0.436530\pi$$
$$312$$ 0 0
$$313$$ −34.9157 −1.97355 −0.986777 0.162087i $$-0.948178\pi$$
−0.986777 + 0.162087i $$0.948178\pi$$
$$314$$ 0 0
$$315$$ 1.16465 0.0656207
$$316$$ 0 0
$$317$$ 4.29043 0.240974 0.120487 0.992715i $$-0.461554\pi$$
0.120487 + 0.992715i $$0.461554\pi$$
$$318$$ 0 0
$$319$$ 4.40659 0.246722
$$320$$ 0 0
$$321$$ 15.2911 0.853466
$$322$$ 0 0
$$323$$ −8.12692 −0.452194
$$324$$ 0 0
$$325$$ 6.48117 0.359511
$$326$$ 0 0
$$327$$ −17.5797 −0.972162
$$328$$ 0 0
$$329$$ −8.81177 −0.485809
$$330$$ 0 0
$$331$$ 18.0319 0.991122 0.495561 0.868573i $$-0.334963\pi$$
0.495561 + 0.868573i $$0.334963\pi$$
$$332$$ 0 0
$$333$$ −0.671372 −0.0367910
$$334$$ 0 0
$$335$$ −8.13585 −0.444509
$$336$$ 0 0
$$337$$ 22.9618 1.25081 0.625404 0.780301i $$-0.284934\pi$$
0.625404 + 0.780301i $$0.284934\pi$$
$$338$$ 0 0
$$339$$ −8.29043 −0.450274
$$340$$ 0 0
$$341$$ 19.3195 1.04621
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 1.16465 0.0627027
$$346$$ 0 0
$$347$$ −10.7196 −0.575461 −0.287730 0.957711i $$-0.592901\pi$$
−0.287730 + 0.957711i $$0.592901\pi$$
$$348$$ 0 0
$$349$$ 20.6235 1.10395 0.551976 0.833860i $$-0.313874\pi$$
0.551976 + 0.833860i $$0.313874\pi$$
$$350$$ 0 0
$$351$$ 1.77879 0.0949447
$$352$$ 0 0
$$353$$ −0.999325 −0.0531887 −0.0265944 0.999646i $$-0.508466\pi$$
−0.0265944 + 0.999646i $$0.508466\pi$$
$$354$$ 0 0
$$355$$ 11.2044 0.594669
$$356$$ 0 0
$$357$$ −4.94344 −0.261635
$$358$$ 0 0
$$359$$ −19.7172 −1.04063 −0.520317 0.853973i $$-0.674186\pi$$
−0.520317 + 0.853973i $$0.674186\pi$$
$$360$$ 0 0
$$361$$ −16.2973 −0.857754
$$362$$ 0 0
$$363$$ −32.0804 −1.68378
$$364$$ 0 0
$$365$$ 11.0397 0.577842
$$366$$ 0 0
$$367$$ −16.9023 −0.882291 −0.441146 0.897436i $$-0.645428\pi$$
−0.441146 + 0.897436i $$0.645428\pi$$
$$368$$ 0 0
$$369$$ 3.91605 0.203861
$$370$$ 0 0
$$371$$ 5.05684 0.262538
$$372$$ 0 0
$$373$$ −21.3505 −1.10548 −0.552742 0.833352i $$-0.686419\pi$$
−0.552742 + 0.833352i $$0.686419\pi$$
$$374$$ 0 0
$$375$$ 10.0668 0.519846
$$376$$ 0 0
$$377$$ 1.19423 0.0615059
$$378$$ 0 0
$$379$$ −1.13552 −0.0583276 −0.0291638 0.999575i $$-0.509284\pi$$
−0.0291638 + 0.999575i $$0.509284\pi$$
$$380$$ 0 0
$$381$$ −5.34413 −0.273788
$$382$$ 0 0
$$383$$ 0.201599 0.0103013 0.00515063 0.999987i $$-0.498360\pi$$
0.00515063 + 0.999987i $$0.498360\pi$$
$$384$$ 0 0
$$385$$ −7.64426 −0.389588
$$386$$ 0 0
$$387$$ 10.3156 0.524373
$$388$$ 0 0
$$389$$ −7.86167 −0.398603 −0.199301 0.979938i $$-0.563867\pi$$
−0.199301 + 0.979938i $$0.563867\pi$$
$$390$$ 0 0
$$391$$ −4.94344 −0.250000
$$392$$ 0 0
$$393$$ 8.46107 0.426804
$$394$$ 0 0
$$395$$ 11.0412 0.555544
$$396$$ 0 0
$$397$$ −14.9506 −0.750348 −0.375174 0.926954i $$-0.622417\pi$$
−0.375174 + 0.926954i $$0.622417\pi$$
$$398$$ 0 0
$$399$$ 1.64398 0.0823020
$$400$$ 0 0
$$401$$ 21.9652 1.09689 0.548445 0.836187i $$-0.315220\pi$$
0.548445 + 0.836187i $$0.315220\pi$$
$$402$$ 0 0
$$403$$ 5.23576 0.260812
$$404$$ 0 0
$$405$$ 1.16465 0.0578720
$$406$$ 0 0
$$407$$ 4.40659 0.218427
$$408$$ 0 0
$$409$$ 13.4334 0.664237 0.332118 0.943238i $$-0.392237\pi$$
0.332118 + 0.943238i $$0.392237\pi$$
$$410$$ 0 0
$$411$$ 5.25879 0.259397
$$412$$ 0 0
$$413$$ 3.05684 0.150417
$$414$$ 0 0
$$415$$ 0.715255 0.0351105
$$416$$ 0 0
$$417$$ −5.96248 −0.291984
$$418$$ 0 0
$$419$$ 17.6348 0.861518 0.430759 0.902467i $$-0.358246\pi$$
0.430759 + 0.902467i $$0.358246\pi$$
$$420$$ 0 0
$$421$$ −21.4428 −1.04506 −0.522528 0.852622i $$-0.675011\pi$$
−0.522528 + 0.852622i $$0.675011\pi$$
$$422$$ 0 0
$$423$$ −8.81177 −0.428443
$$424$$ 0 0
$$425$$ −18.0119 −0.873703
$$426$$ 0 0
$$427$$ −12.2078 −0.590778
$$428$$ 0 0
$$429$$ −11.6752 −0.563683
$$430$$ 0 0
$$431$$ −14.5274 −0.699760 −0.349880 0.936795i $$-0.613778\pi$$
−0.349880 + 0.936795i $$0.613778\pi$$
$$432$$ 0 0
$$433$$ 9.11498 0.438038 0.219019 0.975721i $$-0.429714\pi$$
0.219019 + 0.975721i $$0.429714\pi$$
$$434$$ 0 0
$$435$$ 0.781914 0.0374899
$$436$$ 0 0
$$437$$ 1.64398 0.0786422
$$438$$ 0 0
$$439$$ −21.9574 −1.04797 −0.523986 0.851727i $$-0.675556\pi$$
−0.523986 + 0.851727i $$0.675556\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −8.69523 −0.413123 −0.206561 0.978434i $$-0.566227\pi$$
−0.206561 + 0.978434i $$0.566227\pi$$
$$444$$ 0 0
$$445$$ −1.26197 −0.0598231
$$446$$ 0 0
$$447$$ −10.5397 −0.498511
$$448$$ 0 0
$$449$$ −22.3569 −1.05509 −0.527543 0.849528i $$-0.676887\pi$$
−0.527543 + 0.849528i $$0.676887\pi$$
$$450$$ 0 0
$$451$$ −25.7032 −1.21032
$$452$$ 0 0
$$453$$ −8.55080 −0.401751
$$454$$ 0 0
$$455$$ −2.07167 −0.0971213
$$456$$ 0 0
$$457$$ −3.14429 −0.147084 −0.0735418 0.997292i $$-0.523430\pi$$
−0.0735418 + 0.997292i $$0.523430\pi$$
$$458$$ 0 0
$$459$$ −4.94344 −0.230740
$$460$$ 0 0
$$461$$ 22.0935 1.02900 0.514499 0.857491i $$-0.327978\pi$$
0.514499 + 0.857491i $$0.327978\pi$$
$$462$$ 0 0
$$463$$ 20.2259 0.939976 0.469988 0.882673i $$-0.344258\pi$$
0.469988 + 0.882673i $$0.344258\pi$$
$$464$$ 0 0
$$465$$ 3.42808 0.158973
$$466$$ 0 0
$$467$$ 40.3656 1.86790 0.933948 0.357410i $$-0.116340\pi$$
0.933948 + 0.357410i $$0.116340\pi$$
$$468$$ 0 0
$$469$$ −6.98566 −0.322568
$$470$$ 0 0
$$471$$ −5.41258 −0.249399
$$472$$ 0 0
$$473$$ −67.7073 −3.11319
$$474$$ 0 0
$$475$$ 5.98999 0.274839
$$476$$ 0 0
$$477$$ 5.05684 0.231537
$$478$$ 0 0
$$479$$ 36.9367 1.68768 0.843841 0.536594i $$-0.180289\pi$$
0.843841 + 0.536594i $$0.180289\pi$$
$$480$$ 0 0
$$481$$ 1.19423 0.0544522
$$482$$ 0 0
$$483$$ 1.00000 0.0455016
$$484$$ 0 0
$$485$$ 0.540199 0.0245292
$$486$$ 0 0
$$487$$ 30.0616 1.36222 0.681111 0.732180i $$-0.261497\pi$$
0.681111 + 0.732180i $$0.261497\pi$$
$$488$$ 0 0
$$489$$ 21.1426 0.956103
$$490$$ 0 0
$$491$$ 1.88374 0.0850119 0.0425059 0.999096i $$-0.486466\pi$$
0.0425059 + 0.999096i $$0.486466\pi$$
$$492$$ 0 0
$$493$$ −3.31889 −0.149475
$$494$$ 0 0
$$495$$ −7.64426 −0.343584
$$496$$ 0 0
$$497$$ 9.62041 0.431534
$$498$$ 0 0
$$499$$ −22.0103 −0.985316 −0.492658 0.870223i $$-0.663975\pi$$
−0.492658 + 0.870223i $$0.663975\pi$$
$$500$$ 0 0
$$501$$ −2.35602 −0.105259
$$502$$ 0 0
$$503$$ 41.7473 1.86142 0.930711 0.365757i $$-0.119190\pi$$
0.930711 + 0.365757i $$0.119190\pi$$
$$504$$ 0 0
$$505$$ 13.2385 0.589103
$$506$$ 0 0
$$507$$ 9.83591 0.436828
$$508$$ 0 0
$$509$$ 32.2029 1.42737 0.713684 0.700468i $$-0.247025\pi$$
0.713684 + 0.700468i $$0.247025\pi$$
$$510$$ 0 0
$$511$$ 9.47894 0.419323
$$512$$ 0 0
$$513$$ 1.64398 0.0725835
$$514$$ 0 0
$$515$$ 17.8758 0.787701
$$516$$ 0 0
$$517$$ 57.8366 2.54365
$$518$$ 0 0
$$519$$ −1.85438 −0.0813984
$$520$$ 0 0
$$521$$ 42.3487 1.85533 0.927665 0.373415i $$-0.121813\pi$$
0.927665 + 0.373415i $$0.121813\pi$$
$$522$$ 0 0
$$523$$ 30.1185 1.31699 0.658494 0.752586i $$-0.271194\pi$$
0.658494 + 0.752586i $$0.271194\pi$$
$$524$$ 0 0
$$525$$ 3.64359 0.159019
$$526$$ 0 0
$$527$$ −14.5507 −0.633839
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 3.05684 0.132656
$$532$$ 0 0
$$533$$ −6.96582 −0.301723
$$534$$ 0 0
$$535$$ −17.8088 −0.769942
$$536$$ 0 0
$$537$$ −6.40490 −0.276392
$$538$$ 0 0
$$539$$ −6.56356 −0.282713
$$540$$ 0 0
$$541$$ 17.6549 0.759042 0.379521 0.925183i $$-0.376089\pi$$
0.379521 + 0.925183i $$0.376089\pi$$
$$542$$ 0 0
$$543$$ 12.9354 0.555110
$$544$$ 0 0
$$545$$ 20.4743 0.877021
$$546$$ 0 0
$$547$$ 9.31226 0.398164 0.199082 0.979983i $$-0.436204\pi$$
0.199082 + 0.979983i $$0.436204\pi$$
$$548$$ 0 0
$$549$$ −12.2078 −0.521017
$$550$$ 0 0
$$551$$ 1.10372 0.0470202
$$552$$ 0 0
$$553$$ 9.48029 0.403143
$$554$$ 0 0
$$555$$ 0.781914 0.0331904
$$556$$ 0 0
$$557$$ −18.8575 −0.799018 −0.399509 0.916729i $$-0.630819\pi$$
−0.399509 + 0.916729i $$0.630819\pi$$
$$558$$ 0 0
$$559$$ −18.3493 −0.776094
$$560$$ 0 0
$$561$$ 32.4466 1.36990
$$562$$ 0 0
$$563$$ −11.5942 −0.488636 −0.244318 0.969695i $$-0.578564\pi$$
−0.244318 + 0.969695i $$0.578564\pi$$
$$564$$ 0 0
$$565$$ 9.65546 0.406208
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 27.8808 1.16882 0.584412 0.811457i $$-0.301325\pi$$
0.584412 + 0.811457i $$0.301325\pi$$
$$570$$ 0 0
$$571$$ 18.1923 0.761325 0.380663 0.924714i $$-0.375696\pi$$
0.380663 + 0.924714i $$0.375696\pi$$
$$572$$ 0 0
$$573$$ 1.87403 0.0782888
$$574$$ 0 0
$$575$$ 3.64359 0.151948
$$576$$ 0 0
$$577$$ −29.0990 −1.21141 −0.605704 0.795690i $$-0.707108\pi$$
−0.605704 + 0.795690i $$0.707108\pi$$
$$578$$ 0 0
$$579$$ −24.9062 −1.03507
$$580$$ 0 0
$$581$$ 0.614137 0.0254787
$$582$$ 0 0
$$583$$ −33.1909 −1.37463
$$584$$ 0 0
$$585$$ −2.07167 −0.0856529
$$586$$ 0 0
$$587$$ −10.1911 −0.420632 −0.210316 0.977633i $$-0.567449\pi$$
−0.210316 + 0.977633i $$0.567449\pi$$
$$588$$ 0 0
$$589$$ 4.83896 0.199386
$$590$$ 0 0
$$591$$ 1.41396 0.0581627
$$592$$ 0 0
$$593$$ 10.4629 0.429660 0.214830 0.976651i $$-0.431080\pi$$
0.214830 + 0.976651i $$0.431080\pi$$
$$594$$ 0 0
$$595$$ 5.75738 0.236030
$$596$$ 0 0
$$597$$ −2.92556 −0.119735
$$598$$ 0 0
$$599$$ 17.0401 0.696240 0.348120 0.937450i $$-0.386820\pi$$
0.348120 + 0.937450i $$0.386820\pi$$
$$600$$ 0 0
$$601$$ 24.0270 0.980082 0.490041 0.871699i $$-0.336982\pi$$
0.490041 + 0.871699i $$0.336982\pi$$
$$602$$ 0 0
$$603$$ −6.98566 −0.284478
$$604$$ 0 0
$$605$$ 37.3624 1.51900
$$606$$ 0 0
$$607$$ 17.2171 0.698819 0.349409 0.936970i $$-0.386382\pi$$
0.349409 + 0.936970i $$0.386382\pi$$
$$608$$ 0 0
$$609$$ 0.671372 0.0272054
$$610$$ 0 0
$$611$$ 15.6743 0.634113
$$612$$ 0 0
$$613$$ −3.65928 −0.147797 −0.0738985 0.997266i $$-0.523544\pi$$
−0.0738985 + 0.997266i $$0.523544\pi$$
$$614$$ 0 0
$$615$$ −4.56083 −0.183910
$$616$$ 0 0
$$617$$ 15.5059 0.624245 0.312123 0.950042i $$-0.398960\pi$$
0.312123 + 0.950042i $$0.398960\pi$$
$$618$$ 0 0
$$619$$ 27.4731 1.10424 0.552119 0.833765i $$-0.313819\pi$$
0.552119 + 0.833765i $$0.313819\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ −1.08356 −0.0434119
$$624$$ 0 0
$$625$$ 6.49367 0.259747
$$626$$ 0 0
$$627$$ −10.7904 −0.430926
$$628$$ 0 0
$$629$$ −3.31889 −0.132333
$$630$$ 0 0
$$631$$ −16.1579 −0.643237 −0.321618 0.946869i $$-0.604227\pi$$
−0.321618 + 0.946869i $$0.604227\pi$$
$$632$$ 0 0
$$633$$ −6.56356 −0.260878
$$634$$ 0 0
$$635$$ 6.22405 0.246994
$$636$$ 0 0
$$637$$ −1.77879 −0.0704782
$$638$$ 0 0
$$639$$ 9.62041 0.380577
$$640$$ 0 0
$$641$$ −15.1536 −0.598533 −0.299266 0.954170i $$-0.596742\pi$$
−0.299266 + 0.954170i $$0.596742\pi$$
$$642$$ 0 0
$$643$$ 16.0551 0.633150 0.316575 0.948567i $$-0.397467\pi$$
0.316575 + 0.948567i $$0.397467\pi$$
$$644$$ 0 0
$$645$$ −12.0141 −0.473055
$$646$$ 0 0
$$647$$ 35.5239 1.39659 0.698293 0.715812i $$-0.253943\pi$$
0.698293 + 0.715812i $$0.253943\pi$$
$$648$$ 0 0
$$649$$ −20.0638 −0.787572
$$650$$ 0 0
$$651$$ 2.94344 0.115362
$$652$$ 0 0
$$653$$ 15.5511 0.608560 0.304280 0.952583i $$-0.401584\pi$$
0.304280 + 0.952583i $$0.401584\pi$$
$$654$$ 0 0
$$655$$ −9.85419 −0.385035
$$656$$ 0 0
$$657$$ 9.47894 0.369809
$$658$$ 0 0
$$659$$ −33.5364 −1.30639 −0.653197 0.757188i $$-0.726573\pi$$
−0.653197 + 0.757188i $$0.726573\pi$$
$$660$$ 0 0
$$661$$ 30.6772 1.19321 0.596603 0.802536i $$-0.296517\pi$$
0.596603 + 0.802536i $$0.296517\pi$$
$$662$$ 0 0
$$663$$ 8.79333 0.341505
$$664$$ 0 0
$$665$$ −1.91466 −0.0742475
$$666$$ 0 0
$$667$$ 0.671372 0.0259956
$$668$$ 0 0
$$669$$ 14.5652 0.563123
$$670$$ 0 0
$$671$$ 80.1268 3.09326
$$672$$ 0 0
$$673$$ −2.11360 −0.0814734 −0.0407367 0.999170i $$-0.512970\pi$$
−0.0407367 + 0.999170i $$0.512970\pi$$
$$674$$ 0 0
$$675$$ 3.64359 0.140242
$$676$$ 0 0
$$677$$ −37.0481 −1.42388 −0.711938 0.702242i $$-0.752182\pi$$
−0.711938 + 0.702242i $$0.752182\pi$$
$$678$$ 0 0
$$679$$ 0.463829 0.0178001
$$680$$ 0 0
$$681$$ −18.8153 −0.721004
$$682$$ 0 0
$$683$$ −4.12692 −0.157912 −0.0789561 0.996878i $$-0.525159\pi$$
−0.0789561 + 0.996878i $$0.525159\pi$$
$$684$$ 0 0
$$685$$ −6.12466 −0.234011
$$686$$ 0 0
$$687$$ −9.18899 −0.350582
$$688$$ 0 0
$$689$$ −8.99505 −0.342684
$$690$$ 0 0
$$691$$ 22.4364 0.853520 0.426760 0.904365i $$-0.359655\pi$$
0.426760 + 0.904365i $$0.359655\pi$$
$$692$$ 0 0
$$693$$ −6.56356 −0.249329
$$694$$ 0 0
$$695$$ 6.94420 0.263409
$$696$$ 0 0
$$697$$ 19.3587 0.733265
$$698$$ 0 0
$$699$$ 29.8388 1.12861
$$700$$ 0 0
$$701$$ −47.2272 −1.78375 −0.891874 0.452284i $$-0.850609\pi$$
−0.891874 + 0.452284i $$0.850609\pi$$
$$702$$ 0 0
$$703$$ 1.10372 0.0416277
$$704$$ 0 0
$$705$$ 10.2626 0.386514
$$706$$ 0 0
$$707$$ 11.3669 0.427496
$$708$$ 0 0
$$709$$ 0.595776 0.0223748 0.0111874 0.999937i $$-0.496439\pi$$
0.0111874 + 0.999937i $$0.496439\pi$$
$$710$$ 0 0
$$711$$ 9.48029 0.355538
$$712$$ 0 0
$$713$$ 2.94344 0.110233
$$714$$ 0 0
$$715$$ 13.5975 0.508519
$$716$$ 0 0
$$717$$ 8.87254 0.331351
$$718$$ 0 0
$$719$$ −22.2121 −0.828370 −0.414185 0.910193i $$-0.635933\pi$$
−0.414185 + 0.910193i $$0.635933\pi$$
$$720$$ 0 0
$$721$$ 15.3486 0.571612
$$722$$ 0 0
$$723$$ −29.5203 −1.09787
$$724$$ 0 0
$$725$$ 2.44620 0.0908497
$$726$$ 0 0
$$727$$ 21.6161 0.801696 0.400848 0.916144i $$-0.368715\pi$$
0.400848 + 0.916144i $$0.368715\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 50.9947 1.88611
$$732$$ 0 0
$$733$$ −28.8306 −1.06488 −0.532442 0.846467i $$-0.678725\pi$$
−0.532442 + 0.846467i $$0.678725\pi$$
$$734$$ 0 0
$$735$$ −1.16465 −0.0429588
$$736$$ 0 0
$$737$$ 45.8508 1.68894
$$738$$ 0 0
$$739$$ 52.5595 1.93343 0.966716 0.255853i $$-0.0823562\pi$$
0.966716 + 0.255853i $$0.0823562\pi$$
$$740$$ 0 0
$$741$$ −2.92429 −0.107427
$$742$$ 0 0
$$743$$ −5.00834 −0.183738 −0.0918691 0.995771i $$-0.529284\pi$$
−0.0918691 + 0.995771i $$0.529284\pi$$
$$744$$ 0 0
$$745$$ 12.2751 0.449724
$$746$$ 0 0
$$747$$ 0.614137 0.0224701
$$748$$ 0 0
$$749$$ −15.2911 −0.558725
$$750$$ 0 0
$$751$$ 1.62343 0.0592399 0.0296199 0.999561i $$-0.490570\pi$$
0.0296199 + 0.999561i $$0.490570\pi$$
$$752$$ 0 0
$$753$$ 13.7833 0.502291
$$754$$ 0 0
$$755$$ 9.95870 0.362434
$$756$$ 0 0
$$757$$ 19.8414 0.721147 0.360574 0.932731i $$-0.382581\pi$$
0.360574 + 0.932731i $$0.382581\pi$$
$$758$$ 0 0
$$759$$ −6.56356 −0.238242
$$760$$ 0 0
$$761$$ −25.9904 −0.942152 −0.471076 0.882093i $$-0.656134\pi$$
−0.471076 + 0.882093i $$0.656134\pi$$
$$762$$ 0 0
$$763$$ 17.5797 0.636429
$$764$$ 0 0
$$765$$ 5.75738 0.208159
$$766$$ 0 0
$$767$$ −5.43748 −0.196336
$$768$$ 0 0
$$769$$ 50.8662 1.83428 0.917142 0.398561i $$-0.130490\pi$$
0.917142 + 0.398561i $$0.130490\pi$$
$$770$$ 0 0
$$771$$ 17.2531 0.621355
$$772$$ 0 0
$$773$$ −41.0843 −1.47770 −0.738850 0.673870i $$-0.764631\pi$$
−0.738850 + 0.673870i $$0.764631\pi$$
$$774$$ 0 0
$$775$$ 10.7247 0.385242
$$776$$ 0 0
$$777$$ 0.671372 0.0240853
$$778$$ 0 0
$$779$$ −6.43791 −0.230662
$$780$$ 0 0
$$781$$ −63.1441 −2.25947
$$782$$ 0 0
$$783$$ 0.671372 0.0239929
$$784$$ 0 0
$$785$$ 6.30377 0.224991
$$786$$ 0 0
$$787$$ −7.53400 −0.268558 −0.134279 0.990944i $$-0.542872\pi$$
−0.134279 + 0.990944i $$0.542872\pi$$
$$788$$ 0 0
$$789$$ −22.7629 −0.810382
$$790$$ 0 0
$$791$$ 8.29043 0.294774
$$792$$ 0 0
$$793$$ 21.7151 0.771127
$$794$$ 0 0
$$795$$ −5.88946 −0.208878
$$796$$ 0 0
$$797$$ 15.8202 0.560381 0.280190 0.959944i $$-0.409602\pi$$
0.280190 + 0.959944i $$0.409602\pi$$
$$798$$ 0 0
$$799$$ −43.5605 −1.54106
$$800$$ 0 0
$$801$$ −1.08356 −0.0382857
$$802$$ 0 0
$$803$$ −62.2156 −2.19554
$$804$$ 0 0
$$805$$ −1.16465 −0.0410486
$$806$$ 0 0
$$807$$ 4.80332 0.169085
$$808$$ 0 0
$$809$$ −17.7335 −0.623478 −0.311739 0.950168i $$-0.600911\pi$$
−0.311739 + 0.950168i $$0.600911\pi$$
$$810$$ 0 0
$$811$$ 8.01688 0.281511 0.140755 0.990044i $$-0.455047\pi$$
0.140755 + 0.990044i $$0.455047\pi$$
$$812$$ 0 0
$$813$$ −11.8544 −0.415751
$$814$$ 0 0
$$815$$ −24.6238 −0.862534
$$816$$ 0 0
$$817$$ −16.9587 −0.593310
$$818$$ 0 0
$$819$$ −1.77879 −0.0621559
$$820$$ 0 0
$$821$$ −20.9584 −0.731454 −0.365727 0.930722i $$-0.619180\pi$$
−0.365727 + 0.930722i $$0.619180\pi$$
$$822$$ 0 0
$$823$$ 45.0419 1.57006 0.785032 0.619456i $$-0.212647\pi$$
0.785032 + 0.619456i $$0.212647\pi$$
$$824$$ 0 0
$$825$$ −23.9149 −0.832611
$$826$$ 0 0
$$827$$ 12.1066 0.420987 0.210494 0.977595i $$-0.432493\pi$$
0.210494 + 0.977595i $$0.432493\pi$$
$$828$$ 0 0
$$829$$ −27.8045 −0.965688 −0.482844 0.875706i $$-0.660396\pi$$
−0.482844 + 0.875706i $$0.660396\pi$$
$$830$$ 0 0
$$831$$ −23.7654 −0.824413
$$832$$ 0 0
$$833$$ 4.94344 0.171280
$$834$$ 0 0
$$835$$ 2.74394 0.0949580
$$836$$ 0 0
$$837$$ 2.94344 0.101740
$$838$$ 0 0
$$839$$ 23.3694 0.806802 0.403401 0.915023i $$-0.367828\pi$$
0.403401 + 0.915023i $$0.367828\pi$$
$$840$$ 0 0
$$841$$ −28.5493 −0.984457
$$842$$ 0 0
$$843$$ −18.5534 −0.639014
$$844$$ 0 0
$$845$$ −11.4554 −0.394078
$$846$$ 0 0
$$847$$ 32.0804 1.10229
$$848$$ 0 0
$$849$$ −29.0913 −0.998410
$$850$$ 0 0
$$851$$ 0.671372 0.0230143
$$852$$ 0 0
$$853$$ −16.7158 −0.572339 −0.286170 0.958179i $$-0.592382\pi$$
−0.286170 + 0.958179i $$0.592382\pi$$
$$854$$ 0 0
$$855$$ −1.91466 −0.0654801
$$856$$ 0 0
$$857$$ −18.2207 −0.622408 −0.311204 0.950343i $$-0.600732\pi$$
−0.311204 + 0.950343i $$0.600732\pi$$
$$858$$ 0 0
$$859$$ 9.83298 0.335497 0.167748 0.985830i $$-0.446350\pi$$
0.167748 + 0.985830i $$0.446350\pi$$
$$860$$ 0 0
$$861$$ −3.91605 −0.133459
$$862$$ 0 0
$$863$$ 0.648383 0.0220712 0.0110356 0.999939i $$-0.496487\pi$$
0.0110356 + 0.999939i $$0.496487\pi$$
$$864$$ 0 0
$$865$$ 2.15971 0.0734324
$$866$$ 0 0
$$867$$ −7.43759 −0.252594
$$868$$ 0 0
$$869$$ −62.2245 −2.11082
$$870$$ 0 0
$$871$$ 12.4260 0.421039
$$872$$ 0 0
$$873$$ 0.463829 0.0156982
$$874$$ 0 0
$$875$$ −10.0668 −0.340319
$$876$$ 0 0
$$877$$ 51.2854 1.73179 0.865893 0.500229i $$-0.166751\pi$$
0.865893 + 0.500229i $$0.166751\pi$$
$$878$$ 0 0
$$879$$ 23.5422 0.794059
$$880$$ 0 0
$$881$$ −15.9911 −0.538755 −0.269378 0.963035i $$-0.586818\pi$$
−0.269378 + 0.963035i $$0.586818\pi$$
$$882$$ 0 0
$$883$$ 26.8120 0.902294 0.451147 0.892450i $$-0.351015\pi$$
0.451147 + 0.892450i $$0.351015\pi$$
$$884$$ 0 0
$$885$$ −3.56016 −0.119673
$$886$$ 0 0
$$887$$ 52.6201 1.76681 0.883404 0.468612i $$-0.155246\pi$$
0.883404 + 0.468612i $$0.155246\pi$$
$$888$$ 0 0
$$889$$ 5.34413 0.179236
$$890$$ 0 0
$$891$$ −6.56356 −0.219888
$$892$$ 0 0
$$893$$ 14.4864 0.484768
$$894$$ 0 0
$$895$$ 7.45947 0.249343
$$896$$ 0 0
$$897$$ −1.77879 −0.0593920
$$898$$ 0 0
$$899$$ 1.97614 0.0659081
$$900$$ 0 0
$$901$$ 24.9982 0.832811
$$902$$ 0 0
$$903$$ −10.3156 −0.343283
$$904$$ 0 0
$$905$$ −15.0652 −0.500784
$$906$$ 0 0
$$907$$ 11.3547 0.377028 0.188514 0.982071i $$-0.439633\pi$$
0.188514 + 0.982071i $$0.439633\pi$$
$$908$$ 0 0
$$909$$ 11.3669 0.377016
$$910$$ 0 0
$$911$$ −56.2394 −1.86329 −0.931646 0.363366i $$-0.881627\pi$$
−0.931646 + 0.363366i $$0.881627\pi$$
$$912$$ 0 0
$$913$$ −4.03093 −0.133404
$$914$$ 0 0
$$915$$ 14.2179 0.470028
$$916$$ 0 0
$$917$$ −8.46107 −0.279409
$$918$$ 0 0
$$919$$ −45.2603 −1.49300 −0.746500 0.665385i $$-0.768267\pi$$
−0.746500 + 0.665385i $$0.768267\pi$$
$$920$$ 0 0
$$921$$ −9.08511 −0.299365
$$922$$ 0 0
$$923$$ −17.1127 −0.563270
$$924$$ 0 0
$$925$$ 2.44620 0.0804307
$$926$$ 0 0
$$927$$ 15.3486 0.504115
$$928$$ 0 0
$$929$$ −50.0431 −1.64186 −0.820930 0.571028i $$-0.806545\pi$$
−0.820930 + 0.571028i $$0.806545\pi$$
$$930$$ 0 0
$$931$$ −1.64398 −0.0538793
$$932$$ 0 0
$$933$$ −6.98628 −0.228720
$$934$$ 0 0
$$935$$ −37.7889 −1.23583
$$936$$ 0 0
$$937$$ 30.4113 0.993493 0.496746 0.867896i $$-0.334528\pi$$
0.496746 + 0.867896i $$0.334528\pi$$
$$938$$ 0 0
$$939$$ 34.9157 1.13943
$$940$$ 0 0
$$941$$ −27.8445 −0.907706 −0.453853 0.891077i $$-0.649951\pi$$
−0.453853 + 0.891077i $$0.649951\pi$$
$$942$$ 0 0
$$943$$ −3.91605 −0.127524
$$944$$ 0 0
$$945$$ −1.16465 −0.0378861
$$946$$ 0 0
$$947$$ −4.96761 −0.161426 −0.0807129 0.996737i $$-0.525720\pi$$
−0.0807129 + 0.996737i $$0.525720\pi$$
$$948$$ 0 0
$$949$$ −16.8610 −0.547332
$$950$$ 0 0
$$951$$ −4.29043 −0.139127
$$952$$ 0 0
$$953$$ −37.1476 −1.20333 −0.601664 0.798749i $$-0.705495\pi$$
−0.601664 + 0.798749i $$0.705495\pi$$
$$954$$ 0 0
$$955$$ −2.18259 −0.0706270
$$956$$ 0 0
$$957$$ −4.40659 −0.142445
$$958$$ 0 0
$$959$$ −5.25879 −0.169815
$$960$$ 0 0
$$961$$ −22.3362 −0.720521
$$962$$ 0 0
$$963$$ −15.2911 −0.492749
$$964$$ 0 0
$$965$$ 29.0070 0.933769
$$966$$ 0 0
$$967$$ −30.9579 −0.995538 −0.497769 0.867310i $$-0.665847\pi$$
−0.497769 + 0.867310i $$0.665847\pi$$
$$968$$ 0 0
$$969$$ 8.12692 0.261074
$$970$$ 0 0
$$971$$ −29.6060 −0.950103 −0.475052 0.879958i $$-0.657571\pi$$
−0.475052 + 0.879958i $$0.657571\pi$$
$$972$$ 0 0
$$973$$ 5.96248 0.191148
$$974$$ 0 0
$$975$$ −6.48117 −0.207564
$$976$$ 0 0
$$977$$ −19.2597 −0.616172 −0.308086 0.951358i $$-0.599688\pi$$
−0.308086 + 0.951358i $$0.599688\pi$$
$$978$$ 0 0
$$979$$ 7.11201 0.227301
$$980$$ 0 0
$$981$$ 17.5797 0.561278
$$982$$ 0 0
$$983$$ 47.8490 1.52615 0.763073 0.646312i $$-0.223690\pi$$
0.763073 + 0.646312i $$0.223690\pi$$
$$984$$ 0 0
$$985$$ −1.64678 −0.0524706
$$986$$ 0 0
$$987$$ 8.81177 0.280482
$$988$$ 0 0
$$989$$ −10.3156 −0.328018
$$990$$ 0 0
$$991$$ −27.6629 −0.878741 −0.439371 0.898306i $$-0.644798\pi$$
−0.439371 + 0.898306i $$0.644798\pi$$
$$992$$ 0 0
$$993$$ −18.0319 −0.572224
$$994$$ 0 0
$$995$$ 3.40726 0.108017
$$996$$ 0 0
$$997$$ −11.7434 −0.371917 −0.185959 0.982558i $$-0.559539\pi$$
−0.185959 + 0.982558i $$0.559539\pi$$
$$998$$ 0 0
$$999$$ 0.671372 0.0212413
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.cg.1.4 6
4.3 odd 2 3864.2.a.x.1.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.x.1.4 6 4.3 odd 2
7728.2.a.cg.1.4 6 1.1 even 1 trivial