# Properties

 Label 8112.2.a.by.1.2 Level $8112$ Weight $2$ Character 8112.1 Self dual yes Analytic conductor $64.775$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8112 = 2^{4} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.7746461197$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 507) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.445042$$ of defining polynomial Character $$\chi$$ $$=$$ 8112.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.44504 q^{5} +3.44504 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.44504 q^{5} +3.44504 q^{7} +1.00000 q^{9} -5.18598 q^{11} +1.44504 q^{15} -0.753020 q^{17} +7.96077 q^{19} -3.44504 q^{21} +2.82908 q^{23} -2.91185 q^{25} -1.00000 q^{27} -3.91185 q^{29} -4.89977 q^{31} +5.18598 q^{33} -4.97823 q^{35} -6.24698 q^{37} -1.80194 q^{41} +7.09783 q^{43} -1.44504 q^{45} +10.5526 q^{47} +4.86831 q^{49} +0.753020 q^{51} -3.08815 q^{53} +7.49396 q^{55} -7.96077 q^{57} +1.87800 q^{59} +3.34481 q^{61} +3.44504 q^{63} -4.54288 q^{67} -2.82908 q^{69} -9.11960 q^{71} -2.95108 q^{73} +2.91185 q^{75} -17.8659 q^{77} +9.43296 q^{79} +1.00000 q^{81} -6.46681 q^{83} +1.08815 q^{85} +3.91185 q^{87} -1.15883 q^{89} +4.89977 q^{93} -11.5036 q^{95} -8.65817 q^{97} -5.18598 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} - 4q^{5} + 10q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} - 4q^{5} + 10q^{7} + 3q^{9} - q^{11} + 4q^{15} - 7q^{17} + 11q^{19} - 10q^{21} - 2q^{23} - 5q^{25} - 3q^{27} - 8q^{29} + 8q^{31} + q^{33} - 18q^{35} - 14q^{37} - q^{41} + 3q^{43} - 4q^{45} - 9q^{47} + 17q^{49} + 7q^{51} - 13q^{53} + 13q^{55} - 11q^{57} - 14q^{59} - 13q^{61} + 10q^{63} + 5q^{67} + 2q^{69} - 6q^{71} - 18q^{73} + 5q^{75} - 15q^{77} + 9q^{79} + 3q^{81} - 16q^{83} + 7q^{85} + 8q^{87} + 5q^{89} - 8q^{93} - 3q^{95} - 5q^{97} - q^{99} + O(q^{100})$$

## 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
$$5$$ −1.44504 −0.646242 −0.323121 0.946358i $$-0.604732\pi$$
−0.323121 + 0.946358i $$0.604732\pi$$
$$6$$ 0 0
$$7$$ 3.44504 1.30210 0.651052 0.759033i $$-0.274328\pi$$
0.651052 + 0.759033i $$0.274328\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.18598 −1.56363 −0.781816 0.623509i $$-0.785706\pi$$
−0.781816 + 0.623509i $$0.785706\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 1.44504 0.373108
$$16$$ 0 0
$$17$$ −0.753020 −0.182634 −0.0913171 0.995822i $$-0.529108\pi$$
−0.0913171 + 0.995822i $$0.529108\pi$$
$$18$$ 0 0
$$19$$ 7.96077 1.82633 0.913163 0.407594i $$-0.133632\pi$$
0.913163 + 0.407594i $$0.133632\pi$$
$$20$$ 0 0
$$21$$ −3.44504 −0.751770
$$22$$ 0 0
$$23$$ 2.82908 0.589905 0.294952 0.955512i $$-0.404696\pi$$
0.294952 + 0.955512i $$0.404696\pi$$
$$24$$ 0 0
$$25$$ −2.91185 −0.582371
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −3.91185 −0.726413 −0.363207 0.931709i $$-0.618318\pi$$
−0.363207 + 0.931709i $$0.618318\pi$$
$$30$$ 0 0
$$31$$ −4.89977 −0.880025 −0.440013 0.897992i $$-0.645026\pi$$
−0.440013 + 0.897992i $$0.645026\pi$$
$$32$$ 0 0
$$33$$ 5.18598 0.902763
$$34$$ 0 0
$$35$$ −4.97823 −0.841474
$$36$$ 0 0
$$37$$ −6.24698 −1.02700 −0.513499 0.858090i $$-0.671651\pi$$
−0.513499 + 0.858090i $$0.671651\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.80194 −0.281415 −0.140708 0.990051i $$-0.544938\pi$$
−0.140708 + 0.990051i $$0.544938\pi$$
$$42$$ 0 0
$$43$$ 7.09783 1.08241 0.541205 0.840891i $$-0.317968\pi$$
0.541205 + 0.840891i $$0.317968\pi$$
$$44$$ 0 0
$$45$$ −1.44504 −0.215414
$$46$$ 0 0
$$47$$ 10.5526 1.53925 0.769625 0.638496i $$-0.220443\pi$$
0.769625 + 0.638496i $$0.220443\pi$$
$$48$$ 0 0
$$49$$ 4.86831 0.695473
$$50$$ 0 0
$$51$$ 0.753020 0.105444
$$52$$ 0 0
$$53$$ −3.08815 −0.424189 −0.212095 0.977249i $$-0.568029\pi$$
−0.212095 + 0.977249i $$0.568029\pi$$
$$54$$ 0 0
$$55$$ 7.49396 1.01049
$$56$$ 0 0
$$57$$ −7.96077 −1.05443
$$58$$ 0 0
$$59$$ 1.87800 0.244495 0.122248 0.992500i $$-0.460990\pi$$
0.122248 + 0.992500i $$0.460990\pi$$
$$60$$ 0 0
$$61$$ 3.34481 0.428260 0.214130 0.976805i $$-0.431308\pi$$
0.214130 + 0.976805i $$0.431308\pi$$
$$62$$ 0 0
$$63$$ 3.44504 0.434034
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.54288 −0.555001 −0.277500 0.960726i $$-0.589506\pi$$
−0.277500 + 0.960726i $$0.589506\pi$$
$$68$$ 0 0
$$69$$ −2.82908 −0.340582
$$70$$ 0 0
$$71$$ −9.11960 −1.08230 −0.541149 0.840927i $$-0.682011\pi$$
−0.541149 + 0.840927i $$0.682011\pi$$
$$72$$ 0 0
$$73$$ −2.95108 −0.345398 −0.172699 0.984975i $$-0.555249\pi$$
−0.172699 + 0.984975i $$0.555249\pi$$
$$74$$ 0 0
$$75$$ 2.91185 0.336232
$$76$$ 0 0
$$77$$ −17.8659 −2.03601
$$78$$ 0 0
$$79$$ 9.43296 1.06129 0.530645 0.847594i $$-0.321950\pi$$
0.530645 + 0.847594i $$0.321950\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.46681 −0.709825 −0.354912 0.934900i $$-0.615489\pi$$
−0.354912 + 0.934900i $$0.615489\pi$$
$$84$$ 0 0
$$85$$ 1.08815 0.118026
$$86$$ 0 0
$$87$$ 3.91185 0.419395
$$88$$ 0 0
$$89$$ −1.15883 −0.122836 −0.0614181 0.998112i $$-0.519562\pi$$
−0.0614181 + 0.998112i $$0.519562\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.89977 0.508083
$$94$$ 0 0
$$95$$ −11.5036 −1.18025
$$96$$ 0 0
$$97$$ −8.65817 −0.879104 −0.439552 0.898217i $$-0.644863\pi$$
−0.439552 + 0.898217i $$0.644863\pi$$
$$98$$ 0 0
$$99$$ −5.18598 −0.521211
$$100$$ 0 0
$$101$$ −8.47650 −0.843443 −0.421722 0.906725i $$-0.638574\pi$$
−0.421722 + 0.906725i $$0.638574\pi$$
$$102$$ 0 0
$$103$$ −5.64742 −0.556456 −0.278228 0.960515i $$-0.589747\pi$$
−0.278228 + 0.960515i $$0.589747\pi$$
$$104$$ 0 0
$$105$$ 4.97823 0.485825
$$106$$ 0 0
$$107$$ 6.73556 0.651151 0.325576 0.945516i $$-0.394442\pi$$
0.325576 + 0.945516i $$0.394442\pi$$
$$108$$ 0 0
$$109$$ 2.07606 0.198851 0.0994255 0.995045i $$-0.468300\pi$$
0.0994255 + 0.995045i $$0.468300\pi$$
$$110$$ 0 0
$$111$$ 6.24698 0.592937
$$112$$ 0 0
$$113$$ 6.16852 0.580286 0.290143 0.956983i $$-0.406297\pi$$
0.290143 + 0.956983i $$0.406297\pi$$
$$114$$ 0 0
$$115$$ −4.08815 −0.381222
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −2.59419 −0.237809
$$120$$ 0 0
$$121$$ 15.8944 1.44495
$$122$$ 0 0
$$123$$ 1.80194 0.162475
$$124$$ 0 0
$$125$$ 11.4330 1.02260
$$126$$ 0 0
$$127$$ 14.2620 1.26555 0.632776 0.774335i $$-0.281915\pi$$
0.632776 + 0.774335i $$0.281915\pi$$
$$128$$ 0 0
$$129$$ −7.09783 −0.624929
$$130$$ 0 0
$$131$$ −22.6015 −1.97470 −0.987350 0.158554i $$-0.949317\pi$$
−0.987350 + 0.158554i $$0.949317\pi$$
$$132$$ 0 0
$$133$$ 27.4252 2.37807
$$134$$ 0 0
$$135$$ 1.44504 0.124369
$$136$$ 0 0
$$137$$ 13.6353 1.16495 0.582473 0.812850i $$-0.302085\pi$$
0.582473 + 0.812850i $$0.302085\pi$$
$$138$$ 0 0
$$139$$ −17.6015 −1.49294 −0.746469 0.665420i $$-0.768252\pi$$
−0.746469 + 0.665420i $$0.768252\pi$$
$$140$$ 0 0
$$141$$ −10.5526 −0.888686
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 5.65279 0.469439
$$146$$ 0 0
$$147$$ −4.86831 −0.401532
$$148$$ 0 0
$$149$$ −12.7385 −1.04358 −0.521791 0.853073i $$-0.674736\pi$$
−0.521791 + 0.853073i $$0.674736\pi$$
$$150$$ 0 0
$$151$$ 15.6407 1.27282 0.636412 0.771350i $$-0.280418\pi$$
0.636412 + 0.771350i $$0.280418\pi$$
$$152$$ 0 0
$$153$$ −0.753020 −0.0608781
$$154$$ 0 0
$$155$$ 7.08038 0.568710
$$156$$ 0 0
$$157$$ −0.823708 −0.0657391 −0.0328695 0.999460i $$-0.510465\pi$$
−0.0328695 + 0.999460i $$0.510465\pi$$
$$158$$ 0 0
$$159$$ 3.08815 0.244906
$$160$$ 0 0
$$161$$ 9.74632 0.768117
$$162$$ 0 0
$$163$$ 6.26875 0.491006 0.245503 0.969396i $$-0.421047\pi$$
0.245503 + 0.969396i $$0.421047\pi$$
$$164$$ 0 0
$$165$$ −7.49396 −0.583404
$$166$$ 0 0
$$167$$ −7.45042 −0.576531 −0.288265 0.957551i $$-0.593079\pi$$
−0.288265 + 0.957551i $$0.593079\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 7.96077 0.608775
$$172$$ 0 0
$$173$$ −2.00969 −0.152794 −0.0763969 0.997077i $$-0.524342\pi$$
−0.0763969 + 0.997077i $$0.524342\pi$$
$$174$$ 0 0
$$175$$ −10.0315 −0.758307
$$176$$ 0 0
$$177$$ −1.87800 −0.141159
$$178$$ 0 0
$$179$$ −20.0368 −1.49762 −0.748812 0.662783i $$-0.769375\pi$$
−0.748812 + 0.662783i $$0.769375\pi$$
$$180$$ 0 0
$$181$$ 24.1226 1.79302 0.896509 0.443026i $$-0.146095\pi$$
0.896509 + 0.443026i $$0.146095\pi$$
$$182$$ 0 0
$$183$$ −3.34481 −0.247256
$$184$$ 0 0
$$185$$ 9.02715 0.663689
$$186$$ 0 0
$$187$$ 3.90515 0.285573
$$188$$ 0 0
$$189$$ −3.44504 −0.250590
$$190$$ 0 0
$$191$$ 7.08038 0.512318 0.256159 0.966635i $$-0.417543\pi$$
0.256159 + 0.966635i $$0.417543\pi$$
$$192$$ 0 0
$$193$$ −9.76809 −0.703122 −0.351561 0.936165i $$-0.614349\pi$$
−0.351561 + 0.936165i $$0.614349\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −23.4112 −1.66798 −0.833989 0.551781i $$-0.813948\pi$$
−0.833989 + 0.551781i $$0.813948\pi$$
$$198$$ 0 0
$$199$$ −4.02475 −0.285307 −0.142654 0.989773i $$-0.545563\pi$$
−0.142654 + 0.989773i $$0.545563\pi$$
$$200$$ 0 0
$$201$$ 4.54288 0.320430
$$202$$ 0 0
$$203$$ −13.4765 −0.945865
$$204$$ 0 0
$$205$$ 2.60388 0.181863
$$206$$ 0 0
$$207$$ 2.82908 0.196635
$$208$$ 0 0
$$209$$ −41.2844 −2.85570
$$210$$ 0 0
$$211$$ 3.91185 0.269303 0.134652 0.990893i $$-0.457008\pi$$
0.134652 + 0.990893i $$0.457008\pi$$
$$212$$ 0 0
$$213$$ 9.11960 0.624865
$$214$$ 0 0
$$215$$ −10.2567 −0.699499
$$216$$ 0 0
$$217$$ −16.8799 −1.14588
$$218$$ 0 0
$$219$$ 2.95108 0.199416
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 7.44935 0.498846 0.249423 0.968395i $$-0.419759\pi$$
0.249423 + 0.968395i $$0.419759\pi$$
$$224$$ 0 0
$$225$$ −2.91185 −0.194124
$$226$$ 0 0
$$227$$ −21.2500 −1.41041 −0.705205 0.709004i $$-0.749145\pi$$
−0.705205 + 0.709004i $$0.749145\pi$$
$$228$$ 0 0
$$229$$ −9.29590 −0.614290 −0.307145 0.951663i $$-0.599374\pi$$
−0.307145 + 0.951663i $$0.599374\pi$$
$$230$$ 0 0
$$231$$ 17.8659 1.17549
$$232$$ 0 0
$$233$$ 16.2107 1.06200 0.531000 0.847372i $$-0.321816\pi$$
0.531000 + 0.847372i $$0.321816\pi$$
$$234$$ 0 0
$$235$$ −15.2489 −0.994728
$$236$$ 0 0
$$237$$ −9.43296 −0.612737
$$238$$ 0 0
$$239$$ −13.5090 −0.873826 −0.436913 0.899504i $$-0.643928\pi$$
−0.436913 + 0.899504i $$0.643928\pi$$
$$240$$ 0 0
$$241$$ −6.26875 −0.403806 −0.201903 0.979406i $$-0.564713\pi$$
−0.201903 + 0.979406i $$0.564713\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −7.03492 −0.449444
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 6.46681 0.409818
$$250$$ 0 0
$$251$$ 0.753020 0.0475302 0.0237651 0.999718i $$-0.492435\pi$$
0.0237651 + 0.999718i $$0.492435\pi$$
$$252$$ 0 0
$$253$$ −14.6716 −0.922394
$$254$$ 0 0
$$255$$ −1.08815 −0.0681423
$$256$$ 0 0
$$257$$ 19.7265 1.23050 0.615252 0.788331i $$-0.289054\pi$$
0.615252 + 0.788331i $$0.289054\pi$$
$$258$$ 0 0
$$259$$ −21.5211 −1.33726
$$260$$ 0 0
$$261$$ −3.91185 −0.242138
$$262$$ 0 0
$$263$$ 17.6093 1.08583 0.542917 0.839787i $$-0.317320\pi$$
0.542917 + 0.839787i $$0.317320\pi$$
$$264$$ 0 0
$$265$$ 4.46250 0.274129
$$266$$ 0 0
$$267$$ 1.15883 0.0709195
$$268$$ 0 0
$$269$$ −16.3870 −0.999135 −0.499567 0.866275i $$-0.666508\pi$$
−0.499567 + 0.866275i $$0.666508\pi$$
$$270$$ 0 0
$$271$$ −0.795233 −0.0483070 −0.0241535 0.999708i $$-0.507689\pi$$
−0.0241535 + 0.999708i $$0.507689\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 15.1008 0.910614
$$276$$ 0 0
$$277$$ −4.83340 −0.290411 −0.145205 0.989402i $$-0.546384\pi$$
−0.145205 + 0.989402i $$0.546384\pi$$
$$278$$ 0 0
$$279$$ −4.89977 −0.293342
$$280$$ 0 0
$$281$$ −18.7748 −1.12001 −0.560005 0.828489i $$-0.689201\pi$$
−0.560005 + 0.828489i $$0.689201\pi$$
$$282$$ 0 0
$$283$$ 7.91723 0.470631 0.235315 0.971919i $$-0.424388\pi$$
0.235315 + 0.971919i $$0.424388\pi$$
$$284$$ 0 0
$$285$$ 11.5036 0.681417
$$286$$ 0 0
$$287$$ −6.20775 −0.366432
$$288$$ 0 0
$$289$$ −16.4330 −0.966645
$$290$$ 0 0
$$291$$ 8.65817 0.507551
$$292$$ 0 0
$$293$$ −6.57912 −0.384356 −0.192178 0.981360i $$-0.561555\pi$$
−0.192178 + 0.981360i $$0.561555\pi$$
$$294$$ 0 0
$$295$$ −2.71379 −0.158003
$$296$$ 0 0
$$297$$ 5.18598 0.300921
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 24.4523 1.40941
$$302$$ 0 0
$$303$$ 8.47650 0.486962
$$304$$ 0 0
$$305$$ −4.83340 −0.276759
$$306$$ 0 0
$$307$$ −24.8649 −1.41911 −0.709556 0.704649i $$-0.751105\pi$$
−0.709556 + 0.704649i $$0.751105\pi$$
$$308$$ 0 0
$$309$$ 5.64742 0.321270
$$310$$ 0 0
$$311$$ 17.0804 0.968539 0.484270 0.874919i $$-0.339085\pi$$
0.484270 + 0.874919i $$0.339085\pi$$
$$312$$ 0 0
$$313$$ 15.6974 0.887269 0.443635 0.896208i $$-0.353689\pi$$
0.443635 + 0.896208i $$0.353689\pi$$
$$314$$ 0 0
$$315$$ −4.97823 −0.280491
$$316$$ 0 0
$$317$$ −32.7821 −1.84123 −0.920613 0.390477i $$-0.872310\pi$$
−0.920613 + 0.390477i $$0.872310\pi$$
$$318$$ 0 0
$$319$$ 20.2868 1.13584
$$320$$ 0 0
$$321$$ −6.73556 −0.375942
$$322$$ 0 0
$$323$$ −5.99462 −0.333550
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −2.07606 −0.114807
$$328$$ 0 0
$$329$$ 36.3540 2.00426
$$330$$ 0 0
$$331$$ −29.1618 −1.60288 −0.801439 0.598076i $$-0.795932\pi$$
−0.801439 + 0.598076i $$0.795932\pi$$
$$332$$ 0 0
$$333$$ −6.24698 −0.342332
$$334$$ 0 0
$$335$$ 6.56465 0.358665
$$336$$ 0 0
$$337$$ −33.2911 −1.81348 −0.906741 0.421688i $$-0.861438\pi$$
−0.906741 + 0.421688i $$0.861438\pi$$
$$338$$ 0 0
$$339$$ −6.16852 −0.335028
$$340$$ 0 0
$$341$$ 25.4101 1.37604
$$342$$ 0 0
$$343$$ −7.34375 −0.396525
$$344$$ 0 0
$$345$$ 4.08815 0.220098
$$346$$ 0 0
$$347$$ 0.873690 0.0469022 0.0234511 0.999725i $$-0.492535\pi$$
0.0234511 + 0.999725i $$0.492535\pi$$
$$348$$ 0 0
$$349$$ 3.23191 0.173000 0.0865002 0.996252i $$-0.472432\pi$$
0.0865002 + 0.996252i $$0.472432\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −8.14675 −0.433608 −0.216804 0.976215i $$-0.569563\pi$$
−0.216804 + 0.976215i $$0.569563\pi$$
$$354$$ 0 0
$$355$$ 13.1782 0.699427
$$356$$ 0 0
$$357$$ 2.59419 0.137299
$$358$$ 0 0
$$359$$ 2.64071 0.139371 0.0696857 0.997569i $$-0.477800\pi$$
0.0696857 + 0.997569i $$0.477800\pi$$
$$360$$ 0 0
$$361$$ 44.3739 2.33547
$$362$$ 0 0
$$363$$ −15.8944 −0.834239
$$364$$ 0 0
$$365$$ 4.26444 0.223211
$$366$$ 0 0
$$367$$ 2.90408 0.151592 0.0757960 0.997123i $$-0.475850\pi$$
0.0757960 + 0.997123i $$0.475850\pi$$
$$368$$ 0 0
$$369$$ −1.80194 −0.0938051
$$370$$ 0 0
$$371$$ −10.6388 −0.552339
$$372$$ 0 0
$$373$$ −8.39852 −0.434859 −0.217429 0.976076i $$-0.569767\pi$$
−0.217429 + 0.976076i $$0.569767\pi$$
$$374$$ 0 0
$$375$$ −11.4330 −0.590396
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 15.7482 0.808932 0.404466 0.914553i $$-0.367457\pi$$
0.404466 + 0.914553i $$0.367457\pi$$
$$380$$ 0 0
$$381$$ −14.2620 −0.730667
$$382$$ 0 0
$$383$$ 12.7385 0.650909 0.325455 0.945558i $$-0.394483\pi$$
0.325455 + 0.945558i $$0.394483\pi$$
$$384$$ 0 0
$$385$$ 25.8170 1.31576
$$386$$ 0 0
$$387$$ 7.09783 0.360803
$$388$$ 0 0
$$389$$ −0.310371 −0.0157365 −0.00786823 0.999969i $$-0.502505\pi$$
−0.00786823 + 0.999969i $$0.502505\pi$$
$$390$$ 0 0
$$391$$ −2.13036 −0.107737
$$392$$ 0 0
$$393$$ 22.6015 1.14009
$$394$$ 0 0
$$395$$ −13.6310 −0.685851
$$396$$ 0 0
$$397$$ −1.49098 −0.0748299 −0.0374150 0.999300i $$-0.511912\pi$$
−0.0374150 + 0.999300i $$0.511912\pi$$
$$398$$ 0 0
$$399$$ −27.4252 −1.37298
$$400$$ 0 0
$$401$$ 23.8334 1.19018 0.595092 0.803658i $$-0.297116\pi$$
0.595092 + 0.803658i $$0.297116\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −1.44504 −0.0718047
$$406$$ 0 0
$$407$$ 32.3967 1.60585
$$408$$ 0 0
$$409$$ −4.26742 −0.211010 −0.105505 0.994419i $$-0.533646\pi$$
−0.105505 + 0.994419i $$0.533646\pi$$
$$410$$ 0 0
$$411$$ −13.6353 −0.672581
$$412$$ 0 0
$$413$$ 6.46980 0.318358
$$414$$ 0 0
$$415$$ 9.34481 0.458719
$$416$$ 0 0
$$417$$ 17.6015 0.861948
$$418$$ 0 0
$$419$$ −29.6896 −1.45043 −0.725217 0.688521i $$-0.758260\pi$$
−0.725217 + 0.688521i $$0.758260\pi$$
$$420$$ 0 0
$$421$$ −29.3991 −1.43282 −0.716412 0.697677i $$-0.754217\pi$$
−0.716412 + 0.697677i $$0.754217\pi$$
$$422$$ 0 0
$$423$$ 10.5526 0.513083
$$424$$ 0 0
$$425$$ 2.19269 0.106361
$$426$$ 0 0
$$427$$ 11.5230 0.557638
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −33.0562 −1.59226 −0.796131 0.605124i $$-0.793123\pi$$
−0.796131 + 0.605124i $$0.793123\pi$$
$$432$$ 0 0
$$433$$ −29.2664 −1.40645 −0.703226 0.710967i $$-0.748258\pi$$
−0.703226 + 0.710967i $$0.748258\pi$$
$$434$$ 0 0
$$435$$ −5.65279 −0.271031
$$436$$ 0 0
$$437$$ 22.5217 1.07736
$$438$$ 0 0
$$439$$ 2.13169 0.101740 0.0508699 0.998705i $$-0.483801\pi$$
0.0508699 + 0.998705i $$0.483801\pi$$
$$440$$ 0 0
$$441$$ 4.86831 0.231824
$$442$$ 0 0
$$443$$ 22.9922 1.09239 0.546197 0.837657i $$-0.316075\pi$$
0.546197 + 0.837657i $$0.316075\pi$$
$$444$$ 0 0
$$445$$ 1.67456 0.0793819
$$446$$ 0 0
$$447$$ 12.7385 0.602513
$$448$$ 0 0
$$449$$ 12.9379 0.610579 0.305289 0.952260i $$-0.401247\pi$$
0.305289 + 0.952260i $$0.401247\pi$$
$$450$$ 0 0
$$451$$ 9.34481 0.440030
$$452$$ 0 0
$$453$$ −15.6407 −0.734865
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.85325 0.227025 0.113513 0.993537i $$-0.463790\pi$$
0.113513 + 0.993537i $$0.463790\pi$$
$$458$$ 0 0
$$459$$ 0.753020 0.0351480
$$460$$ 0 0
$$461$$ −18.8345 −0.877208 −0.438604 0.898680i $$-0.644527\pi$$
−0.438604 + 0.898680i $$0.644527\pi$$
$$462$$ 0 0
$$463$$ −22.8767 −1.06317 −0.531585 0.847005i $$-0.678403\pi$$
−0.531585 + 0.847005i $$0.678403\pi$$
$$464$$ 0 0
$$465$$ −7.08038 −0.328345
$$466$$ 0 0
$$467$$ −13.0000 −0.601568 −0.300784 0.953692i $$-0.597248\pi$$
−0.300784 + 0.953692i $$0.597248\pi$$
$$468$$ 0 0
$$469$$ −15.6504 −0.722668
$$470$$ 0 0
$$471$$ 0.823708 0.0379545
$$472$$ 0 0
$$473$$ −36.8092 −1.69249
$$474$$ 0 0
$$475$$ −23.1806 −1.06360
$$476$$ 0 0
$$477$$ −3.08815 −0.141396
$$478$$ 0 0
$$479$$ −38.0901 −1.74038 −0.870190 0.492717i $$-0.836004\pi$$
−0.870190 + 0.492717i $$0.836004\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −9.74632 −0.443473
$$484$$ 0 0
$$485$$ 12.5114 0.568114
$$486$$ 0 0
$$487$$ −21.2500 −0.962928 −0.481464 0.876466i $$-0.659895\pi$$
−0.481464 + 0.876466i $$0.659895\pi$$
$$488$$ 0 0
$$489$$ −6.26875 −0.283483
$$490$$ 0 0
$$491$$ 6.35019 0.286580 0.143290 0.989681i $$-0.454232\pi$$
0.143290 + 0.989681i $$0.454232\pi$$
$$492$$ 0 0
$$493$$ 2.94571 0.132668
$$494$$ 0 0
$$495$$ 7.49396 0.336828
$$496$$ 0 0
$$497$$ −31.4174 −1.40926
$$498$$ 0 0
$$499$$ −4.65087 −0.208202 −0.104101 0.994567i $$-0.533196\pi$$
−0.104101 + 0.994567i $$0.533196\pi$$
$$500$$ 0 0
$$501$$ 7.45042 0.332860
$$502$$ 0 0
$$503$$ −15.4752 −0.690004 −0.345002 0.938602i $$-0.612122\pi$$
−0.345002 + 0.938602i $$0.612122\pi$$
$$504$$ 0 0
$$505$$ 12.2489 0.545069
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 20.5047 0.908855 0.454428 0.890784i $$-0.349844\pi$$
0.454428 + 0.890784i $$0.349844\pi$$
$$510$$ 0 0
$$511$$ −10.1666 −0.449744
$$512$$ 0 0
$$513$$ −7.96077 −0.351477
$$514$$ 0 0
$$515$$ 8.16075 0.359606
$$516$$ 0 0
$$517$$ −54.7254 −2.40682
$$518$$ 0 0
$$519$$ 2.00969 0.0882155
$$520$$ 0 0
$$521$$ −42.0267 −1.84122 −0.920611 0.390481i $$-0.872309\pi$$
−0.920611 + 0.390481i $$0.872309\pi$$
$$522$$ 0 0
$$523$$ 29.9885 1.31131 0.655653 0.755062i $$-0.272393\pi$$
0.655653 + 0.755062i $$0.272393\pi$$
$$524$$ 0 0
$$525$$ 10.0315 0.437809
$$526$$ 0 0
$$527$$ 3.68963 0.160723
$$528$$ 0 0
$$529$$ −14.9963 −0.652012
$$530$$ 0 0
$$531$$ 1.87800 0.0814984
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −9.73317 −0.420802
$$536$$ 0 0
$$537$$ 20.0368 0.864653
$$538$$ 0 0
$$539$$ −25.2470 −1.08746
$$540$$ 0 0
$$541$$ −36.3803 −1.56411 −0.782056 0.623208i $$-0.785829\pi$$
−0.782056 + 0.623208i $$0.785829\pi$$
$$542$$ 0 0
$$543$$ −24.1226 −1.03520
$$544$$ 0 0
$$545$$ −3.00000 −0.128506
$$546$$ 0 0
$$547$$ 25.8159 1.10381 0.551905 0.833907i $$-0.313901\pi$$
0.551905 + 0.833907i $$0.313901\pi$$
$$548$$ 0 0
$$549$$ 3.34481 0.142753
$$550$$ 0 0
$$551$$ −31.1414 −1.32667
$$552$$ 0 0
$$553$$ 32.4969 1.38191
$$554$$ 0 0
$$555$$ −9.02715 −0.383181
$$556$$ 0 0
$$557$$ −17.9903 −0.762274 −0.381137 0.924519i $$-0.624467\pi$$
−0.381137 + 0.924519i $$0.624467\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −3.90515 −0.164876
$$562$$ 0 0
$$563$$ 39.1323 1.64923 0.824614 0.565695i $$-0.191392\pi$$
0.824614 + 0.565695i $$0.191392\pi$$
$$564$$ 0 0
$$565$$ −8.91377 −0.375005
$$566$$ 0 0
$$567$$ 3.44504 0.144678
$$568$$ 0 0
$$569$$ 30.6002 1.28283 0.641413 0.767196i $$-0.278349\pi$$
0.641413 + 0.767196i $$0.278349\pi$$
$$570$$ 0 0
$$571$$ 2.96184 0.123949 0.0619745 0.998078i $$-0.480260\pi$$
0.0619745 + 0.998078i $$0.480260\pi$$
$$572$$ 0 0
$$573$$ −7.08038 −0.295787
$$574$$ 0 0
$$575$$ −8.23788 −0.343543
$$576$$ 0 0
$$577$$ 0.819396 0.0341119 0.0170560 0.999855i $$-0.494571\pi$$
0.0170560 + 0.999855i $$0.494571\pi$$
$$578$$ 0 0
$$579$$ 9.76809 0.405948
$$580$$ 0 0
$$581$$ −22.2784 −0.924265
$$582$$ 0 0
$$583$$ 16.0151 0.663276
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −31.7995 −1.31251 −0.656254 0.754540i $$-0.727860\pi$$
−0.656254 + 0.754540i $$0.727860\pi$$
$$588$$ 0 0
$$589$$ −39.0060 −1.60721
$$590$$ 0 0
$$591$$ 23.4112 0.963008
$$592$$ 0 0
$$593$$ 4.26337 0.175076 0.0875379 0.996161i $$-0.472100\pi$$
0.0875379 + 0.996161i $$0.472100\pi$$
$$594$$ 0 0
$$595$$ 3.74871 0.153682
$$596$$ 0 0
$$597$$ 4.02475 0.164722
$$598$$ 0 0
$$599$$ −24.7278 −1.01035 −0.505175 0.863017i $$-0.668572\pi$$
−0.505175 + 0.863017i $$0.668572\pi$$
$$600$$ 0 0
$$601$$ 6.82371 0.278345 0.139172 0.990268i $$-0.455556\pi$$
0.139172 + 0.990268i $$0.455556\pi$$
$$602$$ 0 0
$$603$$ −4.54288 −0.185000
$$604$$ 0 0
$$605$$ −22.9681 −0.933785
$$606$$ 0 0
$$607$$ −31.9963 −1.29869 −0.649344 0.760494i $$-0.724957\pi$$
−0.649344 + 0.760494i $$0.724957\pi$$
$$608$$ 0 0
$$609$$ 13.4765 0.546095
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −33.5875 −1.35659 −0.678293 0.734792i $$-0.737280\pi$$
−0.678293 + 0.734792i $$0.737280\pi$$
$$614$$ 0 0
$$615$$ −2.60388 −0.104998
$$616$$ 0 0
$$617$$ 26.5870 1.07035 0.535176 0.844740i $$-0.320245\pi$$
0.535176 + 0.844740i $$0.320245\pi$$
$$618$$ 0 0
$$619$$ −9.17928 −0.368946 −0.184473 0.982838i $$-0.559058\pi$$
−0.184473 + 0.982838i $$0.559058\pi$$
$$620$$ 0 0
$$621$$ −2.82908 −0.113527
$$622$$ 0 0
$$623$$ −3.99223 −0.159945
$$624$$ 0 0
$$625$$ −1.96184 −0.0784735
$$626$$ 0 0
$$627$$ 41.2844 1.64874
$$628$$ 0 0
$$629$$ 4.70410 0.187565
$$630$$ 0 0
$$631$$ 17.0043 0.676931 0.338465 0.940979i $$-0.390092\pi$$
0.338465 + 0.940979i $$0.390092\pi$$
$$632$$ 0 0
$$633$$ −3.91185 −0.155482
$$634$$ 0 0
$$635$$ −20.6093 −0.817853
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −9.11960 −0.360766
$$640$$ 0 0
$$641$$ −21.6649 −0.855711 −0.427856 0.903847i $$-0.640731\pi$$
−0.427856 + 0.903847i $$0.640731\pi$$
$$642$$ 0 0
$$643$$ 9.35557 0.368948 0.184474 0.982837i $$-0.440942\pi$$
0.184474 + 0.982837i $$0.440942\pi$$
$$644$$ 0 0
$$645$$ 10.2567 0.403856
$$646$$ 0 0
$$647$$ 0.702775 0.0276289 0.0138145 0.999905i $$-0.495603\pi$$
0.0138145 + 0.999905i $$0.495603\pi$$
$$648$$ 0 0
$$649$$ −9.73928 −0.382300
$$650$$ 0 0
$$651$$ 16.8799 0.661576
$$652$$ 0 0
$$653$$ −37.3411 −1.46127 −0.730635 0.682768i $$-0.760776\pi$$
−0.730635 + 0.682768i $$0.760776\pi$$
$$654$$ 0 0
$$655$$ 32.6601 1.27614
$$656$$ 0 0
$$657$$ −2.95108 −0.115133
$$658$$ 0 0
$$659$$ 0.735562 0.0286534 0.0143267 0.999897i $$-0.495440\pi$$
0.0143267 + 0.999897i $$0.495440\pi$$
$$660$$ 0 0
$$661$$ −13.8485 −0.538643 −0.269321 0.963050i $$-0.586799\pi$$
−0.269321 + 0.963050i $$0.586799\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −39.6305 −1.53681
$$666$$ 0 0
$$667$$ −11.0670 −0.428515
$$668$$ 0 0
$$669$$ −7.44935 −0.288009
$$670$$ 0 0
$$671$$ −17.3461 −0.669640
$$672$$ 0 0
$$673$$ −6.35019 −0.244782 −0.122391 0.992482i $$-0.539056\pi$$
−0.122391 + 0.992482i $$0.539056\pi$$
$$674$$ 0 0
$$675$$ 2.91185 0.112077
$$676$$ 0 0
$$677$$ 33.7241 1.29612 0.648061 0.761589i $$-0.275580\pi$$
0.648061 + 0.761589i $$0.275580\pi$$
$$678$$ 0 0
$$679$$ −29.8278 −1.14468
$$680$$ 0 0
$$681$$ 21.2500 0.814300
$$682$$ 0 0
$$683$$ 19.2687 0.737298 0.368649 0.929569i $$-0.379820\pi$$
0.368649 + 0.929569i $$0.379820\pi$$
$$684$$ 0 0
$$685$$ −19.7036 −0.752837
$$686$$ 0 0
$$687$$ 9.29590 0.354661
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −39.4010 −1.49889 −0.749443 0.662069i $$-0.769679\pi$$
−0.749443 + 0.662069i $$0.769679\pi$$
$$692$$ 0 0
$$693$$ −17.8659 −0.678670
$$694$$ 0 0
$$695$$ 25.4349 0.964800
$$696$$ 0 0
$$697$$ 1.35690 0.0513961
$$698$$ 0 0
$$699$$ −16.2107 −0.613146
$$700$$ 0 0
$$701$$ 18.3985 0.694902 0.347451 0.937698i $$-0.387047\pi$$
0.347451 + 0.937698i $$0.387047\pi$$
$$702$$ 0 0
$$703$$ −49.7308 −1.87563
$$704$$ 0 0
$$705$$ 15.2489 0.574307
$$706$$ 0 0
$$707$$ −29.2019 −1.09825
$$708$$ 0 0
$$709$$ 38.4553 1.44422 0.722110 0.691778i $$-0.243172\pi$$
0.722110 + 0.691778i $$0.243172\pi$$
$$710$$ 0 0
$$711$$ 9.43296 0.353764
$$712$$ 0 0
$$713$$ −13.8619 −0.519131
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 13.5090 0.504504
$$718$$ 0 0
$$719$$ −48.4999 −1.80874 −0.904371 0.426747i $$-0.859659\pi$$
−0.904371 + 0.426747i $$0.859659\pi$$
$$720$$ 0 0
$$721$$ −19.4556 −0.724564
$$722$$ 0 0
$$723$$ 6.26875 0.233137
$$724$$ 0 0
$$725$$ 11.3907 0.423042
$$726$$ 0 0
$$727$$ −19.0344 −0.705948 −0.352974 0.935633i $$-0.614830\pi$$
−0.352974 + 0.935633i $$0.614830\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −5.34481 −0.197685
$$732$$ 0 0
$$733$$ −24.6213 −0.909410 −0.454705 0.890642i $$-0.650255\pi$$
−0.454705 + 0.890642i $$0.650255\pi$$
$$734$$ 0 0
$$735$$ 7.03492 0.259487
$$736$$ 0 0
$$737$$ 23.5593 0.867817
$$738$$ 0 0
$$739$$ −44.5115 −1.63738 −0.818692 0.574233i $$-0.805300\pi$$
−0.818692 + 0.574233i $$0.805300\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 10.4112 0.381950 0.190975 0.981595i $$-0.438835\pi$$
0.190975 + 0.981595i $$0.438835\pi$$
$$744$$ 0 0
$$745$$ 18.4077 0.674407
$$746$$ 0 0
$$747$$ −6.46681 −0.236608
$$748$$ 0 0
$$749$$ 23.2043 0.847866
$$750$$ 0 0
$$751$$ −1.69979 −0.0620263 −0.0310131 0.999519i $$-0.509873\pi$$
−0.0310131 + 0.999519i $$0.509873\pi$$
$$752$$ 0 0
$$753$$ −0.753020 −0.0274416
$$754$$ 0 0
$$755$$ −22.6015 −0.822552
$$756$$ 0 0
$$757$$ 27.4252 0.996786 0.498393 0.866951i $$-0.333924\pi$$
0.498393 + 0.866951i $$0.333924\pi$$
$$758$$ 0 0
$$759$$ 14.6716 0.532545
$$760$$ 0 0
$$761$$ 5.02608 0.182195 0.0910977 0.995842i $$-0.470962\pi$$
0.0910977 + 0.995842i $$0.470962\pi$$
$$762$$ 0 0
$$763$$ 7.15213 0.258924
$$764$$ 0 0
$$765$$ 1.08815 0.0393420
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −42.4456 −1.53063 −0.765314 0.643657i $$-0.777416\pi$$
−0.765314 + 0.643657i $$0.777416\pi$$
$$770$$ 0 0
$$771$$ −19.7265 −0.710431
$$772$$ 0 0
$$773$$ −26.3593 −0.948078 −0.474039 0.880504i $$-0.657204\pi$$
−0.474039 + 0.880504i $$0.657204\pi$$
$$774$$ 0 0
$$775$$ 14.2674 0.512501
$$776$$ 0 0
$$777$$ 21.5211 0.772065
$$778$$ 0 0
$$779$$ −14.3448 −0.513956
$$780$$ 0 0
$$781$$ 47.2941 1.69232
$$782$$ 0 0
$$783$$ 3.91185 0.139798
$$784$$ 0 0
$$785$$ 1.19029 0.0424834
$$786$$ 0 0
$$787$$ 17.1424 0.611062 0.305531 0.952182i $$-0.401166\pi$$
0.305531 + 0.952182i $$0.401166\pi$$
$$788$$ 0 0
$$789$$ −17.6093 −0.626906
$$790$$ 0 0
$$791$$ 21.2508 0.755592
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −4.46250 −0.158269
$$796$$ 0 0
$$797$$ 30.1629 1.06842 0.534212 0.845351i $$-0.320608\pi$$
0.534212 + 0.845351i $$0.320608\pi$$
$$798$$ 0 0
$$799$$ −7.94630 −0.281120
$$800$$ 0 0
$$801$$ −1.15883 −0.0409454
$$802$$ 0 0
$$803$$ 15.3043 0.540076
$$804$$ 0 0
$$805$$ −14.0838 −0.496390
$$806$$ 0 0
$$807$$ 16.3870 0.576851
$$808$$ 0 0
$$809$$ −29.8504 −1.04948 −0.524742 0.851261i $$-0.675838\pi$$
−0.524742 + 0.851261i $$0.675838\pi$$
$$810$$ 0 0
$$811$$ 47.7362 1.67624 0.838122 0.545484i $$-0.183654\pi$$
0.838122 + 0.545484i $$0.183654\pi$$
$$812$$ 0 0
$$813$$ 0.795233 0.0278900
$$814$$ 0 0
$$815$$ −9.05861 −0.317309
$$816$$ 0 0
$$817$$ 56.5042 1.97683
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 17.9299 0.625758 0.312879 0.949793i $$-0.398707\pi$$
0.312879 + 0.949793i $$0.398707\pi$$
$$822$$ 0 0
$$823$$ −54.3196 −1.89346 −0.946731 0.322026i $$-0.895636\pi$$
−0.946731 + 0.322026i $$0.895636\pi$$
$$824$$ 0 0
$$825$$ −15.1008 −0.525743
$$826$$ 0 0
$$827$$ 49.1041 1.70752 0.853758 0.520670i $$-0.174318\pi$$
0.853758 + 0.520670i $$0.174318\pi$$
$$828$$ 0 0
$$829$$ −7.35796 −0.255553 −0.127776 0.991803i $$-0.540784\pi$$
−0.127776 + 0.991803i $$0.540784\pi$$
$$830$$ 0 0
$$831$$ 4.83340 0.167669
$$832$$ 0 0
$$833$$ −3.66594 −0.127017
$$834$$ 0 0
$$835$$ 10.7662 0.372579
$$836$$ 0 0
$$837$$ 4.89977 0.169361
$$838$$ 0 0
$$839$$ −36.5013 −1.26016 −0.630082 0.776529i $$-0.716979\pi$$
−0.630082 + 0.776529i $$0.716979\pi$$
$$840$$ 0 0
$$841$$ −13.6974 −0.472324
$$842$$ 0 0
$$843$$ 18.7748 0.646638
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 54.7569 1.88147
$$848$$ 0 0
$$849$$ −7.91723 −0.271719
$$850$$ 0 0
$$851$$ −17.6732 −0.605831
$$852$$ 0 0
$$853$$ 9.73855 0.333441 0.166721 0.986004i $$-0.446682\pi$$
0.166721 + 0.986004i $$0.446682\pi$$
$$854$$ 0 0
$$855$$ −11.5036 −0.393416
$$856$$ 0 0
$$857$$ −15.2030 −0.519323 −0.259662 0.965700i $$-0.583611\pi$$
−0.259662 + 0.965700i $$0.583611\pi$$
$$858$$ 0 0
$$859$$ 31.9885 1.09143 0.545717 0.837970i $$-0.316257\pi$$
0.545717 + 0.837970i $$0.316257\pi$$
$$860$$ 0 0
$$861$$ 6.20775 0.211560
$$862$$ 0 0
$$863$$ −35.2905 −1.20130 −0.600652 0.799511i $$-0.705092\pi$$
−0.600652 + 0.799511i $$0.705092\pi$$
$$864$$ 0 0
$$865$$ 2.90408 0.0987418
$$866$$ 0 0
$$867$$ 16.4330 0.558093
$$868$$ 0 0
$$869$$ −48.9191 −1.65947
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −8.65817 −0.293035
$$874$$ 0 0
$$875$$ 39.3870 1.33152
$$876$$ 0 0
$$877$$ 15.3263 0.517532 0.258766 0.965940i $$-0.416684\pi$$
0.258766 + 0.965940i $$0.416684\pi$$
$$878$$ 0 0
$$879$$ 6.57912 0.221908
$$880$$ 0 0
$$881$$ −36.4306 −1.22738 −0.613689 0.789548i $$-0.710315\pi$$
−0.613689 + 0.789548i $$0.710315\pi$$
$$882$$ 0 0
$$883$$ 37.6819 1.26810 0.634048 0.773294i $$-0.281392\pi$$
0.634048 + 0.773294i $$0.281392\pi$$
$$884$$ 0 0
$$885$$ 2.71379 0.0912231
$$886$$ 0 0
$$887$$ −4.24890 −0.142664 −0.0713320 0.997453i $$-0.522725\pi$$
−0.0713320 + 0.997453i $$0.522725\pi$$
$$888$$ 0 0
$$889$$ 49.1333 1.64788
$$890$$ 0 0
$$891$$ −5.18598 −0.173737
$$892$$ 0 0
$$893$$ 84.0066 2.81117
$$894$$ 0 0
$$895$$ 28.9541 0.967828
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 19.1672 0.639262
$$900$$ 0 0
$$901$$ 2.32544 0.0774715
$$902$$ 0 0
$$903$$ −24.4523 −0.813723
$$904$$ 0 0
$$905$$ −34.8582 −1.15872
$$906$$ 0 0
$$907$$ 12.6183 0.418985 0.209493 0.977810i $$-0.432819\pi$$
0.209493 + 0.977810i $$0.432819\pi$$
$$908$$ 0 0
$$909$$ −8.47650 −0.281148
$$910$$ 0 0
$$911$$ −6.77777 −0.224558 −0.112279 0.993677i $$-0.535815\pi$$
−0.112279 + 0.993677i $$0.535815\pi$$
$$912$$ 0 0
$$913$$ 33.5368 1.10990
$$914$$ 0 0
$$915$$ 4.83340 0.159787
$$916$$ 0 0
$$917$$ −77.8631 −2.57126
$$918$$ 0 0
$$919$$ 20.4674 0.675157 0.337579 0.941297i $$-0.390392\pi$$
0.337579 + 0.941297i $$0.390392\pi$$
$$920$$ 0 0
$$921$$ 24.8649 0.819325
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 18.1903 0.598093
$$926$$ 0 0
$$927$$ −5.64742 −0.185485
$$928$$ 0 0
$$929$$ 4.65220 0.152634 0.0763169 0.997084i $$-0.475684\pi$$
0.0763169 + 0.997084i $$0.475684\pi$$
$$930$$ 0 0
$$931$$ 38.7555 1.27016
$$932$$ 0 0
$$933$$ −17.0804 −0.559186
$$934$$ 0 0
$$935$$ −5.64310 −0.184549
$$936$$ 0 0
$$937$$ −41.8544 −1.36732 −0.683662 0.729798i $$-0.739614\pi$$
−0.683662 + 0.729798i $$0.739614\pi$$
$$938$$ 0 0
$$939$$ −15.6974 −0.512265
$$940$$ 0 0
$$941$$ −30.3454 −0.989232 −0.494616 0.869112i $$-0.664691\pi$$
−0.494616 + 0.869112i $$0.664691\pi$$
$$942$$ 0 0
$$943$$ −5.09783 −0.166008
$$944$$ 0 0
$$945$$ 4.97823 0.161942
$$946$$ 0 0
$$947$$ 12.0325 0.391004 0.195502 0.980703i $$-0.437366\pi$$
0.195502 + 0.980703i $$0.437366\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 32.7821 1.06303
$$952$$ 0 0
$$953$$ 22.9825 0.744478 0.372239 0.928137i $$-0.378590\pi$$
0.372239 + 0.928137i $$0.378590\pi$$
$$954$$ 0 0
$$955$$ −10.2314 −0.331082
$$956$$ 0 0
$$957$$ −20.2868 −0.655779
$$958$$ 0 0
$$959$$ 46.9743 1.51688
$$960$$ 0 0
$$961$$ −6.99223 −0.225556
$$962$$ 0 0
$$963$$ 6.73556 0.217050
$$964$$ 0 0
$$965$$ 14.1153 0.454387
$$966$$ 0 0
$$967$$ −38.8883 −1.25056 −0.625281 0.780399i $$-0.715016\pi$$
−0.625281 + 0.780399i $$0.715016\pi$$
$$968$$ 0 0
$$969$$ 5.99462 0.192575
$$970$$ 0 0
$$971$$ 57.5133 1.84569 0.922845 0.385171i $$-0.125857\pi$$
0.922845 + 0.385171i $$0.125857\pi$$
$$972$$ 0 0
$$973$$ −60.6378 −1.94396
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −16.3690 −0.523690 −0.261845 0.965110i $$-0.584331\pi$$
−0.261845 + 0.965110i $$0.584331\pi$$
$$978$$ 0 0
$$979$$ 6.00969 0.192070
$$980$$ 0 0
$$981$$ 2.07606 0.0662836
$$982$$ 0 0
$$983$$ −15.6963 −0.500635 −0.250318 0.968164i $$-0.580535\pi$$
−0.250318 + 0.968164i $$0.580535\pi$$
$$984$$ 0 0
$$985$$ 33.8301 1.07792
$$986$$ 0 0
$$987$$ −36.3540 −1.15716
$$988$$ 0 0
$$989$$ 20.0804 0.638519
$$990$$ 0 0
$$991$$ 11.2644 0.357827 0.178913 0.983865i $$-0.442742\pi$$
0.178913 + 0.983865i $$0.442742\pi$$
$$992$$ 0 0
$$993$$ 29.1618 0.925422
$$994$$ 0 0
$$995$$ 5.81594 0.184378
$$996$$ 0 0
$$997$$ −7.70112 −0.243897 −0.121948 0.992536i $$-0.538914\pi$$
−0.121948 + 0.992536i $$0.538914\pi$$
$$998$$ 0 0
$$999$$ 6.24698 0.197646
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 8112.2.a.by.1.2 3
4.3 odd 2 507.2.a.j.1.1 3
12.11 even 2 1521.2.a.q.1.3 3
13.12 even 2 8112.2.a.cf.1.2 3
52.3 odd 6 507.2.e.k.22.3 6
52.7 even 12 507.2.j.h.361.1 12
52.11 even 12 507.2.j.h.316.6 12
52.15 even 12 507.2.j.h.316.1 12
52.19 even 12 507.2.j.h.361.6 12
52.23 odd 6 507.2.e.j.22.1 6
52.31 even 4 507.2.b.g.337.6 6
52.35 odd 6 507.2.e.k.484.3 6
52.43 odd 6 507.2.e.j.484.1 6
52.47 even 4 507.2.b.g.337.1 6
52.51 odd 2 507.2.a.k.1.3 yes 3
156.47 odd 4 1521.2.b.m.1351.6 6
156.83 odd 4 1521.2.b.m.1351.1 6
156.155 even 2 1521.2.a.p.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.1 3 4.3 odd 2
507.2.a.k.1.3 yes 3 52.51 odd 2
507.2.b.g.337.1 6 52.47 even 4
507.2.b.g.337.6 6 52.31 even 4
507.2.e.j.22.1 6 52.23 odd 6
507.2.e.j.484.1 6 52.43 odd 6
507.2.e.k.22.3 6 52.3 odd 6
507.2.e.k.484.3 6 52.35 odd 6
507.2.j.h.316.1 12 52.15 even 12
507.2.j.h.316.6 12 52.11 even 12
507.2.j.h.361.1 12 52.7 even 12
507.2.j.h.361.6 12 52.19 even 12
1521.2.a.p.1.1 3 156.155 even 2
1521.2.a.q.1.3 3 12.11 even 2
1521.2.b.m.1351.1 6 156.83 odd 4
1521.2.b.m.1351.6 6 156.47 odd 4
8112.2.a.by.1.2 3 1.1 even 1 trivial
8112.2.a.cf.1.2 3 13.12 even 2