# Properties

 Label 3856.2.a.j.1.5 Level $3856$ Weight $2$ Character 3856.1 Self dual yes Analytic conductor $30.790$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3856 = 2^{4} \cdot 241$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.7903150194$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: 7.7.31056073.1 Defining polynomial: $$x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 241) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$1.48734$$ of defining polynomial Character $$\chi$$ $$=$$ 3856.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.815004 q^{3} +0.961999 q^{5} +4.61392 q^{7} -2.33577 q^{9} +O(q^{10})$$ $$q+0.815004 q^{3} +0.961999 q^{5} +4.61392 q^{7} -2.33577 q^{9} -1.93974 q^{11} -3.85571 q^{13} +0.784033 q^{15} +5.40289 q^{17} +4.17145 q^{19} +3.76036 q^{21} +1.42545 q^{23} -4.07456 q^{25} -4.34867 q^{27} -4.85744 q^{29} +7.24699 q^{31} -1.58090 q^{33} +4.43859 q^{35} +7.12597 q^{37} -3.14242 q^{39} +9.18955 q^{41} -2.93624 q^{43} -2.24701 q^{45} -2.48170 q^{47} +14.2883 q^{49} +4.40338 q^{51} +5.64997 q^{53} -1.86603 q^{55} +3.39975 q^{57} +11.9783 q^{59} -13.9214 q^{61} -10.7771 q^{63} -3.70919 q^{65} +7.30682 q^{67} +1.16175 q^{69} +14.8844 q^{71} +0.240264 q^{73} -3.32078 q^{75} -8.94983 q^{77} +3.15128 q^{79} +3.46313 q^{81} +2.46821 q^{83} +5.19758 q^{85} -3.95883 q^{87} +5.55181 q^{89} -17.7900 q^{91} +5.90632 q^{93} +4.01293 q^{95} +5.27964 q^{97} +4.53079 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 3q^{3} - 8q^{5} + 7q^{7} - 2q^{9} + O(q^{10})$$ $$7q + 3q^{3} - 8q^{5} + 7q^{7} - 2q^{9} + 18q^{11} - q^{13} + 11q^{15} - 2q^{17} + 6q^{19} - 2q^{21} + 22q^{23} + 5q^{25} - 3q^{27} - 16q^{29} + 18q^{31} + 4q^{33} - 7q^{35} + 8q^{37} + 9q^{39} - 15q^{41} - 14q^{43} + 3q^{45} + 10q^{47} + 6q^{49} - 13q^{51} + 15q^{53} - 29q^{55} + 14q^{57} + 18q^{59} + 4q^{61} + 16q^{63} - 7q^{65} - 18q^{67} + 26q^{69} + 50q^{71} - 16q^{75} + 17q^{77} + 15q^{79} - 9q^{81} + 24q^{83} - 2q^{85} - 12q^{87} - 13q^{89} + 12q^{91} + 14q^{93} + 41q^{95} + q^{97} + 20q^{99} + O(q^{100})$$

## 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
$$3$$ 0.815004 0.470543 0.235271 0.971930i $$-0.424402\pi$$
0.235271 + 0.971930i $$0.424402\pi$$
$$4$$ 0 0
$$5$$ 0.961999 0.430219 0.215110 0.976590i $$-0.430989\pi$$
0.215110 + 0.976590i $$0.430989\pi$$
$$6$$ 0 0
$$7$$ 4.61392 1.74390 0.871950 0.489596i $$-0.162856\pi$$
0.871950 + 0.489596i $$0.162856\pi$$
$$8$$ 0 0
$$9$$ −2.33577 −0.778590
$$10$$ 0 0
$$11$$ −1.93974 −0.584855 −0.292427 0.956288i $$-0.594463\pi$$
−0.292427 + 0.956288i $$0.594463\pi$$
$$12$$ 0 0
$$13$$ −3.85571 −1.06938 −0.534691 0.845048i $$-0.679572\pi$$
−0.534691 + 0.845048i $$0.679572\pi$$
$$14$$ 0 0
$$15$$ 0.784033 0.202436
$$16$$ 0 0
$$17$$ 5.40289 1.31039 0.655197 0.755458i $$-0.272586\pi$$
0.655197 + 0.755458i $$0.272586\pi$$
$$18$$ 0 0
$$19$$ 4.17145 0.956997 0.478498 0.878088i $$-0.341181\pi$$
0.478498 + 0.878088i $$0.341181\pi$$
$$20$$ 0 0
$$21$$ 3.76036 0.820579
$$22$$ 0 0
$$23$$ 1.42545 0.297227 0.148614 0.988895i $$-0.452519\pi$$
0.148614 + 0.988895i $$0.452519\pi$$
$$24$$ 0 0
$$25$$ −4.07456 −0.814911
$$26$$ 0 0
$$27$$ −4.34867 −0.836902
$$28$$ 0 0
$$29$$ −4.85744 −0.902003 −0.451002 0.892523i $$-0.648933\pi$$
−0.451002 + 0.892523i $$0.648933\pi$$
$$30$$ 0 0
$$31$$ 7.24699 1.30160 0.650799 0.759250i $$-0.274434\pi$$
0.650799 + 0.759250i $$0.274434\pi$$
$$32$$ 0 0
$$33$$ −1.58090 −0.275199
$$34$$ 0 0
$$35$$ 4.43859 0.750259
$$36$$ 0 0
$$37$$ 7.12597 1.17150 0.585751 0.810491i $$-0.300800\pi$$
0.585751 + 0.810491i $$0.300800\pi$$
$$38$$ 0 0
$$39$$ −3.14242 −0.503190
$$40$$ 0 0
$$41$$ 9.18955 1.43517 0.717583 0.696473i $$-0.245248\pi$$
0.717583 + 0.696473i $$0.245248\pi$$
$$42$$ 0 0
$$43$$ −2.93624 −0.447772 −0.223886 0.974615i $$-0.571874\pi$$
−0.223886 + 0.974615i $$0.571874\pi$$
$$44$$ 0 0
$$45$$ −2.24701 −0.334964
$$46$$ 0 0
$$47$$ −2.48170 −0.361994 −0.180997 0.983484i $$-0.557932\pi$$
−0.180997 + 0.983484i $$0.557932\pi$$
$$48$$ 0 0
$$49$$ 14.2883 2.04118
$$50$$ 0 0
$$51$$ 4.40338 0.616596
$$52$$ 0 0
$$53$$ 5.64997 0.776083 0.388041 0.921642i $$-0.373152\pi$$
0.388041 + 0.921642i $$0.373152\pi$$
$$54$$ 0 0
$$55$$ −1.86603 −0.251616
$$56$$ 0 0
$$57$$ 3.39975 0.450308
$$58$$ 0 0
$$59$$ 11.9783 1.55944 0.779718 0.626131i $$-0.215362\pi$$
0.779718 + 0.626131i $$0.215362\pi$$
$$60$$ 0 0
$$61$$ −13.9214 −1.78245 −0.891223 0.453565i $$-0.850152\pi$$
−0.891223 + 0.453565i $$0.850152\pi$$
$$62$$ 0 0
$$63$$ −10.7771 −1.35778
$$64$$ 0 0
$$65$$ −3.70919 −0.460069
$$66$$ 0 0
$$67$$ 7.30682 0.892670 0.446335 0.894866i $$-0.352729\pi$$
0.446335 + 0.894866i $$0.352729\pi$$
$$68$$ 0 0
$$69$$ 1.16175 0.139858
$$70$$ 0 0
$$71$$ 14.8844 1.76645 0.883226 0.468948i $$-0.155367\pi$$
0.883226 + 0.468948i $$0.155367\pi$$
$$72$$ 0 0
$$73$$ 0.240264 0.0281208 0.0140604 0.999901i $$-0.495524\pi$$
0.0140604 + 0.999901i $$0.495524\pi$$
$$74$$ 0 0
$$75$$ −3.32078 −0.383450
$$76$$ 0 0
$$77$$ −8.94983 −1.01993
$$78$$ 0 0
$$79$$ 3.15128 0.354546 0.177273 0.984162i $$-0.443272\pi$$
0.177273 + 0.984162i $$0.443272\pi$$
$$80$$ 0 0
$$81$$ 3.46313 0.384792
$$82$$ 0 0
$$83$$ 2.46821 0.270922 0.135461 0.990783i $$-0.456749\pi$$
0.135461 + 0.990783i $$0.456749\pi$$
$$84$$ 0 0
$$85$$ 5.19758 0.563757
$$86$$ 0 0
$$87$$ −3.95883 −0.424431
$$88$$ 0 0
$$89$$ 5.55181 0.588491 0.294246 0.955730i $$-0.404932\pi$$
0.294246 + 0.955730i $$0.404932\pi$$
$$90$$ 0 0
$$91$$ −17.7900 −1.86489
$$92$$ 0 0
$$93$$ 5.90632 0.612457
$$94$$ 0 0
$$95$$ 4.01293 0.411718
$$96$$ 0 0
$$97$$ 5.27964 0.536067 0.268033 0.963410i $$-0.413626\pi$$
0.268033 + 0.963410i $$0.413626\pi$$
$$98$$ 0 0
$$99$$ 4.53079 0.455362
$$100$$ 0 0
$$101$$ −12.4239 −1.23623 −0.618113 0.786089i $$-0.712103\pi$$
−0.618113 + 0.786089i $$0.712103\pi$$
$$102$$ 0 0
$$103$$ 9.80533 0.966148 0.483074 0.875580i $$-0.339520\pi$$
0.483074 + 0.875580i $$0.339520\pi$$
$$104$$ 0 0
$$105$$ 3.61747 0.353029
$$106$$ 0 0
$$107$$ 9.15031 0.884593 0.442297 0.896869i $$-0.354164\pi$$
0.442297 + 0.896869i $$0.354164\pi$$
$$108$$ 0 0
$$109$$ 3.24534 0.310847 0.155424 0.987848i $$-0.450326\pi$$
0.155424 + 0.987848i $$0.450326\pi$$
$$110$$ 0 0
$$111$$ 5.80769 0.551241
$$112$$ 0 0
$$113$$ −0.842874 −0.0792909 −0.0396455 0.999214i $$-0.512623\pi$$
−0.0396455 + 0.999214i $$0.512623\pi$$
$$114$$ 0 0
$$115$$ 1.37128 0.127873
$$116$$ 0 0
$$117$$ 9.00605 0.832610
$$118$$ 0 0
$$119$$ 24.9285 2.28520
$$120$$ 0 0
$$121$$ −7.23740 −0.657945
$$122$$ 0 0
$$123$$ 7.48952 0.675307
$$124$$ 0 0
$$125$$ −8.72972 −0.780810
$$126$$ 0 0
$$127$$ −11.3416 −1.00641 −0.503203 0.864168i $$-0.667845\pi$$
−0.503203 + 0.864168i $$0.667845\pi$$
$$128$$ 0 0
$$129$$ −2.39305 −0.210696
$$130$$ 0 0
$$131$$ −14.1261 −1.23421 −0.617103 0.786882i $$-0.711694\pi$$
−0.617103 + 0.786882i $$0.711694\pi$$
$$132$$ 0 0
$$133$$ 19.2468 1.66891
$$134$$ 0 0
$$135$$ −4.18342 −0.360051
$$136$$ 0 0
$$137$$ −12.9555 −1.10686 −0.553430 0.832896i $$-0.686681\pi$$
−0.553430 + 0.832896i $$0.686681\pi$$
$$138$$ 0 0
$$139$$ −8.23606 −0.698574 −0.349287 0.937016i $$-0.613576\pi$$
−0.349287 + 0.937016i $$0.613576\pi$$
$$140$$ 0 0
$$141$$ −2.02260 −0.170333
$$142$$ 0 0
$$143$$ 7.47909 0.625433
$$144$$ 0 0
$$145$$ −4.67285 −0.388059
$$146$$ 0 0
$$147$$ 11.6450 0.960464
$$148$$ 0 0
$$149$$ 4.48606 0.367513 0.183756 0.982972i $$-0.441174\pi$$
0.183756 + 0.982972i $$0.441174\pi$$
$$150$$ 0 0
$$151$$ −0.0933624 −0.00759773 −0.00379886 0.999993i $$-0.501209\pi$$
−0.00379886 + 0.999993i $$0.501209\pi$$
$$152$$ 0 0
$$153$$ −12.6199 −1.02026
$$154$$ 0 0
$$155$$ 6.97160 0.559972
$$156$$ 0 0
$$157$$ −10.1337 −0.808760 −0.404380 0.914591i $$-0.632513\pi$$
−0.404380 + 0.914591i $$0.632513\pi$$
$$158$$ 0 0
$$159$$ 4.60474 0.365180
$$160$$ 0 0
$$161$$ 6.57693 0.518334
$$162$$ 0 0
$$163$$ −7.49804 −0.587292 −0.293646 0.955914i $$-0.594869\pi$$
−0.293646 + 0.955914i $$0.594869\pi$$
$$164$$ 0 0
$$165$$ −1.52082 −0.118396
$$166$$ 0 0
$$167$$ −17.7922 −1.37680 −0.688399 0.725332i $$-0.741686\pi$$
−0.688399 + 0.725332i $$0.741686\pi$$
$$168$$ 0 0
$$169$$ 1.86650 0.143577
$$170$$ 0 0
$$171$$ −9.74355 −0.745108
$$172$$ 0 0
$$173$$ 18.4927 1.40597 0.702987 0.711203i $$-0.251849\pi$$
0.702987 + 0.711203i $$0.251849\pi$$
$$174$$ 0 0
$$175$$ −18.7997 −1.42112
$$176$$ 0 0
$$177$$ 9.76232 0.733781
$$178$$ 0 0
$$179$$ 17.3387 1.29595 0.647977 0.761660i $$-0.275615\pi$$
0.647977 + 0.761660i $$0.275615\pi$$
$$180$$ 0 0
$$181$$ 17.7124 1.31655 0.658275 0.752778i $$-0.271287\pi$$
0.658275 + 0.752778i $$0.271287\pi$$
$$182$$ 0 0
$$183$$ −11.3460 −0.838717
$$184$$ 0 0
$$185$$ 6.85518 0.504003
$$186$$ 0 0
$$187$$ −10.4802 −0.766390
$$188$$ 0 0
$$189$$ −20.0644 −1.45947
$$190$$ 0 0
$$191$$ −13.2440 −0.958303 −0.479151 0.877732i $$-0.659056\pi$$
−0.479151 + 0.877732i $$0.659056\pi$$
$$192$$ 0 0
$$193$$ −25.5030 −1.83574 −0.917872 0.396877i $$-0.870094\pi$$
−0.917872 + 0.396877i $$0.870094\pi$$
$$194$$ 0 0
$$195$$ −3.02300 −0.216482
$$196$$ 0 0
$$197$$ −15.0303 −1.07087 −0.535433 0.844578i $$-0.679851\pi$$
−0.535433 + 0.844578i $$0.679851\pi$$
$$198$$ 0 0
$$199$$ 19.3801 1.37382 0.686911 0.726742i $$-0.258966\pi$$
0.686911 + 0.726742i $$0.258966\pi$$
$$200$$ 0 0
$$201$$ 5.95508 0.420039
$$202$$ 0 0
$$203$$ −22.4118 −1.57300
$$204$$ 0 0
$$205$$ 8.84034 0.617436
$$206$$ 0 0
$$207$$ −3.32953 −0.231418
$$208$$ 0 0
$$209$$ −8.09154 −0.559704
$$210$$ 0 0
$$211$$ −27.9383 −1.92335 −0.961675 0.274192i $$-0.911590\pi$$
−0.961675 + 0.274192i $$0.911590\pi$$
$$212$$ 0 0
$$213$$ 12.1308 0.831191
$$214$$ 0 0
$$215$$ −2.82466 −0.192640
$$216$$ 0 0
$$217$$ 33.4370 2.26985
$$218$$ 0 0
$$219$$ 0.195816 0.0132320
$$220$$ 0 0
$$221$$ −20.8320 −1.40131
$$222$$ 0 0
$$223$$ 0.387500 0.0259489 0.0129745 0.999916i $$-0.495870\pi$$
0.0129745 + 0.999916i $$0.495870\pi$$
$$224$$ 0 0
$$225$$ 9.51722 0.634482
$$226$$ 0 0
$$227$$ −10.2778 −0.682161 −0.341081 0.940034i $$-0.610793\pi$$
−0.341081 + 0.940034i $$0.610793\pi$$
$$228$$ 0 0
$$229$$ −10.5346 −0.696147 −0.348074 0.937467i $$-0.613164\pi$$
−0.348074 + 0.937467i $$0.613164\pi$$
$$230$$ 0 0
$$231$$ −7.29414 −0.479919
$$232$$ 0 0
$$233$$ 25.6725 1.68186 0.840932 0.541141i $$-0.182007\pi$$
0.840932 + 0.541141i $$0.182007\pi$$
$$234$$ 0 0
$$235$$ −2.38740 −0.155737
$$236$$ 0 0
$$237$$ 2.56830 0.166829
$$238$$ 0 0
$$239$$ 25.1312 1.62560 0.812801 0.582542i $$-0.197942\pi$$
0.812801 + 0.582542i $$0.197942\pi$$
$$240$$ 0 0
$$241$$ −1.00000 −0.0644157
$$242$$ 0 0
$$243$$ 15.8685 1.01796
$$244$$ 0 0
$$245$$ 13.7453 0.878157
$$246$$ 0 0
$$247$$ −16.0839 −1.02339
$$248$$ 0 0
$$249$$ 2.01160 0.127480
$$250$$ 0 0
$$251$$ −2.52238 −0.159211 −0.0796055 0.996826i $$-0.525366\pi$$
−0.0796055 + 0.996826i $$0.525366\pi$$
$$252$$ 0 0
$$253$$ −2.76501 −0.173835
$$254$$ 0 0
$$255$$ 4.23605 0.265272
$$256$$ 0 0
$$257$$ 3.92348 0.244740 0.122370 0.992485i $$-0.460951\pi$$
0.122370 + 0.992485i $$0.460951\pi$$
$$258$$ 0 0
$$259$$ 32.8787 2.04298
$$260$$ 0 0
$$261$$ 11.3458 0.702290
$$262$$ 0 0
$$263$$ −6.24984 −0.385382 −0.192691 0.981260i $$-0.561721\pi$$
−0.192691 + 0.981260i $$0.561721\pi$$
$$264$$ 0 0
$$265$$ 5.43527 0.333886
$$266$$ 0 0
$$267$$ 4.52475 0.276910
$$268$$ 0 0
$$269$$ 20.4943 1.24956 0.624780 0.780801i $$-0.285189\pi$$
0.624780 + 0.780801i $$0.285189\pi$$
$$270$$ 0 0
$$271$$ −15.3179 −0.930499 −0.465249 0.885180i $$-0.654035\pi$$
−0.465249 + 0.885180i $$0.654035\pi$$
$$272$$ 0 0
$$273$$ −14.4989 −0.877512
$$274$$ 0 0
$$275$$ 7.90359 0.476605
$$276$$ 0 0
$$277$$ 3.26388 0.196108 0.0980539 0.995181i $$-0.468738\pi$$
0.0980539 + 0.995181i $$0.468738\pi$$
$$278$$ 0 0
$$279$$ −16.9273 −1.01341
$$280$$ 0 0
$$281$$ −12.4791 −0.744442 −0.372221 0.928144i $$-0.621404\pi$$
−0.372221 + 0.928144i $$0.621404\pi$$
$$282$$ 0 0
$$283$$ 15.8115 0.939899 0.469949 0.882693i $$-0.344272\pi$$
0.469949 + 0.882693i $$0.344272\pi$$
$$284$$ 0 0
$$285$$ 3.27056 0.193731
$$286$$ 0 0
$$287$$ 42.3999 2.50279
$$288$$ 0 0
$$289$$ 12.1913 0.717133
$$290$$ 0 0
$$291$$ 4.30293 0.252242
$$292$$ 0 0
$$293$$ 18.7147 1.09333 0.546663 0.837353i $$-0.315898\pi$$
0.546663 + 0.837353i $$0.315898\pi$$
$$294$$ 0 0
$$295$$ 11.5231 0.670899
$$296$$ 0 0
$$297$$ 8.43530 0.489466
$$298$$ 0 0
$$299$$ −5.49613 −0.317849
$$300$$ 0 0
$$301$$ −13.5476 −0.780870
$$302$$ 0 0
$$303$$ −10.1255 −0.581697
$$304$$ 0 0
$$305$$ −13.3923 −0.766843
$$306$$ 0 0
$$307$$ −5.23477 −0.298764 −0.149382 0.988780i $$-0.547728\pi$$
−0.149382 + 0.988780i $$0.547728\pi$$
$$308$$ 0 0
$$309$$ 7.99138 0.454614
$$310$$ 0 0
$$311$$ 26.6878 1.51332 0.756662 0.653806i $$-0.226829\pi$$
0.756662 + 0.653806i $$0.226829\pi$$
$$312$$ 0 0
$$313$$ 6.24141 0.352786 0.176393 0.984320i $$-0.443557\pi$$
0.176393 + 0.984320i $$0.443557\pi$$
$$314$$ 0 0
$$315$$ −10.3675 −0.584144
$$316$$ 0 0
$$317$$ 6.89526 0.387276 0.193638 0.981073i $$-0.437971\pi$$
0.193638 + 0.981073i $$0.437971\pi$$
$$318$$ 0 0
$$319$$ 9.42218 0.527541
$$320$$ 0 0
$$321$$ 7.45753 0.416239
$$322$$ 0 0
$$323$$ 22.5379 1.25404
$$324$$ 0 0
$$325$$ 15.7103 0.871451
$$326$$ 0 0
$$327$$ 2.64497 0.146267
$$328$$ 0 0
$$329$$ −11.4504 −0.631281
$$330$$ 0 0
$$331$$ 13.5697 0.745860 0.372930 0.927859i $$-0.378353\pi$$
0.372930 + 0.927859i $$0.378353\pi$$
$$332$$ 0 0
$$333$$ −16.6446 −0.912119
$$334$$ 0 0
$$335$$ 7.02916 0.384044
$$336$$ 0 0
$$337$$ −20.2288 −1.10193 −0.550966 0.834528i $$-0.685740\pi$$
−0.550966 + 0.834528i $$0.685740\pi$$
$$338$$ 0 0
$$339$$ −0.686946 −0.0373098
$$340$$ 0 0
$$341$$ −14.0573 −0.761245
$$342$$ 0 0
$$343$$ 33.6276 1.81572
$$344$$ 0 0
$$345$$ 1.11760 0.0601696
$$346$$ 0 0
$$347$$ −1.52458 −0.0818437 −0.0409219 0.999162i $$-0.513029\pi$$
−0.0409219 + 0.999162i $$0.513029\pi$$
$$348$$ 0 0
$$349$$ 16.5585 0.886358 0.443179 0.896433i $$-0.353851\pi$$
0.443179 + 0.896433i $$0.353851\pi$$
$$350$$ 0 0
$$351$$ 16.7672 0.894968
$$352$$ 0 0
$$353$$ 34.6802 1.84584 0.922920 0.384992i $$-0.125796\pi$$
0.922920 + 0.384992i $$0.125796\pi$$
$$354$$ 0 0
$$355$$ 14.3188 0.759961
$$356$$ 0 0
$$357$$ 20.3168 1.07528
$$358$$ 0 0
$$359$$ −18.1341 −0.957084 −0.478542 0.878065i $$-0.658835\pi$$
−0.478542 + 0.878065i $$0.658835\pi$$
$$360$$ 0 0
$$361$$ −1.59899 −0.0841573
$$362$$ 0 0
$$363$$ −5.89850 −0.309591
$$364$$ 0 0
$$365$$ 0.231134 0.0120981
$$366$$ 0 0
$$367$$ 10.9879 0.573566 0.286783 0.957996i $$-0.407414\pi$$
0.286783 + 0.957996i $$0.407414\pi$$
$$368$$ 0 0
$$369$$ −21.4647 −1.11741
$$370$$ 0 0
$$371$$ 26.0685 1.35341
$$372$$ 0 0
$$373$$ 5.23646 0.271134 0.135567 0.990768i $$-0.456714\pi$$
0.135567 + 0.990768i $$0.456714\pi$$
$$374$$ 0 0
$$375$$ −7.11475 −0.367404
$$376$$ 0 0
$$377$$ 18.7289 0.964586
$$378$$ 0 0
$$379$$ −8.07101 −0.414580 −0.207290 0.978280i $$-0.566464\pi$$
−0.207290 + 0.978280i $$0.566464\pi$$
$$380$$ 0 0
$$381$$ −9.24346 −0.473557
$$382$$ 0 0
$$383$$ −2.46503 −0.125957 −0.0629785 0.998015i $$-0.520060\pi$$
−0.0629785 + 0.998015i $$0.520060\pi$$
$$384$$ 0 0
$$385$$ −8.60973 −0.438792
$$386$$ 0 0
$$387$$ 6.85838 0.348631
$$388$$ 0 0
$$389$$ −28.7159 −1.45595 −0.727977 0.685601i $$-0.759539\pi$$
−0.727977 + 0.685601i $$0.759539\pi$$
$$390$$ 0 0
$$391$$ 7.70157 0.389485
$$392$$ 0 0
$$393$$ −11.5128 −0.580746
$$394$$ 0 0
$$395$$ 3.03153 0.152533
$$396$$ 0 0
$$397$$ 12.0605 0.605301 0.302651 0.953102i $$-0.402129\pi$$
0.302651 + 0.953102i $$0.402129\pi$$
$$398$$ 0 0
$$399$$ 15.6862 0.785291
$$400$$ 0 0
$$401$$ −31.7082 −1.58343 −0.791716 0.610890i $$-0.790812\pi$$
−0.791716 + 0.610890i $$0.790812\pi$$
$$402$$ 0 0
$$403$$ −27.9423 −1.39190
$$404$$ 0 0
$$405$$ 3.33152 0.165545
$$406$$ 0 0
$$407$$ −13.8225 −0.685158
$$408$$ 0 0
$$409$$ −14.1893 −0.701617 −0.350809 0.936447i $$-0.614093\pi$$
−0.350809 + 0.936447i $$0.614093\pi$$
$$410$$ 0 0
$$411$$ −10.5587 −0.520825
$$412$$ 0 0
$$413$$ 55.2668 2.71950
$$414$$ 0 0
$$415$$ 2.37442 0.116556
$$416$$ 0 0
$$417$$ −6.71242 −0.328709
$$418$$ 0 0
$$419$$ −31.1014 −1.51941 −0.759703 0.650271i $$-0.774655\pi$$
−0.759703 + 0.650271i $$0.774655\pi$$
$$420$$ 0 0
$$421$$ −27.4415 −1.33742 −0.668709 0.743524i $$-0.733153\pi$$
−0.668709 + 0.743524i $$0.733153\pi$$
$$422$$ 0 0
$$423$$ 5.79669 0.281845
$$424$$ 0 0
$$425$$ −22.0144 −1.06786
$$426$$ 0 0
$$427$$ −64.2321 −3.10841
$$428$$ 0 0
$$429$$ 6.09548 0.294293
$$430$$ 0 0
$$431$$ 14.1005 0.679199 0.339599 0.940570i $$-0.389709\pi$$
0.339599 + 0.940570i $$0.389709\pi$$
$$432$$ 0 0
$$433$$ −17.7403 −0.852546 −0.426273 0.904595i $$-0.640174\pi$$
−0.426273 + 0.904595i $$0.640174\pi$$
$$434$$ 0 0
$$435$$ −3.80839 −0.182598
$$436$$ 0 0
$$437$$ 5.94621 0.284446
$$438$$ 0 0
$$439$$ 2.52447 0.120486 0.0602431 0.998184i $$-0.480812\pi$$
0.0602431 + 0.998184i $$0.480812\pi$$
$$440$$ 0 0
$$441$$ −33.3741 −1.58924
$$442$$ 0 0
$$443$$ 16.8382 0.800008 0.400004 0.916513i $$-0.369009\pi$$
0.400004 + 0.916513i $$0.369009\pi$$
$$444$$ 0 0
$$445$$ 5.34084 0.253180
$$446$$ 0 0
$$447$$ 3.65616 0.172930
$$448$$ 0 0
$$449$$ −30.5688 −1.44263 −0.721314 0.692608i $$-0.756462\pi$$
−0.721314 + 0.692608i $$0.756462\pi$$
$$450$$ 0 0
$$451$$ −17.8254 −0.839364
$$452$$ 0 0
$$453$$ −0.0760907 −0.00357505
$$454$$ 0 0
$$455$$ −17.1139 −0.802313
$$456$$ 0 0
$$457$$ −32.7043 −1.52984 −0.764922 0.644123i $$-0.777223\pi$$
−0.764922 + 0.644123i $$0.777223\pi$$
$$458$$ 0 0
$$459$$ −23.4954 −1.09667
$$460$$ 0 0
$$461$$ 12.3932 0.577210 0.288605 0.957448i $$-0.406809\pi$$
0.288605 + 0.957448i $$0.406809\pi$$
$$462$$ 0 0
$$463$$ −12.1379 −0.564094 −0.282047 0.959401i $$-0.591013\pi$$
−0.282047 + 0.959401i $$0.591013\pi$$
$$464$$ 0 0
$$465$$ 5.68188 0.263491
$$466$$ 0 0
$$467$$ −5.38203 −0.249051 −0.124525 0.992216i $$-0.539741\pi$$
−0.124525 + 0.992216i $$0.539741\pi$$
$$468$$ 0 0
$$469$$ 33.7131 1.55673
$$470$$ 0 0
$$471$$ −8.25903 −0.380556
$$472$$ 0 0
$$473$$ 5.69555 0.261882
$$474$$ 0 0
$$475$$ −16.9968 −0.779868
$$476$$ 0 0
$$477$$ −13.1970 −0.604250
$$478$$ 0 0
$$479$$ 25.3752 1.15942 0.579711 0.814822i $$-0.303166\pi$$
0.579711 + 0.814822i $$0.303166\pi$$
$$480$$ 0 0
$$481$$ −27.4757 −1.25278
$$482$$ 0 0
$$483$$ 5.36022 0.243898
$$484$$ 0 0
$$485$$ 5.07901 0.230626
$$486$$ 0 0
$$487$$ 10.4243 0.472369 0.236185 0.971708i $$-0.424103\pi$$
0.236185 + 0.971708i $$0.424103\pi$$
$$488$$ 0 0
$$489$$ −6.11093 −0.276346
$$490$$ 0 0
$$491$$ −34.2858 −1.54730 −0.773649 0.633614i $$-0.781571\pi$$
−0.773649 + 0.633614i $$0.781571\pi$$
$$492$$ 0 0
$$493$$ −26.2442 −1.18198
$$494$$ 0 0
$$495$$ 4.35862 0.195905
$$496$$ 0 0
$$497$$ 68.6754 3.08051
$$498$$ 0 0
$$499$$ −15.9672 −0.714792 −0.357396 0.933953i $$-0.616335\pi$$
−0.357396 + 0.933953i $$0.616335\pi$$
$$500$$ 0 0
$$501$$ −14.5007 −0.647842
$$502$$ 0 0
$$503$$ 4.53174 0.202060 0.101030 0.994883i $$-0.467786\pi$$
0.101030 + 0.994883i $$0.467786\pi$$
$$504$$ 0 0
$$505$$ −11.9518 −0.531849
$$506$$ 0 0
$$507$$ 1.52121 0.0675591
$$508$$ 0 0
$$509$$ −4.08798 −0.181197 −0.0905983 0.995888i $$-0.528878\pi$$
−0.0905983 + 0.995888i $$0.528878\pi$$
$$510$$ 0 0
$$511$$ 1.10856 0.0490398
$$512$$ 0 0
$$513$$ −18.1403 −0.800913
$$514$$ 0 0
$$515$$ 9.43272 0.415655
$$516$$ 0 0
$$517$$ 4.81387 0.211714
$$518$$ 0 0
$$519$$ 15.0716 0.661571
$$520$$ 0 0
$$521$$ 34.2884 1.50220 0.751102 0.660186i $$-0.229523\pi$$
0.751102 + 0.660186i $$0.229523\pi$$
$$522$$ 0 0
$$523$$ −33.7724 −1.47676 −0.738382 0.674382i $$-0.764410\pi$$
−0.738382 + 0.674382i $$0.764410\pi$$
$$524$$ 0 0
$$525$$ −15.3218 −0.668699
$$526$$ 0 0
$$527$$ 39.1547 1.70561
$$528$$ 0 0
$$529$$ −20.9681 −0.911656
$$530$$ 0 0
$$531$$ −27.9784 −1.21416
$$532$$ 0 0
$$533$$ −35.4322 −1.53474
$$534$$ 0 0
$$535$$ 8.80259 0.380569
$$536$$ 0 0
$$537$$ 14.1311 0.609801
$$538$$ 0 0
$$539$$ −27.7156 −1.19380
$$540$$ 0 0
$$541$$ −31.0330 −1.33421 −0.667106 0.744963i $$-0.732467\pi$$
−0.667106 + 0.744963i $$0.732467\pi$$
$$542$$ 0 0
$$543$$ 14.4356 0.619492
$$544$$ 0 0
$$545$$ 3.12202 0.133733
$$546$$ 0 0
$$547$$ 33.0104 1.41142 0.705712 0.708499i $$-0.250627\pi$$
0.705712 + 0.708499i $$0.250627\pi$$
$$548$$ 0 0
$$549$$ 32.5171 1.38779
$$550$$ 0 0
$$551$$ −20.2626 −0.863214
$$552$$ 0 0
$$553$$ 14.5398 0.618293
$$554$$ 0 0
$$555$$ 5.58699 0.237155
$$556$$ 0 0
$$557$$ 15.4826 0.656019 0.328010 0.944674i $$-0.393622\pi$$
0.328010 + 0.944674i $$0.393622\pi$$
$$558$$ 0 0
$$559$$ 11.3213 0.478840
$$560$$ 0 0
$$561$$ −8.54142 −0.360619
$$562$$ 0 0
$$563$$ −6.21072 −0.261751 −0.130875 0.991399i $$-0.541779\pi$$
−0.130875 + 0.991399i $$0.541779\pi$$
$$564$$ 0 0
$$565$$ −0.810845 −0.0341125
$$566$$ 0 0
$$567$$ 15.9786 0.671038
$$568$$ 0 0
$$569$$ −20.3014 −0.851078 −0.425539 0.904940i $$-0.639916\pi$$
−0.425539 + 0.904940i $$0.639916\pi$$
$$570$$ 0 0
$$571$$ 38.5209 1.61205 0.806024 0.591882i $$-0.201615\pi$$
0.806024 + 0.591882i $$0.201615\pi$$
$$572$$ 0 0
$$573$$ −10.7939 −0.450922
$$574$$ 0 0
$$575$$ −5.80809 −0.242214
$$576$$ 0 0
$$577$$ 19.3481 0.805472 0.402736 0.915316i $$-0.368059\pi$$
0.402736 + 0.915316i $$0.368059\pi$$
$$578$$ 0 0
$$579$$ −20.7850 −0.863795
$$580$$ 0 0
$$581$$ 11.3881 0.472460
$$582$$ 0 0
$$583$$ −10.9595 −0.453896
$$584$$ 0 0
$$585$$ 8.66381 0.358205
$$586$$ 0 0
$$587$$ 20.0183 0.826243 0.413122 0.910676i $$-0.364438\pi$$
0.413122 + 0.910676i $$0.364438\pi$$
$$588$$ 0 0
$$589$$ 30.2305 1.24562
$$590$$ 0 0
$$591$$ −12.2498 −0.503888
$$592$$ 0 0
$$593$$ 12.5904 0.517027 0.258513 0.966008i $$-0.416767\pi$$
0.258513 + 0.966008i $$0.416767\pi$$
$$594$$ 0 0
$$595$$ 23.9812 0.983135
$$596$$ 0 0
$$597$$ 15.7949 0.646441
$$598$$ 0 0
$$599$$ −24.4282 −0.998107 −0.499054 0.866571i $$-0.666319\pi$$
−0.499054 + 0.866571i $$0.666319\pi$$
$$600$$ 0 0
$$601$$ 9.02256 0.368038 0.184019 0.982923i $$-0.441089\pi$$
0.184019 + 0.982923i $$0.441089\pi$$
$$602$$ 0 0
$$603$$ −17.0670 −0.695024
$$604$$ 0 0
$$605$$ −6.96237 −0.283061
$$606$$ 0 0
$$607$$ 19.0044 0.771364 0.385682 0.922632i $$-0.373966\pi$$
0.385682 + 0.922632i $$0.373966\pi$$
$$608$$ 0 0
$$609$$ −18.2657 −0.740165
$$610$$ 0 0
$$611$$ 9.56873 0.387110
$$612$$ 0 0
$$613$$ 3.38617 0.136766 0.0683831 0.997659i $$-0.478216\pi$$
0.0683831 + 0.997659i $$0.478216\pi$$
$$614$$ 0 0
$$615$$ 7.20491 0.290530
$$616$$ 0 0
$$617$$ −33.9920 −1.36847 −0.684234 0.729263i $$-0.739863\pi$$
−0.684234 + 0.729263i $$0.739863\pi$$
$$618$$ 0 0
$$619$$ −29.6477 −1.19164 −0.595822 0.803117i $$-0.703174\pi$$
−0.595822 + 0.803117i $$0.703174\pi$$
$$620$$ 0 0
$$621$$ −6.19882 −0.248750
$$622$$ 0 0
$$623$$ 25.6156 1.02627
$$624$$ 0 0
$$625$$ 11.9748 0.478992
$$626$$ 0 0
$$627$$ −6.59464 −0.263364
$$628$$ 0 0
$$629$$ 38.5008 1.53513
$$630$$ 0 0
$$631$$ −14.6241 −0.582176 −0.291088 0.956696i $$-0.594017\pi$$
−0.291088 + 0.956696i $$0.594017\pi$$
$$632$$ 0 0
$$633$$ −22.7698 −0.905018
$$634$$ 0 0
$$635$$ −10.9106 −0.432975
$$636$$ 0 0
$$637$$ −55.0915 −2.18280
$$638$$ 0 0
$$639$$ −34.7665 −1.37534
$$640$$ 0 0
$$641$$ 4.17547 0.164921 0.0824606 0.996594i $$-0.473722\pi$$
0.0824606 + 0.996594i $$0.473722\pi$$
$$642$$ 0 0
$$643$$ −23.1928 −0.914635 −0.457317 0.889304i $$-0.651190\pi$$
−0.457317 + 0.889304i $$0.651190\pi$$
$$644$$ 0 0
$$645$$ −2.30211 −0.0906455
$$646$$ 0 0
$$647$$ 31.0975 1.22257 0.611284 0.791411i $$-0.290653\pi$$
0.611284 + 0.791411i $$0.290653\pi$$
$$648$$ 0 0
$$649$$ −23.2347 −0.912043
$$650$$ 0 0
$$651$$ 27.2513 1.06806
$$652$$ 0 0
$$653$$ −31.5439 −1.23441 −0.617204 0.786803i $$-0.711735\pi$$
−0.617204 + 0.786803i $$0.711735\pi$$
$$654$$ 0 0
$$655$$ −13.5893 −0.530979
$$656$$ 0 0
$$657$$ −0.561202 −0.0218946
$$658$$ 0 0
$$659$$ −25.2754 −0.984590 −0.492295 0.870428i $$-0.663842\pi$$
−0.492295 + 0.870428i $$0.663842\pi$$
$$660$$ 0 0
$$661$$ −21.7628 −0.846476 −0.423238 0.906018i $$-0.639107\pi$$
−0.423238 + 0.906018i $$0.639107\pi$$
$$662$$ 0 0
$$663$$ −16.9781 −0.659377
$$664$$ 0 0
$$665$$ 18.5154 0.717995
$$666$$ 0 0
$$667$$ −6.92404 −0.268100
$$668$$ 0 0
$$669$$ 0.315814 0.0122101
$$670$$ 0 0
$$671$$ 27.0038 1.04247
$$672$$ 0 0
$$673$$ −37.2710 −1.43669 −0.718346 0.695686i $$-0.755101\pi$$
−0.718346 + 0.695686i $$0.755101\pi$$
$$674$$ 0 0
$$675$$ 17.7189 0.682001
$$676$$ 0 0
$$677$$ −33.5515 −1.28949 −0.644745 0.764398i $$-0.723036\pi$$
−0.644745 + 0.764398i $$0.723036\pi$$
$$678$$ 0 0
$$679$$ 24.3599 0.934846
$$680$$ 0 0
$$681$$ −8.37644 −0.320986
$$682$$ 0 0
$$683$$ −8.94737 −0.342362 −0.171181 0.985240i $$-0.554758\pi$$
−0.171181 + 0.985240i $$0.554758\pi$$
$$684$$ 0 0
$$685$$ −12.4631 −0.476192
$$686$$ 0 0
$$687$$ −8.58575 −0.327567
$$688$$ 0 0
$$689$$ −21.7846 −0.829929
$$690$$ 0 0
$$691$$ −7.04969 −0.268183 −0.134092 0.990969i $$-0.542812\pi$$
−0.134092 + 0.990969i $$0.542812\pi$$
$$692$$ 0 0
$$693$$ 20.9047 0.794105
$$694$$ 0 0
$$695$$ −7.92309 −0.300540
$$696$$ 0 0
$$697$$ 49.6502 1.88063
$$698$$ 0 0
$$699$$ 20.9232 0.791389
$$700$$ 0 0
$$701$$ −5.19179 −0.196091 −0.0980456 0.995182i $$-0.531259\pi$$
−0.0980456 + 0.995182i $$0.531259\pi$$
$$702$$ 0 0
$$703$$ 29.7256 1.12112
$$704$$ 0 0
$$705$$ −1.94574 −0.0732807
$$706$$ 0 0
$$707$$ −57.3230 −2.15586
$$708$$ 0 0
$$709$$ −16.3067 −0.612410 −0.306205 0.951966i $$-0.599059\pi$$
−0.306205 + 0.951966i $$0.599059\pi$$
$$710$$ 0 0
$$711$$ −7.36066 −0.276046
$$712$$ 0 0
$$713$$ 10.3302 0.386870
$$714$$ 0 0
$$715$$ 7.19488 0.269073
$$716$$ 0 0
$$717$$ 20.4820 0.764915
$$718$$ 0 0
$$719$$ 27.4968 1.02546 0.512729 0.858551i $$-0.328635\pi$$
0.512729 + 0.858551i $$0.328635\pi$$
$$720$$ 0 0
$$721$$ 45.2410 1.68486
$$722$$ 0 0
$$723$$ −0.815004 −0.0303103
$$724$$ 0 0
$$725$$ 19.7919 0.735053
$$726$$ 0 0
$$727$$ 15.2319 0.564919 0.282460 0.959279i $$-0.408850\pi$$
0.282460 + 0.959279i $$0.408850\pi$$
$$728$$ 0 0
$$729$$ 2.54349 0.0942032
$$730$$ 0 0
$$731$$ −15.8642 −0.586758
$$732$$ 0 0
$$733$$ 6.76475 0.249862 0.124931 0.992165i $$-0.460129\pi$$
0.124931 + 0.992165i $$0.460129\pi$$
$$734$$ 0 0
$$735$$ 11.2025 0.413210
$$736$$ 0 0
$$737$$ −14.1734 −0.522082
$$738$$ 0 0
$$739$$ −17.9519 −0.660372 −0.330186 0.943916i $$-0.607111\pi$$
−0.330186 + 0.943916i $$0.607111\pi$$
$$740$$ 0 0
$$741$$ −13.1084 −0.481551
$$742$$ 0 0
$$743$$ 17.2096 0.631357 0.315679 0.948866i $$-0.397768\pi$$
0.315679 + 0.948866i $$0.397768\pi$$
$$744$$ 0 0
$$745$$ 4.31559 0.158111
$$746$$ 0 0
$$747$$ −5.76518 −0.210937
$$748$$ 0 0
$$749$$ 42.2188 1.54264
$$750$$ 0 0
$$751$$ −4.98853 −0.182034 −0.0910170 0.995849i $$-0.529012\pi$$
−0.0910170 + 0.995849i $$0.529012\pi$$
$$752$$ 0 0
$$753$$ −2.05575 −0.0749156
$$754$$ 0 0
$$755$$ −0.0898146 −0.00326869
$$756$$ 0 0
$$757$$ −19.5585 −0.710864 −0.355432 0.934702i $$-0.615666\pi$$
−0.355432 + 0.934702i $$0.615666\pi$$
$$758$$ 0 0
$$759$$ −2.25349 −0.0817966
$$760$$ 0 0
$$761$$ −41.0235 −1.48710 −0.743550 0.668681i $$-0.766859\pi$$
−0.743550 + 0.668681i $$0.766859\pi$$
$$762$$ 0 0
$$763$$ 14.9738 0.542087
$$764$$ 0 0
$$765$$ −12.1403 −0.438935
$$766$$ 0 0
$$767$$ −46.1847 −1.66763
$$768$$ 0 0
$$769$$ 11.7068 0.422159 0.211079 0.977469i $$-0.432302\pi$$
0.211079 + 0.977469i $$0.432302\pi$$
$$770$$ 0 0
$$771$$ 3.19765 0.115161
$$772$$ 0 0
$$773$$ −6.66278 −0.239643 −0.119822 0.992795i $$-0.538232\pi$$
−0.119822 + 0.992795i $$0.538232\pi$$
$$774$$ 0 0
$$775$$ −29.5283 −1.06069
$$776$$ 0 0
$$777$$ 26.7962 0.961309
$$778$$ 0 0
$$779$$ 38.3338 1.37345
$$780$$ 0 0
$$781$$ −28.8719 −1.03312
$$782$$ 0 0
$$783$$ 21.1234 0.754888
$$784$$ 0 0
$$785$$ −9.74865 −0.347944
$$786$$ 0 0
$$787$$ −2.88890 −0.102978 −0.0514891 0.998674i $$-0.516397\pi$$
−0.0514891 + 0.998674i $$0.516397\pi$$
$$788$$ 0 0
$$789$$ −5.09364 −0.181338
$$790$$ 0 0
$$791$$ −3.88896 −0.138275
$$792$$ 0 0
$$793$$ 53.6767 1.90612
$$794$$ 0 0
$$795$$ 4.42976 0.157107
$$796$$ 0 0
$$797$$ 18.0912 0.640824 0.320412 0.947278i $$-0.396179\pi$$
0.320412 + 0.947278i $$0.396179\pi$$
$$798$$ 0 0
$$799$$ −13.4084 −0.474355
$$800$$ 0 0
$$801$$ −12.9678 −0.458193
$$802$$ 0 0
$$803$$ −0.466051 −0.0164466
$$804$$ 0 0
$$805$$ 6.32700 0.222997
$$806$$ 0 0
$$807$$ 16.7029 0.587971
$$808$$ 0 0
$$809$$ 8.04121 0.282714 0.141357 0.989959i $$-0.454853\pi$$
0.141357 + 0.989959i $$0.454853\pi$$
$$810$$ 0 0
$$811$$ −24.5111 −0.860701 −0.430350 0.902662i $$-0.641610\pi$$
−0.430350 + 0.902662i $$0.641610\pi$$
$$812$$ 0 0
$$813$$ −12.4842 −0.437839
$$814$$ 0 0
$$815$$ −7.21311 −0.252664
$$816$$ 0 0
$$817$$ −12.2484 −0.428517
$$818$$ 0 0
$$819$$ 41.5532 1.45199
$$820$$ 0 0
$$821$$ −10.6539 −0.371823 −0.185912 0.982566i $$-0.559524\pi$$
−0.185912 + 0.982566i $$0.559524\pi$$
$$822$$ 0 0
$$823$$ −25.6473 −0.894009 −0.447004 0.894532i $$-0.647509\pi$$
−0.447004 + 0.894532i $$0.647509\pi$$
$$824$$ 0 0
$$825$$ 6.44146 0.224263
$$826$$ 0 0
$$827$$ −18.0150 −0.626442 −0.313221 0.949680i $$-0.601408\pi$$
−0.313221 + 0.949680i $$0.601408\pi$$
$$828$$ 0 0
$$829$$ 10.1189 0.351444 0.175722 0.984440i $$-0.443774\pi$$
0.175722 + 0.984440i $$0.443774\pi$$
$$830$$ 0 0
$$831$$ 2.66008 0.0922771
$$832$$ 0 0
$$833$$ 77.1981 2.67476
$$834$$ 0 0
$$835$$ −17.1160 −0.592325
$$836$$ 0 0
$$837$$ −31.5148 −1.08931
$$838$$ 0 0
$$839$$ 46.0365 1.58936 0.794679 0.607030i $$-0.207639\pi$$
0.794679 + 0.607030i $$0.207639\pi$$
$$840$$ 0 0
$$841$$ −5.40531 −0.186390
$$842$$ 0 0
$$843$$ −10.1705 −0.350292
$$844$$ 0 0
$$845$$ 1.79557 0.0617696
$$846$$ 0 0
$$847$$ −33.3928 −1.14739
$$848$$ 0 0
$$849$$ 12.8865 0.442262
$$850$$ 0 0
$$851$$ 10.1577 0.348202
$$852$$ 0 0
$$853$$ −55.0001 −1.88317 −0.941584 0.336780i $$-0.890662\pi$$
−0.941584 + 0.336780i $$0.890662\pi$$
$$854$$ 0 0
$$855$$ −9.37329 −0.320560
$$856$$ 0 0
$$857$$ −31.5202 −1.07671 −0.538354 0.842719i $$-0.680954\pi$$
−0.538354 + 0.842719i $$0.680954\pi$$
$$858$$ 0 0
$$859$$ 37.2208 1.26996 0.634979 0.772529i $$-0.281009\pi$$
0.634979 + 0.772529i $$0.281009\pi$$
$$860$$ 0 0
$$861$$ 34.5561 1.17767
$$862$$ 0 0
$$863$$ −30.3224 −1.03218 −0.516092 0.856533i $$-0.672614\pi$$
−0.516092 + 0.856533i $$0.672614\pi$$
$$864$$ 0 0
$$865$$ 17.7900 0.604877
$$866$$ 0 0
$$867$$ 9.93592 0.337442
$$868$$ 0 0
$$869$$ −6.11267 −0.207358
$$870$$ 0 0
$$871$$ −28.1730 −0.954605
$$872$$ 0 0
$$873$$ −12.3320 −0.417376
$$874$$ 0 0
$$875$$ −40.2783 −1.36165
$$876$$ 0 0
$$877$$ 27.0037 0.911849 0.455925 0.890018i $$-0.349309\pi$$
0.455925 + 0.890018i $$0.349309\pi$$
$$878$$ 0 0
$$879$$ 15.2526 0.514457
$$880$$ 0 0
$$881$$ −0.565506 −0.0190524 −0.00952619 0.999955i $$-0.503032\pi$$
−0.00952619 + 0.999955i $$0.503032\pi$$
$$882$$ 0 0
$$883$$ −23.8838 −0.803752 −0.401876 0.915694i $$-0.631642\pi$$
−0.401876 + 0.915694i $$0.631642\pi$$
$$884$$ 0 0
$$885$$ 9.39135 0.315687
$$886$$ 0 0
$$887$$ 10.1717 0.341533 0.170767 0.985311i $$-0.445376\pi$$
0.170767 + 0.985311i $$0.445376\pi$$
$$888$$ 0 0
$$889$$ −52.3294 −1.75507
$$890$$ 0 0
$$891$$ −6.71757 −0.225047
$$892$$ 0 0
$$893$$ −10.3523 −0.346427
$$894$$ 0 0
$$895$$ 16.6798 0.557544
$$896$$ 0 0
$$897$$ −4.47937 −0.149562
$$898$$ 0 0
$$899$$ −35.2018 −1.17405
$$900$$ 0 0
$$901$$ 30.5262 1.01697
$$902$$ 0 0
$$903$$ −11.0413 −0.367432
$$904$$ 0 0
$$905$$ 17.0393 0.566405
$$906$$ 0 0
$$907$$ 44.1661 1.46651 0.733255 0.679954i $$-0.238000\pi$$
0.733255 + 0.679954i $$0.238000\pi$$
$$908$$ 0 0
$$909$$ 29.0194 0.962514
$$910$$ 0 0
$$911$$ 7.07316 0.234344 0.117172 0.993112i $$-0.462617\pi$$
0.117172 + 0.993112i $$0.462617\pi$$
$$912$$ 0 0
$$913$$ −4.78770 −0.158450
$$914$$ 0 0
$$915$$ −10.9148 −0.360832
$$916$$ 0 0
$$917$$ −65.1769 −2.15233
$$918$$ 0 0
$$919$$ −23.6016 −0.778544 −0.389272 0.921123i $$-0.627273\pi$$
−0.389272 + 0.921123i $$0.627273\pi$$
$$920$$ 0 0
$$921$$ −4.26636 −0.140581
$$922$$ 0 0
$$923$$ −57.3899 −1.88901
$$924$$ 0 0
$$925$$ −29.0352 −0.954670
$$926$$ 0 0
$$927$$ −22.9030 −0.752233
$$928$$ 0 0
$$929$$ 31.8496 1.04495 0.522475 0.852654i $$-0.325009\pi$$
0.522475 + 0.852654i $$0.325009\pi$$
$$930$$ 0 0
$$931$$ 59.6029 1.95341
$$932$$ 0 0
$$933$$ 21.7506 0.712083
$$934$$ 0 0
$$935$$ −10.0820 −0.329716
$$936$$ 0 0
$$937$$ 44.1729 1.44307 0.721533 0.692380i $$-0.243438\pi$$
0.721533 + 0.692380i $$0.243438\pi$$
$$938$$ 0 0
$$939$$ 5.08677 0.166001
$$940$$ 0 0
$$941$$ −34.5303 −1.12566 −0.562828 0.826574i $$-0.690287\pi$$
−0.562828 + 0.826574i $$0.690287\pi$$
$$942$$ 0 0
$$943$$ 13.0993 0.426571
$$944$$ 0 0
$$945$$ −19.3020 −0.627893
$$946$$ 0 0
$$947$$ 3.46191 0.112497 0.0562485 0.998417i $$-0.482086\pi$$
0.0562485 + 0.998417i $$0.482086\pi$$
$$948$$ 0 0
$$949$$ −0.926389 −0.0300719
$$950$$ 0 0
$$951$$ 5.61966 0.182230
$$952$$ 0 0
$$953$$ 60.2596 1.95200 0.976000 0.217771i $$-0.0698785\pi$$
0.976000 + 0.217771i $$0.0698785\pi$$
$$954$$ 0 0
$$955$$ −12.7407 −0.412280
$$956$$ 0 0
$$957$$ 7.67911 0.248230
$$958$$ 0 0
$$959$$ −59.7755 −1.93025
$$960$$ 0 0
$$961$$ 21.5188 0.694156
$$962$$ 0 0
$$963$$ −21.3730 −0.688735
$$964$$ 0 0
$$965$$ −24.5338 −0.789772
$$966$$ 0 0
$$967$$ 39.9168 1.28364 0.641819 0.766856i $$-0.278180\pi$$
0.641819 + 0.766856i $$0.278180\pi$$
$$968$$ 0 0
$$969$$ 18.3685 0.590081
$$970$$ 0 0
$$971$$ −43.7104 −1.40273 −0.701366 0.712801i $$-0.747426\pi$$
−0.701366 + 0.712801i $$0.747426\pi$$
$$972$$ 0 0
$$973$$ −38.0006 −1.21824
$$974$$ 0 0
$$975$$ 12.8040 0.410055
$$976$$ 0 0
$$977$$ −0.955388 −0.0305656 −0.0152828 0.999883i $$-0.504865\pi$$
−0.0152828 + 0.999883i $$0.504865\pi$$
$$978$$ 0 0
$$979$$ −10.7691 −0.344182
$$980$$ 0 0
$$981$$ −7.58037 −0.242023
$$982$$ 0 0
$$983$$ 48.7488 1.55485 0.777423 0.628978i $$-0.216526\pi$$
0.777423 + 0.628978i $$0.216526\pi$$
$$984$$ 0 0
$$985$$ −14.4591 −0.460707
$$986$$ 0 0
$$987$$ −9.33211 −0.297044
$$988$$ 0 0
$$989$$ −4.18547 −0.133090
$$990$$ 0 0
$$991$$ −0.488986 −0.0155332 −0.00776658 0.999970i $$-0.502472\pi$$
−0.00776658 + 0.999970i $$0.502472\pi$$
$$992$$ 0 0
$$993$$ 11.0594 0.350959
$$994$$ 0 0
$$995$$ 18.6437 0.591044
$$996$$ 0 0
$$997$$ 39.5339 1.25205 0.626026 0.779802i $$-0.284680\pi$$
0.626026 + 0.779802i $$0.284680\pi$$
$$998$$ 0 0
$$999$$ −30.9885 −0.980432
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 3856.2.a.j.1.5 7
4.3 odd 2 241.2.a.a.1.6 7
12.11 even 2 2169.2.a.e.1.2 7
20.19 odd 2 6025.2.a.f.1.2 7

By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.6 7 4.3 odd 2
2169.2.a.e.1.2 7 12.11 even 2
3856.2.a.j.1.5 7 1.1 even 1 trivial
6025.2.a.f.1.2 7 20.19 odd 2