# Properties

 Label 2800.2.a.br.1.1 Level $2800$ Weight $2$ Character 2800.1 Self dual yes Analytic conductor $22.358$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.76156$$ of defining polynomial Character $$\chi$$ $$=$$ 2800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.76156 q^{3} -1.00000 q^{7} +0.103084 q^{9} +O(q^{10})$$ $$q-1.76156 q^{3} -1.00000 q^{7} +0.103084 q^{9} +0.626198 q^{11} +5.49084 q^{13} +0.896916 q^{17} -6.38776 q^{19} +1.76156 q^{21} +3.72928 q^{23} +5.10308 q^{27} -7.87859 q^{29} -7.52311 q^{31} -1.10308 q^{33} +6.00000 q^{37} -9.67243 q^{39} +7.72928 q^{41} -1.72928 q^{43} -5.87859 q^{47} +1.00000 q^{49} -1.57997 q^{51} -6.77551 q^{53} +11.2524 q^{57} +0.593923 q^{59} +7.13536 q^{61} -0.103084 q^{63} +5.79383 q^{67} -6.56934 q^{69} -5.52311 q^{71} -3.72928 q^{73} -0.626198 q^{77} +5.67243 q^{79} -9.29862 q^{81} +17.4340 q^{83} +13.8786 q^{87} +14.2986 q^{89} -5.49084 q^{91} +13.2524 q^{93} -10.1493 q^{97} +0.0645508 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{3} - 3q^{7} + 4q^{9} + O(q^{10})$$ $$3q + q^{3} - 3q^{7} + 4q^{9} - 7q^{11} + 5q^{13} - q^{17} - 4q^{19} - q^{21} + 6q^{23} + 19q^{27} + 3q^{29} - 10q^{31} - 7q^{33} + 18q^{37} + 5q^{39} + 18q^{41} + 9q^{47} + 3q^{49} - 21q^{51} + 10q^{53} + 16q^{57} - 6q^{59} + 24q^{61} - 4q^{63} + 10q^{67} + 18q^{69} - 4q^{71} - 6q^{73} + 7q^{77} - 17q^{79} + 15q^{81} + 12q^{83} + 15q^{87} - 5q^{91} + 22q^{93} - 9q^{97} - 2q^{99} + O(q^{100})$$

## 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
$$3$$ −1.76156 −1.01704 −0.508518 0.861052i $$-0.669806\pi$$
−0.508518 + 0.861052i $$0.669806\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0.103084 0.0343612
$$10$$ 0 0
$$11$$ 0.626198 0.188806 0.0944029 0.995534i $$-0.469906\pi$$
0.0944029 + 0.995534i $$0.469906\pi$$
$$12$$ 0 0
$$13$$ 5.49084 1.52288 0.761442 0.648233i $$-0.224492\pi$$
0.761442 + 0.648233i $$0.224492\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.896916 0.217534 0.108767 0.994067i $$-0.465310\pi$$
0.108767 + 0.994067i $$0.465310\pi$$
$$18$$ 0 0
$$19$$ −6.38776 −1.46545 −0.732726 0.680524i $$-0.761752\pi$$
−0.732726 + 0.680524i $$0.761752\pi$$
$$20$$ 0 0
$$21$$ 1.76156 0.384403
$$22$$ 0 0
$$23$$ 3.72928 0.777609 0.388805 0.921320i $$-0.372888\pi$$
0.388805 + 0.921320i $$0.372888\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.10308 0.982089
$$28$$ 0 0
$$29$$ −7.87859 −1.46302 −0.731509 0.681832i $$-0.761184\pi$$
−0.731509 + 0.681832i $$0.761184\pi$$
$$30$$ 0 0
$$31$$ −7.52311 −1.35119 −0.675596 0.737272i $$-0.736113\pi$$
−0.675596 + 0.737272i $$0.736113\pi$$
$$32$$ 0 0
$$33$$ −1.10308 −0.192022
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ −9.67243 −1.54883
$$40$$ 0 0
$$41$$ 7.72928 1.20711 0.603556 0.797321i $$-0.293750\pi$$
0.603556 + 0.797321i $$0.293750\pi$$
$$42$$ 0 0
$$43$$ −1.72928 −0.263713 −0.131856 0.991269i $$-0.542094\pi$$
−0.131856 + 0.991269i $$0.542094\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −5.87859 −0.857481 −0.428741 0.903428i $$-0.641043\pi$$
−0.428741 + 0.903428i $$0.641043\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −1.57997 −0.221240
$$52$$ 0 0
$$53$$ −6.77551 −0.930688 −0.465344 0.885130i $$-0.654069\pi$$
−0.465344 + 0.885130i $$0.654069\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 11.2524 1.49042
$$58$$ 0 0
$$59$$ 0.593923 0.0773221 0.0386611 0.999252i $$-0.487691\pi$$
0.0386611 + 0.999252i $$0.487691\pi$$
$$60$$ 0 0
$$61$$ 7.13536 0.913589 0.456795 0.889572i $$-0.348997\pi$$
0.456795 + 0.889572i $$0.348997\pi$$
$$62$$ 0 0
$$63$$ −0.103084 −0.0129873
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.79383 0.707829 0.353915 0.935278i $$-0.384850\pi$$
0.353915 + 0.935278i $$0.384850\pi$$
$$68$$ 0 0
$$69$$ −6.56934 −0.790856
$$70$$ 0 0
$$71$$ −5.52311 −0.655473 −0.327737 0.944769i $$-0.606286\pi$$
−0.327737 + 0.944769i $$0.606286\pi$$
$$72$$ 0 0
$$73$$ −3.72928 −0.436479 −0.218240 0.975895i $$-0.570031\pi$$
−0.218240 + 0.975895i $$0.570031\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −0.626198 −0.0713619
$$78$$ 0 0
$$79$$ 5.67243 0.638198 0.319099 0.947721i $$-0.396620\pi$$
0.319099 + 0.947721i $$0.396620\pi$$
$$80$$ 0 0
$$81$$ −9.29862 −1.03318
$$82$$ 0 0
$$83$$ 17.4340 1.91363 0.956814 0.290700i $$-0.0938882\pi$$
0.956814 + 0.290700i $$0.0938882\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 13.8786 1.48794
$$88$$ 0 0
$$89$$ 14.2986 1.51565 0.757826 0.652457i $$-0.226262\pi$$
0.757826 + 0.652457i $$0.226262\pi$$
$$90$$ 0 0
$$91$$ −5.49084 −0.575596
$$92$$ 0 0
$$93$$ 13.2524 1.37421
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −10.1493 −1.03051 −0.515253 0.857038i $$-0.672302\pi$$
−0.515253 + 0.857038i $$0.672302\pi$$
$$98$$ 0 0
$$99$$ 0.0645508 0.00648760
$$100$$ 0 0
$$101$$ 9.64015 0.959231 0.479615 0.877479i $$-0.340776\pi$$
0.479615 + 0.877479i $$0.340776\pi$$
$$102$$ 0 0
$$103$$ 0.626198 0.0617011 0.0308506 0.999524i $$-0.490178\pi$$
0.0308506 + 0.999524i $$0.490178\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 18.9248 1.81267 0.906335 0.422561i $$-0.138869\pi$$
0.906335 + 0.422561i $$0.138869\pi$$
$$110$$ 0 0
$$111$$ −10.5693 −1.00320
$$112$$ 0 0
$$113$$ −1.04623 −0.0984209 −0.0492105 0.998788i $$-0.515671\pi$$
−0.0492105 + 0.998788i $$0.515671\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.566016 0.0523282
$$118$$ 0 0
$$119$$ −0.896916 −0.0822202
$$120$$ 0 0
$$121$$ −10.6079 −0.964352
$$122$$ 0 0
$$123$$ −13.6156 −1.22767
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 21.2803 1.88832 0.944161 0.329485i $$-0.106875\pi$$
0.944161 + 0.329485i $$0.106875\pi$$
$$128$$ 0 0
$$129$$ 3.04623 0.268205
$$130$$ 0 0
$$131$$ 9.91087 0.865917 0.432958 0.901414i $$-0.357470\pi$$
0.432958 + 0.901414i $$0.357470\pi$$
$$132$$ 0 0
$$133$$ 6.38776 0.553889
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.45856 0.295485 0.147743 0.989026i $$-0.452799\pi$$
0.147743 + 0.989026i $$0.452799\pi$$
$$138$$ 0 0
$$139$$ 20.4157 1.73163 0.865817 0.500361i $$-0.166799\pi$$
0.865817 + 0.500361i $$0.166799\pi$$
$$140$$ 0 0
$$141$$ 10.3555 0.872089
$$142$$ 0 0
$$143$$ 3.43835 0.287530
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1.76156 −0.145291
$$148$$ 0 0
$$149$$ −17.0462 −1.39648 −0.698241 0.715863i $$-0.746033\pi$$
−0.698241 + 0.715863i $$0.746033\pi$$
$$150$$ 0 0
$$151$$ −2.89692 −0.235748 −0.117874 0.993029i $$-0.537608\pi$$
−0.117874 + 0.993029i $$0.537608\pi$$
$$152$$ 0 0
$$153$$ 0.0924575 0.00747474
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.1170 0.807427 0.403714 0.914885i $$-0.367719\pi$$
0.403714 + 0.914885i $$0.367719\pi$$
$$158$$ 0 0
$$159$$ 11.9354 0.946543
$$160$$ 0 0
$$161$$ −3.72928 −0.293909
$$162$$ 0 0
$$163$$ −0.476886 −0.0373526 −0.0186763 0.999826i $$-0.505945\pi$$
−0.0186763 + 0.999826i $$0.505945\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 13.1676 1.01894 0.509471 0.860488i $$-0.329841\pi$$
0.509471 + 0.860488i $$0.329841\pi$$
$$168$$ 0 0
$$169$$ 17.1493 1.31918
$$170$$ 0 0
$$171$$ −0.658473 −0.0503547
$$172$$ 0 0
$$173$$ 13.9677 1.06195 0.530973 0.847389i $$-0.321826\pi$$
0.530973 + 0.847389i $$0.321826\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1.04623 −0.0786394
$$178$$ 0 0
$$179$$ −7.45856 −0.557479 −0.278740 0.960367i $$-0.589917\pi$$
−0.278740 + 0.960367i $$0.589917\pi$$
$$180$$ 0 0
$$181$$ −14.1170 −1.04931 −0.524656 0.851315i $$-0.675806\pi$$
−0.524656 + 0.851315i $$0.675806\pi$$
$$182$$ 0 0
$$183$$ −12.5693 −0.929153
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.561647 0.0410717
$$188$$ 0 0
$$189$$ −5.10308 −0.371195
$$190$$ 0 0
$$191$$ 8.42003 0.609252 0.304626 0.952472i $$-0.401469\pi$$
0.304626 + 0.952472i $$0.401469\pi$$
$$192$$ 0 0
$$193$$ −22.2986 −1.60509 −0.802545 0.596591i $$-0.796521\pi$$
−0.802545 + 0.596591i $$0.796521\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 26.3632 1.87830 0.939149 0.343509i $$-0.111616\pi$$
0.939149 + 0.343509i $$0.111616\pi$$
$$198$$ 0 0
$$199$$ −22.4402 −1.59075 −0.795373 0.606120i $$-0.792725\pi$$
−0.795373 + 0.606120i $$0.792725\pi$$
$$200$$ 0 0
$$201$$ −10.2062 −0.719888
$$202$$ 0 0
$$203$$ 7.87859 0.552969
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0.384428 0.0267196
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 15.1955 1.04610 0.523052 0.852301i $$-0.324793\pi$$
0.523052 + 0.852301i $$0.324793\pi$$
$$212$$ 0 0
$$213$$ 9.72928 0.666639
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 7.52311 0.510702
$$218$$ 0 0
$$219$$ 6.56934 0.443915
$$220$$ 0 0
$$221$$ 4.92482 0.331279
$$222$$ 0 0
$$223$$ 6.89692 0.461852 0.230926 0.972971i $$-0.425825\pi$$
0.230926 + 0.972971i $$0.425825\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2.77988 0.184507 0.0922535 0.995736i $$-0.470593\pi$$
0.0922535 + 0.995736i $$0.470593\pi$$
$$228$$ 0 0
$$229$$ 18.6585 1.23299 0.616493 0.787360i $$-0.288553\pi$$
0.616493 + 0.787360i $$0.288553\pi$$
$$230$$ 0 0
$$231$$ 1.10308 0.0725776
$$232$$ 0 0
$$233$$ −11.0462 −0.723663 −0.361831 0.932244i $$-0.617848\pi$$
−0.361831 + 0.932244i $$0.617848\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −9.99230 −0.649070
$$238$$ 0 0
$$239$$ 20.1493 1.30335 0.651675 0.758498i $$-0.274066\pi$$
0.651675 + 0.758498i $$0.274066\pi$$
$$240$$ 0 0
$$241$$ 3.72928 0.240224 0.120112 0.992760i $$-0.461675\pi$$
0.120112 + 0.992760i $$0.461675\pi$$
$$242$$ 0 0
$$243$$ 1.07081 0.0686924
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −35.0741 −2.23171
$$248$$ 0 0
$$249$$ −30.7110 −1.94623
$$250$$ 0 0
$$251$$ 21.4985 1.35698 0.678488 0.734612i $$-0.262636\pi$$
0.678488 + 0.734612i $$0.262636\pi$$
$$252$$ 0 0
$$253$$ 2.33527 0.146817
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4.27072 0.266400 0.133200 0.991089i $$-0.457475\pi$$
0.133200 + 0.991089i $$0.457475\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ −0.812155 −0.0502711
$$262$$ 0 0
$$263$$ 11.4586 0.706565 0.353283 0.935517i $$-0.385065\pi$$
0.353283 + 0.935517i $$0.385065\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −25.1878 −1.54147
$$268$$ 0 0
$$269$$ −7.07081 −0.431115 −0.215557 0.976491i $$-0.569157\pi$$
−0.215557 + 0.976491i $$0.569157\pi$$
$$270$$ 0 0
$$271$$ −17.0096 −1.03326 −0.516629 0.856209i $$-0.672813\pi$$
−0.516629 + 0.856209i $$0.672813\pi$$
$$272$$ 0 0
$$273$$ 9.67243 0.585402
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −18.7755 −1.12811 −0.564056 0.825737i $$-0.690760\pi$$
−0.564056 + 0.825737i $$0.690760\pi$$
$$278$$ 0 0
$$279$$ −0.775511 −0.0464286
$$280$$ 0 0
$$281$$ −4.59829 −0.274311 −0.137156 0.990550i $$-0.543796\pi$$
−0.137156 + 0.990550i $$0.543796\pi$$
$$282$$ 0 0
$$283$$ 13.5833 0.807443 0.403722 0.914882i $$-0.367716\pi$$
0.403722 + 0.914882i $$0.367716\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −7.72928 −0.456245
$$288$$ 0 0
$$289$$ −16.1955 −0.952679
$$290$$ 0 0
$$291$$ 17.8786 1.04806
$$292$$ 0 0
$$293$$ 11.8261 0.690889 0.345444 0.938439i $$-0.387728\pi$$
0.345444 + 0.938439i $$0.387728\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.19554 0.185424
$$298$$ 0 0
$$299$$ 20.4769 1.18421
$$300$$ 0 0
$$301$$ 1.72928 0.0996741
$$302$$ 0 0
$$303$$ −16.9817 −0.975572
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.9860 1.19774 0.598868 0.800847i $$-0.295617\pi$$
0.598868 + 0.800847i $$0.295617\pi$$
$$308$$ 0 0
$$309$$ −1.10308 −0.0627522
$$310$$ 0 0
$$311$$ 22.5048 1.27613 0.638065 0.769983i $$-0.279735\pi$$
0.638065 + 0.769983i $$0.279735\pi$$
$$312$$ 0 0
$$313$$ −12.4846 −0.705670 −0.352835 0.935686i $$-0.614782\pi$$
−0.352835 + 0.935686i $$0.614782\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 25.5231 1.43352 0.716760 0.697319i $$-0.245624\pi$$
0.716760 + 0.697319i $$0.245624\pi$$
$$318$$ 0 0
$$319$$ −4.93356 −0.276226
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −5.72928 −0.318786
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −33.3372 −1.84355
$$328$$ 0 0
$$329$$ 5.87859 0.324097
$$330$$ 0 0
$$331$$ 2.68305 0.147474 0.0737370 0.997278i $$-0.476507\pi$$
0.0737370 + 0.997278i $$0.476507\pi$$
$$332$$ 0 0
$$333$$ 0.618502 0.0338937
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 12.5048 0.681179 0.340590 0.940212i $$-0.389373\pi$$
0.340590 + 0.940212i $$0.389373\pi$$
$$338$$ 0 0
$$339$$ 1.84299 0.100098
$$340$$ 0 0
$$341$$ −4.71096 −0.255113
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 28.9450 1.55385 0.776925 0.629593i $$-0.216778\pi$$
0.776925 + 0.629593i $$0.216778\pi$$
$$348$$ 0 0
$$349$$ 32.1449 1.72068 0.860340 0.509721i $$-0.170251\pi$$
0.860340 + 0.509721i $$0.170251\pi$$
$$350$$ 0 0
$$351$$ 28.0202 1.49561
$$352$$ 0 0
$$353$$ 8.17722 0.435229 0.217615 0.976035i $$-0.430172\pi$$
0.217615 + 0.976035i $$0.430172\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1.57997 0.0836208
$$358$$ 0 0
$$359$$ −18.5048 −0.976646 −0.488323 0.872663i $$-0.662391\pi$$
−0.488323 + 0.872663i $$0.662391\pi$$
$$360$$ 0 0
$$361$$ 21.8034 1.14755
$$362$$ 0 0
$$363$$ 18.6864 0.980781
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 27.4942 1.43518 0.717592 0.696464i $$-0.245244\pi$$
0.717592 + 0.696464i $$0.245244\pi$$
$$368$$ 0 0
$$369$$ 0.796763 0.0414778
$$370$$ 0 0
$$371$$ 6.77551 0.351767
$$372$$ 0 0
$$373$$ 6.06455 0.314011 0.157005 0.987598i $$-0.449816\pi$$
0.157005 + 0.987598i $$0.449816\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −43.2601 −2.22801
$$378$$ 0 0
$$379$$ 5.72928 0.294293 0.147147 0.989115i $$-0.452991\pi$$
0.147147 + 0.989115i $$0.452991\pi$$
$$380$$ 0 0
$$381$$ −37.4865 −1.92049
$$382$$ 0 0
$$383$$ 1.72928 0.0883622 0.0441811 0.999024i $$-0.485932\pi$$
0.0441811 + 0.999024i $$0.485932\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −0.178261 −0.00906150
$$388$$ 0 0
$$389$$ −36.7187 −1.86171 −0.930855 0.365389i $$-0.880936\pi$$
−0.930855 + 0.365389i $$0.880936\pi$$
$$390$$ 0 0
$$391$$ 3.34485 0.169157
$$392$$ 0 0
$$393$$ −17.4586 −0.880668
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.03228 −0.101997 −0.0509985 0.998699i $$-0.516240\pi$$
−0.0509985 + 0.998699i $$0.516240\pi$$
$$398$$ 0 0
$$399$$ −11.2524 −0.563324
$$400$$ 0 0
$$401$$ −1.64452 −0.0821234 −0.0410617 0.999157i $$-0.513074\pi$$
−0.0410617 + 0.999157i $$0.513074\pi$$
$$402$$ 0 0
$$403$$ −41.3082 −2.05771
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3.75719 0.186237
$$408$$ 0 0
$$409$$ −4.98168 −0.246328 −0.123164 0.992386i $$-0.539304\pi$$
−0.123164 + 0.992386i $$0.539304\pi$$
$$410$$ 0 0
$$411$$ −6.09246 −0.300519
$$412$$ 0 0
$$413$$ −0.593923 −0.0292250
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −35.9634 −1.76113
$$418$$ 0 0
$$419$$ −22.9205 −1.11974 −0.559869 0.828581i $$-0.689148\pi$$
−0.559869 + 0.828581i $$0.689148\pi$$
$$420$$ 0 0
$$421$$ −20.9527 −1.02117 −0.510587 0.859826i $$-0.670572\pi$$
−0.510587 + 0.859826i $$0.670572\pi$$
$$422$$ 0 0
$$423$$ −0.605987 −0.0294641
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −7.13536 −0.345304
$$428$$ 0 0
$$429$$ −6.05685 −0.292428
$$430$$ 0 0
$$431$$ −33.1589 −1.59721 −0.798604 0.601857i $$-0.794428\pi$$
−0.798604 + 0.601857i $$0.794428\pi$$
$$432$$ 0 0
$$433$$ −18.5414 −0.891045 −0.445522 0.895271i $$-0.646982\pi$$
−0.445522 + 0.895271i $$0.646982\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −23.8217 −1.13955
$$438$$ 0 0
$$439$$ 20.8401 0.994642 0.497321 0.867567i $$-0.334317\pi$$
0.497321 + 0.867567i $$0.334317\pi$$
$$440$$ 0 0
$$441$$ 0.103084 0.00490875
$$442$$ 0 0
$$443$$ −17.5510 −0.833874 −0.416937 0.908935i $$-0.636896\pi$$
−0.416937 + 0.908935i $$0.636896\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 30.0279 1.42027
$$448$$ 0 0
$$449$$ −13.1955 −0.622736 −0.311368 0.950289i $$-0.600787\pi$$
−0.311368 + 0.950289i $$0.600787\pi$$
$$450$$ 0 0
$$451$$ 4.84006 0.227910
$$452$$ 0 0
$$453$$ 5.10308 0.239764
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −29.1020 −1.36134 −0.680668 0.732592i $$-0.738310\pi$$
−0.680668 + 0.732592i $$0.738310\pi$$
$$458$$ 0 0
$$459$$ 4.57704 0.213638
$$460$$ 0 0
$$461$$ −18.8280 −0.876907 −0.438454 0.898754i $$-0.644474\pi$$
−0.438454 + 0.898754i $$0.644474\pi$$
$$462$$ 0 0
$$463$$ 14.0925 0.654932 0.327466 0.944863i $$-0.393805\pi$$
0.327466 + 0.944863i $$0.393805\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 12.2663 0.567619 0.283809 0.958881i $$-0.408402\pi$$
0.283809 + 0.958881i $$0.408402\pi$$
$$468$$ 0 0
$$469$$ −5.79383 −0.267534
$$470$$ 0 0
$$471$$ −17.8217 −0.821182
$$472$$ 0 0
$$473$$ −1.08287 −0.0497905
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −0.698445 −0.0319796
$$478$$ 0 0
$$479$$ 14.1570 0.646850 0.323425 0.946254i $$-0.395166\pi$$
0.323425 + 0.946254i $$0.395166\pi$$
$$480$$ 0 0
$$481$$ 32.9450 1.50216
$$482$$ 0 0
$$483$$ 6.56934 0.298915
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −30.2341 −1.37004 −0.685018 0.728526i $$-0.740206\pi$$
−0.685018 + 0.728526i $$0.740206\pi$$
$$488$$ 0 0
$$489$$ 0.840061 0.0379889
$$490$$ 0 0
$$491$$ −39.9065 −1.80096 −0.900478 0.434902i $$-0.856783\pi$$
−0.900478 + 0.434902i $$0.856783\pi$$
$$492$$ 0 0
$$493$$ −7.06644 −0.318256
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 5.52311 0.247746
$$498$$ 0 0
$$499$$ −27.3246 −1.22322 −0.611610 0.791160i $$-0.709478\pi$$
−0.611610 + 0.791160i $$0.709478\pi$$
$$500$$ 0 0
$$501$$ −23.1955 −1.03630
$$502$$ 0 0
$$503$$ −1.40171 −0.0624991 −0.0312495 0.999512i $$-0.509949\pi$$
−0.0312495 + 0.999512i $$0.509949\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −30.2095 −1.34165
$$508$$ 0 0
$$509$$ 18.6585 0.827022 0.413511 0.910499i $$-0.364302\pi$$
0.413511 + 0.910499i $$0.364302\pi$$
$$510$$ 0 0
$$511$$ 3.72928 0.164974
$$512$$ 0 0
$$513$$ −32.5972 −1.43920
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3.68116 −0.161897
$$518$$ 0 0
$$519$$ −24.6049 −1.08004
$$520$$ 0 0
$$521$$ −4.02791 −0.176466 −0.0882329 0.996100i $$-0.528122\pi$$
−0.0882329 + 0.996100i $$0.528122\pi$$
$$522$$ 0 0
$$523$$ −38.4436 −1.68102 −0.840510 0.541796i $$-0.817744\pi$$
−0.840510 + 0.541796i $$0.817744\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −6.74760 −0.293930
$$528$$ 0 0
$$529$$ −9.09246 −0.395324
$$530$$ 0 0
$$531$$ 0.0612237 0.00265688
$$532$$ 0 0
$$533$$ 42.4402 1.83829
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 13.1387 0.566976
$$538$$ 0 0
$$539$$ 0.626198 0.0269723
$$540$$ 0 0
$$541$$ 12.4846 0.536754 0.268377 0.963314i $$-0.413513\pi$$
0.268377 + 0.963314i $$0.413513\pi$$
$$542$$ 0 0
$$543$$ 24.8680 1.06719
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −37.7851 −1.61557 −0.807787 0.589474i $$-0.799335\pi$$
−0.807787 + 0.589474i $$0.799335\pi$$
$$548$$ 0 0
$$549$$ 0.735539 0.0313921
$$550$$ 0 0
$$551$$ 50.3265 2.14398
$$552$$ 0 0
$$553$$ −5.67243 −0.241216
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1.93545 0.0820076 0.0410038 0.999159i $$-0.486944\pi$$
0.0410038 + 0.999159i $$0.486944\pi$$
$$558$$ 0 0
$$559$$ −9.49521 −0.401604
$$560$$ 0 0
$$561$$ −0.989374 −0.0417714
$$562$$ 0 0
$$563$$ 29.8463 1.25787 0.628936 0.777457i $$-0.283491\pi$$
0.628936 + 0.777457i $$0.283491\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 9.29862 0.390506
$$568$$ 0 0
$$569$$ 15.3449 0.643290 0.321645 0.946860i $$-0.395764\pi$$
0.321645 + 0.946860i $$0.395764\pi$$
$$570$$ 0 0
$$571$$ −35.8217 −1.49909 −0.749547 0.661952i $$-0.769728\pi$$
−0.749547 + 0.661952i $$0.769728\pi$$
$$572$$ 0 0
$$573$$ −14.8324 −0.619631
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.92482 0.121762 0.0608810 0.998145i $$-0.480609\pi$$
0.0608810 + 0.998145i $$0.480609\pi$$
$$578$$ 0 0
$$579$$ 39.2803 1.63243
$$580$$ 0 0
$$581$$ −17.4340 −0.723284
$$582$$ 0 0
$$583$$ −4.24281 −0.175719
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −19.5756 −0.807972 −0.403986 0.914765i $$-0.632375\pi$$
−0.403986 + 0.914765i $$0.632375\pi$$
$$588$$ 0 0
$$589$$ 48.0558 1.98011
$$590$$ 0 0
$$591$$ −46.4402 −1.91030
$$592$$ 0 0
$$593$$ 8.00770 0.328837 0.164418 0.986391i $$-0.447425\pi$$
0.164418 + 0.986391i $$0.447425\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 39.5298 1.61785
$$598$$ 0 0
$$599$$ 29.3005 1.19719 0.598593 0.801053i $$-0.295727\pi$$
0.598593 + 0.801053i $$0.295727\pi$$
$$600$$ 0 0
$$601$$ 21.3449 0.870675 0.435337 0.900267i $$-0.356629\pi$$
0.435337 + 0.900267i $$0.356629\pi$$
$$602$$ 0 0
$$603$$ 0.597250 0.0243219
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −9.53081 −0.386844 −0.193422 0.981116i $$-0.561959\pi$$
−0.193422 + 0.981116i $$0.561959\pi$$
$$608$$ 0 0
$$609$$ −13.8786 −0.562389
$$610$$ 0 0
$$611$$ −32.2784 −1.30584
$$612$$ 0 0
$$613$$ −9.75719 −0.394089 −0.197045 0.980395i $$-0.563134\pi$$
−0.197045 + 0.980395i $$0.563134\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −37.9267 −1.52687 −0.763436 0.645884i $$-0.776489\pi$$
−0.763436 + 0.645884i $$0.776489\pi$$
$$618$$ 0 0
$$619$$ −41.9109 −1.68454 −0.842270 0.539056i $$-0.818781\pi$$
−0.842270 + 0.539056i $$0.818781\pi$$
$$620$$ 0 0
$$621$$ 19.0308 0.763681
$$622$$ 0 0
$$623$$ −14.2986 −0.572862
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 7.04623 0.281399
$$628$$ 0 0
$$629$$ 5.38150 0.214574
$$630$$ 0 0
$$631$$ 8.42003 0.335196 0.167598 0.985855i $$-0.446399\pi$$
0.167598 + 0.985855i $$0.446399\pi$$
$$632$$ 0 0
$$633$$ −26.7678 −1.06393
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5.49084 0.217555
$$638$$ 0 0
$$639$$ −0.569343 −0.0225229
$$640$$ 0 0
$$641$$ −2.07707 −0.0820392 −0.0410196 0.999158i $$-0.513061\pi$$
−0.0410196 + 0.999158i $$0.513061\pi$$
$$642$$ 0 0
$$643$$ 27.2480 1.07456 0.537279 0.843405i $$-0.319452\pi$$
0.537279 + 0.843405i $$0.319452\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12.3632 −0.486047 −0.243023 0.970020i $$-0.578139\pi$$
−0.243023 + 0.970020i $$0.578139\pi$$
$$648$$ 0 0
$$649$$ 0.371913 0.0145989
$$650$$ 0 0
$$651$$ −13.2524 −0.519402
$$652$$ 0 0
$$653$$ 37.3449 1.46142 0.730709 0.682690i $$-0.239190\pi$$
0.730709 + 0.682690i $$0.239190\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −0.384428 −0.0149980
$$658$$ 0 0
$$659$$ 14.1772 0.552266 0.276133 0.961119i $$-0.410947\pi$$
0.276133 + 0.961119i $$0.410947\pi$$
$$660$$ 0 0
$$661$$ −1.76925 −0.0688160 −0.0344080 0.999408i $$-0.510955\pi$$
−0.0344080 + 0.999408i $$0.510955\pi$$
$$662$$ 0 0
$$663$$ −8.67536 −0.336923
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −29.3815 −1.13766
$$668$$ 0 0
$$669$$ −12.1493 −0.469720
$$670$$ 0 0
$$671$$ 4.46815 0.172491
$$672$$ 0 0
$$673$$ 4.05581 0.156340 0.0781701 0.996940i $$-0.475092\pi$$
0.0781701 + 0.996940i $$0.475092\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −9.37713 −0.360392 −0.180196 0.983631i $$-0.557673\pi$$
−0.180196 + 0.983631i $$0.557673\pi$$
$$678$$ 0 0
$$679$$ 10.1493 0.389495
$$680$$ 0 0
$$681$$ −4.89692 −0.187650
$$682$$ 0 0
$$683$$ 5.49521 0.210268 0.105134 0.994458i $$-0.466473\pi$$
0.105134 + 0.994458i $$0.466473\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −32.8680 −1.25399
$$688$$ 0 0
$$689$$ −37.2032 −1.41733
$$690$$ 0 0
$$691$$ 3.03416 0.115425 0.0577125 0.998333i $$-0.481619\pi$$
0.0577125 + 0.998333i $$0.481619\pi$$
$$692$$ 0 0
$$693$$ −0.0645508 −0.00245208
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 6.93252 0.262588
$$698$$ 0 0
$$699$$ 19.4586 0.735990
$$700$$ 0 0
$$701$$ 35.2234 1.33037 0.665186 0.746678i $$-0.268352\pi$$
0.665186 + 0.746678i $$0.268352\pi$$
$$702$$ 0 0
$$703$$ −38.3265 −1.44551
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −9.64015 −0.362555
$$708$$ 0 0
$$709$$ 28.7187 1.07855 0.539276 0.842129i $$-0.318698\pi$$
0.539276 + 0.842129i $$0.318698\pi$$
$$710$$ 0 0
$$711$$ 0.584735 0.0219293
$$712$$ 0 0
$$713$$ −28.0558 −1.05070
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −35.4942 −1.32555
$$718$$ 0 0
$$719$$ 8.60599 0.320949 0.160475 0.987040i $$-0.448698\pi$$
0.160475 + 0.987040i $$0.448698\pi$$
$$720$$ 0 0
$$721$$ −0.626198 −0.0233208
$$722$$ 0 0
$$723$$ −6.56934 −0.244316
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 15.2803 0.566715 0.283358 0.959014i $$-0.408552\pi$$
0.283358 + 0.959014i $$0.408552\pi$$
$$728$$ 0 0
$$729$$ 26.0096 0.963318
$$730$$ 0 0
$$731$$ −1.55102 −0.0573666
$$732$$ 0 0
$$733$$ −46.0235 −1.69992 −0.849959 0.526849i $$-0.823373\pi$$
−0.849959 + 0.526849i $$0.823373\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 3.62809 0.133642
$$738$$ 0 0
$$739$$ −46.3467 −1.70489 −0.852446 0.522815i $$-0.824882\pi$$
−0.852446 + 0.522815i $$0.824882\pi$$
$$740$$ 0 0
$$741$$ 61.7851 2.26973
$$742$$ 0 0
$$743$$ −10.7755 −0.395315 −0.197658 0.980271i $$-0.563333\pi$$
−0.197658 + 0.980271i $$0.563333\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 1.79716 0.0657546
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −2.76781 −0.100999 −0.0504995 0.998724i $$-0.516081\pi$$
−0.0504995 + 0.998724i $$0.516081\pi$$
$$752$$ 0 0
$$753$$ −37.8709 −1.38009
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 14.8401 0.539371 0.269686 0.962948i $$-0.413080\pi$$
0.269686 + 0.962948i $$0.413080\pi$$
$$758$$ 0 0
$$759$$ −4.11371 −0.149318
$$760$$ 0 0
$$761$$ 31.3169 1.13524 0.567619 0.823291i $$-0.307865\pi$$
0.567619 + 0.823291i $$0.307865\pi$$
$$762$$ 0 0
$$763$$ −18.9248 −0.685125
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3.26113 0.117753
$$768$$ 0 0
$$769$$ 8.92586 0.321875 0.160937 0.986965i $$-0.448548\pi$$
0.160937 + 0.986965i $$0.448548\pi$$
$$770$$ 0 0
$$771$$ −7.52311 −0.270938
$$772$$ 0 0
$$773$$ 36.7711 1.32257 0.661283 0.750136i $$-0.270012\pi$$
0.661283 + 0.750136i $$0.270012\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 10.5693 0.379173
$$778$$ 0 0
$$779$$ −49.3728 −1.76896
$$780$$ 0 0
$$781$$ −3.45856 −0.123757
$$782$$ 0 0
$$783$$ −40.2051 −1.43681
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −6.53707 −0.233021 −0.116511 0.993189i $$-0.537171\pi$$
−0.116511 + 0.993189i $$0.537171\pi$$
$$788$$ 0 0
$$789$$ −20.1849 −0.718602
$$790$$ 0 0
$$791$$ 1.04623 0.0371996
$$792$$ 0 0
$$793$$ 39.1791 1.39129
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 7.82611 0.277215 0.138607 0.990347i $$-0.455737\pi$$
0.138607 + 0.990347i $$0.455737\pi$$
$$798$$ 0 0
$$799$$ −5.27261 −0.186531
$$800$$ 0 0
$$801$$ 1.47396 0.0520796
$$802$$ 0 0
$$803$$ −2.33527 −0.0824099
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 12.4556 0.438459
$$808$$ 0 0
$$809$$ 38.4113 1.35047 0.675235 0.737603i $$-0.264042\pi$$
0.675235 + 0.737603i $$0.264042\pi$$
$$810$$ 0 0
$$811$$ −7.64015 −0.268282 −0.134141 0.990962i $$-0.542828\pi$$
−0.134141 + 0.990962i $$0.542828\pi$$
$$812$$ 0 0
$$813$$ 29.9634 1.05086
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 11.0462 0.386459
$$818$$ 0 0
$$819$$ −0.566016 −0.0197782
$$820$$ 0 0
$$821$$ 2.56165 0.0894021 0.0447011 0.999000i $$-0.485766\pi$$
0.0447011 + 0.999000i $$0.485766\pi$$
$$822$$ 0 0
$$823$$ −0.784248 −0.0273372 −0.0136686 0.999907i $$-0.504351\pi$$
−0.0136686 + 0.999907i $$0.504351\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −7.28904 −0.253465 −0.126732 0.991937i $$-0.540449\pi$$
−0.126732 + 0.991937i $$0.540449\pi$$
$$828$$ 0 0
$$829$$ −11.0708 −0.384505 −0.192253 0.981345i $$-0.561579\pi$$
−0.192253 + 0.981345i $$0.561579\pi$$
$$830$$ 0 0
$$831$$ 33.0741 1.14733
$$832$$ 0 0
$$833$$ 0.896916 0.0310763
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −38.3911 −1.32699
$$838$$ 0 0
$$839$$ 5.55976 0.191944 0.0959721 0.995384i $$-0.469404\pi$$
0.0959721 + 0.995384i $$0.469404\pi$$
$$840$$ 0 0
$$841$$ 33.0722 1.14042
$$842$$ 0 0
$$843$$ 8.10015 0.278984
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.6079 0.364491
$$848$$ 0 0
$$849$$ −23.9278 −0.821198
$$850$$ 0 0
$$851$$ 22.3757 0.767029
$$852$$ 0 0
$$853$$ 16.0804 0.550582 0.275291 0.961361i $$-0.411226\pi$$
0.275291 + 0.961361i $$0.411226\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1.58767 0.0542336 0.0271168 0.999632i $$-0.491367\pi$$
0.0271168 + 0.999632i $$0.491367\pi$$
$$858$$ 0 0
$$859$$ −4.94171 −0.168609 −0.0843044 0.996440i $$-0.526867\pi$$
−0.0843044 + 0.996440i $$0.526867\pi$$
$$860$$ 0 0
$$861$$ 13.6156 0.464017
$$862$$ 0 0
$$863$$ 37.6647 1.28212 0.641061 0.767490i $$-0.278494\pi$$
0.641061 + 0.767490i $$0.278494\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 28.5294 0.968908
$$868$$ 0 0
$$869$$ 3.55206 0.120495
$$870$$ 0 0
$$871$$ 31.8130 1.07794
$$872$$ 0 0
$$873$$ −1.04623 −0.0354095
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −3.66473 −0.123749 −0.0618746 0.998084i $$-0.519708\pi$$
−0.0618746 + 0.998084i $$0.519708\pi$$
$$878$$ 0 0
$$879$$ −20.8324 −0.702658
$$880$$ 0 0
$$881$$ −43.4373 −1.46344 −0.731720 0.681605i $$-0.761282\pi$$
−0.731720 + 0.681605i $$0.761282\pi$$
$$882$$ 0 0
$$883$$ 47.3853 1.59464 0.797321 0.603556i $$-0.206250\pi$$
0.797321 + 0.603556i $$0.206250\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −21.3169 −0.715753 −0.357877 0.933769i $$-0.616499\pi$$
−0.357877 + 0.933769i $$0.616499\pi$$
$$888$$ 0 0
$$889$$ −21.2803 −0.713718
$$890$$ 0 0
$$891$$ −5.82278 −0.195071
$$892$$ 0 0
$$893$$ 37.5510 1.25660
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −36.0712 −1.20438
$$898$$ 0 0
$$899$$ 59.2716 1.97682
$$900$$ 0 0
$$901$$ −6.07707 −0.202456
$$902$$ 0 0
$$903$$ −3.04623 −0.101372
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 18.9325 0.628644 0.314322 0.949316i $$-0.398223\pi$$
0.314322 + 0.949316i $$0.398223\pi$$
$$908$$ 0 0
$$909$$ 0.993743 0.0329604
$$910$$ 0 0
$$911$$ 12.6705 0.419794 0.209897 0.977724i $$-0.432687\pi$$
0.209897 + 0.977724i $$0.432687\pi$$
$$912$$ 0 0
$$913$$ 10.9171 0.361304
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −9.91087 −0.327286
$$918$$ 0 0
$$919$$ −22.8690 −0.754379 −0.377190 0.926136i $$-0.623109\pi$$
−0.377190 + 0.926136i $$0.623109\pi$$
$$920$$ 0 0
$$921$$ −36.9681 −1.21814
$$922$$ 0 0
$$923$$ −30.3265 −0.998210
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0.0645508 0.00212013
$$928$$ 0 0
$$929$$ 50.4190 1.65419 0.827097 0.562060i $$-0.189991\pi$$
0.827097 + 0.562060i $$0.189991\pi$$
$$930$$ 0 0
$$931$$ −6.38776 −0.209350
$$932$$ 0 0
$$933$$ −39.6435 −1.29787
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −47.9344 −1.56595 −0.782974 0.622054i $$-0.786298\pi$$
−0.782974 + 0.622054i $$0.786298\pi$$
$$938$$ 0 0
$$939$$ 21.9923 0.717692
$$940$$ 0 0
$$941$$ 59.9667 1.95486 0.977429 0.211264i $$-0.0677580\pi$$
0.977429 + 0.211264i $$0.0677580\pi$$
$$942$$ 0 0
$$943$$ 28.8247 0.938660
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 27.5231 0.894381 0.447191 0.894439i $$-0.352425\pi$$
0.447191 + 0.894439i $$0.352425\pi$$
$$948$$ 0 0
$$949$$ −20.4769 −0.664708
$$950$$ 0 0
$$951$$ −44.9604 −1.45794
$$952$$ 0 0
$$953$$ 0.840061 0.0272123 0.0136061 0.999907i $$-0.495669\pi$$
0.0136061 + 0.999907i $$0.495669\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 8.69075 0.280932
$$958$$ 0 0
$$959$$ −3.45856 −0.111683
$$960$$ 0 0
$$961$$ 25.5972 0.825718
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −31.8988 −1.02580 −0.512898 0.858449i $$-0.671428\pi$$
−0.512898 + 0.858449i $$0.671428\pi$$
$$968$$ 0 0
$$969$$ 10.0925 0.324216
$$970$$ 0 0
$$971$$ −28.4157 −0.911902 −0.455951 0.890005i $$-0.650701\pi$$
−0.455951 + 0.890005i $$0.650701\pi$$
$$972$$ 0 0
$$973$$ −20.4157 −0.654496
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 9.55102 0.305564 0.152782 0.988260i $$-0.451177\pi$$
0.152782 + 0.988260i $$0.451177\pi$$
$$978$$ 0 0
$$979$$ 8.95377 0.286164
$$980$$ 0 0
$$981$$ 1.95084 0.0622856
$$982$$ 0 0
$$983$$ 0.0443400 0.00141423 0.000707113 1.00000i $$-0.499775\pi$$
0.000707113 1.00000i $$0.499775\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −10.3555 −0.329619
$$988$$ 0 0
$$989$$ −6.44898 −0.205066
$$990$$ 0 0
$$991$$ 13.9913 0.444447 0.222224 0.974996i $$-0.428669\pi$$
0.222224 + 0.974996i $$0.428669\pi$$
$$992$$ 0 0
$$993$$ −4.72635 −0.149986
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −19.0419 −0.603062 −0.301531 0.953456i $$-0.597498\pi$$
−0.301531 + 0.953456i $$0.597498\pi$$
$$998$$ 0 0
$$999$$ 30.6185 0.968727
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 2800.2.a.br.1.1 3
4.3 odd 2 1400.2.a.s.1.3 3
5.2 odd 4 560.2.g.f.449.5 6
5.3 odd 4 560.2.g.f.449.2 6
5.4 even 2 2800.2.a.bq.1.3 3
15.2 even 4 5040.2.t.y.1009.4 6
15.8 even 4 5040.2.t.y.1009.3 6
20.3 even 4 280.2.g.b.169.5 yes 6
20.7 even 4 280.2.g.b.169.2 6
20.19 odd 2 1400.2.a.t.1.1 3
28.27 even 2 9800.2.a.cg.1.1 3
40.3 even 4 2240.2.g.l.449.2 6
40.13 odd 4 2240.2.g.m.449.5 6
40.27 even 4 2240.2.g.l.449.5 6
40.37 odd 4 2240.2.g.m.449.2 6
60.23 odd 4 2520.2.t.g.1009.3 6
60.47 odd 4 2520.2.t.g.1009.4 6
140.27 odd 4 1960.2.g.c.1569.5 6
140.83 odd 4 1960.2.g.c.1569.2 6
140.139 even 2 9800.2.a.cd.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.g.b.169.2 6 20.7 even 4
280.2.g.b.169.5 yes 6 20.3 even 4
560.2.g.f.449.2 6 5.3 odd 4
560.2.g.f.449.5 6 5.2 odd 4
1400.2.a.s.1.3 3 4.3 odd 2
1400.2.a.t.1.1 3 20.19 odd 2
1960.2.g.c.1569.2 6 140.83 odd 4
1960.2.g.c.1569.5 6 140.27 odd 4
2240.2.g.l.449.2 6 40.3 even 4
2240.2.g.l.449.5 6 40.27 even 4
2240.2.g.m.449.2 6 40.37 odd 4
2240.2.g.m.449.5 6 40.13 odd 4
2520.2.t.g.1009.3 6 60.23 odd 4
2520.2.t.g.1009.4 6 60.47 odd 4
2800.2.a.bq.1.3 3 5.4 even 2
2800.2.a.br.1.1 3 1.1 even 1 trivial
5040.2.t.y.1009.3 6 15.8 even 4
5040.2.t.y.1009.4 6 15.2 even 4
9800.2.a.cd.1.3 3 140.139 even 2
9800.2.a.cg.1.1 3 28.27 even 2