# Properties

 Label 5733.2.a.w.1.2 Level $5733$ Weight $2$ Character 5733.1 Self dual yes Analytic conductor $45.778$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5733 = 3^{2} \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5733.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$45.7782354788$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 5733.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.61803 q^{2} +4.85410 q^{4} +2.23607 q^{5} +7.47214 q^{8} +O(q^{10})$$ $$q+2.61803 q^{2} +4.85410 q^{4} +2.23607 q^{5} +7.47214 q^{8} +5.85410 q^{10} +3.00000 q^{11} +1.00000 q^{13} +9.85410 q^{16} +1.47214 q^{17} -3.00000 q^{19} +10.8541 q^{20} +7.85410 q^{22} +8.23607 q^{23} +2.61803 q^{26} -4.47214 q^{29} -5.00000 q^{31} +10.8541 q^{32} +3.85410 q^{34} +4.70820 q^{37} -7.85410 q^{38} +16.7082 q^{40} -4.47214 q^{41} -8.00000 q^{43} +14.5623 q^{44} +21.5623 q^{46} -7.47214 q^{47} +4.85410 q^{52} +7.47214 q^{53} +6.70820 q^{55} -11.7082 q^{58} -1.47214 q^{59} -3.00000 q^{61} -13.0902 q^{62} +8.70820 q^{64} +2.23607 q^{65} -3.00000 q^{67} +7.14590 q^{68} +8.94427 q^{71} +2.70820 q^{73} +12.3262 q^{74} -14.5623 q^{76} -2.70820 q^{79} +22.0344 q^{80} -11.7082 q^{82} +3.29180 q^{85} -20.9443 q^{86} +22.4164 q^{88} +2.23607 q^{89} +39.9787 q^{92} -19.5623 q^{94} -6.70820 q^{95} -9.41641 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} + O(q^{10})$$ $$2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} + 5 q^{10} + 6 q^{11} + 2 q^{13} + 13 q^{16} - 6 q^{17} - 6 q^{19} + 15 q^{20} + 9 q^{22} + 12 q^{23} + 3 q^{26} - 10 q^{31} + 15 q^{32} + q^{34} - 4 q^{37} - 9 q^{38} + 20 q^{40} - 16 q^{43} + 9 q^{44} + 23 q^{46} - 6 q^{47} + 3 q^{52} + 6 q^{53} - 10 q^{58} + 6 q^{59} - 6 q^{61} - 15 q^{62} + 4 q^{64} - 6 q^{67} + 21 q^{68} - 8 q^{73} + 9 q^{74} - 9 q^{76} + 8 q^{79} + 15 q^{80} - 10 q^{82} + 20 q^{85} - 24 q^{86} + 18 q^{88} + 33 q^{92} - 19 q^{94} + 8 q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.61803 1.85123 0.925615 0.378467i $$-0.123549\pi$$
0.925615 + 0.378467i $$0.123549\pi$$
$$3$$ 0 0
$$4$$ 4.85410 2.42705
$$5$$ 2.23607 1.00000 0.500000 0.866025i $$-0.333333\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 7.47214 2.64180
$$9$$ 0 0
$$10$$ 5.85410 1.85123
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 9.85410 2.46353
$$17$$ 1.47214 0.357045 0.178523 0.983936i $$-0.442868\pi$$
0.178523 + 0.983936i $$0.442868\pi$$
$$18$$ 0 0
$$19$$ −3.00000 −0.688247 −0.344124 0.938924i $$-0.611824\pi$$
−0.344124 + 0.938924i $$0.611824\pi$$
$$20$$ 10.8541 2.42705
$$21$$ 0 0
$$22$$ 7.85410 1.67450
$$23$$ 8.23607 1.71734 0.858669 0.512530i $$-0.171292\pi$$
0.858669 + 0.512530i $$0.171292\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.61803 0.513439
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −4.47214 −0.830455 −0.415227 0.909718i $$-0.636298\pi$$
−0.415227 + 0.909718i $$0.636298\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 10.8541 1.91875
$$33$$ 0 0
$$34$$ 3.85410 0.660973
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.70820 0.774024 0.387012 0.922075i $$-0.373507\pi$$
0.387012 + 0.922075i $$0.373507\pi$$
$$38$$ −7.85410 −1.27410
$$39$$ 0 0
$$40$$ 16.7082 2.64180
$$41$$ −4.47214 −0.698430 −0.349215 0.937043i $$-0.613552\pi$$
−0.349215 + 0.937043i $$0.613552\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 14.5623 2.19535
$$45$$ 0 0
$$46$$ 21.5623 3.17919
$$47$$ −7.47214 −1.08992 −0.544962 0.838461i $$-0.683456\pi$$
−0.544962 + 0.838461i $$0.683456\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 4.85410 0.673143
$$53$$ 7.47214 1.02638 0.513188 0.858276i $$-0.328464\pi$$
0.513188 + 0.858276i $$0.328464\pi$$
$$54$$ 0 0
$$55$$ 6.70820 0.904534
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −11.7082 −1.53736
$$59$$ −1.47214 −0.191656 −0.0958279 0.995398i $$-0.530550\pi$$
−0.0958279 + 0.995398i $$0.530550\pi$$
$$60$$ 0 0
$$61$$ −3.00000 −0.384111 −0.192055 0.981384i $$-0.561515\pi$$
−0.192055 + 0.981384i $$0.561515\pi$$
$$62$$ −13.0902 −1.66245
$$63$$ 0 0
$$64$$ 8.70820 1.08853
$$65$$ 2.23607 0.277350
$$66$$ 0 0
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ 7.14590 0.866567
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.94427 1.06149 0.530745 0.847532i $$-0.321912\pi$$
0.530745 + 0.847532i $$0.321912\pi$$
$$72$$ 0 0
$$73$$ 2.70820 0.316971 0.158486 0.987361i $$-0.449339\pi$$
0.158486 + 0.987361i $$0.449339\pi$$
$$74$$ 12.3262 1.43290
$$75$$ 0 0
$$76$$ −14.5623 −1.67041
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −2.70820 −0.304697 −0.152348 0.988327i $$-0.548684\pi$$
−0.152348 + 0.988327i $$0.548684\pi$$
$$80$$ 22.0344 2.46353
$$81$$ 0 0
$$82$$ −11.7082 −1.29295
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 3.29180 0.357045
$$86$$ −20.9443 −2.25848
$$87$$ 0 0
$$88$$ 22.4164 2.38960
$$89$$ 2.23607 0.237023 0.118511 0.992953i $$-0.462188\pi$$
0.118511 + 0.992953i $$0.462188\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 39.9787 4.16807
$$93$$ 0 0
$$94$$ −19.5623 −2.01770
$$95$$ −6.70820 −0.688247
$$96$$ 0 0
$$97$$ −9.41641 −0.956091 −0.478046 0.878335i $$-0.658655\pi$$
−0.478046 + 0.878335i $$0.658655\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 0 0
$$103$$ 2.70820 0.266847 0.133424 0.991059i $$-0.457403\pi$$
0.133424 + 0.991059i $$0.457403\pi$$
$$104$$ 7.47214 0.732703
$$105$$ 0 0
$$106$$ 19.5623 1.90006
$$107$$ 9.76393 0.943915 0.471957 0.881621i $$-0.343548\pi$$
0.471957 + 0.881621i $$0.343548\pi$$
$$108$$ 0 0
$$109$$ −2.70820 −0.259399 −0.129699 0.991553i $$-0.541401\pi$$
−0.129699 + 0.991553i $$0.541401\pi$$
$$110$$ 17.5623 1.67450
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.94427 −0.276974 −0.138487 0.990364i $$-0.544224\pi$$
−0.138487 + 0.990364i $$0.544224\pi$$
$$114$$ 0 0
$$115$$ 18.4164 1.71734
$$116$$ −21.7082 −2.01556
$$117$$ 0 0
$$118$$ −3.85410 −0.354799
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −7.85410 −0.711077
$$123$$ 0 0
$$124$$ −24.2705 −2.17956
$$125$$ −11.1803 −1.00000
$$126$$ 0 0
$$127$$ −11.4164 −1.01304 −0.506521 0.862228i $$-0.669069\pi$$
−0.506521 + 0.862228i $$0.669069\pi$$
$$128$$ 1.09017 0.0963583
$$129$$ 0 0
$$130$$ 5.85410 0.513439
$$131$$ −8.23607 −0.719589 −0.359794 0.933032i $$-0.617153\pi$$
−0.359794 + 0.933032i $$0.617153\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −7.85410 −0.678491
$$135$$ 0 0
$$136$$ 11.0000 0.943242
$$137$$ 8.23607 0.703655 0.351827 0.936065i $$-0.385560\pi$$
0.351827 + 0.936065i $$0.385560\pi$$
$$138$$ 0 0
$$139$$ 23.4164 1.98615 0.993077 0.117466i $$-0.0374771\pi$$
0.993077 + 0.117466i $$0.0374771\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 23.4164 1.96506
$$143$$ 3.00000 0.250873
$$144$$ 0 0
$$145$$ −10.0000 −0.830455
$$146$$ 7.09017 0.586787
$$147$$ 0 0
$$148$$ 22.8541 1.87860
$$149$$ −0.708204 −0.0580183 −0.0290092 0.999579i $$-0.509235\pi$$
−0.0290092 + 0.999579i $$0.509235\pi$$
$$150$$ 0 0
$$151$$ 20.4164 1.66146 0.830732 0.556673i $$-0.187922\pi$$
0.830732 + 0.556673i $$0.187922\pi$$
$$152$$ −22.4164 −1.81821
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −11.1803 −0.898027
$$156$$ 0 0
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ −7.09017 −0.564064
$$159$$ 0 0
$$160$$ 24.2705 1.91875
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −16.4164 −1.28583 −0.642916 0.765937i $$-0.722276\pi$$
−0.642916 + 0.765937i $$0.722276\pi$$
$$164$$ −21.7082 −1.69513
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 22.4721 1.73895 0.869473 0.493980i $$-0.164459\pi$$
0.869473 + 0.493980i $$0.164459\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 8.61803 0.660973
$$171$$ 0 0
$$172$$ −38.8328 −2.96097
$$173$$ −16.4164 −1.24812 −0.624058 0.781378i $$-0.714517\pi$$
−0.624058 + 0.781378i $$0.714517\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 29.5623 2.22834
$$177$$ 0 0
$$178$$ 5.85410 0.438783
$$179$$ 20.1246 1.50418 0.752092 0.659058i $$-0.229045\pi$$
0.752092 + 0.659058i $$0.229045\pi$$
$$180$$ 0 0
$$181$$ 25.4164 1.88919 0.944593 0.328243i $$-0.106456\pi$$
0.944593 + 0.328243i $$0.106456\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 61.5410 4.53686
$$185$$ 10.5279 0.774024
$$186$$ 0 0
$$187$$ 4.41641 0.322960
$$188$$ −36.2705 −2.64530
$$189$$ 0 0
$$190$$ −17.5623 −1.27410
$$191$$ −11.1803 −0.808981 −0.404491 0.914542i $$-0.632551\pi$$
−0.404491 + 0.914542i $$0.632551\pi$$
$$192$$ 0 0
$$193$$ 0.708204 0.0509776 0.0254888 0.999675i $$-0.491886\pi$$
0.0254888 + 0.999675i $$0.491886\pi$$
$$194$$ −24.6525 −1.76994
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 9.05573 0.645194 0.322597 0.946536i $$-0.395444\pi$$
0.322597 + 0.946536i $$0.395444\pi$$
$$198$$ 0 0
$$199$$ 20.7082 1.46797 0.733983 0.679168i $$-0.237659\pi$$
0.733983 + 0.679168i $$0.237659\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −23.5623 −1.65784
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −10.0000 −0.698430
$$206$$ 7.09017 0.493996
$$207$$ 0 0
$$208$$ 9.85410 0.683259
$$209$$ −9.00000 −0.622543
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 36.2705 2.49107
$$213$$ 0 0
$$214$$ 25.5623 1.74740
$$215$$ −17.8885 −1.21999
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −7.09017 −0.480207
$$219$$ 0 0
$$220$$ 32.5623 2.19535
$$221$$ 1.47214 0.0990266
$$222$$ 0 0
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −7.70820 −0.512742
$$227$$ 5.94427 0.394535 0.197268 0.980350i $$-0.436793\pi$$
0.197268 + 0.980350i $$0.436793\pi$$
$$228$$ 0 0
$$229$$ −24.1246 −1.59420 −0.797100 0.603848i $$-0.793633\pi$$
−0.797100 + 0.603848i $$0.793633\pi$$
$$230$$ 48.2148 3.17919
$$231$$ 0 0
$$232$$ −33.4164 −2.19389
$$233$$ −11.9443 −0.782495 −0.391248 0.920285i $$-0.627956\pi$$
−0.391248 + 0.920285i $$0.627956\pi$$
$$234$$ 0 0
$$235$$ −16.7082 −1.08992
$$236$$ −7.14590 −0.465158
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −19.4164 −1.25594 −0.627972 0.778236i $$-0.716115\pi$$
−0.627972 + 0.778236i $$0.716115\pi$$
$$240$$ 0 0
$$241$$ −4.70820 −0.303282 −0.151641 0.988436i $$-0.548456\pi$$
−0.151641 + 0.988436i $$0.548456\pi$$
$$242$$ −5.23607 −0.336587
$$243$$ 0 0
$$244$$ −14.5623 −0.932256
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.00000 −0.190885
$$248$$ −37.3607 −2.37241
$$249$$ 0 0
$$250$$ −29.2705 −1.85123
$$251$$ 1.52786 0.0964379 0.0482190 0.998837i $$-0.484645\pi$$
0.0482190 + 0.998837i $$0.484645\pi$$
$$252$$ 0 0
$$253$$ 24.7082 1.55339
$$254$$ −29.8885 −1.87537
$$255$$ 0 0
$$256$$ −14.5623 −0.910144
$$257$$ −0.0557281 −0.00347622 −0.00173811 0.999998i $$-0.500553\pi$$
−0.00173811 + 0.999998i $$0.500553\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 10.8541 0.673143
$$261$$ 0 0
$$262$$ −21.5623 −1.33212
$$263$$ −26.1246 −1.61091 −0.805456 0.592655i $$-0.798080\pi$$
−0.805456 + 0.592655i $$0.798080\pi$$
$$264$$ 0 0
$$265$$ 16.7082 1.02638
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −14.5623 −0.889534
$$269$$ 13.4721 0.821411 0.410705 0.911768i $$-0.365283\pi$$
0.410705 + 0.911768i $$0.365283\pi$$
$$270$$ 0 0
$$271$$ 20.4164 1.24021 0.620104 0.784519i $$-0.287090\pi$$
0.620104 + 0.784519i $$0.287090\pi$$
$$272$$ 14.5066 0.879590
$$273$$ 0 0
$$274$$ 21.5623 1.30263
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −0.416408 −0.0250195 −0.0125098 0.999922i $$-0.503982\pi$$
−0.0125098 + 0.999922i $$0.503982\pi$$
$$278$$ 61.3050 3.67683
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −26.9443 −1.60736 −0.803680 0.595061i $$-0.797128\pi$$
−0.803680 + 0.595061i $$0.797128\pi$$
$$282$$ 0 0
$$283$$ −26.1246 −1.55295 −0.776473 0.630150i $$-0.782993\pi$$
−0.776473 + 0.630150i $$0.782993\pi$$
$$284$$ 43.4164 2.57629
$$285$$ 0 0
$$286$$ 7.85410 0.464423
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −14.8328 −0.872519
$$290$$ −26.1803 −1.53736
$$291$$ 0 0
$$292$$ 13.1459 0.769305
$$293$$ −14.9443 −0.873054 −0.436527 0.899691i $$-0.643792\pi$$
−0.436527 + 0.899691i $$0.643792\pi$$
$$294$$ 0 0
$$295$$ −3.29180 −0.191656
$$296$$ 35.1803 2.04482
$$297$$ 0 0
$$298$$ −1.85410 −0.107405
$$299$$ 8.23607 0.476304
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 53.4508 3.07575
$$303$$ 0 0
$$304$$ −29.5623 −1.69551
$$305$$ −6.70820 −0.384111
$$306$$ 0 0
$$307$$ −19.4164 −1.10815 −0.554076 0.832466i $$-0.686928\pi$$
−0.554076 + 0.832466i $$0.686928\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −29.2705 −1.66245
$$311$$ −27.7639 −1.57435 −0.787174 0.616731i $$-0.788457\pi$$
−0.787174 + 0.616731i $$0.788457\pi$$
$$312$$ 0 0
$$313$$ 5.58359 0.315603 0.157802 0.987471i $$-0.449559\pi$$
0.157802 + 0.987471i $$0.449559\pi$$
$$314$$ −18.3262 −1.03421
$$315$$ 0 0
$$316$$ −13.1459 −0.739515
$$317$$ 8.23607 0.462584 0.231292 0.972884i $$-0.425705\pi$$
0.231292 + 0.972884i $$0.425705\pi$$
$$318$$ 0 0
$$319$$ −13.4164 −0.751175
$$320$$ 19.4721 1.08853
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4.41641 −0.245736
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −42.9787 −2.38037
$$327$$ 0 0
$$328$$ −33.4164 −1.84511
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.58359 −0.0870421 −0.0435210 0.999053i $$-0.513858\pi$$
−0.0435210 + 0.999053i $$0.513858\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 58.8328 3.21919
$$335$$ −6.70820 −0.366508
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ 2.61803 0.142402
$$339$$ 0 0
$$340$$ 15.9787 0.866567
$$341$$ −15.0000 −0.812296
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −59.7771 −3.22296
$$345$$ 0 0
$$346$$ −42.9787 −2.31055
$$347$$ −23.0689 −1.23840 −0.619201 0.785232i $$-0.712544\pi$$
−0.619201 + 0.785232i $$0.712544\pi$$
$$348$$ 0 0
$$349$$ 29.4164 1.57462 0.787312 0.616555i $$-0.211472\pi$$
0.787312 + 0.616555i $$0.211472\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 32.5623 1.73558
$$353$$ 17.2918 0.920349 0.460175 0.887828i $$-0.347787\pi$$
0.460175 + 0.887828i $$0.347787\pi$$
$$354$$ 0 0
$$355$$ 20.0000 1.06149
$$356$$ 10.8541 0.575266
$$357$$ 0 0
$$358$$ 52.6869 2.78459
$$359$$ −11.9443 −0.630395 −0.315197 0.949026i $$-0.602071\pi$$
−0.315197 + 0.949026i $$0.602071\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 66.5410 3.49732
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 6.05573 0.316971
$$366$$ 0 0
$$367$$ 12.7082 0.663363 0.331681 0.943391i $$-0.392384\pi$$
0.331681 + 0.943391i $$0.392384\pi$$
$$368$$ 81.1591 4.23071
$$369$$ 0 0
$$370$$ 27.5623 1.43290
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.58359 −0.0819953 −0.0409976 0.999159i $$-0.513054\pi$$
−0.0409976 + 0.999159i $$0.513054\pi$$
$$374$$ 11.5623 0.597873
$$375$$ 0 0
$$376$$ −55.8328 −2.87936
$$377$$ −4.47214 −0.230327
$$378$$ 0 0
$$379$$ −15.4164 −0.791888 −0.395944 0.918275i $$-0.629583\pi$$
−0.395944 + 0.918275i $$0.629583\pi$$
$$380$$ −32.5623 −1.67041
$$381$$ 0 0
$$382$$ −29.2705 −1.49761
$$383$$ 15.0000 0.766464 0.383232 0.923652i $$-0.374811\pi$$
0.383232 + 0.923652i $$0.374811\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 1.85410 0.0943713
$$387$$ 0 0
$$388$$ −45.7082 −2.32048
$$389$$ −1.47214 −0.0746403 −0.0373201 0.999303i $$-0.511882\pi$$
−0.0373201 + 0.999303i $$0.511882\pi$$
$$390$$ 0 0
$$391$$ 12.1246 0.613168
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 23.7082 1.19440
$$395$$ −6.05573 −0.304697
$$396$$ 0 0
$$397$$ 26.1246 1.31116 0.655578 0.755127i $$-0.272425\pi$$
0.655578 + 0.755127i $$0.272425\pi$$
$$398$$ 54.2148 2.71754
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.2361 −0.710915 −0.355458 0.934692i $$-0.615675\pi$$
−0.355458 + 0.934692i $$0.615675\pi$$
$$402$$ 0 0
$$403$$ −5.00000 −0.249068
$$404$$ −43.6869 −2.17351
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 14.1246 0.700131
$$408$$ 0 0
$$409$$ −8.70820 −0.430593 −0.215296 0.976549i $$-0.569072\pi$$
−0.215296 + 0.976549i $$0.569072\pi$$
$$410$$ −26.1803 −1.29295
$$411$$ 0 0
$$412$$ 13.1459 0.647652
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 10.8541 0.532166
$$417$$ 0 0
$$418$$ −23.5623 −1.15247
$$419$$ 32.9443 1.60943 0.804717 0.593659i $$-0.202317\pi$$
0.804717 + 0.593659i $$0.202317\pi$$
$$420$$ 0 0
$$421$$ 13.4164 0.653876 0.326938 0.945046i $$-0.393983\pi$$
0.326938 + 0.945046i $$0.393983\pi$$
$$422$$ 10.4721 0.509776
$$423$$ 0 0
$$424$$ 55.8328 2.71148
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 47.3951 2.29093
$$429$$ 0 0
$$430$$ −46.8328 −2.25848
$$431$$ 31.3607 1.51059 0.755295 0.655385i $$-0.227493\pi$$
0.755295 + 0.655385i $$0.227493\pi$$
$$432$$ 0 0
$$433$$ −29.4164 −1.41366 −0.706831 0.707382i $$-0.749876\pi$$
−0.706831 + 0.707382i $$0.749876\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −13.1459 −0.629574
$$437$$ −24.7082 −1.18195
$$438$$ 0 0
$$439$$ −24.1246 −1.15140 −0.575702 0.817659i $$-0.695271\pi$$
−0.575702 + 0.817659i $$0.695271\pi$$
$$440$$ 50.1246 2.38960
$$441$$ 0 0
$$442$$ 3.85410 0.183321
$$443$$ −2.23607 −0.106239 −0.0531194 0.998588i $$-0.516916\pi$$
−0.0531194 + 0.998588i $$0.516916\pi$$
$$444$$ 0 0
$$445$$ 5.00000 0.237023
$$446$$ 10.4721 0.495870
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 34.3607 1.62158 0.810790 0.585337i $$-0.199038\pi$$
0.810790 + 0.585337i $$0.199038\pi$$
$$450$$ 0 0
$$451$$ −13.4164 −0.631754
$$452$$ −14.2918 −0.672230
$$453$$ 0 0
$$454$$ 15.5623 0.730375
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.12461 −0.286497 −0.143249 0.989687i $$-0.545755\pi$$
−0.143249 + 0.989687i $$0.545755\pi$$
$$458$$ −63.1591 −2.95123
$$459$$ 0 0
$$460$$ 89.3951 4.16807
$$461$$ 34.3607 1.60034 0.800168 0.599776i $$-0.204744\pi$$
0.800168 + 0.599776i $$0.204744\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ −44.0689 −2.04585
$$465$$ 0 0
$$466$$ −31.2705 −1.44858
$$467$$ 9.65248 0.446663 0.223332 0.974743i $$-0.428307\pi$$
0.223332 + 0.974743i $$0.428307\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −43.7426 −2.01770
$$471$$ 0 0
$$472$$ −11.0000 −0.506316
$$473$$ −24.0000 −1.10352
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −50.8328 −2.32504
$$479$$ 23.8328 1.08895 0.544475 0.838777i $$-0.316729\pi$$
0.544475 + 0.838777i $$0.316729\pi$$
$$480$$ 0 0
$$481$$ 4.70820 0.214676
$$482$$ −12.3262 −0.561445
$$483$$ 0 0
$$484$$ −9.70820 −0.441282
$$485$$ −21.0557 −0.956091
$$486$$ 0 0
$$487$$ −21.8328 −0.989339 −0.494670 0.869081i $$-0.664711\pi$$
−0.494670 + 0.869081i $$0.664711\pi$$
$$488$$ −22.4164 −1.01474
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 25.5279 1.15206 0.576028 0.817430i $$-0.304602\pi$$
0.576028 + 0.817430i $$0.304602\pi$$
$$492$$ 0 0
$$493$$ −6.58359 −0.296510
$$494$$ −7.85410 −0.353373
$$495$$ 0 0
$$496$$ −49.2705 −2.21231
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 26.4164 1.18256 0.591280 0.806466i $$-0.298623\pi$$
0.591280 + 0.806466i $$0.298623\pi$$
$$500$$ −54.2705 −2.42705
$$501$$ 0 0
$$502$$ 4.00000 0.178529
$$503$$ 20.9443 0.933859 0.466929 0.884295i $$-0.345360\pi$$
0.466929 + 0.884295i $$0.345360\pi$$
$$504$$ 0 0
$$505$$ −20.1246 −0.895533
$$506$$ 64.6869 2.87568
$$507$$ 0 0
$$508$$ −55.4164 −2.45871
$$509$$ −20.2361 −0.896948 −0.448474 0.893796i $$-0.648032\pi$$
−0.448474 + 0.893796i $$0.648032\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −40.3050 −1.78124
$$513$$ 0 0
$$514$$ −0.145898 −0.00643529
$$515$$ 6.05573 0.266847
$$516$$ 0 0
$$517$$ −22.4164 −0.985872
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 16.7082 0.732703
$$521$$ −17.9443 −0.786153 −0.393076 0.919506i $$-0.628589\pi$$
−0.393076 + 0.919506i $$0.628589\pi$$
$$522$$ 0 0
$$523$$ −32.7082 −1.43023 −0.715115 0.699007i $$-0.753626\pi$$
−0.715115 + 0.699007i $$0.753626\pi$$
$$524$$ −39.9787 −1.74648
$$525$$ 0 0
$$526$$ −68.3951 −2.98217
$$527$$ −7.36068 −0.320636
$$528$$ 0 0
$$529$$ 44.8328 1.94925
$$530$$ 43.7426 1.90006
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −4.47214 −0.193710
$$534$$ 0 0
$$535$$ 21.8328 0.943915
$$536$$ −22.4164 −0.968241
$$537$$ 0 0
$$538$$ 35.2705 1.52062
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1.29180 0.0555387 0.0277693 0.999614i $$-0.491160\pi$$
0.0277693 + 0.999614i $$0.491160\pi$$
$$542$$ 53.4508 2.29591
$$543$$ 0 0
$$544$$ 15.9787 0.685082
$$545$$ −6.05573 −0.259399
$$546$$ 0 0
$$547$$ −4.58359 −0.195980 −0.0979901 0.995187i $$-0.531241\pi$$
−0.0979901 + 0.995187i $$0.531241\pi$$
$$548$$ 39.9787 1.70781
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 13.4164 0.571558
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −1.09017 −0.0463169
$$555$$ 0 0
$$556$$ 113.666 4.82050
$$557$$ 18.7082 0.792692 0.396346 0.918101i $$-0.370278\pi$$
0.396346 + 0.918101i $$0.370278\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −70.5410 −2.97559
$$563$$ 12.5967 0.530890 0.265445 0.964126i $$-0.414481\pi$$
0.265445 + 0.964126i $$0.414481\pi$$
$$564$$ 0 0
$$565$$ −6.58359 −0.276974
$$566$$ −68.3951 −2.87486
$$567$$ 0 0
$$568$$ 66.8328 2.80424
$$569$$ −25.4721 −1.06785 −0.533924 0.845533i $$-0.679283\pi$$
−0.533924 + 0.845533i $$0.679283\pi$$
$$570$$ 0 0
$$571$$ −36.1246 −1.51177 −0.755884 0.654706i $$-0.772793\pi$$
−0.755884 + 0.654706i $$0.772793\pi$$
$$572$$ 14.5623 0.608881
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 19.2918 0.803128 0.401564 0.915831i $$-0.368467\pi$$
0.401564 + 0.915831i $$0.368467\pi$$
$$578$$ −38.8328 −1.61523
$$579$$ 0 0
$$580$$ −48.5410 −2.01556
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 22.4164 0.928393
$$584$$ 20.2361 0.837374
$$585$$ 0 0
$$586$$ −39.1246 −1.61622
$$587$$ −6.11146 −0.252247 −0.126123 0.992015i $$-0.540254\pi$$
−0.126123 + 0.992015i $$0.540254\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ −8.61803 −0.354799
$$591$$ 0 0
$$592$$ 46.3951 1.90683
$$593$$ 27.7639 1.14013 0.570064 0.821600i $$-0.306918\pi$$
0.570064 + 0.821600i $$0.306918\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −3.43769 −0.140813
$$597$$ 0 0
$$598$$ 21.5623 0.881748
$$599$$ 17.0689 0.697416 0.348708 0.937231i $$-0.386621\pi$$
0.348708 + 0.937231i $$0.386621\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 99.1033 4.03246
$$605$$ −4.47214 −0.181818
$$606$$ 0 0
$$607$$ −24.1246 −0.979188 −0.489594 0.871951i $$-0.662855\pi$$
−0.489594 + 0.871951i $$0.662855\pi$$
$$608$$ −32.5623 −1.32058
$$609$$ 0 0
$$610$$ −17.5623 −0.711077
$$611$$ −7.47214 −0.302290
$$612$$ 0 0
$$613$$ −18.1246 −0.732046 −0.366023 0.930606i $$-0.619281\pi$$
−0.366023 + 0.930606i $$0.619281\pi$$
$$614$$ −50.8328 −2.05145
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4.47214 0.180041 0.0900207 0.995940i $$-0.471307\pi$$
0.0900207 + 0.995940i $$0.471307\pi$$
$$618$$ 0 0
$$619$$ −17.0000 −0.683288 −0.341644 0.939829i $$-0.610984\pi$$
−0.341644 + 0.939829i $$0.610984\pi$$
$$620$$ −54.2705 −2.17956
$$621$$ 0 0
$$622$$ −72.6869 −2.91448
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −25.0000 −1.00000
$$626$$ 14.6180 0.584254
$$627$$ 0 0
$$628$$ −33.9787 −1.35590
$$629$$ 6.93112 0.276362
$$630$$ 0 0
$$631$$ 22.8328 0.908960 0.454480 0.890757i $$-0.349825\pi$$
0.454480 + 0.890757i $$0.349825\pi$$
$$632$$ −20.2361 −0.804948
$$633$$ 0 0
$$634$$ 21.5623 0.856349
$$635$$ −25.5279 −1.01304
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −35.1246 −1.39060
$$639$$ 0 0
$$640$$ 2.43769 0.0963583
$$641$$ −11.9443 −0.471770 −0.235885 0.971781i $$-0.575799\pi$$
−0.235885 + 0.971781i $$0.575799\pi$$
$$642$$ 0 0
$$643$$ −34.8328 −1.37367 −0.686836 0.726812i $$-0.741001\pi$$
−0.686836 + 0.726812i $$0.741001\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −11.5623 −0.454913
$$647$$ 20.2361 0.795562 0.397781 0.917480i $$-0.369780\pi$$
0.397781 + 0.917480i $$0.369780\pi$$
$$648$$ 0 0
$$649$$ −4.41641 −0.173359
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −79.6869 −3.12078
$$653$$ −4.52786 −0.177189 −0.0885945 0.996068i $$-0.528238\pi$$
−0.0885945 + 0.996068i $$0.528238\pi$$
$$654$$ 0 0
$$655$$ −18.4164 −0.719589
$$656$$ −44.0689 −1.72060
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 8.94427 0.348419 0.174210 0.984709i $$-0.444263\pi$$
0.174210 + 0.984709i $$0.444263\pi$$
$$660$$ 0 0
$$661$$ 6.70820 0.260919 0.130459 0.991454i $$-0.458355\pi$$
0.130459 + 0.991454i $$0.458355\pi$$
$$662$$ −4.14590 −0.161135
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −36.8328 −1.42617
$$668$$ 109.082 4.22051
$$669$$ 0 0
$$670$$ −17.5623 −0.678491
$$671$$ −9.00000 −0.347441
$$672$$ 0 0
$$673$$ −9.41641 −0.362976 −0.181488 0.983393i $$-0.558091\pi$$
−0.181488 + 0.983393i $$0.558091\pi$$
$$674$$ 47.1246 1.81517
$$675$$ 0 0
$$676$$ 4.85410 0.186696
$$677$$ −2.88854 −0.111016 −0.0555079 0.998458i $$-0.517678\pi$$
−0.0555079 + 0.998458i $$0.517678\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 24.5967 0.943242
$$681$$ 0 0
$$682$$ −39.2705 −1.50375
$$683$$ 13.4721 0.515497 0.257748 0.966212i $$-0.417019\pi$$
0.257748 + 0.966212i $$0.417019\pi$$
$$684$$ 0 0
$$685$$ 18.4164 0.703655
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −78.8328 −3.00547
$$689$$ 7.47214 0.284666
$$690$$ 0 0
$$691$$ 51.8328 1.97181 0.985907 0.167297i $$-0.0535037\pi$$
0.985907 + 0.167297i $$0.0535037\pi$$
$$692$$ −79.6869 −3.02924
$$693$$ 0 0
$$694$$ −60.3951 −2.29257
$$695$$ 52.3607 1.98615
$$696$$ 0 0
$$697$$ −6.58359 −0.249371
$$698$$ 77.0132 2.91499
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22.3607 0.844551 0.422276 0.906467i $$-0.361231\pi$$
0.422276 + 0.906467i $$0.361231\pi$$
$$702$$ 0 0
$$703$$ −14.1246 −0.532720
$$704$$ 26.1246 0.984608
$$705$$ 0 0
$$706$$ 45.2705 1.70378
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −50.1246 −1.88247 −0.941235 0.337753i $$-0.890333\pi$$
−0.941235 + 0.337753i $$0.890333\pi$$
$$710$$ 52.3607 1.96506
$$711$$ 0 0
$$712$$ 16.7082 0.626166
$$713$$ −41.1803 −1.54222
$$714$$ 0 0
$$715$$ 6.70820 0.250873
$$716$$ 97.6869 3.65073
$$717$$ 0 0
$$718$$ −31.2705 −1.16701
$$719$$ 24.7082 0.921461 0.460730 0.887540i $$-0.347588\pi$$
0.460730 + 0.887540i $$0.347588\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −26.1803 −0.974331
$$723$$ 0 0
$$724$$ 123.374 4.58515
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 38.8328 1.44023 0.720115 0.693855i $$-0.244089\pi$$
0.720115 + 0.693855i $$0.244089\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 15.8541 0.586787
$$731$$ −11.7771 −0.435591
$$732$$ 0 0
$$733$$ −28.7082 −1.06036 −0.530181 0.847885i $$-0.677876\pi$$
−0.530181 + 0.847885i $$0.677876\pi$$
$$734$$ 33.2705 1.22804
$$735$$ 0 0
$$736$$ 89.3951 3.29515
$$737$$ −9.00000 −0.331519
$$738$$ 0 0
$$739$$ −17.8328 −0.655991 −0.327995 0.944679i $$-0.606373\pi$$
−0.327995 + 0.944679i $$0.606373\pi$$
$$740$$ 51.1033 1.87860
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −32.9443 −1.20861 −0.604304 0.796754i $$-0.706549\pi$$
−0.604304 + 0.796754i $$0.706549\pi$$
$$744$$ 0 0
$$745$$ −1.58359 −0.0580183
$$746$$ −4.14590 −0.151792
$$747$$ 0 0
$$748$$ 21.4377 0.783840
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.1246 −0.369452 −0.184726 0.982790i $$-0.559140\pi$$
−0.184726 + 0.982790i $$0.559140\pi$$
$$752$$ −73.6312 −2.68505
$$753$$ 0 0
$$754$$ −11.7082 −0.426388
$$755$$ 45.6525 1.66146
$$756$$ 0 0
$$757$$ −52.8328 −1.92024 −0.960121 0.279586i $$-0.909803\pi$$
−0.960121 + 0.279586i $$0.909803\pi$$
$$758$$ −40.3607 −1.46597
$$759$$ 0 0
$$760$$ −50.1246 −1.81821
$$761$$ −33.5410 −1.21586 −0.607931 0.793990i $$-0.708000\pi$$
−0.607931 + 0.793990i $$0.708000\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −54.2705 −1.96344
$$765$$ 0 0
$$766$$ 39.2705 1.41890
$$767$$ −1.47214 −0.0531557
$$768$$ 0 0
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 3.43769 0.123725
$$773$$ −47.0689 −1.69295 −0.846475 0.532428i $$-0.821280\pi$$
−0.846475 + 0.532428i $$0.821280\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −70.3607 −2.52580
$$777$$ 0 0
$$778$$ −3.85410 −0.138176
$$779$$ 13.4164 0.480693
$$780$$ 0 0
$$781$$ 26.8328 0.960154
$$782$$ 31.7426 1.13511
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −15.6525 −0.558661
$$786$$ 0 0
$$787$$ −20.4164 −0.727766 −0.363883 0.931445i $$-0.618549\pi$$
−0.363883 + 0.931445i $$0.618549\pi$$
$$788$$ 43.9574 1.56592
$$789$$ 0 0
$$790$$ −15.8541 −0.564064
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −3.00000 −0.106533
$$794$$ 68.3951 2.42725
$$795$$ 0 0
$$796$$ 100.520 3.56283
$$797$$ 9.05573 0.320770 0.160385 0.987055i $$-0.448726\pi$$
0.160385 + 0.987055i $$0.448726\pi$$
$$798$$ 0 0
$$799$$ −11.0000 −0.389152
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −37.2705 −1.31607
$$803$$ 8.12461 0.286711
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −13.0902 −0.461082
$$807$$ 0 0
$$808$$ −67.2492 −2.36582
$$809$$ −22.4164 −0.788119 −0.394059 0.919085i $$-0.628930\pi$$
−0.394059 + 0.919085i $$0.628930\pi$$
$$810$$ 0 0
$$811$$ 14.8328 0.520851 0.260425 0.965494i $$-0.416137\pi$$
0.260425 + 0.965494i $$0.416137\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 36.9787 1.29610
$$815$$ −36.7082 −1.28583
$$816$$ 0 0
$$817$$ 24.0000 0.839654
$$818$$ −22.7984 −0.797126
$$819$$ 0 0
$$820$$ −48.5410 −1.69513
$$821$$ −38.2361 −1.33445 −0.667224 0.744857i $$-0.732518\pi$$
−0.667224 + 0.744857i $$0.732518\pi$$
$$822$$ 0 0
$$823$$ 34.1246 1.18951 0.594755 0.803907i $$-0.297249\pi$$
0.594755 + 0.803907i $$0.297249\pi$$
$$824$$ 20.2361 0.704957
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −26.8328 −0.933068 −0.466534 0.884503i $$-0.654498\pi$$
−0.466534 + 0.884503i $$0.654498\pi$$
$$828$$ 0 0
$$829$$ 25.0000 0.868286 0.434143 0.900844i $$-0.357051\pi$$
0.434143 + 0.900844i $$0.357051\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 8.70820 0.301903
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 50.2492 1.73895
$$836$$ −43.6869 −1.51094
$$837$$ 0 0
$$838$$ 86.2492 2.97943
$$839$$ −5.88854 −0.203295 −0.101648 0.994820i $$-0.532411\pi$$
−0.101648 + 0.994820i $$0.532411\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 35.1246 1.21047
$$843$$ 0 0
$$844$$ 19.4164 0.668340
$$845$$ 2.23607 0.0769231
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 73.6312 2.52851
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 38.7771 1.32926
$$852$$ 0 0
$$853$$ 52.2492 1.78898 0.894490 0.447089i $$-0.147539\pi$$
0.894490 + 0.447089i $$0.147539\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 72.9574 2.49363
$$857$$ −1.36068 −0.0464799 −0.0232400 0.999730i $$-0.507398\pi$$
−0.0232400 + 0.999730i $$0.507398\pi$$
$$858$$ 0 0
$$859$$ −20.7082 −0.706555 −0.353277 0.935519i $$-0.614933\pi$$
−0.353277 + 0.935519i $$0.614933\pi$$
$$860$$ −86.8328 −2.96097
$$861$$ 0 0
$$862$$ 82.1033 2.79645
$$863$$ −23.9443 −0.815072 −0.407536 0.913189i $$-0.633612\pi$$
−0.407536 + 0.913189i $$0.633612\pi$$
$$864$$ 0 0
$$865$$ −36.7082 −1.24812
$$866$$ −77.0132 −2.61701
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −8.12461 −0.275609
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ −20.2361 −0.685280
$$873$$ 0 0
$$874$$ −64.6869 −2.18807
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.1246 1.08477 0.542386 0.840130i $$-0.317521\pi$$
0.542386 + 0.840130i $$0.317521\pi$$
$$878$$ −63.1591 −2.13151
$$879$$ 0 0
$$880$$ 66.1033 2.22834
$$881$$ 43.3050 1.45898 0.729490 0.683991i $$-0.239757\pi$$
0.729490 + 0.683991i $$0.239757\pi$$
$$882$$ 0 0
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 7.14590 0.240343
$$885$$ 0 0
$$886$$ −5.85410 −0.196672
$$887$$ −20.2361 −0.679461 −0.339730 0.940523i $$-0.610336\pi$$
−0.339730 + 0.940523i $$0.610336\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 13.0902 0.438783
$$891$$ 0 0
$$892$$ 19.4164 0.650109
$$893$$ 22.4164 0.750136
$$894$$ 0 0
$$895$$ 45.0000 1.50418
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 89.9574 3.00192
$$899$$ 22.3607 0.745770
$$900$$ 0 0
$$901$$ 11.0000 0.366463
$$902$$ −35.1246 −1.16952
$$903$$ 0 0
$$904$$ −22.0000 −0.731709
$$905$$ 56.8328 1.88919
$$906$$ 0 0
$$907$$ 18.7082 0.621196 0.310598 0.950541i $$-0.399471\pi$$
0.310598 + 0.950541i $$0.399471\pi$$
$$908$$ 28.8541 0.957557
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 34.2492 1.13473 0.567364 0.823467i $$-0.307963\pi$$
0.567364 + 0.823467i $$0.307963\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −16.0344 −0.530372
$$915$$ 0 0
$$916$$ −117.103 −3.86920
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 20.1246 0.663850 0.331925 0.943306i $$-0.392302\pi$$
0.331925 + 0.943306i $$0.392302\pi$$
$$920$$ 137.610 4.53686
$$921$$ 0 0
$$922$$ 89.9574 2.96259
$$923$$ 8.94427 0.294404
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −62.8328 −2.06481
$$927$$ 0 0
$$928$$ −48.5410 −1.59344
$$929$$ −24.8197 −0.814307 −0.407153 0.913360i $$-0.633479\pi$$
−0.407153 + 0.913360i $$0.633479\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −57.9787 −1.89916
$$933$$ 0 0
$$934$$ 25.2705 0.826876
$$935$$ 9.87539 0.322960
$$936$$ 0 0
$$937$$ −25.4164 −0.830318 −0.415159 0.909749i $$-0.636274\pi$$
−0.415159 + 0.909749i $$0.636274\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −81.1033 −2.64530
$$941$$ 50.2361 1.63765 0.818825 0.574044i $$-0.194626\pi$$
0.818825 + 0.574044i $$0.194626\pi$$
$$942$$ 0 0
$$943$$ −36.8328 −1.19944
$$944$$ −14.5066 −0.472149
$$945$$ 0 0
$$946$$ −62.8328 −2.04287
$$947$$ 22.5279 0.732057 0.366029 0.930604i $$-0.380717\pi$$
0.366029 + 0.930604i $$0.380717\pi$$
$$948$$ 0 0
$$949$$ 2.70820 0.0879120
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 41.7771 1.35329 0.676646 0.736308i $$-0.263433\pi$$
0.676646 + 0.736308i $$0.263433\pi$$
$$954$$ 0 0
$$955$$ −25.0000 −0.808981
$$956$$ −94.2492 −3.04824
$$957$$ 0 0
$$958$$ 62.3951 2.01589
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 12.3262 0.397414
$$963$$ 0 0
$$964$$ −22.8541 −0.736081
$$965$$ 1.58359 0.0509776
$$966$$ 0 0
$$967$$ 16.5836 0.533292 0.266646 0.963794i $$-0.414084\pi$$
0.266646 + 0.963794i $$0.414084\pi$$
$$968$$ −14.9443 −0.480327
$$969$$ 0 0
$$970$$ −55.1246 −1.76994
$$971$$ −11.2918 −0.362371 −0.181185 0.983449i $$-0.557993\pi$$
−0.181185 + 0.983449i $$0.557993\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −57.1591 −1.83149
$$975$$ 0 0
$$976$$ −29.5623 −0.946266
$$977$$ 2.34752 0.0751040 0.0375520 0.999295i $$-0.488044\pi$$
0.0375520 + 0.999295i $$0.488044\pi$$
$$978$$ 0 0
$$979$$ 6.70820 0.214395
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 66.8328 2.13272
$$983$$ 7.47214 0.238324 0.119162 0.992875i $$-0.461979\pi$$
0.119162 + 0.992875i $$0.461979\pi$$
$$984$$ 0 0
$$985$$ 20.2492 0.645194
$$986$$ −17.2361 −0.548908
$$987$$ 0 0
$$988$$ −14.5623 −0.463289
$$989$$ −65.8885 −2.09513
$$990$$ 0 0
$$991$$ 30.7082 0.975478 0.487739 0.872989i $$-0.337822\pi$$
0.487739 + 0.872989i $$0.337822\pi$$
$$992$$ −54.2705 −1.72309
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 46.3050 1.46797
$$996$$ 0 0
$$997$$ −26.4164 −0.836616 −0.418308 0.908305i $$-0.637377\pi$$
−0.418308 + 0.908305i $$0.637377\pi$$
$$998$$ 69.1591 2.18919
$$999$$ 0 0
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 5733.2.a.w.1.2 2
3.2 odd 2 637.2.a.e.1.1 2
7.3 odd 6 819.2.j.c.352.1 4
7.5 odd 6 819.2.j.c.235.1 4
7.6 odd 2 5733.2.a.v.1.2 2
21.2 odd 6 637.2.e.h.508.2 4
21.5 even 6 91.2.e.b.53.2 4
21.11 odd 6 637.2.e.h.79.2 4
21.17 even 6 91.2.e.b.79.2 yes 4
21.20 even 2 637.2.a.f.1.1 2
39.38 odd 2 8281.2.a.ba.1.2 2
84.47 odd 6 1456.2.r.j.417.1 4
84.59 odd 6 1456.2.r.j.625.1 4
273.38 even 6 1183.2.e.d.170.1 4
273.194 even 6 1183.2.e.d.508.1 4
273.272 even 2 8281.2.a.z.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.b.53.2 4 21.5 even 6
91.2.e.b.79.2 yes 4 21.17 even 6
637.2.a.e.1.1 2 3.2 odd 2
637.2.a.f.1.1 2 21.20 even 2
637.2.e.h.79.2 4 21.11 odd 6
637.2.e.h.508.2 4 21.2 odd 6
819.2.j.c.235.1 4 7.5 odd 6
819.2.j.c.352.1 4 7.3 odd 6
1183.2.e.d.170.1 4 273.38 even 6
1183.2.e.d.508.1 4 273.194 even 6
1456.2.r.j.417.1 4 84.47 odd 6
1456.2.r.j.625.1 4 84.59 odd 6
5733.2.a.v.1.2 2 7.6 odd 2
5733.2.a.w.1.2 2 1.1 even 1 trivial
8281.2.a.z.1.2 2 273.272 even 2
8281.2.a.ba.1.2 2 39.38 odd 2