# Properties

 Label 5472.2.a.bf.1.1 Level $5472$ Weight $2$ Character 5472.1 Self dual yes Analytic conductor $43.694$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5472 = 2^{5} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5472.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.6941399860$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 5472.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.56155 q^{5} +3.00000 q^{7} +O(q^{10})$$ $$q-1.56155 q^{5} +3.00000 q^{7} +3.56155 q^{11} +2.56155 q^{13} +8.12311 q^{17} +1.00000 q^{19} -1.43845 q^{23} -2.56155 q^{25} +7.68466 q^{29} +0.876894 q^{31} -4.68466 q^{35} -1.12311 q^{37} -4.00000 q^{41} +9.56155 q^{43} -8.68466 q^{47} +2.00000 q^{49} +8.56155 q^{53} -5.56155 q^{55} +8.56155 q^{59} -5.80776 q^{61} -4.00000 q^{65} -4.56155 q^{67} -12.2462 q^{71} +7.24621 q^{73} +10.6847 q^{77} +10.0000 q^{79} -7.36932 q^{83} -12.6847 q^{85} -9.36932 q^{89} +7.68466 q^{91} -1.56155 q^{95} -1.12311 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{5} + 6q^{7} + O(q^{10})$$ $$2q + q^{5} + 6q^{7} + 3q^{11} + q^{13} + 8q^{17} + 2q^{19} - 7q^{23} - q^{25} + 3q^{29} + 10q^{31} + 3q^{35} + 6q^{37} - 8q^{41} + 15q^{43} - 5q^{47} + 4q^{49} + 13q^{53} - 7q^{55} + 13q^{59} + 9q^{61} - 8q^{65} - 5q^{67} - 8q^{71} - 2q^{73} + 9q^{77} + 20q^{79} + 10q^{83} - 13q^{85} + 6q^{89} + 3q^{91} + q^{95} + 6q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.56155 −0.698348 −0.349174 0.937058i $$-0.613538\pi$$
−0.349174 + 0.937058i $$0.613538\pi$$
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.56155 1.07385 0.536924 0.843630i $$-0.319586\pi$$
0.536924 + 0.843630i $$0.319586\pi$$
$$12$$ 0 0
$$13$$ 2.56155 0.710447 0.355223 0.934781i $$-0.384405\pi$$
0.355223 + 0.934781i $$0.384405\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 8.12311 1.97014 0.985071 0.172147i $$-0.0550704\pi$$
0.985071 + 0.172147i $$0.0550704\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.43845 −0.299937 −0.149968 0.988691i $$-0.547917\pi$$
−0.149968 + 0.988691i $$0.547917\pi$$
$$24$$ 0 0
$$25$$ −2.56155 −0.512311
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7.68466 1.42701 0.713503 0.700653i $$-0.247108\pi$$
0.713503 + 0.700653i $$0.247108\pi$$
$$30$$ 0 0
$$31$$ 0.876894 0.157495 0.0787474 0.996895i $$-0.474908\pi$$
0.0787474 + 0.996895i $$0.474908\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.68466 −0.791852
$$36$$ 0 0
$$37$$ −1.12311 −0.184637 −0.0923187 0.995730i $$-0.529428\pi$$
−0.0923187 + 0.995730i $$0.529428\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 0 0
$$43$$ 9.56155 1.45812 0.729062 0.684448i $$-0.239957\pi$$
0.729062 + 0.684448i $$0.239957\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.68466 −1.26679 −0.633394 0.773830i $$-0.718339\pi$$
−0.633394 + 0.773830i $$0.718339\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 8.56155 1.17602 0.588010 0.808854i $$-0.299912\pi$$
0.588010 + 0.808854i $$0.299912\pi$$
$$54$$ 0 0
$$55$$ −5.56155 −0.749920
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.56155 1.11462 0.557310 0.830305i $$-0.311834\pi$$
0.557310 + 0.830305i $$0.311834\pi$$
$$60$$ 0 0
$$61$$ −5.80776 −0.743608 −0.371804 0.928311i $$-0.621261\pi$$
−0.371804 + 0.928311i $$0.621261\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ −4.56155 −0.557282 −0.278641 0.960395i $$-0.589884\pi$$
−0.278641 + 0.960395i $$0.589884\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.2462 −1.45336 −0.726679 0.686977i $$-0.758937\pi$$
−0.726679 + 0.686977i $$0.758937\pi$$
$$72$$ 0 0
$$73$$ 7.24621 0.848105 0.424052 0.905638i $$-0.360607\pi$$
0.424052 + 0.905638i $$0.360607\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 10.6847 1.21763
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −7.36932 −0.808888 −0.404444 0.914563i $$-0.632535\pi$$
−0.404444 + 0.914563i $$0.632535\pi$$
$$84$$ 0 0
$$85$$ −12.6847 −1.37584
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −9.36932 −0.993146 −0.496573 0.867995i $$-0.665408\pi$$
−0.496573 + 0.867995i $$0.665408\pi$$
$$90$$ 0 0
$$91$$ 7.68466 0.805571
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.56155 −0.160212
$$96$$ 0 0
$$97$$ −1.12311 −0.114034 −0.0570170 0.998373i $$-0.518159\pi$$
−0.0570170 + 0.998373i $$0.518159\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −16.2462 −1.61656 −0.808279 0.588799i $$-0.799601\pi$$
−0.808279 + 0.588799i $$0.799601\pi$$
$$102$$ 0 0
$$103$$ −18.4924 −1.82211 −0.911056 0.412282i $$-0.864732\pi$$
−0.911056 + 0.412282i $$0.864732\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.9309 1.15340 0.576700 0.816956i $$-0.304340\pi$$
0.576700 + 0.816956i $$0.304340\pi$$
$$108$$ 0 0
$$109$$ 0.561553 0.0537870 0.0268935 0.999638i $$-0.491439\pi$$
0.0268935 + 0.999638i $$0.491439\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −13.1231 −1.23452 −0.617259 0.786760i $$-0.711757\pi$$
−0.617259 + 0.786760i $$0.711757\pi$$
$$114$$ 0 0
$$115$$ 2.24621 0.209460
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 24.3693 2.23393
$$120$$ 0 0
$$121$$ 1.68466 0.153151
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 11.8078 1.05612
$$126$$ 0 0
$$127$$ 6.87689 0.610226 0.305113 0.952316i $$-0.401306\pi$$
0.305113 + 0.952316i $$0.401306\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 17.5616 1.53436 0.767180 0.641432i $$-0.221659\pi$$
0.767180 + 0.641432i $$0.221659\pi$$
$$132$$ 0 0
$$133$$ 3.00000 0.260133
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 10.3693 0.885911 0.442955 0.896544i $$-0.353930\pi$$
0.442955 + 0.896544i $$0.353930\pi$$
$$138$$ 0 0
$$139$$ 9.56155 0.811000 0.405500 0.914095i $$-0.367097\pi$$
0.405500 + 0.914095i $$0.367097\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 9.12311 0.762912
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.43845 −0.527458 −0.263729 0.964597i $$-0.584952\pi$$
−0.263729 + 0.964597i $$0.584952\pi$$
$$150$$ 0 0
$$151$$ 10.4924 0.853861 0.426931 0.904284i $$-0.359595\pi$$
0.426931 + 0.904284i $$0.359595\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1.36932 −0.109986
$$156$$ 0 0
$$157$$ −24.2462 −1.93506 −0.967529 0.252759i $$-0.918662\pi$$
−0.967529 + 0.252759i $$0.918662\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.31534 −0.340097
$$162$$ 0 0
$$163$$ −16.4924 −1.29179 −0.645893 0.763428i $$-0.723515\pi$$
−0.645893 + 0.763428i $$0.723515\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −6.43845 −0.495265
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 3.75379 0.285395 0.142698 0.989766i $$-0.454422\pi$$
0.142698 + 0.989766i $$0.454422\pi$$
$$174$$ 0 0
$$175$$ −7.68466 −0.580906
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 14.2462 1.06481 0.532406 0.846489i $$-0.321288\pi$$
0.532406 + 0.846489i $$0.321288\pi$$
$$180$$ 0 0
$$181$$ 16.2462 1.20757 0.603786 0.797147i $$-0.293658\pi$$
0.603786 + 0.797147i $$0.293658\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.75379 0.128941
$$186$$ 0 0
$$187$$ 28.9309 2.11563
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1.00000 0.0723575 0.0361787 0.999345i $$-0.488481\pi$$
0.0361787 + 0.999345i $$0.488481\pi$$
$$192$$ 0 0
$$193$$ −14.4924 −1.04319 −0.521594 0.853194i $$-0.674662\pi$$
−0.521594 + 0.853194i $$0.674662\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 13.3693 0.952524 0.476262 0.879303i $$-0.341991\pi$$
0.476262 + 0.879303i $$0.341991\pi$$
$$198$$ 0 0
$$199$$ 25.7386 1.82456 0.912282 0.409563i $$-0.134319\pi$$
0.912282 + 0.409563i $$0.134319\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 23.0540 1.61807
$$204$$ 0 0
$$205$$ 6.24621 0.436254
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 3.56155 0.246358
$$210$$ 0 0
$$211$$ 11.4384 0.787455 0.393728 0.919227i $$-0.371185\pi$$
0.393728 + 0.919227i $$0.371185\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −14.9309 −1.01828
$$216$$ 0 0
$$217$$ 2.63068 0.178582
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 20.8078 1.39968
$$222$$ 0 0
$$223$$ 23.3693 1.56493 0.782463 0.622698i $$-0.213963\pi$$
0.782463 + 0.622698i $$0.213963\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −4.31534 −0.286419 −0.143210 0.989692i $$-0.545742\pi$$
−0.143210 + 0.989692i $$0.545742\pi$$
$$228$$ 0 0
$$229$$ −8.43845 −0.557628 −0.278814 0.960345i $$-0.589941\pi$$
−0.278814 + 0.960345i $$0.589941\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −8.93087 −0.585081 −0.292540 0.956253i $$-0.594501\pi$$
−0.292540 + 0.956253i $$0.594501\pi$$
$$234$$ 0 0
$$235$$ 13.5616 0.884658
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ 2.24621 0.144691 0.0723456 0.997380i $$-0.476952\pi$$
0.0723456 + 0.997380i $$0.476952\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −3.12311 −0.199528
$$246$$ 0 0
$$247$$ 2.56155 0.162988
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6.68466 0.421932 0.210966 0.977493i $$-0.432339\pi$$
0.210966 + 0.977493i $$0.432339\pi$$
$$252$$ 0 0
$$253$$ −5.12311 −0.322087
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −9.75379 −0.608425 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$258$$ 0 0
$$259$$ −3.36932 −0.209359
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 26.0540 1.60656 0.803278 0.595604i $$-0.203087\pi$$
0.803278 + 0.595604i $$0.203087\pi$$
$$264$$ 0 0
$$265$$ −13.3693 −0.821271
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −14.4924 −0.883619 −0.441809 0.897109i $$-0.645663\pi$$
−0.441809 + 0.897109i $$0.645663\pi$$
$$270$$ 0 0
$$271$$ −27.0540 −1.64341 −0.821706 0.569912i $$-0.806977\pi$$
−0.821706 + 0.569912i $$0.806977\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −9.12311 −0.550144
$$276$$ 0 0
$$277$$ 0.684658 0.0411371 0.0205686 0.999788i $$-0.493452\pi$$
0.0205686 + 0.999788i $$0.493452\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 25.3693 1.51341 0.756703 0.653758i $$-0.226809\pi$$
0.756703 + 0.653758i $$0.226809\pi$$
$$282$$ 0 0
$$283$$ 24.4384 1.45271 0.726357 0.687317i $$-0.241212\pi$$
0.726357 + 0.687317i $$0.241212\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ 48.9848 2.88146
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −27.6847 −1.61736 −0.808678 0.588252i $$-0.799816\pi$$
−0.808678 + 0.588252i $$0.799816\pi$$
$$294$$ 0 0
$$295$$ −13.3693 −0.778392
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −3.68466 −0.213089
$$300$$ 0 0
$$301$$ 28.6847 1.65336
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 9.06913 0.519297
$$306$$ 0 0
$$307$$ 29.3693 1.67620 0.838098 0.545520i $$-0.183668\pi$$
0.838098 + 0.545520i $$0.183668\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2.12311 −0.120390 −0.0601951 0.998187i $$-0.519172\pi$$
−0.0601951 + 0.998187i $$0.519172\pi$$
$$312$$ 0 0
$$313$$ 27.9309 1.57875 0.789373 0.613914i $$-0.210406\pi$$
0.789373 + 0.613914i $$0.210406\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −20.5616 −1.15485 −0.577426 0.816443i $$-0.695943\pi$$
−0.577426 + 0.816443i $$0.695943\pi$$
$$318$$ 0 0
$$319$$ 27.3693 1.53239
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 8.12311 0.451982
$$324$$ 0 0
$$325$$ −6.56155 −0.363969
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −26.0540 −1.43640
$$330$$ 0 0
$$331$$ 8.56155 0.470586 0.235293 0.971925i $$-0.424395\pi$$
0.235293 + 0.971925i $$0.424395\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 7.12311 0.389177
$$336$$ 0 0
$$337$$ 12.4924 0.680506 0.340253 0.940334i $$-0.389487\pi$$
0.340253 + 0.940334i $$0.389487\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3.12311 0.169126
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0.438447 0.0235371 0.0117685 0.999931i $$-0.496254\pi$$
0.0117685 + 0.999931i $$0.496254\pi$$
$$348$$ 0 0
$$349$$ 1.31534 0.0704086 0.0352043 0.999380i $$-0.488792\pi$$
0.0352043 + 0.999380i $$0.488792\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0.0691303 0.00367944 0.00183972 0.999998i $$-0.499414\pi$$
0.00183972 + 0.999998i $$0.499414\pi$$
$$354$$ 0 0
$$355$$ 19.1231 1.01495
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −36.1231 −1.90650 −0.953252 0.302176i $$-0.902287\pi$$
−0.953252 + 0.302176i $$0.902287\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −11.3153 −0.592272
$$366$$ 0 0
$$367$$ −20.0000 −1.04399 −0.521996 0.852948i $$-0.674812\pi$$
−0.521996 + 0.852948i $$0.674812\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 25.6847 1.33348
$$372$$ 0 0
$$373$$ 13.9309 0.721313 0.360657 0.932699i $$-0.382553\pi$$
0.360657 + 0.932699i $$0.382553\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 19.6847 1.01381
$$378$$ 0 0
$$379$$ −21.0540 −1.08147 −0.540735 0.841193i $$-0.681854\pi$$
−0.540735 + 0.841193i $$0.681854\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −7.75379 −0.396200 −0.198100 0.980182i $$-0.563477\pi$$
−0.198100 + 0.980182i $$0.563477\pi$$
$$384$$ 0 0
$$385$$ −16.6847 −0.850329
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −6.19224 −0.313959 −0.156979 0.987602i $$-0.550176\pi$$
−0.156979 + 0.987602i $$0.550176\pi$$
$$390$$ 0 0
$$391$$ −11.6847 −0.590919
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −15.6155 −0.785702
$$396$$ 0 0
$$397$$ −9.56155 −0.479881 −0.239940 0.970788i $$-0.577128\pi$$
−0.239940 + 0.970788i $$0.577128\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1.36932 0.0683804 0.0341902 0.999415i $$-0.489115\pi$$
0.0341902 + 0.999415i $$0.489115\pi$$
$$402$$ 0 0
$$403$$ 2.24621 0.111892
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.00000 −0.198273
$$408$$ 0 0
$$409$$ 27.1231 1.34115 0.670576 0.741841i $$-0.266047\pi$$
0.670576 + 0.741841i $$0.266047\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 25.6847 1.26386
$$414$$ 0 0
$$415$$ 11.5076 0.564885
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 9.75379 0.476504 0.238252 0.971203i $$-0.423426\pi$$
0.238252 + 0.971203i $$0.423426\pi$$
$$420$$ 0 0
$$421$$ −17.0540 −0.831160 −0.415580 0.909557i $$-0.636421\pi$$
−0.415580 + 0.909557i $$0.636421\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −20.8078 −1.00932
$$426$$ 0 0
$$427$$ −17.4233 −0.843172
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 22.8769 1.10194 0.550971 0.834525i $$-0.314258\pi$$
0.550971 + 0.834525i $$0.314258\pi$$
$$432$$ 0 0
$$433$$ −1.61553 −0.0776373 −0.0388187 0.999246i $$-0.512359\pi$$
−0.0388187 + 0.999246i $$0.512359\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1.43845 −0.0688103
$$438$$ 0 0
$$439$$ 18.2462 0.870844 0.435422 0.900226i $$-0.356599\pi$$
0.435422 + 0.900226i $$0.356599\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 10.6847 0.507643 0.253822 0.967251i $$-0.418312\pi$$
0.253822 + 0.967251i $$0.418312\pi$$
$$444$$ 0 0
$$445$$ 14.6307 0.693561
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6.87689 0.324541 0.162270 0.986746i $$-0.448118\pi$$
0.162270 + 0.986746i $$0.448118\pi$$
$$450$$ 0 0
$$451$$ −14.2462 −0.670828
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ −27.8769 −1.30403 −0.652013 0.758208i $$-0.726075\pi$$
−0.652013 + 0.758208i $$0.726075\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 5.06913 0.236093 0.118046 0.993008i $$-0.462337\pi$$
0.118046 + 0.993008i $$0.462337\pi$$
$$462$$ 0 0
$$463$$ −22.0540 −1.02494 −0.512468 0.858707i $$-0.671269\pi$$
−0.512468 + 0.858707i $$0.671269\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −5.31534 −0.245965 −0.122982 0.992409i $$-0.539246\pi$$
−0.122982 + 0.992409i $$0.539246\pi$$
$$468$$ 0 0
$$469$$ −13.6847 −0.631899
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 34.0540 1.56580
$$474$$ 0 0
$$475$$ −2.56155 −0.117532
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −2.24621 −0.102632 −0.0513160 0.998682i $$-0.516342\pi$$
−0.0513160 + 0.998682i $$0.516342\pi$$
$$480$$ 0 0
$$481$$ −2.87689 −0.131175
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1.75379 0.0796354
$$486$$ 0 0
$$487$$ 26.4924 1.20049 0.600243 0.799818i $$-0.295070\pi$$
0.600243 + 0.799818i $$0.295070\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −0.492423 −0.0222227 −0.0111114 0.999938i $$-0.503537\pi$$
−0.0111114 + 0.999938i $$0.503537\pi$$
$$492$$ 0 0
$$493$$ 62.4233 2.81140
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −36.7386 −1.64795
$$498$$ 0 0
$$499$$ 8.68466 0.388779 0.194389 0.980924i $$-0.437727\pi$$
0.194389 + 0.980924i $$0.437727\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 6.56155 0.292565 0.146283 0.989243i $$-0.453269\pi$$
0.146283 + 0.989243i $$0.453269\pi$$
$$504$$ 0 0
$$505$$ 25.3693 1.12892
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −21.6155 −0.958091 −0.479046 0.877790i $$-0.659017\pi$$
−0.479046 + 0.877790i $$0.659017\pi$$
$$510$$ 0 0
$$511$$ 21.7386 0.961661
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 28.8769 1.27247
$$516$$ 0 0
$$517$$ −30.9309 −1.36034
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 1.36932 0.0599909 0.0299954 0.999550i $$-0.490451\pi$$
0.0299954 + 0.999550i $$0.490451\pi$$
$$522$$ 0 0
$$523$$ 37.5464 1.64179 0.820895 0.571080i $$-0.193475\pi$$
0.820895 + 0.571080i $$0.193475\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 7.12311 0.310287
$$528$$ 0 0
$$529$$ −20.9309 −0.910038
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −10.2462 −0.443813
$$534$$ 0 0
$$535$$ −18.6307 −0.805475
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 7.12311 0.306814
$$540$$ 0 0
$$541$$ −39.5616 −1.70088 −0.850442 0.526069i $$-0.823665\pi$$
−0.850442 + 0.526069i $$0.823665\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −0.876894 −0.0375620
$$546$$ 0 0
$$547$$ −8.49242 −0.363110 −0.181555 0.983381i $$-0.558113\pi$$
−0.181555 + 0.983381i $$0.558113\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 7.68466 0.327377
$$552$$ 0 0
$$553$$ 30.0000 1.27573
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −10.9309 −0.463156 −0.231578 0.972816i $$-0.574389\pi$$
−0.231578 + 0.972816i $$0.574389\pi$$
$$558$$ 0 0
$$559$$ 24.4924 1.03592
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 13.7538 0.579653 0.289827 0.957079i $$-0.406402\pi$$
0.289827 + 0.957079i $$0.406402\pi$$
$$564$$ 0 0
$$565$$ 20.4924 0.862123
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −22.7386 −0.953253 −0.476627 0.879106i $$-0.658141\pi$$
−0.476627 + 0.879106i $$0.658141\pi$$
$$570$$ 0 0
$$571$$ −8.00000 −0.334790 −0.167395 0.985890i $$-0.553535\pi$$
−0.167395 + 0.985890i $$0.553535\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3.68466 0.153661
$$576$$ 0 0
$$577$$ −11.2462 −0.468186 −0.234093 0.972214i $$-0.575212\pi$$
−0.234093 + 0.972214i $$0.575212\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −22.1080 −0.917192
$$582$$ 0 0
$$583$$ 30.4924 1.26287
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 5.06913 0.209225 0.104613 0.994513i $$-0.466640\pi$$
0.104613 + 0.994513i $$0.466640\pi$$
$$588$$ 0 0
$$589$$ 0.876894 0.0361318
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 16.7386 0.687373 0.343687 0.939084i $$-0.388324\pi$$
0.343687 + 0.939084i $$0.388324\pi$$
$$594$$ 0 0
$$595$$ −38.0540 −1.56006
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 13.8617 0.566375 0.283188 0.959065i $$-0.408608\pi$$
0.283188 + 0.959065i $$0.408608\pi$$
$$600$$ 0 0
$$601$$ −29.3693 −1.19800 −0.599000 0.800749i $$-0.704435\pi$$
−0.599000 + 0.800749i $$0.704435\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −2.63068 −0.106952
$$606$$ 0 0
$$607$$ −21.1231 −0.857360 −0.428680 0.903456i $$-0.641021\pi$$
−0.428680 + 0.903456i $$0.641021\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −22.2462 −0.899985
$$612$$ 0 0
$$613$$ 28.5464 1.15298 0.576489 0.817105i $$-0.304422\pi$$
0.576489 + 0.817105i $$0.304422\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −9.31534 −0.375022 −0.187511 0.982263i $$-0.560042\pi$$
−0.187511 + 0.982263i $$0.560042\pi$$
$$618$$ 0 0
$$619$$ −38.8769 −1.56259 −0.781297 0.624159i $$-0.785442\pi$$
−0.781297 + 0.624159i $$0.785442\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −28.1080 −1.12612
$$624$$ 0 0
$$625$$ −5.63068 −0.225227
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −9.12311 −0.363762
$$630$$ 0 0
$$631$$ 8.30019 0.330425 0.165213 0.986258i $$-0.447169\pi$$
0.165213 + 0.986258i $$0.447169\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −10.7386 −0.426150
$$636$$ 0 0
$$637$$ 5.12311 0.202985
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 50.1080 1.97915 0.989573 0.144035i $$-0.0460079\pi$$
0.989573 + 0.144035i $$0.0460079\pi$$
$$642$$ 0 0
$$643$$ −26.3002 −1.03718 −0.518589 0.855024i $$-0.673543\pi$$
−0.518589 + 0.855024i $$0.673543\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −47.0000 −1.84776 −0.923880 0.382682i $$-0.875001\pi$$
−0.923880 + 0.382682i $$0.875001\pi$$
$$648$$ 0 0
$$649$$ 30.4924 1.19693
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 11.8078 0.462074 0.231037 0.972945i $$-0.425788\pi$$
0.231037 + 0.972945i $$0.425788\pi$$
$$654$$ 0 0
$$655$$ −27.4233 −1.07152
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −37.5464 −1.46260 −0.731300 0.682056i $$-0.761086\pi$$
−0.731300 + 0.682056i $$0.761086\pi$$
$$660$$ 0 0
$$661$$ 24.1771 0.940379 0.470190 0.882565i $$-0.344185\pi$$
0.470190 + 0.882565i $$0.344185\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −4.68466 −0.181663
$$666$$ 0 0
$$667$$ −11.0540 −0.428012
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −20.6847 −0.798522
$$672$$ 0 0
$$673$$ −12.8769 −0.496368 −0.248184 0.968713i $$-0.579834\pi$$
−0.248184 + 0.968713i $$0.579834\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.56155 0.252181 0.126090 0.992019i $$-0.459757\pi$$
0.126090 + 0.992019i $$0.459757\pi$$
$$678$$ 0 0
$$679$$ −3.36932 −0.129303
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 34.2462 1.31039 0.655197 0.755458i $$-0.272585\pi$$
0.655197 + 0.755458i $$0.272585\pi$$
$$684$$ 0 0
$$685$$ −16.1922 −0.618674
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 21.9309 0.835500
$$690$$ 0 0
$$691$$ −7.94602 −0.302281 −0.151141 0.988512i $$-0.548295\pi$$
−0.151141 + 0.988512i $$0.548295\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −14.9309 −0.566360
$$696$$ 0 0
$$697$$ −32.4924 −1.23074
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 27.6155 1.04302 0.521512 0.853244i $$-0.325368\pi$$
0.521512 + 0.853244i $$0.325368\pi$$
$$702$$ 0 0
$$703$$ −1.12311 −0.0423587
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −48.7386 −1.83300
$$708$$ 0 0
$$709$$ 17.5076 0.657511 0.328755 0.944415i $$-0.393371\pi$$
0.328755 + 0.944415i $$0.393371\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −1.26137 −0.0472385
$$714$$ 0 0
$$715$$ −14.2462 −0.532778
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 9.49242 0.354008 0.177004 0.984210i $$-0.443360\pi$$
0.177004 + 0.984210i $$0.443360\pi$$
$$720$$ 0 0
$$721$$ −55.4773 −2.06608
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −19.6847 −0.731070
$$726$$ 0 0
$$727$$ −2.36932 −0.0878731 −0.0439365 0.999034i $$-0.513990\pi$$
−0.0439365 + 0.999034i $$0.513990\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 77.6695 2.87271
$$732$$ 0 0
$$733$$ −21.3693 −0.789294 −0.394647 0.918833i $$-0.629133\pi$$
−0.394647 + 0.918833i $$0.629133\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.2462 −0.598437
$$738$$ 0 0
$$739$$ 5.56155 0.204585 0.102293 0.994754i $$-0.467382\pi$$
0.102293 + 0.994754i $$0.467382\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −31.1231 −1.14180 −0.570898 0.821021i $$-0.693405\pi$$
−0.570898 + 0.821021i $$0.693405\pi$$
$$744$$ 0 0
$$745$$ 10.0540 0.368349
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 35.7926 1.30783
$$750$$ 0 0
$$751$$ −12.0000 −0.437886 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −16.3845 −0.596292
$$756$$ 0 0
$$757$$ −16.0540 −0.583492 −0.291746 0.956496i $$-0.594236\pi$$
−0.291746 + 0.956496i $$0.594236\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −16.8617 −0.611238 −0.305619 0.952154i $$-0.598863\pi$$
−0.305619 + 0.952154i $$0.598863\pi$$
$$762$$ 0 0
$$763$$ 1.68466 0.0609887
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 21.9309 0.791878
$$768$$ 0 0
$$769$$ −12.3693 −0.446049 −0.223024 0.974813i $$-0.571593\pi$$
−0.223024 + 0.974813i $$0.571593\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −5.05398 −0.181779 −0.0908894 0.995861i $$-0.528971\pi$$
−0.0908894 + 0.995861i $$0.528971\pi$$
$$774$$ 0 0
$$775$$ −2.24621 −0.0806863
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ −43.6155 −1.56069
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 37.8617 1.35134
$$786$$ 0 0
$$787$$ 26.5616 0.946817 0.473409 0.880843i $$-0.343023\pi$$
0.473409 + 0.880843i $$0.343023\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −39.3693 −1.39981
$$792$$ 0 0
$$793$$ −14.8769 −0.528294
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −9.68466 −0.343048 −0.171524 0.985180i $$-0.554869\pi$$
−0.171524 + 0.985180i $$0.554869\pi$$
$$798$$ 0 0
$$799$$ −70.5464 −2.49575
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 25.8078 0.910736
$$804$$ 0 0
$$805$$ 6.73863 0.237506
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 2.12311 0.0746444 0.0373222 0.999303i $$-0.488117\pi$$
0.0373222 + 0.999303i $$0.488117\pi$$
$$810$$ 0 0
$$811$$ 50.4233 1.77060 0.885301 0.465019i $$-0.153953\pi$$
0.885301 + 0.465019i $$0.153953\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 25.7538 0.902116
$$816$$ 0 0
$$817$$ 9.56155 0.334516
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16.6847 −0.582299 −0.291149 0.956678i $$-0.594038\pi$$
−0.291149 + 0.956678i $$0.594038\pi$$
$$822$$ 0 0
$$823$$ −4.36932 −0.152305 −0.0761524 0.997096i $$-0.524264\pi$$
−0.0761524 + 0.997096i $$0.524264\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −15.4384 −0.536847 −0.268424 0.963301i $$-0.586503\pi$$
−0.268424 + 0.963301i $$0.586503\pi$$
$$828$$ 0 0
$$829$$ −51.5464 −1.79028 −0.895140 0.445785i $$-0.852925\pi$$
−0.895140 + 0.445785i $$0.852925\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 16.2462 0.562898
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 11.6155 0.401013 0.200506 0.979692i $$-0.435741\pi$$
0.200506 + 0.979692i $$0.435741\pi$$
$$840$$ 0 0
$$841$$ 30.0540 1.03634
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 10.0540 0.345867
$$846$$ 0 0
$$847$$ 5.05398 0.173657
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 1.61553 0.0553796
$$852$$ 0 0
$$853$$ 15.7538 0.539399 0.269700 0.962944i $$-0.413076\pi$$
0.269700 + 0.962944i $$0.413076\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −37.3693 −1.27651 −0.638256 0.769824i $$-0.720344\pi$$
−0.638256 + 0.769824i $$0.720344\pi$$
$$858$$ 0 0
$$859$$ −7.06913 −0.241196 −0.120598 0.992701i $$-0.538481\pi$$
−0.120598 + 0.992701i $$0.538481\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −47.3693 −1.61247 −0.806235 0.591595i $$-0.798498\pi$$
−0.806235 + 0.591595i $$0.798498\pi$$
$$864$$ 0 0
$$865$$ −5.86174 −0.199305
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 35.6155 1.20817
$$870$$ 0 0
$$871$$ −11.6847 −0.395920
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 35.4233 1.19753
$$876$$ 0 0
$$877$$ −9.68466 −0.327028 −0.163514 0.986541i $$-0.552283\pi$$
−0.163514 + 0.986541i $$0.552283\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −37.3153 −1.25719 −0.628593 0.777735i $$-0.716369\pi$$
−0.628593 + 0.777735i $$0.716369\pi$$
$$882$$ 0 0
$$883$$ 28.9309 0.973601 0.486801 0.873513i $$-0.338164\pi$$
0.486801 + 0.873513i $$0.338164\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −45.2311 −1.51871 −0.759355 0.650676i $$-0.774485\pi$$
−0.759355 + 0.650676i $$0.774485\pi$$
$$888$$ 0 0
$$889$$ 20.6307 0.691931
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −8.68466 −0.290621
$$894$$ 0 0
$$895$$ −22.2462 −0.743609
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 6.73863 0.224746
$$900$$ 0 0
$$901$$ 69.5464 2.31693
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −25.3693 −0.843305
$$906$$ 0 0
$$907$$ −11.4384 −0.379807 −0.189904 0.981803i $$-0.560818\pi$$
−0.189904 + 0.981803i $$0.560818\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0.876894 0.0290528 0.0145264 0.999894i $$-0.495376\pi$$
0.0145264 + 0.999894i $$0.495376\pi$$
$$912$$ 0 0
$$913$$ −26.2462 −0.868623
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 52.6847 1.73980
$$918$$ 0 0
$$919$$ 29.3002 0.966524 0.483262 0.875476i $$-0.339452\pi$$
0.483262 + 0.875476i $$0.339452\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −31.3693 −1.03253
$$924$$ 0 0
$$925$$ 2.87689 0.0945917
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 9.82292 0.322280 0.161140 0.986932i $$-0.448483\pi$$
0.161140 + 0.986932i $$0.448483\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −45.1771 −1.47745
$$936$$ 0 0
$$937$$ −29.2462 −0.955432 −0.477716 0.878514i $$-0.658535\pi$$
−0.477716 + 0.878514i $$0.658535\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 35.3002 1.15075 0.575377 0.817889i $$-0.304856\pi$$
0.575377 + 0.817889i $$0.304856\pi$$
$$942$$ 0 0
$$943$$ 5.75379 0.187369
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 28.9848 0.941881 0.470940 0.882165i $$-0.343915\pi$$
0.470940 + 0.882165i $$0.343915\pi$$
$$948$$ 0 0
$$949$$ 18.5616 0.602534
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −8.24621 −0.267121 −0.133560 0.991041i $$-0.542641\pi$$
−0.133560 + 0.991041i $$0.542641\pi$$
$$954$$ 0 0
$$955$$ −1.56155 −0.0505307
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 31.1080 1.00453
$$960$$ 0 0
$$961$$ −30.2311 −0.975195
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 22.6307 0.728507
$$966$$ 0 0
$$967$$ −24.0000 −0.771788 −0.385894 0.922543i $$-0.626107\pi$$
−0.385894 + 0.922543i $$0.626107\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 56.4924 1.81293 0.906464 0.422283i $$-0.138771\pi$$
0.906464 + 0.422283i $$0.138771\pi$$
$$972$$ 0 0
$$973$$ 28.6847 0.919588
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 12.3845 0.396214 0.198107 0.980180i $$-0.436521\pi$$
0.198107 + 0.980180i $$0.436521\pi$$
$$978$$ 0 0
$$979$$ −33.3693 −1.06649
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 29.6155 0.944589 0.472294 0.881441i $$-0.343426\pi$$
0.472294 + 0.881441i $$0.343426\pi$$
$$984$$ 0 0
$$985$$ −20.8769 −0.665193
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −13.7538 −0.437345
$$990$$ 0 0
$$991$$ −1.75379 −0.0557109 −0.0278555 0.999612i $$-0.508868\pi$$
−0.0278555 + 0.999612i $$0.508868\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −40.1922 −1.27418
$$996$$ 0 0
$$997$$ 47.9157 1.51751 0.758753 0.651379i $$-0.225809\pi$$
0.758753 + 0.651379i $$0.225809\pi$$
$$998$$ 0 0
$$999$$ 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 5472.2.a.bf.1.1 2
3.2 odd 2 608.2.a.h.1.2 yes 2
4.3 odd 2 5472.2.a.bc.1.1 2
12.11 even 2 608.2.a.g.1.1 2
24.5 odd 2 1216.2.a.s.1.1 2
24.11 even 2 1216.2.a.t.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.g.1.1 2 12.11 even 2
608.2.a.h.1.2 yes 2 3.2 odd 2
1216.2.a.s.1.1 2 24.5 odd 2
1216.2.a.t.1.2 2 24.11 even 2
5472.2.a.bc.1.1 2 4.3 odd 2
5472.2.a.bf.1.1 2 1.1 even 1 trivial