Properties

 Label 392.2.a.e.1.1 Level $392$ Weight $2$ Character 392.1 Self dual yes Analytic conductor $3.130$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 392.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$3.13013575923$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 392.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} -2.00000 q^{9} +3.00000 q^{11} +6.00000 q^{13} +1.00000 q^{15} +5.00000 q^{17} -1.00000 q^{19} -7.00000 q^{23} -4.00000 q^{25} -5.00000 q^{27} +2.00000 q^{29} +5.00000 q^{31} +3.00000 q^{33} +3.00000 q^{37} +6.00000 q^{39} +2.00000 q^{41} -4.00000 q^{43} -2.00000 q^{45} -5.00000 q^{47} +5.00000 q^{51} -1.00000 q^{53} +3.00000 q^{55} -1.00000 q^{57} -15.0000 q^{59} +5.00000 q^{61} +6.00000 q^{65} -9.00000 q^{67} -7.00000 q^{69} -7.00000 q^{73} -4.00000 q^{75} +1.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +5.00000 q^{85} +2.00000 q^{87} -7.00000 q^{89} +5.00000 q^{93} -1.00000 q^{95} +2.00000 q^{97} -6.00000 q^{99} +O(q^{100})$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −7.00000 −1.45960 −0.729800 0.683660i $$-0.760387\pi$$
−0.729800 + 0.683660i $$0.760387\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 0 0
$$33$$ 3.00000 0.522233
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 0 0
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ −5.00000 −0.729325 −0.364662 0.931140i $$-0.618816\pi$$
−0.364662 + 0.931140i $$0.618816\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 5.00000 0.700140
$$52$$ 0 0
$$53$$ −1.00000 −0.137361 −0.0686803 0.997639i $$-0.521879\pi$$
−0.0686803 + 0.997639i $$0.521879\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 0 0
$$59$$ −15.0000 −1.95283 −0.976417 0.215894i $$-0.930733\pi$$
−0.976417 + 0.215894i $$0.930733\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ −9.00000 −1.09952 −0.549762 0.835321i $$-0.685282\pi$$
−0.549762 + 0.835321i $$0.685282\pi$$
$$68$$ 0 0
$$69$$ −7.00000 −0.842701
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −7.00000 −0.819288 −0.409644 0.912245i $$-0.634347\pi$$
−0.409644 + 0.912245i $$0.634347\pi$$
$$74$$ 0 0
$$75$$ −4.00000 −0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 5.00000 0.542326
$$86$$ 0 0
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −7.00000 −0.741999 −0.370999 0.928633i $$-0.620985\pi$$
−0.370999 + 0.928633i $$0.620985\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 5.00000 0.518476
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ 15.0000 1.47799 0.738997 0.673709i $$-0.235300\pi$$
0.738997 + 0.673709i $$0.235300\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 9.00000 0.870063 0.435031 0.900415i $$-0.356737\pi$$
0.435031 + 0.900415i $$0.356737\pi$$
$$108$$ 0 0
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 0 0
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 0 0
$$115$$ −7.00000 −0.652753
$$116$$ 0 0
$$117$$ −12.0000 −1.10940
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 2.00000 0.180334
$$124$$ 0 0
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −5.00000 −0.436852 −0.218426 0.975854i $$-0.570092\pi$$
−0.218426 + 0.975854i $$0.570092\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −5.00000 −0.430331
$$136$$ 0 0
$$137$$ 11.0000 0.939793 0.469897 0.882721i $$-0.344291\pi$$
0.469897 + 0.882721i $$0.344291\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ −5.00000 −0.421076
$$142$$ 0 0
$$143$$ 18.0000 1.50524
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −17.0000 −1.39269 −0.696347 0.717705i $$-0.745193\pi$$
−0.696347 + 0.717705i $$0.745193\pi$$
$$150$$ 0 0
$$151$$ −5.00000 −0.406894 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$152$$ 0 0
$$153$$ −10.0000 −0.808452
$$154$$ 0 0
$$155$$ 5.00000 0.401610
$$156$$ 0 0
$$157$$ 9.00000 0.718278 0.359139 0.933284i $$-0.383070\pi$$
0.359139 + 0.933284i $$0.383070\pi$$
$$158$$ 0 0
$$159$$ −1.00000 −0.0793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 13.0000 1.01824 0.509119 0.860696i $$-0.329971\pi$$
0.509119 + 0.860696i $$0.329971\pi$$
$$164$$ 0 0
$$165$$ 3.00000 0.233550
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 0 0
$$173$$ 13.0000 0.988372 0.494186 0.869356i $$-0.335466\pi$$
0.494186 + 0.869356i $$0.335466\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −15.0000 −1.12747
$$178$$ 0 0
$$179$$ −13.0000 −0.971666 −0.485833 0.874052i $$-0.661484\pi$$
−0.485833 + 0.874052i $$0.661484\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 5.00000 0.369611
$$184$$ 0 0
$$185$$ 3.00000 0.220564
$$186$$ 0 0
$$187$$ 15.0000 1.09691
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −11.0000 −0.795932 −0.397966 0.917400i $$-0.630284\pi$$
−0.397966 + 0.917400i $$0.630284\pi$$
$$192$$ 0 0
$$193$$ 3.00000 0.215945 0.107972 0.994154i $$-0.465564\pi$$
0.107972 + 0.994154i $$0.465564\pi$$
$$194$$ 0 0
$$195$$ 6.00000 0.429669
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 13.0000 0.921546 0.460773 0.887518i $$-0.347572\pi$$
0.460773 + 0.887518i $$0.347572\pi$$
$$200$$ 0 0
$$201$$ −9.00000 −0.634811
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2.00000 0.139686
$$206$$ 0 0
$$207$$ 14.0000 0.973067
$$208$$ 0 0
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −7.00000 −0.473016
$$220$$ 0 0
$$221$$ 30.0000 2.01802
$$222$$ 0 0
$$223$$ 24.0000 1.60716 0.803579 0.595198i $$-0.202926\pi$$
0.803579 + 0.595198i $$0.202926\pi$$
$$224$$ 0 0
$$225$$ 8.00000 0.533333
$$226$$ 0 0
$$227$$ −11.0000 −0.730096 −0.365048 0.930989i $$-0.618947\pi$$
−0.365048 + 0.930989i $$0.618947\pi$$
$$228$$ 0 0
$$229$$ −23.0000 −1.51988 −0.759941 0.649992i $$-0.774772\pi$$
−0.759941 + 0.649992i $$0.774772\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 11.0000 0.720634 0.360317 0.932830i $$-0.382669\pi$$
0.360317 + 0.932830i $$0.382669\pi$$
$$234$$ 0 0
$$235$$ −5.00000 −0.326164
$$236$$ 0 0
$$237$$ 1.00000 0.0649570
$$238$$ 0 0
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ 17.0000 1.09507 0.547533 0.836784i $$-0.315567\pi$$
0.547533 + 0.836784i $$0.315567\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.00000 −0.381771
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 0 0
$$253$$ −21.0000 −1.32026
$$254$$ 0 0
$$255$$ 5.00000 0.313112
$$256$$ 0 0
$$257$$ 21.0000 1.30994 0.654972 0.755653i $$-0.272680\pi$$
0.654972 + 0.755653i $$0.272680\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 0 0
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ 0 0
$$265$$ −1.00000 −0.0614295
$$266$$ 0 0
$$267$$ −7.00000 −0.428393
$$268$$ 0 0
$$269$$ 9.00000 0.548740 0.274370 0.961624i $$-0.411531\pi$$
0.274370 + 0.961624i $$0.411531\pi$$
$$270$$ 0 0
$$271$$ 7.00000 0.425220 0.212610 0.977137i $$-0.431804\pi$$
0.212610 + 0.977137i $$0.431804\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −12.0000 −0.723627
$$276$$ 0 0
$$277$$ −17.0000 −1.02143 −0.510716 0.859750i $$-0.670619\pi$$
−0.510716 + 0.859750i $$0.670619\pi$$
$$278$$ 0 0
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 13.0000 0.772770 0.386385 0.922338i $$-0.373724\pi$$
0.386385 + 0.922338i $$0.373724\pi$$
$$284$$ 0 0
$$285$$ −1.00000 −0.0592349
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ −15.0000 −0.873334
$$296$$ 0 0
$$297$$ −15.0000 −0.870388
$$298$$ 0 0
$$299$$ −42.0000 −2.42892
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −3.00000 −0.172345
$$304$$ 0 0
$$305$$ 5.00000 0.286299
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ 15.0000 0.853320
$$310$$ 0 0
$$311$$ −15.0000 −0.850572 −0.425286 0.905059i $$-0.639826\pi$$
−0.425286 + 0.905059i $$0.639826\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.00000 0.168497 0.0842484 0.996445i $$-0.473151\pi$$
0.0842484 + 0.996445i $$0.473151\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 9.00000 0.502331
$$322$$ 0 0
$$323$$ −5.00000 −0.278207
$$324$$ 0 0
$$325$$ −24.0000 −1.33128
$$326$$ 0 0
$$327$$ −5.00000 −0.276501
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 29.0000 1.59398 0.796992 0.603990i $$-0.206423\pi$$
0.796992 + 0.603990i $$0.206423\pi$$
$$332$$ 0 0
$$333$$ −6.00000 −0.328798
$$334$$ 0 0
$$335$$ −9.00000 −0.491723
$$336$$ 0 0
$$337$$ −10.0000 −0.544735 −0.272367 0.962193i $$-0.587807\pi$$
−0.272367 + 0.962193i $$0.587807\pi$$
$$338$$ 0 0
$$339$$ −18.0000 −0.977626
$$340$$ 0 0
$$341$$ 15.0000 0.812296
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −7.00000 −0.376867
$$346$$ 0 0
$$347$$ 15.0000 0.805242 0.402621 0.915367i $$-0.368099\pi$$
0.402621 + 0.915367i $$0.368099\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ −30.0000 −1.60128
$$352$$ 0 0
$$353$$ −3.00000 −0.159674 −0.0798369 0.996808i $$-0.525440\pi$$
−0.0798369 + 0.996808i $$0.525440\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 21.0000 1.10834 0.554169 0.832404i $$-0.313036\pi$$
0.554169 + 0.832404i $$0.313036\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ −7.00000 −0.366397
$$366$$ 0 0
$$367$$ −23.0000 −1.20059 −0.600295 0.799779i $$-0.704950\pi$$
−0.600295 + 0.799779i $$0.704950\pi$$
$$368$$ 0 0
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 11.0000 0.569558 0.284779 0.958593i $$-0.408080\pi$$
0.284779 + 0.958593i $$0.408080\pi$$
$$374$$ 0 0
$$375$$ −9.00000 −0.464758
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 24.0000 1.23280 0.616399 0.787434i $$-0.288591\pi$$
0.616399 + 0.787434i $$0.288591\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 3.00000 0.153293 0.0766464 0.997058i $$-0.475579\pi$$
0.0766464 + 0.997058i $$0.475579\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.00000 0.406663
$$388$$ 0 0
$$389$$ −29.0000 −1.47036 −0.735179 0.677873i $$-0.762902\pi$$
−0.735179 + 0.677873i $$0.762902\pi$$
$$390$$ 0 0
$$391$$ −35.0000 −1.77003
$$392$$ 0 0
$$393$$ −5.00000 −0.252217
$$394$$ 0 0
$$395$$ 1.00000 0.0503155
$$396$$ 0 0
$$397$$ 17.0000 0.853206 0.426603 0.904439i $$-0.359710\pi$$
0.426603 + 0.904439i $$0.359710\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 27.0000 1.34832 0.674158 0.738587i $$-0.264507\pi$$
0.674158 + 0.738587i $$0.264507\pi$$
$$402$$ 0 0
$$403$$ 30.0000 1.49441
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 9.00000 0.446113
$$408$$ 0 0
$$409$$ −27.0000 −1.33506 −0.667532 0.744581i $$-0.732649\pi$$
−0.667532 + 0.744581i $$0.732649\pi$$
$$410$$ 0 0
$$411$$ 11.0000 0.542590
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ 0 0
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 0 0
$$423$$ 10.0000 0.486217
$$424$$ 0 0
$$425$$ −20.0000 −0.970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 18.0000 0.869048
$$430$$ 0 0
$$431$$ 3.00000 0.144505 0.0722525 0.997386i $$-0.476981\pi$$
0.0722525 + 0.997386i $$0.476981\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 2.00000 0.0958927
$$436$$ 0 0
$$437$$ 7.00000 0.334855
$$438$$ 0 0
$$439$$ −21.0000 −1.00228 −0.501138 0.865368i $$-0.667085\pi$$
−0.501138 + 0.865368i $$0.667085\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −31.0000 −1.47285 −0.736427 0.676517i $$-0.763489\pi$$
−0.736427 + 0.676517i $$0.763489\pi$$
$$444$$ 0 0
$$445$$ −7.00000 −0.331832
$$446$$ 0 0
$$447$$ −17.0000 −0.804072
$$448$$ 0 0
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ 0 0
$$453$$ −5.00000 −0.234920
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −17.0000 −0.795226 −0.397613 0.917553i $$-0.630161\pi$$
−0.397613 + 0.917553i $$0.630161\pi$$
$$458$$ 0 0
$$459$$ −25.0000 −1.16690
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 0 0
$$465$$ 5.00000 0.231869
$$466$$ 0 0
$$467$$ 3.00000 0.138823 0.0694117 0.997588i $$-0.477888\pi$$
0.0694117 + 0.997588i $$0.477888\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 9.00000 0.414698
$$472$$ 0 0
$$473$$ −12.0000 −0.551761
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 2.00000 0.0915737
$$478$$ 0 0
$$479$$ −31.0000 −1.41643 −0.708213 0.705999i $$-0.750498\pi$$
−0.708213 + 0.705999i $$0.750498\pi$$
$$480$$ 0 0
$$481$$ 18.0000 0.820729
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2.00000 0.0908153
$$486$$ 0 0
$$487$$ 7.00000 0.317200 0.158600 0.987343i $$-0.449302\pi$$
0.158600 + 0.987343i $$0.449302\pi$$
$$488$$ 0 0
$$489$$ 13.0000 0.587880
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ 0 0
$$493$$ 10.0000 0.450377
$$494$$ 0 0
$$495$$ −6.00000 −0.269680
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −35.0000 −1.56682 −0.783408 0.621508i $$-0.786520\pi$$
−0.783408 + 0.621508i $$0.786520\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 32.0000 1.42681 0.713405 0.700752i $$-0.247152\pi$$
0.713405 + 0.700752i $$0.247152\pi$$
$$504$$ 0 0
$$505$$ −3.00000 −0.133498
$$506$$ 0 0
$$507$$ 23.0000 1.02147
$$508$$ 0 0
$$509$$ 9.00000 0.398918 0.199459 0.979906i $$-0.436082\pi$$
0.199459 + 0.979906i $$0.436082\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 5.00000 0.220755
$$514$$ 0 0
$$515$$ 15.0000 0.660979
$$516$$ 0 0
$$517$$ −15.0000 −0.659699
$$518$$ 0 0
$$519$$ 13.0000 0.570637
$$520$$ 0 0
$$521$$ 33.0000 1.44576 0.722878 0.690976i $$-0.242819\pi$$
0.722878 + 0.690976i $$0.242819\pi$$
$$522$$ 0 0
$$523$$ 7.00000 0.306089 0.153044 0.988219i $$-0.451092\pi$$
0.153044 + 0.988219i $$0.451092\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 25.0000 1.08902
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 0 0
$$531$$ 30.0000 1.30189
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ 9.00000 0.389104
$$536$$ 0 0
$$537$$ −13.0000 −0.560991
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −41.0000 −1.76273 −0.881364 0.472438i $$-0.843374\pi$$
−0.881364 + 0.472438i $$0.843374\pi$$
$$542$$ 0 0
$$543$$ 10.0000 0.429141
$$544$$ 0 0
$$545$$ −5.00000 −0.214176
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ −2.00000 −0.0852029
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 3.00000 0.127343
$$556$$ 0 0
$$557$$ −5.00000 −0.211857 −0.105928 0.994374i $$-0.533781\pi$$
−0.105928 + 0.994374i $$0.533781\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 15.0000 0.633300
$$562$$ 0 0
$$563$$ 41.0000 1.72794 0.863972 0.503540i $$-0.167969\pi$$
0.863972 + 0.503540i $$0.167969\pi$$
$$564$$ 0 0
$$565$$ −18.0000 −0.757266
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −33.0000 −1.38343 −0.691716 0.722170i $$-0.743145\pi$$
−0.691716 + 0.722170i $$0.743145\pi$$
$$570$$ 0 0
$$571$$ −13.0000 −0.544033 −0.272017 0.962293i $$-0.587691\pi$$
−0.272017 + 0.962293i $$0.587691\pi$$
$$572$$ 0 0
$$573$$ −11.0000 −0.459532
$$574$$ 0 0
$$575$$ 28.0000 1.16768
$$576$$ 0 0
$$577$$ 33.0000 1.37381 0.686904 0.726748i $$-0.258969\pi$$
0.686904 + 0.726748i $$0.258969\pi$$
$$578$$ 0 0
$$579$$ 3.00000 0.124676
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −3.00000 −0.124247
$$584$$ 0 0
$$585$$ −12.0000 −0.496139
$$586$$ 0 0
$$587$$ 20.0000 0.825488 0.412744 0.910847i $$-0.364570\pi$$
0.412744 + 0.910847i $$0.364570\pi$$
$$588$$ 0 0
$$589$$ −5.00000 −0.206021
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 45.0000 1.84793 0.923964 0.382479i $$-0.124930\pi$$
0.923964 + 0.382479i $$0.124930\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 13.0000 0.532055
$$598$$ 0 0
$$599$$ −17.0000 −0.694601 −0.347301 0.937754i $$-0.612902\pi$$
−0.347301 + 0.937754i $$0.612902\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ 18.0000 0.733017
$$604$$ 0 0
$$605$$ −2.00000 −0.0813116
$$606$$ 0 0
$$607$$ −13.0000 −0.527654 −0.263827 0.964570i $$-0.584985\pi$$
−0.263827 + 0.964570i $$0.584985\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −30.0000 −1.21367
$$612$$ 0 0
$$613$$ 43.0000 1.73675 0.868377 0.495905i $$-0.165164\pi$$
0.868377 + 0.495905i $$0.165164\pi$$
$$614$$ 0 0
$$615$$ 2.00000 0.0806478
$$616$$ 0 0
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 0 0
$$619$$ 9.00000 0.361741 0.180870 0.983507i $$-0.442109\pi$$
0.180870 + 0.983507i $$0.442109\pi$$
$$620$$ 0 0
$$621$$ 35.0000 1.40450
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ −3.00000 −0.119808
$$628$$ 0 0
$$629$$ 15.0000 0.598089
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −17.0000 −0.671460 −0.335730 0.941958i $$-0.608983\pi$$
−0.335730 + 0.941958i $$0.608983\pi$$
$$642$$ 0 0
$$643$$ 44.0000 1.73519 0.867595 0.497271i $$-0.165665\pi$$
0.867595 + 0.497271i $$0.165665\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 0 0
$$647$$ −15.0000 −0.589711 −0.294855 0.955542i $$-0.595271\pi$$
−0.294855 + 0.955542i $$0.595271\pi$$
$$648$$ 0 0
$$649$$ −45.0000 −1.76640
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 35.0000 1.36966 0.684828 0.728705i $$-0.259877\pi$$
0.684828 + 0.728705i $$0.259877\pi$$
$$654$$ 0 0
$$655$$ −5.00000 −0.195366
$$656$$ 0 0
$$657$$ 14.0000 0.546192
$$658$$ 0 0
$$659$$ 44.0000 1.71400 0.856998 0.515319i $$-0.172327\pi$$
0.856998 + 0.515319i $$0.172327\pi$$
$$660$$ 0 0
$$661$$ −23.0000 −0.894596 −0.447298 0.894385i $$-0.647614\pi$$
−0.447298 + 0.894385i $$0.647614\pi$$
$$662$$ 0 0
$$663$$ 30.0000 1.16510
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −14.0000 −0.542082
$$668$$ 0 0
$$669$$ 24.0000 0.927894
$$670$$ 0 0
$$671$$ 15.0000 0.579069
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 0 0
$$675$$ 20.0000 0.769800
$$676$$ 0 0
$$677$$ −39.0000 −1.49889 −0.749446 0.662066i $$-0.769680\pi$$
−0.749446 + 0.662066i $$0.769680\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −11.0000 −0.421521
$$682$$ 0 0
$$683$$ 27.0000 1.03313 0.516563 0.856249i $$-0.327211\pi$$
0.516563 + 0.856249i $$0.327211\pi$$
$$684$$ 0 0
$$685$$ 11.0000 0.420288
$$686$$ 0 0
$$687$$ −23.0000 −0.877505
$$688$$ 0 0
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ −5.00000 −0.190209 −0.0951045 0.995467i $$-0.530319\pi$$
−0.0951045 + 0.995467i $$0.530319\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ 10.0000 0.378777
$$698$$ 0 0
$$699$$ 11.0000 0.416058
$$700$$ 0 0
$$701$$ −50.0000 −1.88847 −0.944237 0.329267i $$-0.893198\pi$$
−0.944237 + 0.329267i $$0.893198\pi$$
$$702$$ 0 0
$$703$$ −3.00000 −0.113147
$$704$$ 0 0
$$705$$ −5.00000 −0.188311
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −29.0000 −1.08912 −0.544559 0.838723i $$-0.683303\pi$$
−0.544559 + 0.838723i $$0.683303\pi$$
$$710$$ 0 0
$$711$$ −2.00000 −0.0750059
$$712$$ 0 0
$$713$$ −35.0000 −1.31076
$$714$$ 0 0
$$715$$ 18.0000 0.673162
$$716$$ 0 0
$$717$$ 20.0000 0.746914
$$718$$ 0 0
$$719$$ 39.0000 1.45445 0.727227 0.686397i $$-0.240809\pi$$
0.727227 + 0.686397i $$0.240809\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 17.0000 0.632237
$$724$$ 0 0
$$725$$ −8.00000 −0.297113
$$726$$ 0 0
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −20.0000 −0.739727
$$732$$ 0 0
$$733$$ −19.0000 −0.701781 −0.350891 0.936416i $$-0.614121\pi$$
−0.350891 + 0.936416i $$0.614121\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −27.0000 −0.994558
$$738$$ 0 0
$$739$$ −5.00000 −0.183928 −0.0919640 0.995762i $$-0.529314\pi$$
−0.0919640 + 0.995762i $$0.529314\pi$$
$$740$$ 0 0
$$741$$ −6.00000 −0.220416
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 0 0
$$745$$ −17.0000 −0.622832
$$746$$ 0 0
$$747$$ 24.0000 0.878114
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 53.0000 1.93400 0.966999 0.254781i $$-0.0820034\pi$$
0.966999 + 0.254781i $$0.0820034\pi$$
$$752$$ 0 0
$$753$$ −16.0000 −0.583072
$$754$$ 0 0
$$755$$ −5.00000 −0.181969
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ −21.0000 −0.762252
$$760$$ 0 0
$$761$$ −3.00000 −0.108750 −0.0543750 0.998521i $$-0.517317\pi$$
−0.0543750 + 0.998521i $$0.517317\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −10.0000 −0.361551
$$766$$ 0 0
$$767$$ −90.0000 −3.24971
$$768$$ 0 0
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 21.0000 0.756297
$$772$$ 0 0
$$773$$ −19.0000 −0.683383 −0.341691 0.939812i $$-0.611000\pi$$
−0.341691 + 0.939812i $$0.611000\pi$$
$$774$$ 0 0
$$775$$ −20.0000 −0.718421
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2.00000 −0.0716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −10.0000 −0.357371
$$784$$ 0 0
$$785$$ 9.00000 0.321224
$$786$$ 0 0
$$787$$ 17.0000 0.605985 0.302992 0.952993i $$-0.402014\pi$$
0.302992 + 0.952993i $$0.402014\pi$$
$$788$$ 0 0
$$789$$ −9.00000 −0.320408
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 30.0000 1.06533
$$794$$ 0 0
$$795$$ −1.00000 −0.0354663
$$796$$ 0 0
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ −25.0000 −0.884436
$$800$$ 0 0
$$801$$ 14.0000 0.494666
$$802$$ 0 0
$$803$$ −21.0000 −0.741074
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 9.00000 0.316815
$$808$$ 0 0
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 0 0
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 0 0
$$813$$ 7.00000 0.245501
$$814$$ 0 0
$$815$$ 13.0000 0.455370
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −9.00000 −0.314102 −0.157051 0.987590i $$-0.550199\pi$$
−0.157051 + 0.987590i $$0.550199\pi$$
$$822$$ 0 0
$$823$$ 47.0000 1.63832 0.819159 0.573567i $$-0.194441\pi$$
0.819159 + 0.573567i $$0.194441\pi$$
$$824$$ 0 0
$$825$$ −12.0000 −0.417786
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 0 0
$$831$$ −17.0000 −0.589723
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 12.0000 0.415277
$$836$$ 0 0
$$837$$ −25.0000 −0.864126
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 6.00000 0.206651
$$844$$ 0 0
$$845$$ 23.0000 0.791224
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 13.0000 0.446159
$$850$$ 0 0
$$851$$ −21.0000 −0.719871
$$852$$ 0 0
$$853$$ −34.0000 −1.16414 −0.582069 0.813139i $$-0.697757\pi$$
−0.582069 + 0.813139i $$0.697757\pi$$
$$854$$ 0 0
$$855$$ 2.00000 0.0683986
$$856$$ 0 0
$$857$$ −39.0000 −1.33221 −0.666107 0.745856i $$-0.732041\pi$$
−0.666107 + 0.745856i $$0.732041\pi$$
$$858$$ 0 0
$$859$$ −1.00000 −0.0341196 −0.0170598 0.999854i $$-0.505431\pi$$
−0.0170598 + 0.999854i $$0.505431\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 17.0000 0.578687 0.289343 0.957225i $$-0.406563\pi$$
0.289343 + 0.957225i $$0.406563\pi$$
$$864$$ 0 0
$$865$$ 13.0000 0.442013
$$866$$ 0 0
$$867$$ 8.00000 0.271694
$$868$$ 0 0
$$869$$ 3.00000 0.101768
$$870$$ 0 0
$$871$$ −54.0000 −1.82972
$$872$$ 0 0
$$873$$ −4.00000 −0.135379
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 23.0000 0.776655 0.388327 0.921521i $$-0.373053\pi$$
0.388327 + 0.921521i $$0.373053\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ −15.0000 −0.504219
$$886$$ 0 0
$$887$$ 3.00000 0.100730 0.0503651 0.998731i $$-0.483962\pi$$
0.0503651 + 0.998731i $$0.483962\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ 0 0
$$893$$ 5.00000 0.167319
$$894$$ 0 0
$$895$$ −13.0000 −0.434542
$$896$$ 0 0
$$897$$ −42.0000 −1.40234
$$898$$ 0 0
$$899$$ 10.0000 0.333519
$$900$$ 0 0
$$901$$ −5.00000 −0.166574
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ −1.00000 −0.0332045 −0.0166022 0.999862i $$-0.505285\pi$$
−0.0166022 + 0.999862i $$0.505285\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ 0 0
$$915$$ 5.00000 0.165295
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −11.0000 −0.362857 −0.181428 0.983404i $$-0.558072\pi$$
−0.181428 + 0.983404i $$0.558072\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −12.0000 −0.394558
$$926$$ 0 0
$$927$$ −30.0000 −0.985329
$$928$$ 0 0
$$929$$ −7.00000 −0.229663 −0.114831 0.993385i $$-0.536633\pi$$
−0.114831 + 0.993385i $$0.536633\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −15.0000 −0.491078
$$934$$ 0 0
$$935$$ 15.0000 0.490552
$$936$$ 0 0
$$937$$ 6.00000 0.196011 0.0980057 0.995186i $$-0.468754\pi$$
0.0980057 + 0.995186i $$0.468754\pi$$
$$938$$ 0 0
$$939$$ 1.00000 0.0326338
$$940$$ 0 0
$$941$$ 33.0000 1.07577 0.537885 0.843018i $$-0.319224\pi$$
0.537885 + 0.843018i $$0.319224\pi$$
$$942$$ 0 0
$$943$$ −14.0000 −0.455903
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 37.0000 1.20234 0.601169 0.799122i $$-0.294702\pi$$
0.601169 + 0.799122i $$0.294702\pi$$
$$948$$ 0 0
$$949$$ −42.0000 −1.36338
$$950$$ 0 0
$$951$$ 3.00000 0.0972817
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 0 0
$$955$$ −11.0000 −0.355952
$$956$$ 0 0
$$957$$ 6.00000 0.193952
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ −18.0000 −0.580042
$$964$$ 0 0
$$965$$ 3.00000 0.0965734
$$966$$ 0 0
$$967$$ 48.0000 1.54358 0.771788 0.635880i $$-0.219363\pi$$
0.771788 + 0.635880i $$0.219363\pi$$
$$968$$ 0 0
$$969$$ −5.00000 −0.160623
$$970$$ 0 0
$$971$$ 35.0000 1.12320 0.561602 0.827408i $$-0.310185\pi$$
0.561602 + 0.827408i $$0.310185\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −24.0000 −0.768615
$$976$$ 0 0
$$977$$ 3.00000 0.0959785 0.0479893 0.998848i $$-0.484719\pi$$
0.0479893 + 0.998848i $$0.484719\pi$$
$$978$$ 0 0
$$979$$ −21.0000 −0.671163
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 0 0
$$983$$ 21.0000 0.669796 0.334898 0.942254i $$-0.391298\pi$$
0.334898 + 0.942254i $$0.391298\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 28.0000 0.890348
$$990$$ 0 0
$$991$$ 15.0000 0.476491 0.238245 0.971205i $$-0.423428\pi$$
0.238245 + 0.971205i $$0.423428\pi$$
$$992$$ 0 0
$$993$$ 29.0000 0.920287
$$994$$ 0 0
$$995$$ 13.0000 0.412128
$$996$$ 0 0
$$997$$ 41.0000 1.29848 0.649242 0.760582i $$-0.275086\pi$$
0.649242 + 0.760582i $$0.275086\pi$$
$$998$$ 0 0
$$999$$ −15.0000 −0.474579
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 392.2.a.e.1.1 1
3.2 odd 2 3528.2.a.j.1.1 1
4.3 odd 2 784.2.a.c.1.1 1
5.4 even 2 9800.2.a.s.1.1 1
7.2 even 3 392.2.i.b.361.1 2
7.3 odd 6 56.2.i.b.9.1 2
7.4 even 3 392.2.i.b.177.1 2
7.5 odd 6 56.2.i.b.25.1 yes 2
7.6 odd 2 392.2.a.c.1.1 1
8.3 odd 2 3136.2.a.t.1.1 1
8.5 even 2 3136.2.a.i.1.1 1
12.11 even 2 7056.2.a.u.1.1 1
21.2 odd 6 3528.2.s.q.361.1 2
21.5 even 6 504.2.s.c.361.1 2
21.11 odd 6 3528.2.s.q.3313.1 2
21.17 even 6 504.2.s.c.289.1 2
21.20 even 2 3528.2.a.p.1.1 1
28.3 even 6 112.2.i.a.65.1 2
28.11 odd 6 784.2.i.h.177.1 2
28.19 even 6 112.2.i.a.81.1 2
28.23 odd 6 784.2.i.h.753.1 2
28.27 even 2 784.2.a.h.1.1 1
35.3 even 12 1400.2.bh.a.849.1 4
35.12 even 12 1400.2.bh.a.249.1 4
35.17 even 12 1400.2.bh.a.849.2 4
35.19 odd 6 1400.2.q.d.1201.1 2
35.24 odd 6 1400.2.q.d.401.1 2
35.33 even 12 1400.2.bh.a.249.2 4
35.34 odd 2 9800.2.a.be.1.1 1
56.3 even 6 448.2.i.d.65.1 2
56.5 odd 6 448.2.i.b.193.1 2
56.13 odd 2 3136.2.a.u.1.1 1
56.19 even 6 448.2.i.d.193.1 2
56.27 even 2 3136.2.a.j.1.1 1
56.45 odd 6 448.2.i.b.65.1 2
84.47 odd 6 1008.2.s.g.865.1 2
84.59 odd 6 1008.2.s.g.289.1 2
84.83 odd 2 7056.2.a.bj.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.i.b.9.1 2 7.3 odd 6
56.2.i.b.25.1 yes 2 7.5 odd 6
112.2.i.a.65.1 2 28.3 even 6
112.2.i.a.81.1 2 28.19 even 6
392.2.a.c.1.1 1 7.6 odd 2
392.2.a.e.1.1 1 1.1 even 1 trivial
392.2.i.b.177.1 2 7.4 even 3
392.2.i.b.361.1 2 7.2 even 3
448.2.i.b.65.1 2 56.45 odd 6
448.2.i.b.193.1 2 56.5 odd 6
448.2.i.d.65.1 2 56.3 even 6
448.2.i.d.193.1 2 56.19 even 6
504.2.s.c.289.1 2 21.17 even 6
504.2.s.c.361.1 2 21.5 even 6
784.2.a.c.1.1 1 4.3 odd 2
784.2.a.h.1.1 1 28.27 even 2
784.2.i.h.177.1 2 28.11 odd 6
784.2.i.h.753.1 2 28.23 odd 6
1008.2.s.g.289.1 2 84.59 odd 6
1008.2.s.g.865.1 2 84.47 odd 6
1400.2.q.d.401.1 2 35.24 odd 6
1400.2.q.d.1201.1 2 35.19 odd 6
1400.2.bh.a.249.1 4 35.12 even 12
1400.2.bh.a.249.2 4 35.33 even 12
1400.2.bh.a.849.1 4 35.3 even 12
1400.2.bh.a.849.2 4 35.17 even 12
3136.2.a.i.1.1 1 8.5 even 2
3136.2.a.j.1.1 1 56.27 even 2
3136.2.a.t.1.1 1 8.3 odd 2
3136.2.a.u.1.1 1 56.13 odd 2
3528.2.a.j.1.1 1 3.2 odd 2
3528.2.a.p.1.1 1 21.20 even 2
3528.2.s.q.361.1 2 21.2 odd 6
3528.2.s.q.3313.1 2 21.11 odd 6
7056.2.a.u.1.1 1 12.11 even 2
7056.2.a.bj.1.1 1 84.83 odd 2
9800.2.a.s.1.1 1 5.4 even 2
9800.2.a.be.1.1 1 35.34 odd 2