Properties

 Label 2736.2.a.bf.1.1 Level $2736$ Weight $2$ Character 2736.1 Self dual yes Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13068.1 Defining polynomial: $$x^{4} - x^{3} - 6x^{2} - x + 1$$ x^4 - x^3 - 6*x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 171) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$3.04374$$ of defining polynomial Character $$\chi$$ $$=$$ 2736.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-3.22060 q^{5} +2.37228 q^{7} +O(q^{10})$$ $$q-3.22060 q^{5} +2.37228 q^{7} +2.20979 q^{11} +2.00000 q^{13} -3.22060 q^{17} -1.00000 q^{19} -1.01082 q^{23} +5.37228 q^{25} +1.01082 q^{29} -4.74456 q^{31} -7.64018 q^{35} +10.7446 q^{37} -5.43039 q^{41} +11.1168 q^{43} +4.23142 q^{47} -1.37228 q^{49} +9.84996 q^{53} -7.11684 q^{55} -10.8608 q^{59} -5.11684 q^{61} -6.44121 q^{65} +4.00000 q^{67} -2.02163 q^{71} -5.11684 q^{73} +5.24224 q^{77} +4.00000 q^{79} +11.8716 q^{83} +10.3723 q^{85} +9.84996 q^{89} +4.74456 q^{91} +3.22060 q^{95} +7.48913 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{7}+O(q^{10})$$ 4 * q - 2 * q^7 $$4 q - 2 q^{7} + 8 q^{13} - 4 q^{19} + 10 q^{25} + 4 q^{31} + 20 q^{37} + 10 q^{43} + 6 q^{49} + 6 q^{55} + 14 q^{61} + 16 q^{67} + 14 q^{73} + 16 q^{79} + 30 q^{85} - 4 q^{91} - 16 q^{97}+O(q^{100})$$ 4 * q - 2 * q^7 + 8 * q^13 - 4 * q^19 + 10 * q^25 + 4 * q^31 + 20 * q^37 + 10 * q^43 + 6 * q^49 + 6 * q^55 + 14 * q^61 + 16 * q^67 + 14 * q^73 + 16 * q^79 + 30 * q^85 - 4 * q^91 - 16 * q^97

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
$$5$$ −3.22060 −1.44030 −0.720149 0.693820i $$-0.755926\pi$$
−0.720149 + 0.693820i $$0.755926\pi$$
$$6$$ 0 0
$$7$$ 2.37228 0.896638 0.448319 0.893874i $$-0.352023\pi$$
0.448319 + 0.893874i $$0.352023\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.20979 0.666276 0.333138 0.942878i $$-0.391893\pi$$
0.333138 + 0.942878i $$0.391893\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.22060 −0.781111 −0.390555 0.920579i $$-0.627717\pi$$
−0.390555 + 0.920579i $$0.627717\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.01082 −0.210770 −0.105385 0.994432i $$-0.533607\pi$$
−0.105385 + 0.994432i $$0.533607\pi$$
$$24$$ 0 0
$$25$$ 5.37228 1.07446
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.01082 0.187704 0.0938519 0.995586i $$-0.470082\pi$$
0.0938519 + 0.995586i $$0.470082\pi$$
$$30$$ 0 0
$$31$$ −4.74456 −0.852149 −0.426074 0.904688i $$-0.640104\pi$$
−0.426074 + 0.904688i $$0.640104\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −7.64018 −1.29143
$$36$$ 0 0
$$37$$ 10.7446 1.76640 0.883198 0.469001i $$-0.155386\pi$$
0.883198 + 0.469001i $$0.155386\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.43039 −0.848084 −0.424042 0.905642i $$-0.639389\pi$$
−0.424042 + 0.905642i $$0.639389\pi$$
$$42$$ 0 0
$$43$$ 11.1168 1.69530 0.847651 0.530554i $$-0.178016\pi$$
0.847651 + 0.530554i $$0.178016\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.23142 0.617216 0.308608 0.951189i $$-0.400137\pi$$
0.308608 + 0.951189i $$0.400137\pi$$
$$48$$ 0 0
$$49$$ −1.37228 −0.196040
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 9.84996 1.35300 0.676498 0.736444i $$-0.263497\pi$$
0.676498 + 0.736444i $$0.263497\pi$$
$$54$$ 0 0
$$55$$ −7.11684 −0.959635
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −10.8608 −1.41395 −0.706976 0.707237i $$-0.749941\pi$$
−0.706976 + 0.707237i $$0.749941\pi$$
$$60$$ 0 0
$$61$$ −5.11684 −0.655145 −0.327572 0.944826i $$-0.606231\pi$$
−0.327572 + 0.944826i $$0.606231\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −6.44121 −0.798933
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −2.02163 −0.239924 −0.119962 0.992779i $$-0.538277\pi$$
−0.119962 + 0.992779i $$0.538277\pi$$
$$72$$ 0 0
$$73$$ −5.11684 −0.598881 −0.299441 0.954115i $$-0.596800\pi$$
−0.299441 + 0.954115i $$0.596800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 5.24224 0.597408
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 11.8716 1.30308 0.651538 0.758616i $$-0.274124\pi$$
0.651538 + 0.758616i $$0.274124\pi$$
$$84$$ 0 0
$$85$$ 10.3723 1.12503
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 9.84996 1.04409 0.522047 0.852917i $$-0.325169\pi$$
0.522047 + 0.852917i $$0.325169\pi$$
$$90$$ 0 0
$$91$$ 4.74456 0.497365
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 3.22060 0.330427
$$96$$ 0 0
$$97$$ 7.48913 0.760405 0.380203 0.924903i $$-0.375854\pi$$
0.380203 + 0.924903i $$0.375854\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.8824 1.28185 0.640924 0.767604i $$-0.278551\pi$$
0.640924 + 0.767604i $$0.278551\pi$$
$$102$$ 0 0
$$103$$ 7.25544 0.714899 0.357450 0.933932i $$-0.383646\pi$$
0.357450 + 0.933932i $$0.383646\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −4.41957 −0.427256 −0.213628 0.976915i $$-0.568528\pi$$
−0.213628 + 0.976915i $$0.568528\pi$$
$$108$$ 0 0
$$109$$ 5.25544 0.503380 0.251690 0.967808i $$-0.419014\pi$$
0.251690 + 0.967808i $$0.419014\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −11.8716 −1.11679 −0.558393 0.829577i $$-0.688582\pi$$
−0.558393 + 0.829577i $$0.688582\pi$$
$$114$$ 0 0
$$115$$ 3.25544 0.303571
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −7.64018 −0.700374
$$120$$ 0 0
$$121$$ −6.11684 −0.556077
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.19897 −0.107239
$$126$$ 0 0
$$127$$ 12.7446 1.13090 0.565449 0.824784i $$-0.308703\pi$$
0.565449 + 0.824784i $$0.308703\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 15.0922 1.31861 0.659306 0.751875i $$-0.270850\pi$$
0.659306 + 0.751875i $$0.270850\pi$$
$$132$$ 0 0
$$133$$ −2.37228 −0.205703
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0597 1.03033 0.515167 0.857090i $$-0.327730\pi$$
0.515167 + 0.857090i $$0.327730\pi$$
$$138$$ 0 0
$$139$$ −3.11684 −0.264367 −0.132184 0.991225i $$-0.542199\pi$$
−0.132184 + 0.991225i $$0.542199\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.41957 0.369583
$$144$$ 0 0
$$145$$ −3.25544 −0.270349
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 20.5226 1.68128 0.840638 0.541598i $$-0.182180\pi$$
0.840638 + 0.541598i $$0.182180\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 15.2804 1.22735
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −2.39794 −0.188984
$$162$$ 0 0
$$163$$ 21.4891 1.68316 0.841579 0.540134i $$-0.181626\pi$$
0.841579 + 0.540134i $$0.181626\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.44121 0.498435 0.249218 0.968447i $$-0.419827\pi$$
0.249218 + 0.968447i $$0.419827\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −1.01082 −0.0768509 −0.0384255 0.999261i $$-0.512234\pi$$
−0.0384255 + 0.999261i $$0.512234\pi$$
$$174$$ 0 0
$$175$$ 12.7446 0.963398
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4.41957 0.330334 0.165167 0.986266i $$-0.447184\pi$$
0.165167 + 0.986266i $$0.447184\pi$$
$$180$$ 0 0
$$181$$ 19.4891 1.44862 0.724308 0.689477i $$-0.242160\pi$$
0.724308 + 0.689477i $$0.242160\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −34.6040 −2.54413
$$186$$ 0 0
$$187$$ −7.11684 −0.520435
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 21.5334 1.55810 0.779051 0.626960i $$-0.215701\pi$$
0.779051 + 0.626960i $$0.215701\pi$$
$$192$$ 0 0
$$193$$ 16.2337 1.16853 0.584263 0.811564i $$-0.301384\pi$$
0.584263 + 0.811564i $$0.301384\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −23.7432 −1.69163 −0.845816 0.533475i $$-0.820886\pi$$
−0.845816 + 0.533475i $$0.820886\pi$$
$$198$$ 0 0
$$199$$ −0.883156 −0.0626053 −0.0313026 0.999510i $$-0.509966\pi$$
−0.0313026 + 0.999510i $$0.509966\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 2.39794 0.168302
$$204$$ 0 0
$$205$$ 17.4891 1.22149
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2.20979 −0.152854
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −35.8029 −2.44174
$$216$$ 0 0
$$217$$ −11.2554 −0.764069
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.44121 −0.433282
$$222$$ 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
$$226$$ 0 0
$$227$$ −19.3236 −1.28255 −0.641277 0.767310i $$-0.721595\pi$$
−0.641277 + 0.767310i $$0.721595\pi$$
$$228$$ 0 0
$$229$$ 15.6277 1.03271 0.516354 0.856375i $$-0.327289\pi$$
0.516354 + 0.856375i $$0.327289\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −7.64018 −0.500525 −0.250262 0.968178i $$-0.580517\pi$$
−0.250262 + 0.968178i $$0.580517\pi$$
$$234$$ 0 0
$$235$$ −13.6277 −0.888974
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.6943 0.821123 0.410562 0.911833i $$-0.365333\pi$$
0.410562 + 0.911833i $$0.365333\pi$$
$$240$$ 0 0
$$241$$ −24.2337 −1.56103 −0.780515 0.625138i $$-0.785043\pi$$
−0.780515 + 0.625138i $$0.785043\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 4.41957 0.282356
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −23.9313 −1.51053 −0.755266 0.655418i $$-0.772493\pi$$
−0.755266 + 0.655418i $$0.772493\pi$$
$$252$$ 0 0
$$253$$ −2.23369 −0.140431
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 5.43039 0.338738 0.169369 0.985553i $$-0.445827\pi$$
0.169369 + 0.985553i $$0.445827\pi$$
$$258$$ 0 0
$$259$$ 25.4891 1.58382
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −13.0706 −0.805966 −0.402983 0.915208i $$-0.632027\pi$$
−0.402983 + 0.915208i $$0.632027\pi$$
$$264$$ 0 0
$$265$$ −31.7228 −1.94872
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 7.82833 0.477302 0.238651 0.971105i $$-0.423295\pi$$
0.238651 + 0.971105i $$0.423295\pi$$
$$270$$ 0 0
$$271$$ −25.4891 −1.54835 −0.774177 0.632969i $$-0.781836\pi$$
−0.774177 + 0.632969i $$0.781836\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 11.8716 0.715884
$$276$$ 0 0
$$277$$ 9.11684 0.547778 0.273889 0.961761i $$-0.411690\pi$$
0.273889 + 0.961761i $$0.411690\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 24.7540 1.47670 0.738350 0.674418i $$-0.235605\pi$$
0.738350 + 0.674418i $$0.235605\pi$$
$$282$$ 0 0
$$283$$ −6.37228 −0.378793 −0.189396 0.981901i $$-0.560653\pi$$
−0.189396 + 0.981901i $$0.560653\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.8824 −0.760425
$$288$$ 0 0
$$289$$ −6.62772 −0.389866
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −25.1303 −1.46813 −0.734064 0.679080i $$-0.762379\pi$$
−0.734064 + 0.679080i $$0.762379\pi$$
$$294$$ 0 0
$$295$$ 34.9783 2.03651
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2.02163 −0.116914
$$300$$ 0 0
$$301$$ 26.3723 1.52007
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 16.4793 0.943603
$$306$$ 0 0
$$307$$ −25.4891 −1.45474 −0.727371 0.686245i $$-0.759258\pi$$
−0.727371 + 0.686245i $$0.759258\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −12.6943 −0.719825 −0.359913 0.932986i $$-0.617194\pi$$
−0.359913 + 0.932986i $$0.617194\pi$$
$$312$$ 0 0
$$313$$ 7.48913 0.423310 0.211655 0.977344i $$-0.432115\pi$$
0.211655 + 0.977344i $$0.432115\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −12.2479 −0.687911 −0.343955 0.938986i $$-0.611767\pi$$
−0.343955 + 0.938986i $$0.611767\pi$$
$$318$$ 0 0
$$319$$ 2.23369 0.125063
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.22060 0.179199
$$324$$ 0 0
$$325$$ 10.7446 0.596001
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 10.0381 0.553419
$$330$$ 0 0
$$331$$ −18.9783 −1.04314 −0.521569 0.853209i $$-0.674653\pi$$
−0.521569 + 0.853209i $$0.674653\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −12.8824 −0.703841
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −10.4845 −0.567766
$$342$$ 0 0
$$343$$ −19.8614 −1.07242
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −32.3942 −1.73901 −0.869505 0.493924i $$-0.835562\pi$$
−0.869505 + 0.493924i $$0.835562\pi$$
$$348$$ 0 0
$$349$$ 17.8614 0.956099 0.478050 0.878333i $$-0.341344\pi$$
0.478050 + 0.878333i $$0.341344\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 14.9040 0.793262 0.396631 0.917978i $$-0.370179\pi$$
0.396631 + 0.917978i $$0.370179\pi$$
$$354$$ 0 0
$$355$$ 6.51087 0.345561
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4.60773 0.243187 0.121593 0.992580i $$-0.461200\pi$$
0.121593 + 0.992580i $$0.461200\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 16.4793 0.862567
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 23.3669 1.21315
$$372$$ 0 0
$$373$$ −24.2337 −1.25477 −0.627386 0.778708i $$-0.715875\pi$$
−0.627386 + 0.778708i $$0.715875\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 2.02163 0.104119
$$378$$ 0 0
$$379$$ −28.7446 −1.47651 −0.738255 0.674522i $$-0.764350\pi$$
−0.738255 + 0.674522i $$0.764350\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 17.3020 0.884090 0.442045 0.896993i $$-0.354253\pi$$
0.442045 + 0.896993i $$0.354253\pi$$
$$384$$ 0 0
$$385$$ −16.8832 −0.860445
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 12.0597 0.611454 0.305727 0.952119i $$-0.401101\pi$$
0.305727 + 0.952119i $$0.401101\pi$$
$$390$$ 0 0
$$391$$ 3.25544 0.164635
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −12.8824 −0.648184
$$396$$ 0 0
$$397$$ −17.1168 −0.859070 −0.429535 0.903050i $$-0.641322\pi$$
−0.429535 + 0.903050i $$0.641322\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.40876 0.170225 0.0851126 0.996371i $$-0.472875\pi$$
0.0851126 + 0.996371i $$0.472875\pi$$
$$402$$ 0 0
$$403$$ −9.48913 −0.472687
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 23.7432 1.17691
$$408$$ 0 0
$$409$$ 7.48913 0.370313 0.185157 0.982709i $$-0.440721\pi$$
0.185157 + 0.982709i $$0.440721\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −25.7648 −1.26780
$$414$$ 0 0
$$415$$ −38.2337 −1.87682
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −11.8716 −0.579965 −0.289983 0.957032i $$-0.593650\pi$$
−0.289983 + 0.957032i $$0.593650\pi$$
$$420$$ 0 0
$$421$$ −8.97825 −0.437573 −0.218787 0.975773i $$-0.570210\pi$$
−0.218787 + 0.975773i $$0.570210\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −17.3020 −0.839269
$$426$$ 0 0
$$427$$ −12.1386 −0.587428
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.1844 1.45393 0.726966 0.686674i $$-0.240930\pi$$
0.726966 + 0.686674i $$0.240930\pi$$
$$432$$ 0 0
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.01082 0.0483539
$$438$$ 0 0
$$439$$ 18.2337 0.870246 0.435123 0.900371i $$-0.356705\pi$$
0.435123 + 0.900371i $$0.356705\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 27.9746 1.32911 0.664557 0.747238i $$-0.268620\pi$$
0.664557 + 0.747238i $$0.268620\pi$$
$$444$$ 0 0
$$445$$ −31.7228 −1.50381
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 22.3561 1.05505 0.527524 0.849540i $$-0.323120\pi$$
0.527524 + 0.849540i $$0.323120\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −15.2804 −0.716354
$$456$$ 0 0
$$457$$ −8.37228 −0.391639 −0.195819 0.980640i $$-0.562737\pi$$
−0.195819 + 0.980640i $$0.562737\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −26.9638 −1.25583 −0.627914 0.778282i $$-0.716091\pi$$
−0.627914 + 0.778282i $$0.716091\pi$$
$$462$$ 0 0
$$463$$ 2.37228 0.110249 0.0551246 0.998479i $$-0.482444\pi$$
0.0551246 + 0.998479i $$0.482444\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 36.4374 1.68612 0.843062 0.537816i $$-0.180751\pi$$
0.843062 + 0.537816i $$0.180751\pi$$
$$468$$ 0 0
$$469$$ 9.48913 0.438167
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 24.5659 1.12954
$$474$$ 0 0
$$475$$ −5.37228 −0.246497
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −33.5932 −1.53491 −0.767455 0.641103i $$-0.778477\pi$$
−0.767455 + 0.641103i $$0.778477\pi$$
$$480$$ 0 0
$$481$$ 21.4891 0.979820
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −24.1195 −1.09521
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1.01082 0.0456175 0.0228087 0.999740i $$-0.492739\pi$$
0.0228087 + 0.999740i $$0.492739\pi$$
$$492$$ 0 0
$$493$$ −3.25544 −0.146618
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −4.79588 −0.215125
$$498$$ 0 0
$$499$$ 11.1168 0.497658 0.248829 0.968547i $$-0.419954\pi$$
0.248829 + 0.968547i $$0.419954\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 31.5715 1.40770 0.703852 0.710346i $$-0.251462\pi$$
0.703852 + 0.710346i $$0.251462\pi$$
$$504$$ 0 0
$$505$$ −41.4891 −1.84624
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 24.7540 1.09720 0.548601 0.836084i $$-0.315161\pi$$
0.548601 + 0.836084i $$0.315161\pi$$
$$510$$ 0 0
$$511$$ −12.1386 −0.536980
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −23.3669 −1.02967
$$516$$ 0 0
$$517$$ 9.35053 0.411236
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 16.2912 0.713729 0.356864 0.934156i $$-0.383846\pi$$
0.356864 + 0.934156i $$0.383846\pi$$
$$522$$ 0 0
$$523$$ 18.2337 0.797304 0.398652 0.917102i $$-0.369478\pi$$
0.398652 + 0.917102i $$0.369478\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 15.2804 0.665623
$$528$$ 0 0
$$529$$ −21.9783 −0.955576
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −10.8608 −0.470433
$$534$$ 0 0
$$535$$ 14.2337 0.615376
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −3.03245 −0.130617
$$540$$ 0 0
$$541$$ 21.1168 0.907884 0.453942 0.891031i $$-0.350017\pi$$
0.453942 + 0.891031i $$0.350017\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −16.9257 −0.725016
$$546$$ 0 0
$$547$$ −22.2337 −0.950644 −0.475322 0.879812i $$-0.657668\pi$$
−0.475322 + 0.879812i $$0.657668\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.01082 −0.0430622
$$552$$ 0 0
$$553$$ 9.48913 0.403519
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −20.5226 −0.869570 −0.434785 0.900534i $$-0.643176\pi$$
−0.434785 + 0.900534i $$0.643176\pi$$
$$558$$ 0 0
$$559$$ 22.2337 0.940385
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 38.6472 1.62879 0.814393 0.580313i $$-0.197070\pi$$
0.814393 + 0.580313i $$0.197070\pi$$
$$564$$ 0 0
$$565$$ 38.2337 1.60850
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 3.03245 0.127127 0.0635634 0.997978i $$-0.479753\pi$$
0.0635634 + 0.997978i $$0.479753\pi$$
$$570$$ 0 0
$$571$$ −2.51087 −0.105077 −0.0525384 0.998619i $$-0.516731\pi$$
−0.0525384 + 0.998619i $$0.516731\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −5.43039 −0.226463
$$576$$ 0 0
$$577$$ −41.1168 −1.71172 −0.855858 0.517210i $$-0.826970\pi$$
−0.855858 + 0.517210i $$0.826970\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 28.1628 1.16839
$$582$$ 0 0
$$583$$ 21.7663 0.901469
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1.83348 0.0756758 0.0378379 0.999284i $$-0.487953\pi$$
0.0378379 + 0.999284i $$0.487953\pi$$
$$588$$ 0 0
$$589$$ 4.74456 0.195496
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 4.04326 0.166037 0.0830185 0.996548i $$-0.473544\pi$$
0.0830185 + 0.996548i $$0.473544\pi$$
$$594$$ 0 0
$$595$$ 24.6060 1.00875
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −12.8824 −0.526361 −0.263181 0.964747i $$-0.584771\pi$$
−0.263181 + 0.964747i $$0.584771\pi$$
$$600$$ 0 0
$$601$$ −25.2554 −1.03019 −0.515095 0.857133i $$-0.672244\pi$$
−0.515095 + 0.857133i $$0.672244\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 19.6999 0.800916
$$606$$ 0 0
$$607$$ −28.7446 −1.16671 −0.583353 0.812219i $$-0.698260\pi$$
−0.583353 + 0.812219i $$0.698260\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.46284 0.342370
$$612$$ 0 0
$$613$$ 2.60597 0.105254 0.0526271 0.998614i $$-0.483241\pi$$
0.0526271 + 0.998614i $$0.483241\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 3.22060 0.129657 0.0648283 0.997896i $$-0.479350\pi$$
0.0648283 + 0.997896i $$0.479350\pi$$
$$618$$ 0 0
$$619$$ −6.97825 −0.280480 −0.140240 0.990118i $$-0.544787\pi$$
−0.140240 + 0.990118i $$0.544787\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 23.3669 0.936174
$$624$$ 0 0
$$625$$ −23.0000 −0.920000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −34.6040 −1.37975
$$630$$ 0 0
$$631$$ −15.1168 −0.601792 −0.300896 0.953657i $$-0.597286\pi$$
−0.300896 + 0.953657i $$0.597286\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −41.0452 −1.62883
$$636$$ 0 0
$$637$$ −2.74456 −0.108744
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −24.7540 −0.977724 −0.488862 0.872361i $$-0.662588\pi$$
−0.488862 + 0.872361i $$0.662588\pi$$
$$642$$ 0 0
$$643$$ 35.1168 1.38487 0.692437 0.721479i $$-0.256537\pi$$
0.692437 + 0.721479i $$0.256537\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17.1138 0.672814 0.336407 0.941717i $$-0.390788\pi$$
0.336407 + 0.941717i $$0.390788\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 14.0814 0.551047 0.275524 0.961294i $$-0.411149\pi$$
0.275524 + 0.961294i $$0.411149\pi$$
$$654$$ 0 0
$$655$$ −48.6060 −1.89919
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −11.2371 −0.437735 −0.218867 0.975755i $$-0.570236\pi$$
−0.218867 + 0.975755i $$0.570236\pi$$
$$660$$ 0 0
$$661$$ −6.74456 −0.262333 −0.131167 0.991360i $$-0.541872\pi$$
−0.131167 + 0.991360i $$0.541872\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 7.64018 0.296273
$$666$$ 0 0
$$667$$ −1.02175 −0.0395623
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −11.3071 −0.436507
$$672$$ 0 0
$$673$$ −25.2554 −0.973526 −0.486763 0.873534i $$-0.661822\pi$$
−0.486763 + 0.873534i $$0.661822\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 44.4539 1.70850 0.854252 0.519860i $$-0.174016\pi$$
0.854252 + 0.519860i $$0.174016\pi$$
$$678$$ 0 0
$$679$$ 17.7663 0.681808
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −34.2277 −1.30968 −0.654842 0.755765i $$-0.727265\pi$$
−0.654842 + 0.755765i $$0.727265\pi$$
$$684$$ 0 0
$$685$$ −38.8397 −1.48399
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 19.6999 0.750507
$$690$$ 0 0
$$691$$ −27.1168 −1.03157 −0.515787 0.856717i $$-0.672500\pi$$
−0.515787 + 0.856717i $$0.672500\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 10.0381 0.380767
$$696$$ 0 0
$$697$$ 17.4891 0.662448
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.5607 −1.15426 −0.577131 0.816652i $$-0.695828\pi$$
−0.577131 + 0.816652i $$0.695828\pi$$
$$702$$ 0 0
$$703$$ −10.7446 −0.405239
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 30.5607 1.14935
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 4.79588 0.179607
$$714$$ 0 0
$$715$$ −14.2337 −0.532310
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 38.8354 1.44832 0.724158 0.689634i $$-0.242229\pi$$
0.724158 + 0.689634i $$0.242229\pi$$
$$720$$ 0 0
$$721$$ 17.2119 0.641006
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 5.43039 0.201680
$$726$$ 0 0
$$727$$ 37.3505 1.38525 0.692627 0.721296i $$-0.256453\pi$$
0.692627 + 0.721296i $$0.256453\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −35.8029 −1.32422
$$732$$ 0 0
$$733$$ 18.4674 0.682108 0.341054 0.940044i $$-0.389216\pi$$
0.341054 + 0.940044i $$0.389216\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8.83915 0.325594
$$738$$ 0 0
$$739$$ −27.1168 −0.997509 −0.498755 0.866743i $$-0.666209\pi$$
−0.498755 + 0.866743i $$0.666209\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −28.5391 −1.04700 −0.523498 0.852027i $$-0.675373\pi$$
−0.523498 + 0.852027i $$0.675373\pi$$
$$744$$ 0 0
$$745$$ −66.0951 −2.42154
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −10.4845 −0.383094
$$750$$ 0 0
$$751$$ −24.4674 −0.892827 −0.446414 0.894827i $$-0.647299\pi$$
−0.446414 + 0.894827i $$0.647299\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −12.8824 −0.468839
$$756$$ 0 0
$$757$$ −5.11684 −0.185975 −0.0929874 0.995667i $$-0.529642\pi$$
−0.0929874 + 0.995667i $$0.529642\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0.822662 0.0298215 0.0149107 0.999889i $$-0.495254\pi$$
0.0149107 + 0.999889i $$0.495254\pi$$
$$762$$ 0 0
$$763$$ 12.4674 0.451349
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −21.7216 −0.784320
$$768$$ 0 0
$$769$$ 6.88316 0.248213 0.124106 0.992269i $$-0.460394\pi$$
0.124106 + 0.992269i $$0.460394\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 5.43039 0.195318 0.0976588 0.995220i $$-0.468865\pi$$
0.0976588 + 0.995220i $$0.468865\pi$$
$$774$$ 0 0
$$775$$ −25.4891 −0.915596
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 5.43039 0.194564
$$780$$ 0 0
$$781$$ −4.46738 −0.159855
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −6.44121 −0.229896
$$786$$ 0 0
$$787$$ 49.9565 1.78076 0.890378 0.455221i $$-0.150440\pi$$
0.890378 + 0.455221i $$0.150440\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −28.1628 −1.00135
$$792$$ 0 0
$$793$$ −10.2337 −0.363409
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −7.45202 −0.263964 −0.131982 0.991252i $$-0.542134\pi$$
−0.131982 + 0.991252i $$0.542134\pi$$
$$798$$ 0 0
$$799$$ −13.6277 −0.482114
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −11.3071 −0.399020
$$804$$ 0 0
$$805$$ 7.72281 0.272193
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −3.59691 −0.126461 −0.0632303 0.997999i $$-0.520140\pi$$
−0.0632303 + 0.997999i $$0.520140\pi$$
$$810$$ 0 0
$$811$$ 36.7446 1.29028 0.645138 0.764066i $$-0.276800\pi$$
0.645138 + 0.764066i $$0.276800\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −69.2079 −2.42425
$$816$$ 0 0
$$817$$ −11.1168 −0.388929
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −29.3617 −1.02473 −0.512366 0.858767i $$-0.671231\pi$$
−0.512366 + 0.858767i $$0.671231\pi$$
$$822$$ 0 0
$$823$$ −8.60597 −0.299985 −0.149993 0.988687i $$-0.547925\pi$$
−0.149993 + 0.988687i $$0.547925\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.5391 0.992401 0.496200 0.868208i $$-0.334728\pi$$
0.496200 + 0.868208i $$0.334728\pi$$
$$828$$ 0 0
$$829$$ −1.25544 −0.0436031 −0.0218016 0.999762i $$-0.506940\pi$$
−0.0218016 + 0.999762i $$0.506940\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 4.41957 0.153129
$$834$$ 0 0
$$835$$ −20.7446 −0.717895
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −30.1844 −1.04208 −0.521041 0.853532i $$-0.674456\pi$$
−0.521041 + 0.853532i $$0.674456\pi$$
$$840$$ 0 0
$$841$$ −27.9783 −0.964767
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 28.9854 0.997129
$$846$$ 0 0
$$847$$ −14.5109 −0.498600
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −10.8608 −0.372303
$$852$$ 0 0
$$853$$ −38.4674 −1.31710 −0.658549 0.752538i $$-0.728829\pi$$
−0.658549 + 0.752538i $$0.728829\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −35.2385 −1.20372 −0.601862 0.798600i $$-0.705574\pi$$
−0.601862 + 0.798600i $$0.705574\pi$$
$$858$$ 0 0
$$859$$ −3.11684 −0.106345 −0.0531727 0.998585i $$-0.516933\pi$$
−0.0531727 + 0.998585i $$0.516933\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 14.9040 0.507340 0.253670 0.967291i $$-0.418362\pi$$
0.253670 + 0.967291i $$0.418362\pi$$
$$864$$ 0 0
$$865$$ 3.25544 0.110688
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 8.83915 0.299847
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.84429 −0.0961547
$$876$$ 0 0
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −0.822662 −0.0277162 −0.0138581 0.999904i $$-0.504411\pi$$
−0.0138581 + 0.999904i $$0.504411\pi$$
$$882$$ 0 0
$$883$$ −3.11684 −0.104890 −0.0524451 0.998624i $$-0.516701\pi$$
−0.0524451 + 0.998624i $$0.516701\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −21.7216 −0.729338 −0.364669 0.931137i $$-0.618818\pi$$
−0.364669 + 0.931137i $$0.618818\pi$$
$$888$$ 0 0
$$889$$ 30.2337 1.01401
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −4.23142 −0.141599
$$894$$ 0 0
$$895$$ −14.2337 −0.475780
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −4.79588 −0.159952
$$900$$ 0 0
$$901$$ −31.7228 −1.05684
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −62.7667 −2.08644
$$906$$ 0 0
$$907$$ −23.2554 −0.772184 −0.386092 0.922460i $$-0.626175\pi$$
−0.386092 + 0.922460i $$0.626175\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 37.0019 1.22593 0.612964 0.790111i $$-0.289977\pi$$
0.612964 + 0.790111i $$0.289977\pi$$
$$912$$ 0 0
$$913$$ 26.2337 0.868208
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 35.8029 1.18232
$$918$$ 0 0
$$919$$ −18.9783 −0.626035 −0.313017 0.949747i $$-0.601340\pi$$
−0.313017 + 0.949747i $$0.601340\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −4.04326 −0.133086
$$924$$ 0 0
$$925$$ 57.7228 1.89791
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 4.79588 0.157348 0.0786739 0.996900i $$-0.474931\pi$$
0.0786739 + 0.996900i $$0.474931\pi$$
$$930$$ 0 0
$$931$$ 1.37228 0.0449747
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 22.9205 0.749581
$$936$$ 0 0
$$937$$ −5.11684 −0.167160 −0.0835800 0.996501i $$-0.526635\pi$$
−0.0835800 + 0.996501i $$0.526635\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 24.7540 0.806957 0.403479 0.914989i $$-0.367801\pi$$
0.403479 + 0.914989i $$0.367801\pi$$
$$942$$ 0 0
$$943$$ 5.48913 0.178751
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 9.84996 0.320081 0.160040 0.987110i $$-0.448838\pi$$
0.160040 + 0.987110i $$0.448838\pi$$
$$948$$ 0 0
$$949$$ −10.2337 −0.332200
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 9.47365 0.306882 0.153441 0.988158i $$-0.450965\pi$$
0.153441 + 0.988158i $$0.450965\pi$$
$$954$$ 0 0
$$955$$ −69.3505 −2.24413
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 28.6091 0.923837
$$960$$ 0 0
$$961$$ −8.48913 −0.273843
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −52.2823 −1.68303
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 51.5296 1.65366 0.826832 0.562448i $$-0.190140\pi$$
0.826832 + 0.562448i $$0.190140\pi$$
$$972$$ 0 0
$$973$$ −7.39403 −0.237042
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −20.7107 −0.662595 −0.331298 0.943526i $$-0.607486\pi$$
−0.331298 + 0.943526i $$0.607486\pi$$
$$978$$ 0 0
$$979$$ 21.7663 0.695654
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −52.2823 −1.66755 −0.833773 0.552108i $$-0.813824\pi$$
−0.833773 + 0.552108i $$0.813824\pi$$
$$984$$ 0 0
$$985$$ 76.4674 2.43645
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −11.2371 −0.357319
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 2.84429 0.0901702
$$996$$ 0 0
$$997$$ −19.3505 −0.612837 −0.306419 0.951897i $$-0.599131\pi$$
−0.306419 + 0.951897i $$0.599131\pi$$
$$998$$ 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 2736.2.a.bf.1.1 4
3.2 odd 2 inner 2736.2.a.bf.1.4 4
4.3 odd 2 171.2.a.e.1.4 yes 4
12.11 even 2 171.2.a.e.1.1 4
20.19 odd 2 4275.2.a.bp.1.1 4
28.27 even 2 8379.2.a.bw.1.4 4
60.59 even 2 4275.2.a.bp.1.4 4
76.75 even 2 3249.2.a.bf.1.1 4
84.83 odd 2 8379.2.a.bw.1.1 4
228.227 odd 2 3249.2.a.bf.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.1 4 12.11 even 2
171.2.a.e.1.4 yes 4 4.3 odd 2
2736.2.a.bf.1.1 4 1.1 even 1 trivial
2736.2.a.bf.1.4 4 3.2 odd 2 inner
3249.2.a.bf.1.1 4 76.75 even 2
3249.2.a.bf.1.4 4 228.227 odd 2
4275.2.a.bp.1.1 4 20.19 odd 2
4275.2.a.bp.1.4 4 60.59 even 2
8379.2.a.bw.1.1 4 84.83 odd 2
8379.2.a.bw.1.4 4 28.27 even 2