# Properties

 Label 3800.2.a.ba.1.4 Level $3800$ Weight $2$ Character 3800.1 Self dual yes Analytic conductor $30.343$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.3431527681$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3$$ x^6 - 2*x^5 - 10*x^4 + 16*x^3 + 15*x^2 - 14*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-0.185519$$ of defining polynomial Character $$\chi$$ $$=$$ 3800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.185519 q^{3} +4.45651 q^{7} -2.96558 q^{9} +O(q^{10})$$ $$q+0.185519 q^{3} +4.45651 q^{7} -2.96558 q^{9} +2.64623 q^{11} -1.30142 q^{13} +3.51716 q^{17} -1.00000 q^{19} +0.826767 q^{21} +6.52882 q^{23} -1.10673 q^{27} -5.20946 q^{29} +10.8219 q^{31} +0.490925 q^{33} -2.04607 q^{37} -0.241439 q^{39} -3.80044 q^{41} -4.77089 q^{43} -1.49093 q^{47} +12.8605 q^{49} +0.652501 q^{51} -0.225583 q^{53} -0.185519 q^{57} -2.86056 q^{59} -6.31449 q^{61} -13.2161 q^{63} +13.1831 q^{67} +1.21122 q^{69} +12.3310 q^{71} +5.42276 q^{73} +11.7929 q^{77} +14.9688 q^{79} +8.69143 q^{81} -3.84470 q^{83} -0.966455 q^{87} +1.67666 q^{89} -5.79981 q^{91} +2.00768 q^{93} -9.48523 q^{97} -7.84760 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 $$6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} + 3 q^{11} + q^{13} + 14 q^{17} - 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} + 5 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} + 33 q^{51} - 13 q^{53} + 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} + 19 q^{71} - 3 q^{73} + 36 q^{77} - 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} + 42 q^{91} - 39 q^{93} + 11 q^{97} - 6 q^{99}+O(q^{100})$$ 6 * q - 2 * q^3 - 2 * q^7 + 6 * q^9 + 3 * q^11 + q^13 + 14 * q^17 - 6 * q^19 + 15 * q^21 - 12 * q^23 - 8 * q^27 + 9 * q^29 + 5 * q^31 - 2 * q^33 + 8 * q^37 + 12 * q^39 + 3 * q^41 - 15 * q^43 - 4 * q^47 + 22 * q^49 + 33 * q^51 - 13 * q^53 + 2 * q^57 + 9 * q^61 - 21 * q^63 + 3 * q^67 - 11 * q^69 + 19 * q^71 - 3 * q^73 + 36 * q^77 - 16 * q^79 + 26 * q^81 - 31 * q^83 + 25 * q^87 + 14 * q^89 + 42 * q^91 - 39 * q^93 + 11 * q^97 - 6 * q^99

## 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
$$3$$ 0.185519 0.107109 0.0535547 0.998565i $$-0.482945\pi$$
0.0535547 + 0.998565i $$0.482945\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.45651 1.68440 0.842201 0.539164i $$-0.181260\pi$$
0.842201 + 0.539164i $$0.181260\pi$$
$$8$$ 0 0
$$9$$ −2.96558 −0.988528
$$10$$ 0 0
$$11$$ 2.64623 0.797867 0.398934 0.916980i $$-0.369380\pi$$
0.398934 + 0.916980i $$0.369380\pi$$
$$12$$ 0 0
$$13$$ −1.30142 −0.360950 −0.180475 0.983580i $$-0.557764\pi$$
−0.180475 + 0.983580i $$0.557764\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.51716 0.853037 0.426519 0.904479i $$-0.359740\pi$$
0.426519 + 0.904479i $$0.359740\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0.826767 0.180415
$$22$$ 0 0
$$23$$ 6.52882 1.36135 0.680677 0.732584i $$-0.261686\pi$$
0.680677 + 0.732584i $$0.261686\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.10673 −0.212990
$$28$$ 0 0
$$29$$ −5.20946 −0.967373 −0.483686 0.875241i $$-0.660702\pi$$
−0.483686 + 0.875241i $$0.660702\pi$$
$$30$$ 0 0
$$31$$ 10.8219 1.94368 0.971839 0.235646i $$-0.0757207\pi$$
0.971839 + 0.235646i $$0.0757207\pi$$
$$32$$ 0 0
$$33$$ 0.490925 0.0854591
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.04607 −0.336373 −0.168186 0.985755i $$-0.553791\pi$$
−0.168186 + 0.985755i $$0.553791\pi$$
$$38$$ 0 0
$$39$$ −0.241439 −0.0386612
$$40$$ 0 0
$$41$$ −3.80044 −0.593529 −0.296764 0.954951i $$-0.595908\pi$$
−0.296764 + 0.954951i $$0.595908\pi$$
$$42$$ 0 0
$$43$$ −4.77089 −0.727554 −0.363777 0.931486i $$-0.618513\pi$$
−0.363777 + 0.931486i $$0.618513\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −1.49093 −0.217474 −0.108737 0.994071i $$-0.534681\pi$$
−0.108737 + 0.994071i $$0.534681\pi$$
$$48$$ 0 0
$$49$$ 12.8605 1.83721
$$50$$ 0 0
$$51$$ 0.652501 0.0913684
$$52$$ 0 0
$$53$$ −0.225583 −0.0309862 −0.0154931 0.999880i $$-0.504932\pi$$
−0.0154931 + 0.999880i $$0.504932\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.185519 −0.0245726
$$58$$ 0 0
$$59$$ −2.86056 −0.372413 −0.186206 0.982511i $$-0.559619\pi$$
−0.186206 + 0.982511i $$0.559619\pi$$
$$60$$ 0 0
$$61$$ −6.31449 −0.808488 −0.404244 0.914651i $$-0.632465\pi$$
−0.404244 + 0.914651i $$0.632465\pi$$
$$62$$ 0 0
$$63$$ −13.2161 −1.66508
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 13.1831 1.61058 0.805288 0.592884i $$-0.202011\pi$$
0.805288 + 0.592884i $$0.202011\pi$$
$$68$$ 0 0
$$69$$ 1.21122 0.145814
$$70$$ 0 0
$$71$$ 12.3310 1.46342 0.731711 0.681615i $$-0.238722\pi$$
0.731711 + 0.681615i $$0.238722\pi$$
$$72$$ 0 0
$$73$$ 5.42276 0.634686 0.317343 0.948311i $$-0.397209\pi$$
0.317343 + 0.948311i $$0.397209\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 11.7929 1.34393
$$78$$ 0 0
$$79$$ 14.9688 1.68413 0.842063 0.539379i $$-0.181341\pi$$
0.842063 + 0.539379i $$0.181341\pi$$
$$80$$ 0 0
$$81$$ 8.69143 0.965714
$$82$$ 0 0
$$83$$ −3.84470 −0.422011 −0.211005 0.977485i $$-0.567674\pi$$
−0.211005 + 0.977485i $$0.567674\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −0.966455 −0.103615
$$88$$ 0 0
$$89$$ 1.67666 0.177726 0.0888629 0.996044i $$-0.471677\pi$$
0.0888629 + 0.996044i $$0.471677\pi$$
$$90$$ 0 0
$$91$$ −5.79981 −0.607985
$$92$$ 0 0
$$93$$ 2.00768 0.208186
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −9.48523 −0.963079 −0.481539 0.876424i $$-0.659922\pi$$
−0.481539 + 0.876424i $$0.659922\pi$$
$$98$$ 0 0
$$99$$ −7.84760 −0.788714
$$100$$ 0 0
$$101$$ −6.39100 −0.635928 −0.317964 0.948103i $$-0.602999\pi$$
−0.317964 + 0.948103i $$0.602999\pi$$
$$102$$ 0 0
$$103$$ −4.14301 −0.408223 −0.204112 0.978948i $$-0.565431\pi$$
−0.204112 + 0.978948i $$0.565431\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −7.27597 −0.703394 −0.351697 0.936114i $$-0.614395\pi$$
−0.351697 + 0.936114i $$0.614395\pi$$
$$108$$ 0 0
$$109$$ 14.1077 1.35127 0.675637 0.737235i $$-0.263869\pi$$
0.675637 + 0.737235i $$0.263869\pi$$
$$110$$ 0 0
$$111$$ −0.379586 −0.0360287
$$112$$ 0 0
$$113$$ −7.03290 −0.661600 −0.330800 0.943701i $$-0.607319\pi$$
−0.330800 + 0.943701i $$0.607319\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 3.85948 0.356809
$$118$$ 0 0
$$119$$ 15.6743 1.43686
$$120$$ 0 0
$$121$$ −3.99749 −0.363408
$$122$$ 0 0
$$123$$ −0.705053 −0.0635725
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −13.6240 −1.20893 −0.604467 0.796630i $$-0.706614\pi$$
−0.604467 + 0.796630i $$0.706614\pi$$
$$128$$ 0 0
$$129$$ −0.885090 −0.0779279
$$130$$ 0 0
$$131$$ 22.3345 1.95137 0.975685 0.219177i $$-0.0703373\pi$$
0.975685 + 0.219177i $$0.0703373\pi$$
$$132$$ 0 0
$$133$$ −4.45651 −0.386428
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 5.55578 0.474662 0.237331 0.971429i $$-0.423727\pi$$
0.237331 + 0.971429i $$0.423727\pi$$
$$138$$ 0 0
$$139$$ −11.0452 −0.936841 −0.468420 0.883506i $$-0.655177\pi$$
−0.468420 + 0.883506i $$0.655177\pi$$
$$140$$ 0 0
$$141$$ −0.276595 −0.0232935
$$142$$ 0 0
$$143$$ −3.44386 −0.287990
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2.38586 0.196782
$$148$$ 0 0
$$149$$ 17.6003 1.44188 0.720938 0.693000i $$-0.243711\pi$$
0.720938 + 0.693000i $$0.243711\pi$$
$$150$$ 0 0
$$151$$ 5.24140 0.426539 0.213269 0.976993i $$-0.431589\pi$$
0.213269 + 0.976993i $$0.431589\pi$$
$$152$$ 0 0
$$153$$ −10.4304 −0.843251
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.7965 1.50012 0.750061 0.661368i $$-0.230024\pi$$
0.750061 + 0.661368i $$0.230024\pi$$
$$158$$ 0 0
$$159$$ −0.0418499 −0.00331891
$$160$$ 0 0
$$161$$ 29.0957 2.29307
$$162$$ 0 0
$$163$$ −12.1669 −0.952982 −0.476491 0.879179i $$-0.658092\pi$$
−0.476491 + 0.879179i $$0.658092\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −7.46393 −0.577576 −0.288788 0.957393i $$-0.593252\pi$$
−0.288788 + 0.957393i $$0.593252\pi$$
$$168$$ 0 0
$$169$$ −11.3063 −0.869715
$$170$$ 0 0
$$171$$ 2.96558 0.226784
$$172$$ 0 0
$$173$$ −17.8070 −1.35384 −0.676921 0.736055i $$-0.736686\pi$$
−0.676921 + 0.736055i $$0.736686\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −0.530688 −0.0398889
$$178$$ 0 0
$$179$$ −9.48154 −0.708684 −0.354342 0.935116i $$-0.615295\pi$$
−0.354342 + 0.935116i $$0.615295\pi$$
$$180$$ 0 0
$$181$$ 4.32832 0.321721 0.160861 0.986977i $$-0.448573\pi$$
0.160861 + 0.986977i $$0.448573\pi$$
$$182$$ 0 0
$$183$$ −1.17146 −0.0865967
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.30720 0.680610
$$188$$ 0 0
$$189$$ −4.93215 −0.358761
$$190$$ 0 0
$$191$$ −8.21414 −0.594355 −0.297177 0.954822i $$-0.596045\pi$$
−0.297177 + 0.954822i $$0.596045\pi$$
$$192$$ 0 0
$$193$$ −2.89738 −0.208558 −0.104279 0.994548i $$-0.533253\pi$$
−0.104279 + 0.994548i $$0.533253\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 22.4118 1.59678 0.798388 0.602143i $$-0.205686\pi$$
0.798388 + 0.602143i $$0.205686\pi$$
$$198$$ 0 0
$$199$$ 2.68542 0.190364 0.0951821 0.995460i $$-0.469657\pi$$
0.0951821 + 0.995460i $$0.469657\pi$$
$$200$$ 0 0
$$201$$ 2.44572 0.172508
$$202$$ 0 0
$$203$$ −23.2160 −1.62944
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −19.3618 −1.34573
$$208$$ 0 0
$$209$$ −2.64623 −0.183043
$$210$$ 0 0
$$211$$ 7.22869 0.497644 0.248822 0.968549i $$-0.419957\pi$$
0.248822 + 0.968549i $$0.419957\pi$$
$$212$$ 0 0
$$213$$ 2.28764 0.156746
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 48.2281 3.27393
$$218$$ 0 0
$$219$$ 1.00603 0.0679809
$$220$$ 0 0
$$221$$ −4.57732 −0.307904
$$222$$ 0 0
$$223$$ −6.86629 −0.459801 −0.229900 0.973214i $$-0.573840\pi$$
−0.229900 + 0.973214i $$0.573840\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −19.2176 −1.27552 −0.637758 0.770237i $$-0.720138\pi$$
−0.637758 + 0.770237i $$0.720138\pi$$
$$228$$ 0 0
$$229$$ 25.2205 1.66661 0.833307 0.552810i $$-0.186445\pi$$
0.833307 + 0.552810i $$0.186445\pi$$
$$230$$ 0 0
$$231$$ 2.18781 0.143947
$$232$$ 0 0
$$233$$ 24.1123 1.57965 0.789825 0.613332i $$-0.210171\pi$$
0.789825 + 0.613332i $$0.210171\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 2.77701 0.180386
$$238$$ 0 0
$$239$$ 4.34082 0.280784 0.140392 0.990096i $$-0.455164\pi$$
0.140392 + 0.990096i $$0.455164\pi$$
$$240$$ 0 0
$$241$$ 21.9630 1.41476 0.707380 0.706834i $$-0.249877\pi$$
0.707380 + 0.706834i $$0.249877\pi$$
$$242$$ 0 0
$$243$$ 4.93261 0.316427
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.30142 0.0828077
$$248$$ 0 0
$$249$$ −0.713265 −0.0452013
$$250$$ 0 0
$$251$$ 4.41214 0.278492 0.139246 0.990258i $$-0.455532\pi$$
0.139246 + 0.990258i $$0.455532\pi$$
$$252$$ 0 0
$$253$$ 17.2767 1.08618
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 11.9112 0.743000 0.371500 0.928433i $$-0.378844\pi$$
0.371500 + 0.928433i $$0.378844\pi$$
$$258$$ 0 0
$$259$$ −9.11835 −0.566587
$$260$$ 0 0
$$261$$ 15.4491 0.956275
$$262$$ 0 0
$$263$$ 13.7779 0.849581 0.424790 0.905292i $$-0.360348\pi$$
0.424790 + 0.905292i $$0.360348\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0.311053 0.0190361
$$268$$ 0 0
$$269$$ −2.30474 −0.140522 −0.0702612 0.997529i $$-0.522383\pi$$
−0.0702612 + 0.997529i $$0.522383\pi$$
$$270$$ 0 0
$$271$$ −20.1878 −1.22632 −0.613161 0.789958i $$-0.710102\pi$$
−0.613161 + 0.789958i $$0.710102\pi$$
$$272$$ 0 0
$$273$$ −1.07597 −0.0651210
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.7021 0.643024 0.321512 0.946905i $$-0.395809\pi$$
0.321512 + 0.946905i $$0.395809\pi$$
$$278$$ 0 0
$$279$$ −32.0934 −1.92138
$$280$$ 0 0
$$281$$ −2.28603 −0.136373 −0.0681865 0.997673i $$-0.521721\pi$$
−0.0681865 + 0.997673i $$0.521721\pi$$
$$282$$ 0 0
$$283$$ 25.6515 1.52482 0.762411 0.647093i $$-0.224016\pi$$
0.762411 + 0.647093i $$0.224016\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −16.9367 −0.999741
$$288$$ 0 0
$$289$$ −4.62957 −0.272328
$$290$$ 0 0
$$291$$ −1.75969 −0.103155
$$292$$ 0 0
$$293$$ −19.6027 −1.14520 −0.572602 0.819833i $$-0.694066\pi$$
−0.572602 + 0.819833i $$0.694066\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2.92866 −0.169938
$$298$$ 0 0
$$299$$ −8.49677 −0.491381
$$300$$ 0 0
$$301$$ −21.2615 −1.22549
$$302$$ 0 0
$$303$$ −1.18565 −0.0681139
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 31.6679 1.80738 0.903692 0.428183i $$-0.140846\pi$$
0.903692 + 0.428183i $$0.140846\pi$$
$$308$$ 0 0
$$309$$ −0.768608 −0.0437246
$$310$$ 0 0
$$311$$ 14.8957 0.844655 0.422328 0.906443i $$-0.361213\pi$$
0.422328 + 0.906443i $$0.361213\pi$$
$$312$$ 0 0
$$313$$ 0.288235 0.0162920 0.00814601 0.999967i $$-0.497407\pi$$
0.00814601 + 0.999967i $$0.497407\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −21.9612 −1.23347 −0.616733 0.787173i $$-0.711544\pi$$
−0.616733 + 0.787173i $$0.711544\pi$$
$$318$$ 0 0
$$319$$ −13.7854 −0.771835
$$320$$ 0 0
$$321$$ −1.34983 −0.0753402
$$322$$ 0 0
$$323$$ −3.51716 −0.195700
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.61725 0.144734
$$328$$ 0 0
$$329$$ −6.64432 −0.366313
$$330$$ 0 0
$$331$$ 24.0905 1.32413 0.662066 0.749445i $$-0.269680\pi$$
0.662066 + 0.749445i $$0.269680\pi$$
$$332$$ 0 0
$$333$$ 6.06780 0.332514
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −17.7090 −0.964673 −0.482337 0.875986i $$-0.660212\pi$$
−0.482337 + 0.875986i $$0.660212\pi$$
$$338$$ 0 0
$$339$$ −1.30474 −0.0708636
$$340$$ 0 0
$$341$$ 28.6373 1.55080
$$342$$ 0 0
$$343$$ 26.1172 1.41020
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −0.158465 −0.00850687 −0.00425344 0.999991i $$-0.501354\pi$$
−0.00425344 + 0.999991i $$0.501354\pi$$
$$348$$ 0 0
$$349$$ 27.3776 1.46549 0.732745 0.680503i $$-0.238239\pi$$
0.732745 + 0.680503i $$0.238239\pi$$
$$350$$ 0 0
$$351$$ 1.44032 0.0768788
$$352$$ 0 0
$$353$$ 18.3221 0.975185 0.487592 0.873071i $$-0.337875\pi$$
0.487592 + 0.873071i $$0.337875\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 2.90787 0.153901
$$358$$ 0 0
$$359$$ 35.6265 1.88030 0.940148 0.340767i $$-0.110687\pi$$
0.940148 + 0.340767i $$0.110687\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −0.741610 −0.0389245
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3.03379 −0.158362 −0.0791812 0.996860i $$-0.525231\pi$$
−0.0791812 + 0.996860i $$0.525231\pi$$
$$368$$ 0 0
$$369$$ 11.2705 0.586719
$$370$$ 0 0
$$371$$ −1.00531 −0.0521932
$$372$$ 0 0
$$373$$ −9.43451 −0.488500 −0.244250 0.969712i $$-0.578542\pi$$
−0.244250 + 0.969712i $$0.578542\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.77972 0.349173
$$378$$ 0 0
$$379$$ −10.1522 −0.521486 −0.260743 0.965408i $$-0.583967\pi$$
−0.260743 + 0.965408i $$0.583967\pi$$
$$380$$ 0 0
$$381$$ −2.52751 −0.129488
$$382$$ 0 0
$$383$$ −19.0699 −0.974428 −0.487214 0.873283i $$-0.661987\pi$$
−0.487214 + 0.873283i $$0.661987\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 14.1485 0.719207
$$388$$ 0 0
$$389$$ −38.6448 −1.95937 −0.979685 0.200544i $$-0.935729\pi$$
−0.979685 + 0.200544i $$0.935729\pi$$
$$390$$ 0 0
$$391$$ 22.9629 1.16128
$$392$$ 0 0
$$393$$ 4.14347 0.209010
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −10.5339 −0.528679 −0.264339 0.964430i $$-0.585154\pi$$
−0.264339 + 0.964430i $$0.585154\pi$$
$$398$$ 0 0
$$399$$ −0.826767 −0.0413901
$$400$$ 0 0
$$401$$ −10.6994 −0.534304 −0.267152 0.963654i $$-0.586083\pi$$
−0.267152 + 0.963654i $$0.586083\pi$$
$$402$$ 0 0
$$403$$ −14.0839 −0.701571
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −5.41438 −0.268381
$$408$$ 0 0
$$409$$ 39.1036 1.93355 0.966775 0.255627i $$-0.0822819\pi$$
0.966775 + 0.255627i $$0.0822819\pi$$
$$410$$ 0 0
$$411$$ 1.03070 0.0508408
$$412$$ 0 0
$$413$$ −12.7481 −0.627292
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −2.04909 −0.100345
$$418$$ 0 0
$$419$$ 5.00833 0.244673 0.122336 0.992489i $$-0.460961\pi$$
0.122336 + 0.992489i $$0.460961\pi$$
$$420$$ 0 0
$$421$$ −12.7557 −0.621674 −0.310837 0.950463i $$-0.600609\pi$$
−0.310837 + 0.950463i $$0.600609\pi$$
$$422$$ 0 0
$$423$$ 4.42146 0.214979
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −28.1406 −1.36182
$$428$$ 0 0
$$429$$ −0.638902 −0.0308465
$$430$$ 0 0
$$431$$ 13.7140 0.660580 0.330290 0.943879i $$-0.392853\pi$$
0.330290 + 0.943879i $$0.392853\pi$$
$$432$$ 0 0
$$433$$ −1.49408 −0.0718007 −0.0359003 0.999355i $$-0.511430\pi$$
−0.0359003 + 0.999355i $$0.511430\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.52882 −0.312316
$$438$$ 0 0
$$439$$ 5.78568 0.276136 0.138068 0.990423i $$-0.455911\pi$$
0.138068 + 0.990423i $$0.455911\pi$$
$$440$$ 0 0
$$441$$ −38.1388 −1.81613
$$442$$ 0 0
$$443$$ −9.78426 −0.464864 −0.232432 0.972613i $$-0.574668\pi$$
−0.232432 + 0.972613i $$0.574668\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 3.26520 0.154439
$$448$$ 0 0
$$449$$ −38.0631 −1.79631 −0.898154 0.439682i $$-0.855091\pi$$
−0.898154 + 0.439682i $$0.855091\pi$$
$$450$$ 0 0
$$451$$ −10.0568 −0.473557
$$452$$ 0 0
$$453$$ 0.972379 0.0456864
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18.3179 −0.856874 −0.428437 0.903572i $$-0.640936\pi$$
−0.428437 + 0.903572i $$0.640936\pi$$
$$458$$ 0 0
$$459$$ −3.89255 −0.181688
$$460$$ 0 0
$$461$$ −35.8641 −1.67036 −0.835179 0.549977i $$-0.814636\pi$$
−0.835179 + 0.549977i $$0.814636\pi$$
$$462$$ 0 0
$$463$$ 35.3540 1.64304 0.821519 0.570181i $$-0.193127\pi$$
0.821519 + 0.570181i $$0.193127\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −33.5118 −1.55074 −0.775371 0.631506i $$-0.782437\pi$$
−0.775371 + 0.631506i $$0.782437\pi$$
$$468$$ 0 0
$$469$$ 58.7507 2.71286
$$470$$ 0 0
$$471$$ 3.48711 0.160677
$$472$$ 0 0
$$473$$ −12.6248 −0.580491
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0.668984 0.0306307
$$478$$ 0 0
$$479$$ −21.1784 −0.967667 −0.483834 0.875160i $$-0.660756\pi$$
−0.483834 + 0.875160i $$0.660756\pi$$
$$480$$ 0 0
$$481$$ 2.66281 0.121414
$$482$$ 0 0
$$483$$ 5.39781 0.245609
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −35.2032 −1.59521 −0.797605 0.603180i $$-0.793900\pi$$
−0.797605 + 0.603180i $$0.793900\pi$$
$$488$$ 0 0
$$489$$ −2.25719 −0.102073
$$490$$ 0 0
$$491$$ −42.7908 −1.93112 −0.965561 0.260177i $$-0.916219\pi$$
−0.965561 + 0.260177i $$0.916219\pi$$
$$492$$ 0 0
$$493$$ −18.3225 −0.825205
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 54.9533 2.46499
$$498$$ 0 0
$$499$$ 18.3564 0.821747 0.410874 0.911692i $$-0.365224\pi$$
0.410874 + 0.911692i $$0.365224\pi$$
$$500$$ 0 0
$$501$$ −1.38470 −0.0618639
$$502$$ 0 0
$$503$$ −38.3650 −1.71061 −0.855305 0.518125i $$-0.826630\pi$$
−0.855305 + 0.518125i $$0.826630\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −2.09753 −0.0931547
$$508$$ 0 0
$$509$$ 1.73845 0.0770556 0.0385278 0.999258i $$-0.487733\pi$$
0.0385278 + 0.999258i $$0.487733\pi$$
$$510$$ 0 0
$$511$$ 24.1666 1.06907
$$512$$ 0 0
$$513$$ 1.10673 0.0488633
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3.94532 −0.173515
$$518$$ 0 0
$$519$$ −3.30354 −0.145009
$$520$$ 0 0
$$521$$ −39.3764 −1.72511 −0.862556 0.505961i $$-0.831138\pi$$
−0.862556 + 0.505961i $$0.831138\pi$$
$$522$$ 0 0
$$523$$ −43.7296 −1.91216 −0.956080 0.293105i $$-0.905311\pi$$
−0.956080 + 0.293105i $$0.905311\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 38.0625 1.65803
$$528$$ 0 0
$$529$$ 19.6255 0.853282
$$530$$ 0 0
$$531$$ 8.48321 0.368140
$$532$$ 0 0
$$533$$ 4.94598 0.214234
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −1.75901 −0.0759067
$$538$$ 0 0
$$539$$ 34.0317 1.46585
$$540$$ 0 0
$$541$$ −17.8936 −0.769305 −0.384652 0.923061i $$-0.625679\pi$$
−0.384652 + 0.923061i $$0.625679\pi$$
$$542$$ 0 0
$$543$$ 0.802985 0.0344594
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4.91310 0.210069 0.105034 0.994469i $$-0.466505\pi$$
0.105034 + 0.994469i $$0.466505\pi$$
$$548$$ 0 0
$$549$$ 18.7261 0.799212
$$550$$ 0 0
$$551$$ 5.20946 0.221931
$$552$$ 0 0
$$553$$ 66.7088 2.83675
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −38.6114 −1.63602 −0.818008 0.575206i $$-0.804922\pi$$
−0.818008 + 0.575206i $$0.804922\pi$$
$$558$$ 0 0
$$559$$ 6.20895 0.262611
$$560$$ 0 0
$$561$$ 1.72666 0.0728998
$$562$$ 0 0
$$563$$ −12.2697 −0.517105 −0.258553 0.965997i $$-0.583246\pi$$
−0.258553 + 0.965997i $$0.583246\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 38.7334 1.62665
$$568$$ 0 0
$$569$$ −32.0919 −1.34536 −0.672682 0.739932i $$-0.734858\pi$$
−0.672682 + 0.739932i $$0.734858\pi$$
$$570$$ 0 0
$$571$$ 6.85246 0.286767 0.143383 0.989667i $$-0.454202\pi$$
0.143383 + 0.989667i $$0.454202\pi$$
$$572$$ 0 0
$$573$$ −1.52388 −0.0636610
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 20.8983 0.870006 0.435003 0.900429i $$-0.356747\pi$$
0.435003 + 0.900429i $$0.356747\pi$$
$$578$$ 0 0
$$579$$ −0.537519 −0.0223385
$$580$$ 0 0
$$581$$ −17.1339 −0.710835
$$582$$ 0 0
$$583$$ −0.596943 −0.0247228
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −16.5692 −0.683883 −0.341942 0.939721i $$-0.611085\pi$$
−0.341942 + 0.939721i $$0.611085\pi$$
$$588$$ 0 0
$$589$$ −10.8219 −0.445910
$$590$$ 0 0
$$591$$ 4.15782 0.171030
$$592$$ 0 0
$$593$$ −6.96253 −0.285917 −0.142958 0.989729i $$-0.545662\pi$$
−0.142958 + 0.989729i $$0.545662\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0.498196 0.0203898
$$598$$ 0 0
$$599$$ −23.8154 −0.973072 −0.486536 0.873660i $$-0.661740\pi$$
−0.486536 + 0.873660i $$0.661740\pi$$
$$600$$ 0 0
$$601$$ 10.1339 0.413372 0.206686 0.978407i $$-0.433732\pi$$
0.206686 + 0.978407i $$0.433732\pi$$
$$602$$ 0 0
$$603$$ −39.0957 −1.59210
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −37.1179 −1.50657 −0.753286 0.657694i $$-0.771532\pi$$
−0.753286 + 0.657694i $$0.771532\pi$$
$$608$$ 0 0
$$609$$ −4.30701 −0.174529
$$610$$ 0 0
$$611$$ 1.94033 0.0784972
$$612$$ 0 0
$$613$$ −3.21621 −0.129901 −0.0649507 0.997888i $$-0.520689\pi$$
−0.0649507 + 0.997888i $$0.520689\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −4.80395 −0.193400 −0.0967000 0.995314i $$-0.530829\pi$$
−0.0967000 + 0.995314i $$0.530829\pi$$
$$618$$ 0 0
$$619$$ −10.3640 −0.416564 −0.208282 0.978069i $$-0.566787\pi$$
−0.208282 + 0.978069i $$0.566787\pi$$
$$620$$ 0 0
$$621$$ −7.22564 −0.289955
$$622$$ 0 0
$$623$$ 7.47205 0.299362
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −0.490925 −0.0196057
$$628$$ 0 0
$$629$$ −7.19638 −0.286938
$$630$$ 0 0
$$631$$ −0.213288 −0.00849084 −0.00424542 0.999991i $$-0.501351\pi$$
−0.00424542 + 0.999991i $$0.501351\pi$$
$$632$$ 0 0
$$633$$ 1.34106 0.0533024
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −16.7369 −0.663141
$$638$$ 0 0
$$639$$ −36.5686 −1.44663
$$640$$ 0 0
$$641$$ 4.44567 0.175593 0.0877967 0.996138i $$-0.472017\pi$$
0.0877967 + 0.996138i $$0.472017\pi$$
$$642$$ 0 0
$$643$$ 20.0788 0.791830 0.395915 0.918287i $$-0.370427\pi$$
0.395915 + 0.918287i $$0.370427\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 26.5734 1.04471 0.522353 0.852729i $$-0.325054\pi$$
0.522353 + 0.852729i $$0.325054\pi$$
$$648$$ 0 0
$$649$$ −7.56968 −0.297136
$$650$$ 0 0
$$651$$ 8.94722 0.350669
$$652$$ 0 0
$$653$$ 0.0209494 0.000819814 0 0.000409907 1.00000i $$-0.499870\pi$$
0.000409907 1.00000i $$0.499870\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −16.0816 −0.627405
$$658$$ 0 0
$$659$$ −30.9236 −1.20461 −0.602307 0.798265i $$-0.705752\pi$$
−0.602307 + 0.798265i $$0.705752\pi$$
$$660$$ 0 0
$$661$$ −26.7194 −1.03927 −0.519633 0.854390i $$-0.673931\pi$$
−0.519633 + 0.854390i $$0.673931\pi$$
$$662$$ 0 0
$$663$$ −0.849180 −0.0329794
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −34.0116 −1.31694
$$668$$ 0 0
$$669$$ −1.27383 −0.0492490
$$670$$ 0 0
$$671$$ −16.7096 −0.645066
$$672$$ 0 0
$$673$$ 17.3676 0.669472 0.334736 0.942312i $$-0.391353\pi$$
0.334736 + 0.942312i $$0.391353\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 45.7712 1.75913 0.879565 0.475779i $$-0.157834\pi$$
0.879565 + 0.475779i $$0.157834\pi$$
$$678$$ 0 0
$$679$$ −42.2710 −1.62221
$$680$$ 0 0
$$681$$ −3.56523 −0.136620
$$682$$ 0 0
$$683$$ 36.4020 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 4.67887 0.178510
$$688$$ 0 0
$$689$$ 0.293579 0.0111845
$$690$$ 0 0
$$691$$ −20.3322 −0.773472 −0.386736 0.922190i $$-0.626398\pi$$
−0.386736 + 0.922190i $$0.626398\pi$$
$$692$$ 0 0
$$693$$ −34.9729 −1.32851
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −13.3668 −0.506302
$$698$$ 0 0
$$699$$ 4.47329 0.169196
$$700$$ 0 0
$$701$$ −19.4294 −0.733840 −0.366920 0.930253i $$-0.619588\pi$$
−0.366920 + 0.930253i $$0.619588\pi$$
$$702$$ 0 0
$$703$$ 2.04607 0.0771692
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −28.4815 −1.07116
$$708$$ 0 0
$$709$$ −13.0978 −0.491899 −0.245950 0.969283i $$-0.579100\pi$$
−0.245950 + 0.969283i $$0.579100\pi$$
$$710$$ 0 0
$$711$$ −44.3913 −1.66481
$$712$$ 0 0
$$713$$ 70.6545 2.64603
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0.805305 0.0300747
$$718$$ 0 0
$$719$$ 0.214142 0.00798615 0.00399308 0.999992i $$-0.498729\pi$$
0.00399308 + 0.999992i $$0.498729\pi$$
$$720$$ 0 0
$$721$$ −18.4634 −0.687612
$$722$$ 0 0
$$723$$ 4.07455 0.151534
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −10.2004 −0.378310 −0.189155 0.981947i $$-0.560575\pi$$
−0.189155 + 0.981947i $$0.560575\pi$$
$$728$$ 0 0
$$729$$ −25.1592 −0.931822
$$730$$ 0 0
$$731$$ −16.7800 −0.620630
$$732$$ 0 0
$$733$$ 31.0210 1.14579 0.572894 0.819630i $$-0.305821\pi$$
0.572894 + 0.819630i $$0.305821\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 34.8855 1.28503
$$738$$ 0 0
$$739$$ 35.2371 1.29622 0.648108 0.761548i $$-0.275560\pi$$
0.648108 + 0.761548i $$0.275560\pi$$
$$740$$ 0 0
$$741$$ 0.241439 0.00886948
$$742$$ 0 0
$$743$$ −7.82541 −0.287086 −0.143543 0.989644i $$-0.545850\pi$$
−0.143543 + 0.989644i $$0.545850\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 11.4018 0.417169
$$748$$ 0 0
$$749$$ −32.4254 −1.18480
$$750$$ 0 0
$$751$$ −24.2499 −0.884892 −0.442446 0.896795i $$-0.645889\pi$$
−0.442446 + 0.896795i $$0.645889\pi$$
$$752$$ 0 0
$$753$$ 0.818535 0.0298291
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −36.4717 −1.32559 −0.662793 0.748802i $$-0.730629\pi$$
−0.662793 + 0.748802i $$0.730629\pi$$
$$758$$ 0 0
$$759$$ 3.20516 0.116340
$$760$$ 0 0
$$761$$ 24.6495 0.893544 0.446772 0.894648i $$-0.352573\pi$$
0.446772 + 0.894648i $$0.352573\pi$$
$$762$$ 0 0
$$763$$ 62.8711 2.27609
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3.72280 0.134422
$$768$$ 0 0
$$769$$ 14.3625 0.517927 0.258963 0.965887i $$-0.416619\pi$$
0.258963 + 0.965887i $$0.416619\pi$$
$$770$$ 0 0
$$771$$ 2.20976 0.0795824
$$772$$ 0 0
$$773$$ 24.1630 0.869081 0.434541 0.900652i $$-0.356911\pi$$
0.434541 + 0.900652i $$0.356911\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −1.69163 −0.0606868
$$778$$ 0 0
$$779$$ 3.80044 0.136165
$$780$$ 0 0
$$781$$ 32.6306 1.16762
$$782$$ 0 0
$$783$$ 5.76546 0.206041
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 29.6852 1.05816 0.529082 0.848571i $$-0.322537\pi$$
0.529082 + 0.848571i $$0.322537\pi$$
$$788$$ 0 0
$$789$$ 2.55606 0.0909981
$$790$$ 0 0
$$791$$ −31.3422 −1.11440
$$792$$ 0 0
$$793$$ 8.21783 0.291824
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −42.0735 −1.49032 −0.745160 0.666886i $$-0.767627\pi$$
−0.745160 + 0.666886i $$0.767627\pi$$
$$798$$ 0 0
$$799$$ −5.24383 −0.185513
$$800$$ 0 0
$$801$$ −4.97228 −0.175687
$$802$$ 0 0
$$803$$ 14.3498 0.506395
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −0.427573 −0.0150513
$$808$$ 0 0
$$809$$ 27.8348 0.978620 0.489310 0.872110i $$-0.337249\pi$$
0.489310 + 0.872110i $$0.337249\pi$$
$$810$$ 0 0
$$811$$ 14.9245 0.524069 0.262035 0.965059i $$-0.415607\pi$$
0.262035 + 0.965059i $$0.415607\pi$$
$$812$$ 0 0
$$813$$ −3.74522 −0.131351
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.77089 0.166912
$$818$$ 0 0
$$819$$ 17.1998 0.601010
$$820$$ 0 0
$$821$$ 0.503399 0.0175688 0.00878438 0.999961i $$-0.497204\pi$$
0.00878438 + 0.999961i $$0.497204\pi$$
$$822$$ 0 0
$$823$$ 20.7905 0.724713 0.362356 0.932040i $$-0.381972\pi$$
0.362356 + 0.932040i $$0.381972\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −37.3359 −1.29829 −0.649147 0.760663i $$-0.724874\pi$$
−0.649147 + 0.760663i $$0.724874\pi$$
$$828$$ 0 0
$$829$$ −20.5586 −0.714029 −0.357014 0.934099i $$-0.616205\pi$$
−0.357014 + 0.934099i $$0.616205\pi$$
$$830$$ 0 0
$$831$$ 1.98543 0.0688740
$$832$$ 0 0
$$833$$ 45.2323 1.56721
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −11.9770 −0.413984
$$838$$ 0 0
$$839$$ −8.13209 −0.280751 −0.140375 0.990098i $$-0.544831\pi$$
−0.140375 + 0.990098i $$0.544831\pi$$
$$840$$ 0 0
$$841$$ −1.86150 −0.0641896
$$842$$ 0 0
$$843$$ −0.424102 −0.0146068
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −17.8148 −0.612125
$$848$$ 0 0
$$849$$ 4.75884 0.163323
$$850$$ 0 0
$$851$$ −13.3585 −0.457922
$$852$$ 0 0
$$853$$ −32.7714 −1.12207 −0.561035 0.827792i $$-0.689597\pi$$
−0.561035 + 0.827792i $$0.689597\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −6.21619 −0.212341 −0.106170 0.994348i $$-0.533859\pi$$
−0.106170 + 0.994348i $$0.533859\pi$$
$$858$$ 0 0
$$859$$ 21.2194 0.723996 0.361998 0.932179i $$-0.382095\pi$$
0.361998 + 0.932179i $$0.382095\pi$$
$$860$$ 0 0
$$861$$ −3.14208 −0.107082
$$862$$ 0 0
$$863$$ −19.4411 −0.661784 −0.330892 0.943669i $$-0.607350\pi$$
−0.330892 + 0.943669i $$0.607350\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −0.858873 −0.0291689
$$868$$ 0 0
$$869$$ 39.6109 1.34371
$$870$$ 0 0
$$871$$ −17.1569 −0.581338
$$872$$ 0 0
$$873$$ 28.1292 0.952030
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −41.4953 −1.40120 −0.700599 0.713555i $$-0.747084\pi$$
−0.700599 + 0.713555i $$0.747084\pi$$
$$878$$ 0 0
$$879$$ −3.63668 −0.122662
$$880$$ 0 0
$$881$$ −15.6473 −0.527171 −0.263585 0.964636i $$-0.584905\pi$$
−0.263585 + 0.964636i $$0.584905\pi$$
$$882$$ 0 0
$$883$$ 45.4821 1.53059 0.765297 0.643677i $$-0.222592\pi$$
0.765297 + 0.643677i $$0.222592\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 12.9690 0.435457 0.217729 0.976009i $$-0.430135\pi$$
0.217729 + 0.976009i $$0.430135\pi$$
$$888$$ 0 0
$$889$$ −60.7155 −2.03633
$$890$$ 0 0
$$891$$ 22.9995 0.770512
$$892$$ 0 0
$$893$$ 1.49093 0.0498919
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −1.57631 −0.0526315
$$898$$ 0 0
$$899$$ −56.3765 −1.88026
$$900$$ 0 0
$$901$$ −0.793411 −0.0264324
$$902$$ 0 0
$$903$$ −3.94441 −0.131262
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −2.07053 −0.0687508 −0.0343754 0.999409i $$-0.510944\pi$$
−0.0343754 + 0.999409i $$0.510944\pi$$
$$908$$ 0 0
$$909$$ 18.9530 0.628633
$$910$$ 0 0
$$911$$ 25.5409 0.846207 0.423104 0.906081i $$-0.360941\pi$$
0.423104 + 0.906081i $$0.360941\pi$$
$$912$$ 0 0
$$913$$ −10.1739 −0.336708
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 99.5337 3.28689
$$918$$ 0 0
$$919$$ 25.0300 0.825664 0.412832 0.910807i $$-0.364540\pi$$
0.412832 + 0.910807i $$0.364540\pi$$
$$920$$ 0 0
$$921$$ 5.87500 0.193588
$$922$$ 0 0
$$923$$ −16.0479 −0.528223
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 12.2865 0.403540
$$928$$ 0 0
$$929$$ −16.4704 −0.540377 −0.270188 0.962807i $$-0.587086\pi$$
−0.270188 + 0.962807i $$0.587086\pi$$
$$930$$ 0 0
$$931$$ −12.8605 −0.421485
$$932$$ 0 0
$$933$$ 2.76343 0.0904706
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −4.56119 −0.149008 −0.0745039 0.997221i $$-0.523737\pi$$
−0.0745039 + 0.997221i $$0.523737\pi$$
$$938$$ 0 0
$$939$$ 0.0534732 0.00174503
$$940$$ 0 0
$$941$$ −10.3721 −0.338119 −0.169060 0.985606i $$-0.554073\pi$$
−0.169060 + 0.985606i $$0.554073\pi$$
$$942$$ 0 0
$$943$$ −24.8124 −0.808002
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1.70431 −0.0553828 −0.0276914 0.999617i $$-0.508816\pi$$
−0.0276914 + 0.999617i $$0.508816\pi$$
$$948$$ 0 0
$$949$$ −7.05731 −0.229090
$$950$$ 0 0
$$951$$ −4.07422 −0.132116
$$952$$ 0 0
$$953$$ −18.5324 −0.600322 −0.300161 0.953888i $$-0.597040\pi$$
−0.300161 + 0.953888i $$0.597040\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −2.55746 −0.0826708
$$958$$ 0 0
$$959$$ 24.7594 0.799522
$$960$$ 0 0
$$961$$ 86.1144 2.77788
$$962$$ 0 0
$$963$$ 21.5775 0.695325
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −43.7127 −1.40570 −0.702852 0.711336i $$-0.748091\pi$$
−0.702852 + 0.711336i $$0.748091\pi$$
$$968$$ 0 0
$$969$$ −0.652501 −0.0209613
$$970$$ 0 0
$$971$$ −32.5273 −1.04385 −0.521925 0.852992i $$-0.674786\pi$$
−0.521925 + 0.852992i $$0.674786\pi$$
$$972$$ 0 0
$$973$$ −49.2230 −1.57802
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −2.03542 −0.0651187 −0.0325594 0.999470i $$-0.510366\pi$$
−0.0325594 + 0.999470i $$0.510366\pi$$
$$978$$ 0 0
$$979$$ 4.43682 0.141802
$$980$$ 0 0
$$981$$ −41.8376 −1.33577
$$982$$ 0 0
$$983$$ 19.2872 0.615166 0.307583 0.951521i $$-0.400480\pi$$
0.307583 + 0.951521i $$0.400480\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −1.23265 −0.0392356
$$988$$ 0 0
$$989$$ −31.1483 −0.990457
$$990$$ 0 0
$$991$$ −0.611835 −0.0194356 −0.00971778 0.999953i $$-0.503093\pi$$
−0.00971778 + 0.999953i $$0.503093\pi$$
$$992$$ 0 0
$$993$$ 4.46924 0.141827
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −58.8380 −1.86342 −0.931709 0.363207i $$-0.881682\pi$$
−0.931709 + 0.363207i $$0.881682\pi$$
$$998$$ 0 0
$$999$$ 2.26445 0.0716441
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 3800.2.a.ba.1.4 6
4.3 odd 2 7600.2.a.cl.1.3 6
5.2 odd 4 3800.2.d.q.3649.6 12
5.3 odd 4 3800.2.d.q.3649.7 12
5.4 even 2 3800.2.a.bc.1.3 yes 6
20.19 odd 2 7600.2.a.ch.1.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.4 6 1.1 even 1 trivial
3800.2.a.bc.1.3 yes 6 5.4 even 2
3800.2.d.q.3649.6 12 5.2 odd 4
3800.2.d.q.3649.7 12 5.3 odd 4
7600.2.a.ch.1.4 6 20.19 odd 2
7600.2.a.cl.1.3 6 4.3 odd 2