# Properties

 Label 378.5.b.a.323.3 Level $378$ Weight $5$ Character 378.323 Analytic conductor $39.074$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 378.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.0738460457$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.5443747577856.29 Defining polynomial: $$x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324$$ x^8 + 24*x^6 + 180*x^4 + 488*x^2 + 324 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3^{4}\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 323.3 Root $$-3.56118i$$ of defining polynomial Character $$\chi$$ $$=$$ 378.323 Dual form 378.5.b.a.323.6

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.82843i q^{2} -8.00000 q^{4} -3.12468i q^{5} +18.5203 q^{7} +22.6274i q^{8} +O(q^{10})$$ $$q-2.82843i q^{2} -8.00000 q^{4} -3.12468i q^{5} +18.5203 q^{7} +22.6274i q^{8} -8.83793 q^{10} +141.266i q^{11} -237.623 q^{13} -52.3832i q^{14} +64.0000 q^{16} -394.261i q^{17} +535.891 q^{19} +24.9974i q^{20} +399.561 q^{22} -487.858i q^{23} +615.236 q^{25} +672.101i q^{26} -148.162 q^{28} +1201.77i q^{29} +296.662 q^{31} -181.019i q^{32} -1115.14 q^{34} -57.8699i q^{35} -380.977 q^{37} -1515.73i q^{38} +70.7034 q^{40} +73.6638i q^{41} +2912.26 q^{43} -1130.13i q^{44} -1379.87 q^{46} -2845.13i q^{47} +343.000 q^{49} -1740.15i q^{50} +1900.99 q^{52} -2125.45i q^{53} +441.411 q^{55} +419.066i q^{56} +3399.11 q^{58} -5486.16i q^{59} -5032.38 q^{61} -839.086i q^{62} -512.000 q^{64} +742.497i q^{65} +7342.60 q^{67} +3154.09i q^{68} -163.681 q^{70} -8600.64i q^{71} +2165.28 q^{73} +1077.57i q^{74} -4287.13 q^{76} +2616.28i q^{77} -5796.36 q^{79} -199.979i q^{80} +208.353 q^{82} -2410.24i q^{83} -1231.94 q^{85} -8237.10i q^{86} -3196.49 q^{88} -13791.5i q^{89} -4400.85 q^{91} +3902.87i q^{92} -8047.25 q^{94} -1674.49i q^{95} +10613.0 q^{97} -970.151i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 64 q^{4}+O(q^{10})$$ 8 * q - 64 * q^4 $$8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100})$$ 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.82843i − 0.707107i
$$3$$ 0 0
$$4$$ −8.00000 −0.500000
$$5$$ − 3.12468i − 0.124987i −0.998045 0.0624936i $$-0.980095\pi$$
0.998045 0.0624936i $$-0.0199053\pi$$
$$6$$ 0 0
$$7$$ 18.5203 0.377964
$$8$$ 22.6274i 0.353553i
$$9$$ 0 0
$$10$$ −8.83793 −0.0883793
$$11$$ 141.266i 1.16749i 0.811938 + 0.583744i $$0.198413\pi$$
−0.811938 + 0.583744i $$0.801587\pi$$
$$12$$ 0 0
$$13$$ −237.623 −1.40606 −0.703028 0.711162i $$-0.748169\pi$$
−0.703028 + 0.711162i $$0.748169\pi$$
$$14$$ − 52.3832i − 0.267261i
$$15$$ 0 0
$$16$$ 64.0000 0.250000
$$17$$ − 394.261i − 1.36423i −0.731247 0.682113i $$-0.761061\pi$$
0.731247 0.682113i $$-0.238939\pi$$
$$18$$ 0 0
$$19$$ 535.891 1.48446 0.742232 0.670143i $$-0.233767\pi$$
0.742232 + 0.670143i $$0.233767\pi$$
$$20$$ 24.9974i 0.0624936i
$$21$$ 0 0
$$22$$ 399.561 0.825539
$$23$$ − 487.858i − 0.922227i −0.887341 0.461114i $$-0.847450\pi$$
0.887341 0.461114i $$-0.152550\pi$$
$$24$$ 0 0
$$25$$ 615.236 0.984378
$$26$$ 672.101i 0.994232i
$$27$$ 0 0
$$28$$ −148.162 −0.188982
$$29$$ 1201.77i 1.42897i 0.699649 + 0.714487i $$0.253340\pi$$
−0.699649 + 0.714487i $$0.746660\pi$$
$$30$$ 0 0
$$31$$ 296.662 0.308701 0.154350 0.988016i $$-0.450672\pi$$
0.154350 + 0.988016i $$0.450672\pi$$
$$32$$ − 181.019i − 0.176777i
$$33$$ 0 0
$$34$$ −1115.14 −0.964653
$$35$$ − 57.8699i − 0.0472407i
$$36$$ 0 0
$$37$$ −380.977 −0.278289 −0.139144 0.990272i $$-0.544435\pi$$
−0.139144 + 0.990272i $$0.544435\pi$$
$$38$$ − 1515.73i − 1.04967i
$$39$$ 0 0
$$40$$ 70.7034 0.0441896
$$41$$ 73.6638i 0.0438214i 0.999760 + 0.0219107i $$0.00697495\pi$$
−0.999760 + 0.0219107i $$0.993025\pi$$
$$42$$ 0 0
$$43$$ 2912.26 1.57504 0.787522 0.616287i $$-0.211364\pi$$
0.787522 + 0.616287i $$0.211364\pi$$
$$44$$ − 1130.13i − 0.583744i
$$45$$ 0 0
$$46$$ −1379.87 −0.652113
$$47$$ − 2845.13i − 1.28797i −0.765037 0.643986i $$-0.777279\pi$$
0.765037 0.643986i $$-0.222721\pi$$
$$48$$ 0 0
$$49$$ 343.000 0.142857
$$50$$ − 1740.15i − 0.696061i
$$51$$ 0 0
$$52$$ 1900.99 0.703028
$$53$$ − 2125.45i − 0.756657i −0.925671 0.378328i $$-0.876499\pi$$
0.925671 0.378328i $$-0.123501\pi$$
$$54$$ 0 0
$$55$$ 441.411 0.145921
$$56$$ 419.066i 0.133631i
$$57$$ 0 0
$$58$$ 3399.11 1.01044
$$59$$ − 5486.16i − 1.57603i −0.615656 0.788015i $$-0.711109\pi$$
0.615656 0.788015i $$-0.288891\pi$$
$$60$$ 0 0
$$61$$ −5032.38 −1.35243 −0.676214 0.736705i $$-0.736381\pi$$
−0.676214 + 0.736705i $$0.736381\pi$$
$$62$$ − 839.086i − 0.218284i
$$63$$ 0 0
$$64$$ −512.000 −0.125000
$$65$$ 742.497i 0.175739i
$$66$$ 0 0
$$67$$ 7342.60 1.63569 0.817844 0.575440i $$-0.195169\pi$$
0.817844 + 0.575440i $$0.195169\pi$$
$$68$$ 3154.09i 0.682113i
$$69$$ 0 0
$$70$$ −163.681 −0.0334042
$$71$$ − 8600.64i − 1.70614i −0.521799 0.853068i $$-0.674739\pi$$
0.521799 0.853068i $$-0.325261\pi$$
$$72$$ 0 0
$$73$$ 2165.28 0.406320 0.203160 0.979146i $$-0.434879\pi$$
0.203160 + 0.979146i $$0.434879\pi$$
$$74$$ 1077.57i 0.196780i
$$75$$ 0 0
$$76$$ −4287.13 −0.742232
$$77$$ 2616.28i 0.441269i
$$78$$ 0 0
$$79$$ −5796.36 −0.928756 −0.464378 0.885637i $$-0.653722\pi$$
−0.464378 + 0.885637i $$0.653722\pi$$
$$80$$ − 199.979i − 0.0312468i
$$81$$ 0 0
$$82$$ 208.353 0.0309864
$$83$$ − 2410.24i − 0.349868i −0.984580 0.174934i $$-0.944029\pi$$
0.984580 0.174934i $$-0.0559713\pi$$
$$84$$ 0 0
$$85$$ −1231.94 −0.170511
$$86$$ − 8237.10i − 1.11372i
$$87$$ 0 0
$$88$$ −3196.49 −0.412769
$$89$$ − 13791.5i − 1.74113i −0.492055 0.870564i $$-0.663754\pi$$
0.492055 0.870564i $$-0.336246\pi$$
$$90$$ 0 0
$$91$$ −4400.85 −0.531439
$$92$$ 3902.87i 0.461114i
$$93$$ 0 0
$$94$$ −8047.25 −0.910734
$$95$$ − 1674.49i − 0.185539i
$$96$$ 0 0
$$97$$ 10613.0 1.12797 0.563984 0.825786i $$-0.309268\pi$$
0.563984 + 0.825786i $$0.309268\pi$$
$$98$$ − 970.151i − 0.101015i
$$99$$ 0 0
$$100$$ −4921.89 −0.492189
$$101$$ 4424.36i 0.433718i 0.976203 + 0.216859i $$0.0695812\pi$$
−0.976203 + 0.216859i $$0.930419\pi$$
$$102$$ 0 0
$$103$$ 11498.7 1.08387 0.541933 0.840422i $$-0.317693\pi$$
0.541933 + 0.840422i $$0.317693\pi$$
$$104$$ − 5376.81i − 0.497116i
$$105$$ 0 0
$$106$$ −6011.68 −0.535037
$$107$$ 13254.7i 1.15771i 0.815429 + 0.578857i $$0.196501\pi$$
−0.815429 + 0.578857i $$0.803499\pi$$
$$108$$ 0 0
$$109$$ 7021.44 0.590981 0.295490 0.955346i $$-0.404517\pi$$
0.295490 + 0.955346i $$0.404517\pi$$
$$110$$ − 1248.50i − 0.103182i
$$111$$ 0 0
$$112$$ 1185.30 0.0944911
$$113$$ − 16163.8i − 1.26586i −0.774208 0.632932i $$-0.781851\pi$$
0.774208 0.632932i $$-0.218149\pi$$
$$114$$ 0 0
$$115$$ −1524.40 −0.115267
$$116$$ − 9614.14i − 0.714487i
$$117$$ 0 0
$$118$$ −15517.2 −1.11442
$$119$$ − 7301.82i − 0.515629i
$$120$$ 0 0
$$121$$ −5315.10 −0.363028
$$122$$ 14233.7i 0.956311i
$$123$$ 0 0
$$124$$ −2373.29 −0.154350
$$125$$ − 3875.34i − 0.248022i
$$126$$ 0 0
$$127$$ −29047.0 −1.80092 −0.900458 0.434942i $$-0.856769\pi$$
−0.900458 + 0.434942i $$0.856769\pi$$
$$128$$ 1448.15i 0.0883883i
$$129$$ 0 0
$$130$$ 2100.10 0.124266
$$131$$ 33352.2i 1.94349i 0.236032 + 0.971745i $$0.424153\pi$$
−0.236032 + 0.971745i $$0.575847\pi$$
$$132$$ 0 0
$$133$$ 9924.85 0.561075
$$134$$ − 20768.0i − 1.15661i
$$135$$ 0 0
$$136$$ 8921.11 0.482326
$$137$$ − 3627.12i − 0.193251i −0.995321 0.0966254i $$-0.969195\pi$$
0.995321 0.0966254i $$-0.0308049\pi$$
$$138$$ 0 0
$$139$$ 9107.22 0.471364 0.235682 0.971830i $$-0.424268\pi$$
0.235682 + 0.971830i $$0.424268\pi$$
$$140$$ 462.959i 0.0236204i
$$141$$ 0 0
$$142$$ −24326.3 −1.20642
$$143$$ − 33568.1i − 1.64155i
$$144$$ 0 0
$$145$$ 3755.14 0.178603
$$146$$ − 6124.33i − 0.287312i
$$147$$ 0 0
$$148$$ 3047.82 0.139144
$$149$$ − 9668.97i − 0.435520i −0.976002 0.217760i $$-0.930125\pi$$
0.976002 0.217760i $$-0.0698749\pi$$
$$150$$ 0 0
$$151$$ −21411.0 −0.939039 −0.469520 0.882922i $$-0.655573\pi$$
−0.469520 + 0.882922i $$0.655573\pi$$
$$152$$ 12125.8i 0.524837i
$$153$$ 0 0
$$154$$ 7399.97 0.312024
$$155$$ − 926.972i − 0.0385836i
$$156$$ 0 0
$$157$$ 26796.2 1.08711 0.543555 0.839374i $$-0.317078\pi$$
0.543555 + 0.839374i $$0.317078\pi$$
$$158$$ 16394.6i 0.656729i
$$159$$ 0 0
$$160$$ −565.627 −0.0220948
$$161$$ − 9035.26i − 0.348569i
$$162$$ 0 0
$$163$$ 32250.2 1.21383 0.606914 0.794768i $$-0.292407\pi$$
0.606914 + 0.794768i $$0.292407\pi$$
$$164$$ − 589.310i − 0.0219107i
$$165$$ 0 0
$$166$$ −6817.20 −0.247394
$$167$$ 6114.20i 0.219233i 0.993974 + 0.109617i $$0.0349623\pi$$
−0.993974 + 0.109617i $$0.965038\pi$$
$$168$$ 0 0
$$169$$ 27903.9 0.976994
$$170$$ 3484.45i 0.120569i
$$171$$ 0 0
$$172$$ −23298.0 −0.787522
$$173$$ 29551.1i 0.987372i 0.869640 + 0.493686i $$0.164351\pi$$
−0.869640 + 0.493686i $$0.835649\pi$$
$$174$$ 0 0
$$175$$ 11394.3 0.372060
$$176$$ 9041.03i 0.291872i
$$177$$ 0 0
$$178$$ −39008.2 −1.23116
$$179$$ − 3311.38i − 0.103348i −0.998664 0.0516741i $$-0.983544\pi$$
0.998664 0.0516741i $$-0.0164557\pi$$
$$180$$ 0 0
$$181$$ −38821.0 −1.18497 −0.592487 0.805580i $$-0.701854\pi$$
−0.592487 + 0.805580i $$0.701854\pi$$
$$182$$ 12447.5i 0.375784i
$$183$$ 0 0
$$184$$ 11039.0 0.326057
$$185$$ 1190.43i 0.0347825i
$$186$$ 0 0
$$187$$ 55695.7 1.59272
$$188$$ 22761.1i 0.643986i
$$189$$ 0 0
$$190$$ −4736.17 −0.131196
$$191$$ 40870.6i 1.12033i 0.828382 + 0.560163i $$0.189262\pi$$
−0.828382 + 0.560163i $$0.810738\pi$$
$$192$$ 0 0
$$193$$ 38211.7 1.02585 0.512923 0.858435i $$-0.328563\pi$$
0.512923 + 0.858435i $$0.328563\pi$$
$$194$$ − 30018.2i − 0.797593i
$$195$$ 0 0
$$196$$ −2744.00 −0.0714286
$$197$$ 21225.2i 0.546915i 0.961884 + 0.273457i $$0.0881673\pi$$
−0.961884 + 0.273457i $$0.911833\pi$$
$$198$$ 0 0
$$199$$ 44448.3 1.12240 0.561202 0.827679i $$-0.310339\pi$$
0.561202 + 0.827679i $$0.310339\pi$$
$$200$$ 13921.2i 0.348030i
$$201$$ 0 0
$$202$$ 12514.0 0.306685
$$203$$ 22257.0i 0.540102i
$$204$$ 0 0
$$205$$ 230.176 0.00547712
$$206$$ − 32523.3i − 0.766409i
$$207$$ 0 0
$$208$$ −15207.9 −0.351514
$$209$$ 75703.3i 1.73309i
$$210$$ 0 0
$$211$$ 902.367 0.0202683 0.0101342 0.999949i $$-0.496774\pi$$
0.0101342 + 0.999949i $$0.496774\pi$$
$$212$$ 17003.6i 0.378328i
$$213$$ 0 0
$$214$$ 37489.9 0.818628
$$215$$ − 9099.87i − 0.196860i
$$216$$ 0 0
$$217$$ 5494.25 0.116678
$$218$$ − 19859.6i − 0.417886i
$$219$$ 0 0
$$220$$ −3531.29 −0.0729605
$$221$$ 93685.7i 1.91818i
$$222$$ 0 0
$$223$$ 2439.54 0.0490566 0.0245283 0.999699i $$-0.492192\pi$$
0.0245283 + 0.999699i $$0.492192\pi$$
$$224$$ − 3352.53i − 0.0668153i
$$225$$ 0 0
$$226$$ −45718.2 −0.895101
$$227$$ 3408.89i 0.0661548i 0.999453 + 0.0330774i $$0.0105308\pi$$
−0.999453 + 0.0330774i $$0.989469\pi$$
$$228$$ 0 0
$$229$$ −10788.1 −0.205719 −0.102860 0.994696i $$-0.532799\pi$$
−0.102860 + 0.994696i $$0.532799\pi$$
$$230$$ 4311.66i 0.0815058i
$$231$$ 0 0
$$232$$ −27192.9 −0.505219
$$233$$ − 63204.8i − 1.16423i −0.813107 0.582114i $$-0.802226\pi$$
0.813107 0.582114i $$-0.197774\pi$$
$$234$$ 0 0
$$235$$ −8890.12 −0.160980
$$236$$ 43889.3i 0.788015i
$$237$$ 0 0
$$238$$ −20652.7 −0.364604
$$239$$ 42989.7i 0.752607i 0.926496 + 0.376304i $$0.122805\pi$$
−0.926496 + 0.376304i $$0.877195\pi$$
$$240$$ 0 0
$$241$$ 37709.8 0.649262 0.324631 0.945841i $$-0.394760\pi$$
0.324631 + 0.945841i $$0.394760\pi$$
$$242$$ 15033.4i 0.256700i
$$243$$ 0 0
$$244$$ 40259.1 0.676214
$$245$$ − 1071.76i − 0.0178553i
$$246$$ 0 0
$$247$$ −127340. −2.08724
$$248$$ 6712.68i 0.109142i
$$249$$ 0 0
$$250$$ −10961.1 −0.175378
$$251$$ − 113426.i − 1.80038i −0.435497 0.900190i $$-0.643427\pi$$
0.435497 0.900190i $$-0.356573\pi$$
$$252$$ 0 0
$$253$$ 68917.8 1.07669
$$254$$ 82157.3i 1.27344i
$$255$$ 0 0
$$256$$ 4096.00 0.0625000
$$257$$ 8447.17i 0.127892i 0.997953 + 0.0639462i $$0.0203686\pi$$
−0.997953 + 0.0639462i $$0.979631\pi$$
$$258$$ 0 0
$$259$$ −7055.80 −0.105183
$$260$$ − 5939.98i − 0.0878695i
$$261$$ 0 0
$$262$$ 94334.4 1.37426
$$263$$ 115681.i 1.67244i 0.548391 + 0.836222i $$0.315240\pi$$
−0.548391 + 0.836222i $$0.684760\pi$$
$$264$$ 0 0
$$265$$ −6641.35 −0.0945724
$$266$$ − 28071.7i − 0.396740i
$$267$$ 0 0
$$268$$ −58740.8 −0.817844
$$269$$ 34096.2i 0.471195i 0.971851 + 0.235598i $$0.0757048\pi$$
−0.971851 + 0.235598i $$0.924295\pi$$
$$270$$ 0 0
$$271$$ 68752.9 0.936165 0.468083 0.883685i $$-0.344945\pi$$
0.468083 + 0.883685i $$0.344945\pi$$
$$272$$ − 25232.7i − 0.341056i
$$273$$ 0 0
$$274$$ −10259.1 −0.136649
$$275$$ 86912.0i 1.14925i
$$276$$ 0 0
$$277$$ 115701. 1.50792 0.753958 0.656923i $$-0.228142\pi$$
0.753958 + 0.656923i $$0.228142\pi$$
$$278$$ − 25759.1i − 0.333304i
$$279$$ 0 0
$$280$$ 1309.45 0.0167021
$$281$$ − 9710.36i − 0.122977i −0.998108 0.0614883i $$-0.980415\pi$$
0.998108 0.0614883i $$-0.0195847\pi$$
$$282$$ 0 0
$$283$$ 16971.5 0.211907 0.105954 0.994371i $$-0.466210\pi$$
0.105954 + 0.994371i $$0.466210\pi$$
$$284$$ 68805.1i 0.853068i
$$285$$ 0 0
$$286$$ −94945.0 −1.16075
$$287$$ 1364.27i 0.0165629i
$$288$$ 0 0
$$289$$ −71920.8 −0.861110
$$290$$ − 10621.1i − 0.126292i
$$291$$ 0 0
$$292$$ −17322.2 −0.203160
$$293$$ − 46472.1i − 0.541324i −0.962674 0.270662i $$-0.912757\pi$$
0.962674 0.270662i $$-0.0872425\pi$$
$$294$$ 0 0
$$295$$ −17142.5 −0.196983
$$296$$ − 8620.53i − 0.0983899i
$$297$$ 0 0
$$298$$ −27348.0 −0.307959
$$299$$ 115927.i 1.29670i
$$300$$ 0 0
$$301$$ 53935.7 0.595311
$$302$$ 60559.5i 0.664001i
$$303$$ 0 0
$$304$$ 34297.0 0.371116
$$305$$ 15724.6i 0.169036i
$$306$$ 0 0
$$307$$ −104419. −1.10790 −0.553951 0.832549i $$-0.686880\pi$$
−0.553951 + 0.832549i $$0.686880\pi$$
$$308$$ − 20930.3i − 0.220635i
$$309$$ 0 0
$$310$$ −2621.87 −0.0272828
$$311$$ − 114779.i − 1.18671i −0.804943 0.593353i $$-0.797804\pi$$
0.804943 0.593353i $$-0.202196\pi$$
$$312$$ 0 0
$$313$$ −7552.66 −0.0770924 −0.0385462 0.999257i $$-0.512273\pi$$
−0.0385462 + 0.999257i $$0.512273\pi$$
$$314$$ − 75791.0i − 0.768702i
$$315$$ 0 0
$$316$$ 46370.9 0.464378
$$317$$ − 95323.4i − 0.948595i −0.880365 0.474298i $$-0.842702\pi$$
0.880365 0.474298i $$-0.157298\pi$$
$$318$$ 0 0
$$319$$ −169769. −1.66831
$$320$$ 1599.84i 0.0156234i
$$321$$ 0 0
$$322$$ −25555.6 −0.246476
$$323$$ − 211281.i − 2.02514i
$$324$$ 0 0
$$325$$ −146195. −1.38409
$$326$$ − 91217.3i − 0.858306i
$$327$$ 0 0
$$328$$ −1666.82 −0.0154932
$$329$$ − 52692.6i − 0.486808i
$$330$$ 0 0
$$331$$ −66158.3 −0.603849 −0.301924 0.953332i $$-0.597629\pi$$
−0.301924 + 0.953332i $$0.597629\pi$$
$$332$$ 19281.9i 0.174934i
$$333$$ 0 0
$$334$$ 17293.6 0.155021
$$335$$ − 22943.3i − 0.204440i
$$336$$ 0 0
$$337$$ −140142. −1.23398 −0.616990 0.786971i $$-0.711648\pi$$
−0.616990 + 0.786971i $$0.711648\pi$$
$$338$$ − 78924.2i − 0.690839i
$$339$$ 0 0
$$340$$ 9855.51 0.0852553
$$341$$ 41908.2i 0.360405i
$$342$$ 0 0
$$343$$ 6352.45 0.0539949
$$344$$ 65896.8i 0.556862i
$$345$$ 0 0
$$346$$ 83583.0 0.698178
$$347$$ − 28633.6i − 0.237803i −0.992906 0.118901i $$-0.962063\pi$$
0.992906 0.118901i $$-0.0379372\pi$$
$$348$$ 0 0
$$349$$ 82009.8 0.673310 0.336655 0.941628i $$-0.390704\pi$$
0.336655 + 0.941628i $$0.390704\pi$$
$$350$$ − 32228.1i − 0.263086i
$$351$$ 0 0
$$352$$ 25571.9 0.206385
$$353$$ 58668.4i 0.470820i 0.971896 + 0.235410i $$0.0756433\pi$$
−0.971896 + 0.235410i $$0.924357\pi$$
$$354$$ 0 0
$$355$$ −26874.2 −0.213245
$$356$$ 110332.i 0.870564i
$$357$$ 0 0
$$358$$ −9365.99 −0.0730782
$$359$$ 37840.3i 0.293606i 0.989166 + 0.146803i $$0.0468984\pi$$
−0.989166 + 0.146803i $$0.953102\pi$$
$$360$$ 0 0
$$361$$ 156859. 1.20363
$$362$$ 109802.i 0.837904i
$$363$$ 0 0
$$364$$ 35206.8 0.265720
$$365$$ − 6765.80i − 0.0507848i
$$366$$ 0 0
$$367$$ 76141.5 0.565313 0.282657 0.959221i $$-0.408784\pi$$
0.282657 + 0.959221i $$0.408784\pi$$
$$368$$ − 31222.9i − 0.230557i
$$369$$ 0 0
$$370$$ 3367.05 0.0245950
$$371$$ − 39363.9i − 0.285989i
$$372$$ 0 0
$$373$$ 85550.6 0.614901 0.307451 0.951564i $$-0.400524\pi$$
0.307451 + 0.951564i $$0.400524\pi$$
$$374$$ − 157531.i − 1.12622i
$$375$$ 0 0
$$376$$ 64378.0 0.455367
$$377$$ − 285568.i − 2.00922i
$$378$$ 0 0
$$379$$ −136931. −0.953287 −0.476644 0.879097i $$-0.658147\pi$$
−0.476644 + 0.879097i $$0.658147\pi$$
$$380$$ 13395.9i 0.0927695i
$$381$$ 0 0
$$382$$ 115600. 0.792190
$$383$$ − 42414.5i − 0.289146i −0.989494 0.144573i $$-0.953819\pi$$
0.989494 0.144573i $$-0.0461808\pi$$
$$384$$ 0 0
$$385$$ 8175.05 0.0551530
$$386$$ − 108079.i − 0.725383i
$$387$$ 0 0
$$388$$ −84904.4 −0.563984
$$389$$ 258455.i 1.70799i 0.520277 + 0.853997i $$0.325829\pi$$
−0.520277 + 0.853997i $$0.674171\pi$$
$$390$$ 0 0
$$391$$ −192343. −1.25813
$$392$$ 7761.20i 0.0505076i
$$393$$ 0 0
$$394$$ 60034.0 0.386727
$$395$$ 18111.8i 0.116083i
$$396$$ 0 0
$$397$$ 114157. 0.724303 0.362152 0.932119i $$-0.382042\pi$$
0.362152 + 0.932119i $$0.382042\pi$$
$$398$$ − 125719.i − 0.793660i
$$399$$ 0 0
$$400$$ 39375.1 0.246095
$$401$$ 140116.i 0.871362i 0.900101 + 0.435681i $$0.143492\pi$$
−0.900101 + 0.435681i $$0.856508\pi$$
$$402$$ 0 0
$$403$$ −70493.8 −0.434051
$$404$$ − 35394.9i − 0.216859i
$$405$$ 0 0
$$406$$ 62952.4 0.381909
$$407$$ − 53819.1i − 0.324899i
$$408$$ 0 0
$$409$$ −165912. −0.991817 −0.495908 0.868375i $$-0.665165\pi$$
−0.495908 + 0.868375i $$0.665165\pi$$
$$410$$ − 651.035i − 0.00387291i
$$411$$ 0 0
$$412$$ −91989.9 −0.541933
$$413$$ − 101605.i − 0.595683i
$$414$$ 0 0
$$415$$ −7531.24 −0.0437291
$$416$$ 43014.4i 0.248558i
$$417$$ 0 0
$$418$$ 214121. 1.22548
$$419$$ − 130600.i − 0.743903i −0.928252 0.371952i $$-0.878689\pi$$
0.928252 0.371952i $$-0.121311\pi$$
$$420$$ 0 0
$$421$$ −62110.0 −0.350427 −0.175213 0.984530i $$-0.556062\pi$$
−0.175213 + 0.984530i $$0.556062\pi$$
$$422$$ − 2552.28i − 0.0143319i
$$423$$ 0 0
$$424$$ 48093.4 0.267519
$$425$$ − 242564.i − 1.34291i
$$426$$ 0 0
$$427$$ −93201.1 −0.511170
$$428$$ − 106037.i − 0.578857i
$$429$$ 0 0
$$430$$ −25738.3 −0.139201
$$431$$ 112404.i 0.605100i 0.953133 + 0.302550i $$0.0978380\pi$$
−0.953133 + 0.302550i $$0.902162\pi$$
$$432$$ 0 0
$$433$$ 64860.0 0.345940 0.172970 0.984927i $$-0.444664\pi$$
0.172970 + 0.984927i $$0.444664\pi$$
$$434$$ − 15540.1i − 0.0825038i
$$435$$ 0 0
$$436$$ −56171.5 −0.295490
$$437$$ − 261439.i − 1.36901i
$$438$$ 0 0
$$439$$ 146510. 0.760218 0.380109 0.924942i $$-0.375887\pi$$
0.380109 + 0.924942i $$0.375887\pi$$
$$440$$ 9987.99i 0.0515909i
$$441$$ 0 0
$$442$$ 264983. 1.35636
$$443$$ − 130291.i − 0.663908i −0.943296 0.331954i $$-0.892292\pi$$
0.943296 0.331954i $$-0.107708\pi$$
$$444$$ 0 0
$$445$$ −43094.0 −0.217619
$$446$$ − 6900.05i − 0.0346883i
$$447$$ 0 0
$$448$$ −9482.37 −0.0472456
$$449$$ 93245.1i 0.462523i 0.972892 + 0.231262i $$0.0742853\pi$$
−0.972892 + 0.231262i $$0.925715\pi$$
$$450$$ 0 0
$$451$$ −10406.2 −0.0511610
$$452$$ 129311.i 0.632932i
$$453$$ 0 0
$$454$$ 9641.80 0.0467785
$$455$$ 13751.2i 0.0664231i
$$456$$ 0 0
$$457$$ −137893. −0.660252 −0.330126 0.943937i $$-0.607091\pi$$
−0.330126 + 0.943937i $$0.607091\pi$$
$$458$$ 30513.4i 0.145465i
$$459$$ 0 0
$$460$$ 12195.2 0.0576333
$$461$$ − 324498.i − 1.52690i −0.645869 0.763448i $$-0.723505\pi$$
0.645869 0.763448i $$-0.276495\pi$$
$$462$$ 0 0
$$463$$ −267434. −1.24754 −0.623770 0.781608i $$-0.714400\pi$$
−0.623770 + 0.781608i $$0.714400\pi$$
$$464$$ 76913.1i 0.357244i
$$465$$ 0 0
$$466$$ −178770. −0.823234
$$467$$ 77333.7i 0.354597i 0.984157 + 0.177299i $$0.0567358\pi$$
−0.984157 + 0.177299i $$0.943264\pi$$
$$468$$ 0 0
$$469$$ 135987. 0.618232
$$470$$ 25145.1i 0.113830i
$$471$$ 0 0
$$472$$ 124138. 0.557211
$$473$$ 411403.i 1.83884i
$$474$$ 0 0
$$475$$ 329700. 1.46127
$$476$$ 58414.5i 0.257814i
$$477$$ 0 0
$$478$$ 121593. 0.532174
$$479$$ − 291569.i − 1.27078i −0.772191 0.635390i $$-0.780839\pi$$
0.772191 0.635390i $$-0.219161\pi$$
$$480$$ 0 0
$$481$$ 90529.1 0.391290
$$482$$ − 106659.i − 0.459098i
$$483$$ 0 0
$$484$$ 42520.8 0.181514
$$485$$ − 33162.4i − 0.140981i
$$486$$ 0 0
$$487$$ −98706.0 −0.416184 −0.208092 0.978109i $$-0.566725\pi$$
−0.208092 + 0.978109i $$0.566725\pi$$
$$488$$ − 113870.i − 0.478155i
$$489$$ 0 0
$$490$$ −3031.41 −0.0126256
$$491$$ 68699.3i 0.284963i 0.989797 + 0.142482i $$0.0455082\pi$$
−0.989797 + 0.142482i $$0.954492\pi$$
$$492$$ 0 0
$$493$$ 473810. 1.94944
$$494$$ 360173.i 1.47590i
$$495$$ 0 0
$$496$$ 18986.3 0.0771752
$$497$$ − 159286.i − 0.644859i
$$498$$ 0 0
$$499$$ −180484. −0.724834 −0.362417 0.932016i $$-0.618048\pi$$
−0.362417 + 0.932016i $$0.618048\pi$$
$$500$$ 31002.7i 0.124011i
$$501$$ 0 0
$$502$$ −320817. −1.27306
$$503$$ 170691.i 0.674645i 0.941389 + 0.337323i $$0.109521\pi$$
−0.941389 + 0.337323i $$0.890479\pi$$
$$504$$ 0 0
$$505$$ 13824.7 0.0542092
$$506$$ − 194929.i − 0.761334i
$$507$$ 0 0
$$508$$ 232376. 0.900458
$$509$$ − 87777.9i − 0.338805i −0.985547 0.169403i $$-0.945816\pi$$
0.985547 0.169403i $$-0.0541838\pi$$
$$510$$ 0 0
$$511$$ 40101.5 0.153574
$$512$$ − 11585.2i − 0.0441942i
$$513$$ 0 0
$$514$$ 23892.2 0.0904336
$$515$$ − 35929.9i − 0.135469i
$$516$$ 0 0
$$517$$ 401921. 1.50369
$$518$$ 19956.8i 0.0743758i
$$519$$ 0 0
$$520$$ −16800.8 −0.0621331
$$521$$ − 433477.i − 1.59695i −0.602030 0.798473i $$-0.705641\pi$$
0.602030 0.798473i $$-0.294359\pi$$
$$522$$ 0 0
$$523$$ −380650. −1.39163 −0.695813 0.718223i $$-0.744956\pi$$
−0.695813 + 0.718223i $$0.744956\pi$$
$$524$$ − 266818.i − 0.971745i
$$525$$ 0 0
$$526$$ 327196. 1.18260
$$527$$ − 116962.i − 0.421137i
$$528$$ 0 0
$$529$$ 41835.4 0.149497
$$530$$ 18784.6i 0.0668728i
$$531$$ 0 0
$$532$$ −79398.8 −0.280537
$$533$$ − 17504.3i − 0.0616154i
$$534$$ 0 0
$$535$$ 41416.6 0.144699
$$536$$ 166144.i 0.578303i
$$537$$ 0 0
$$538$$ 96438.5 0.333185
$$539$$ 48454.3i 0.166784i
$$540$$ 0 0
$$541$$ −158944. −0.543061 −0.271531 0.962430i $$-0.587530\pi$$
−0.271531 + 0.962430i $$0.587530\pi$$
$$542$$ − 194463.i − 0.661969i
$$543$$ 0 0
$$544$$ −71368.9 −0.241163
$$545$$ − 21939.8i − 0.0738650i
$$546$$ 0 0
$$547$$ −500492. −1.67272 −0.836358 0.548183i $$-0.815320\pi$$
−0.836358 + 0.548183i $$0.815320\pi$$
$$548$$ 29017.0i 0.0966254i
$$549$$ 0 0
$$550$$ 245824. 0.812642
$$551$$ 644017.i 2.12126i
$$552$$ 0 0
$$553$$ −107350. −0.351037
$$554$$ − 327251.i − 1.06626i
$$555$$ 0 0
$$556$$ −72857.7 −0.235682
$$557$$ 411990.i 1.32793i 0.747762 + 0.663967i $$0.231128\pi$$
−0.747762 + 0.663967i $$0.768872\pi$$
$$558$$ 0 0
$$559$$ −692020. −2.21460
$$560$$ − 3703.67i − 0.0118102i
$$561$$ 0 0
$$562$$ −27465.0 −0.0869576
$$563$$ − 237439.i − 0.749091i −0.927209 0.374545i $$-0.877799\pi$$
0.927209 0.374545i $$-0.122201\pi$$
$$564$$ 0 0
$$565$$ −50506.7 −0.158217
$$566$$ − 48002.5i − 0.149841i
$$567$$ 0 0
$$568$$ 194610. 0.603210
$$569$$ 548677.i 1.69470i 0.531037 + 0.847349i $$0.321803\pi$$
−0.531037 + 0.847349i $$0.678197\pi$$
$$570$$ 0 0
$$571$$ 205016. 0.628806 0.314403 0.949290i $$-0.398196\pi$$
0.314403 + 0.949290i $$0.398196\pi$$
$$572$$ 268545.i 0.820777i
$$573$$ 0 0
$$574$$ 3858.75 0.0117118
$$575$$ − 300148.i − 0.907820i
$$576$$ 0 0
$$577$$ −292763. −0.879356 −0.439678 0.898155i $$-0.644907\pi$$
−0.439678 + 0.898155i $$0.644907\pi$$
$$578$$ 203423.i 0.608897i
$$579$$ 0 0
$$580$$ −30041.1 −0.0893017
$$581$$ − 44638.3i − 0.132238i
$$582$$ 0 0
$$583$$ 300254. 0.883388
$$584$$ 48994.7i 0.143656i
$$585$$ 0 0
$$586$$ −131443. −0.382774
$$587$$ 75489.1i 0.219083i 0.993982 + 0.109541i $$0.0349382\pi$$
−0.993982 + 0.109541i $$0.965062\pi$$
$$588$$ 0 0
$$589$$ 158978. 0.458255
$$590$$ 48486.3i 0.139288i
$$591$$ 0 0
$$592$$ −24382.5 −0.0695722
$$593$$ 664448.i 1.88952i 0.327764 + 0.944760i $$0.393705\pi$$
−0.327764 + 0.944760i $$0.606295\pi$$
$$594$$ 0 0
$$595$$ −22815.8 −0.0644470
$$596$$ 77351.8i 0.217760i
$$597$$ 0 0
$$598$$ 327890. 0.916908
$$599$$ 521123.i 1.45240i 0.687483 + 0.726201i $$0.258716\pi$$
−0.687483 + 0.726201i $$0.741284\pi$$
$$600$$ 0 0
$$601$$ 52148.0 0.144374 0.0721869 0.997391i $$-0.477002\pi$$
0.0721869 + 0.997391i $$0.477002\pi$$
$$602$$ − 152553.i − 0.420948i
$$603$$ 0 0
$$604$$ 171288. 0.469520
$$605$$ 16608.0i 0.0453739i
$$606$$ 0 0
$$607$$ −342719. −0.930168 −0.465084 0.885267i $$-0.653976\pi$$
−0.465084 + 0.885267i $$0.653976\pi$$
$$608$$ − 97006.7i − 0.262419i
$$609$$ 0 0
$$610$$ 44475.8 0.119527
$$611$$ 676070.i 1.81096i
$$612$$ 0 0
$$613$$ −702183. −1.86866 −0.934328 0.356415i $$-0.883999\pi$$
−0.934328 + 0.356415i $$0.883999\pi$$
$$614$$ 295340.i 0.783405i
$$615$$ 0 0
$$616$$ −59199.7 −0.156012
$$617$$ − 5722.23i − 0.0150312i −0.999972 0.00751562i $$-0.997608\pi$$
0.999972 0.00751562i $$-0.00239232\pi$$
$$618$$ 0 0
$$619$$ −493596. −1.28822 −0.644110 0.764933i $$-0.722772\pi$$
−0.644110 + 0.764933i $$0.722772\pi$$
$$620$$ 7415.78i 0.0192918i
$$621$$ 0 0
$$622$$ −324645. −0.839127
$$623$$ − 255422.i − 0.658085i
$$624$$ 0 0
$$625$$ 372414. 0.953379
$$626$$ 21362.2i 0.0545126i
$$627$$ 0 0
$$628$$ −214369. −0.543555
$$629$$ 150204.i 0.379648i
$$630$$ 0 0
$$631$$ 369529. 0.928089 0.464045 0.885812i $$-0.346398\pi$$
0.464045 + 0.885812i $$0.346398\pi$$
$$632$$ − 131157.i − 0.328365i
$$633$$ 0 0
$$634$$ −269615. −0.670758
$$635$$ 90762.5i 0.225091i
$$636$$ 0 0
$$637$$ −81504.9 −0.200865
$$638$$ 480179.i 1.17967i
$$639$$ 0 0
$$640$$ 4525.02 0.0110474
$$641$$ − 685619.i − 1.66866i −0.551269 0.834328i $$-0.685856\pi$$
0.551269 0.834328i $$-0.314144\pi$$
$$642$$ 0 0
$$643$$ 90659.6 0.219276 0.109638 0.993972i $$-0.465031\pi$$
0.109638 + 0.993972i $$0.465031\pi$$
$$644$$ 72282.1i 0.174285i
$$645$$ 0 0
$$646$$ −597593. −1.43199
$$647$$ 758270.i 1.81140i 0.423916 + 0.905702i $$0.360655\pi$$
−0.423916 + 0.905702i $$0.639345\pi$$
$$648$$ 0 0
$$649$$ 775008. 1.84000
$$650$$ 413501.i 0.978700i
$$651$$ 0 0
$$652$$ −258001. −0.606914
$$653$$ − 261552.i − 0.613383i −0.951809 0.306692i $$-0.900778\pi$$
0.951809 0.306692i $$-0.0992220\pi$$
$$654$$ 0 0
$$655$$ 104215. 0.242911
$$656$$ 4714.48i 0.0109554i
$$657$$ 0 0
$$658$$ −149037. −0.344225
$$659$$ − 517272.i − 1.19110i −0.803319 0.595549i $$-0.796934\pi$$
0.803319 0.595549i $$-0.203066\pi$$
$$660$$ 0 0
$$661$$ −626741. −1.43445 −0.717224 0.696842i $$-0.754588\pi$$
−0.717224 + 0.696842i $$0.754588\pi$$
$$662$$ 187124.i 0.426985i
$$663$$ 0 0
$$664$$ 54537.6 0.123697
$$665$$ − 31012.0i − 0.0701271i
$$666$$ 0 0
$$667$$ 586292. 1.31784
$$668$$ − 48913.6i − 0.109617i
$$669$$ 0 0
$$670$$ −64893.4 −0.144561
$$671$$ − 710905.i − 1.57894i
$$672$$ 0 0
$$673$$ 742305. 1.63890 0.819450 0.573151i $$-0.194279\pi$$
0.819450 + 0.573151i $$0.194279\pi$$
$$674$$ 396381.i 0.872555i
$$675$$ 0 0
$$676$$ −223231. −0.488497
$$677$$ 604523.i 1.31897i 0.751717 + 0.659486i $$0.229226\pi$$
−0.751717 + 0.659486i $$0.770774\pi$$
$$678$$ 0 0
$$679$$ 196556. 0.426332
$$680$$ − 27875.6i − 0.0602846i
$$681$$ 0 0
$$682$$ 118534. 0.254845
$$683$$ 720939.i 1.54546i 0.634737 + 0.772728i $$0.281109\pi$$
−0.634737 + 0.772728i $$0.718891\pi$$
$$684$$ 0 0
$$685$$ −11333.6 −0.0241539
$$686$$ − 17967.4i − 0.0381802i
$$687$$ 0 0
$$688$$ 186384. 0.393761
$$689$$ 505057.i 1.06390i
$$690$$ 0 0
$$691$$ −588601. −1.23272 −0.616360 0.787464i $$-0.711393\pi$$
−0.616360 + 0.787464i $$0.711393\pi$$
$$692$$ − 236409.i − 0.493686i
$$693$$ 0 0
$$694$$ −80988.0 −0.168152
$$695$$ − 28457.1i − 0.0589144i
$$696$$ 0 0
$$697$$ 29042.8 0.0597823
$$698$$ − 231959.i − 0.476102i
$$699$$ 0 0
$$700$$ −91154.7 −0.186030
$$701$$ 154697.i 0.314809i 0.987534 + 0.157404i $$0.0503126\pi$$
−0.987534 + 0.157404i $$0.949687\pi$$
$$702$$ 0 0
$$703$$ −204162. −0.413109
$$704$$ − 72328.2i − 0.145936i
$$705$$ 0 0
$$706$$ 165939. 0.332920
$$707$$ 81940.3i 0.163930i
$$708$$ 0 0
$$709$$ 715382. 1.42313 0.711567 0.702619i $$-0.247986\pi$$
0.711567 + 0.702619i $$0.247986\pi$$
$$710$$ 76011.8i 0.150787i
$$711$$ 0 0
$$712$$ 312066. 0.615582
$$713$$ − 144729.i − 0.284692i
$$714$$ 0 0
$$715$$ −104890. −0.205173
$$716$$ 26491.0i 0.0516741i
$$717$$ 0 0
$$718$$ 107028. 0.207611
$$719$$ − 371799.i − 0.719202i −0.933106 0.359601i $$-0.882913\pi$$
0.933106 0.359601i $$-0.117087\pi$$
$$720$$ 0 0
$$721$$ 212960. 0.409663
$$722$$ − 443663.i − 0.851097i
$$723$$ 0 0
$$724$$ 310568. 0.592487
$$725$$ 739371.i 1.40665i
$$726$$ 0 0
$$727$$ 52138.1 0.0986476 0.0493238 0.998783i $$-0.484293\pi$$
0.0493238 + 0.998783i $$0.484293\pi$$
$$728$$ − 99579.8i − 0.187892i
$$729$$ 0 0
$$730$$ −19136.6 −0.0359103
$$731$$ − 1.14819e6i − 2.14871i
$$732$$ 0 0
$$733$$ 324955. 0.604805 0.302403 0.953180i $$-0.402211\pi$$
0.302403 + 0.953180i $$0.402211\pi$$
$$734$$ − 215361.i − 0.399737i
$$735$$ 0 0
$$736$$ −88311.8 −0.163028
$$737$$ 1.03726e6i 1.90965i
$$738$$ 0 0
$$739$$ −971247. −1.77845 −0.889224 0.457473i $$-0.848755\pi$$
−0.889224 + 0.457473i $$0.848755\pi$$
$$740$$ − 9523.45i − 0.0173913i
$$741$$ 0 0
$$742$$ −111338. −0.202225
$$743$$ − 681732.i − 1.23491i −0.786605 0.617456i $$-0.788163\pi$$
0.786605 0.617456i $$-0.211837\pi$$
$$744$$ 0 0
$$745$$ −30212.4 −0.0544344
$$746$$ − 241974.i − 0.434801i
$$747$$ 0 0
$$748$$ −445566. −0.796358
$$749$$ 245480.i 0.437575i
$$750$$ 0 0
$$751$$ −317894. −0.563641 −0.281821 0.959467i $$-0.590938\pi$$
−0.281821 + 0.959467i $$0.590938\pi$$
$$752$$ − 182088.i − 0.321993i
$$753$$ 0 0
$$754$$ −807709. −1.42073
$$755$$ 66902.6i 0.117368i
$$756$$ 0 0
$$757$$ −700233. −1.22194 −0.610971 0.791653i $$-0.709221\pi$$
−0.610971 + 0.791653i $$0.709221\pi$$
$$758$$ 387300.i 0.674076i
$$759$$ 0 0
$$760$$ 37889.4 0.0655979
$$761$$ − 305818.i − 0.528072i −0.964513 0.264036i $$-0.914946\pi$$
0.964513 0.264036i $$-0.0850538\pi$$
$$762$$ 0 0
$$763$$ 130039. 0.223370
$$764$$ − 326965.i − 0.560163i
$$765$$ 0 0
$$766$$ −119966. −0.204457
$$767$$ 1.30364e6i 2.21599i
$$768$$ 0 0
$$769$$ 820424. 1.38735 0.693675 0.720289i $$-0.255991\pi$$
0.693675 + 0.720289i $$0.255991\pi$$
$$770$$ − 23122.5i − 0.0389990i
$$771$$ 0 0
$$772$$ −305694. −0.512923
$$773$$ − 483522.i − 0.809202i −0.914493 0.404601i $$-0.867410\pi$$
0.914493 0.404601i $$-0.132590\pi$$
$$774$$ 0 0
$$775$$ 182517. 0.303878
$$776$$ 240146.i 0.398797i
$$777$$ 0 0
$$778$$ 731022. 1.20773
$$779$$ 39475.8i 0.0650513i
$$780$$ 0 0
$$781$$ 1.21498e6 1.99189
$$782$$ 544030.i 0.889629i
$$783$$ 0 0
$$784$$ 21952.0 0.0357143
$$785$$ − 83729.4i − 0.135875i
$$786$$ 0 0
$$787$$ −185741. −0.299888 −0.149944 0.988695i $$-0.547909\pi$$
−0.149944 + 0.988695i $$0.547909\pi$$
$$788$$ − 169802.i − 0.273457i
$$789$$ 0 0
$$790$$ 51227.8 0.0820827
$$791$$ − 299358.i − 0.478452i
$$792$$ 0 0
$$793$$ 1.19581e6 1.90159
$$794$$ − 322884.i − 0.512160i
$$795$$ 0 0
$$796$$ −355587. −0.561202
$$797$$ − 720505.i − 1.13428i −0.823621 0.567140i $$-0.808050\pi$$
0.823621 0.567140i $$-0.191950\pi$$
$$798$$ 0 0
$$799$$ −1.12172e6 −1.75708
$$800$$ − 111370.i − 0.174015i
$$801$$ 0 0
$$802$$ 396307. 0.616146
$$803$$ 305880.i 0.474374i
$$804$$ 0 0
$$805$$ −28232.3 −0.0435667
$$806$$ 199386.i 0.306920i
$$807$$ 0 0
$$808$$ −100112. −0.153343
$$809$$ 686330.i 1.04866i 0.851514 + 0.524331i $$0.175685\pi$$
−0.851514 + 0.524331i $$0.824315\pi$$
$$810$$ 0 0
$$811$$ 1.24830e6 1.89791 0.948955 0.315412i $$-0.102143\pi$$
0.948955 + 0.315412i $$0.102143\pi$$
$$812$$ − 178056.i − 0.270051i
$$813$$ 0 0
$$814$$ −152224. −0.229738
$$815$$ − 100771.i − 0.151713i
$$816$$ 0 0
$$817$$ 1.56065e6 2.33810
$$818$$ 469270.i 0.701320i
$$819$$ 0 0
$$820$$ −1841.41 −0.00273856
$$821$$ − 842160.i − 1.24942i −0.780857 0.624710i $$-0.785217\pi$$
0.780857 0.624710i $$-0.214783\pi$$
$$822$$ 0 0
$$823$$ −105260. −0.155404 −0.0777019 0.996977i $$-0.524758\pi$$
−0.0777019 + 0.996977i $$0.524758\pi$$
$$824$$ 260187.i 0.383204i
$$825$$ 0 0
$$826$$ −287383. −0.421212
$$827$$ − 373318.i − 0.545844i −0.962036 0.272922i $$-0.912010\pi$$
0.962036 0.272922i $$-0.0879900\pi$$
$$828$$ 0 0
$$829$$ 867025. 1.26160 0.630801 0.775944i $$-0.282726\pi$$
0.630801 + 0.775944i $$0.282726\pi$$
$$830$$ 21301.6i 0.0309211i
$$831$$ 0 0
$$832$$ 121663. 0.175757
$$833$$ − 135232.i − 0.194889i
$$834$$ 0 0
$$835$$ 19104.9 0.0274014
$$836$$ − 605626.i − 0.866547i
$$837$$ 0 0
$$838$$ −369394. −0.526019
$$839$$ − 686493.i − 0.975242i −0.873055 0.487621i $$-0.837865\pi$$
0.873055 0.487621i $$-0.162135\pi$$
$$840$$ 0 0
$$841$$ −736964. −1.04197
$$842$$ 175674.i 0.247789i
$$843$$ 0 0
$$844$$ −7218.94 −0.0101342
$$845$$ − 87190.8i − 0.122112i
$$846$$ 0 0
$$847$$ −98437.0 −0.137212
$$848$$ − 136029.i − 0.189164i
$$849$$ 0 0
$$850$$ −686074. −0.949583
$$851$$ 185863.i 0.256645i
$$852$$ 0 0
$$853$$ 974427. 1.33922 0.669609 0.742714i $$-0.266462\pi$$
0.669609 + 0.742714i $$0.266462\pi$$
$$854$$ 263612.i 0.361452i
$$855$$ 0 0
$$856$$ −299919. −0.409314
$$857$$ − 275668.i − 0.375339i −0.982232 0.187670i $$-0.939907\pi$$
0.982232 0.187670i $$-0.0600934\pi$$
$$858$$ 0 0
$$859$$ 176529. 0.239237 0.119619 0.992820i $$-0.461833\pi$$
0.119619 + 0.992820i $$0.461833\pi$$
$$860$$ 72798.9i 0.0984301i
$$861$$ 0 0
$$862$$ 317927. 0.427871
$$863$$ − 856520.i − 1.15005i −0.818136 0.575024i $$-0.804993\pi$$
0.818136 0.575024i $$-0.195007\pi$$
$$864$$ 0 0
$$865$$ 92337.6 0.123409
$$866$$ − 183452.i − 0.244617i
$$867$$ 0 0
$$868$$ −43954.0 −0.0583390
$$869$$ − 818829.i − 1.08431i
$$870$$ 0 0
$$871$$ −1.74478e6 −2.29987
$$872$$ 158877.i 0.208943i
$$873$$ 0 0
$$874$$ −739461. −0.968038
$$875$$ − 71772.3i − 0.0937434i
$$876$$ 0 0
$$877$$ 1.00077e6 1.30118 0.650589 0.759430i $$-0.274522\pi$$
0.650589 + 0.759430i $$0.274522\pi$$
$$878$$ − 414393.i − 0.537555i
$$879$$ 0 0
$$880$$ 28250.3 0.0364803
$$881$$ − 775427.i − 0.999055i −0.866298 0.499527i $$-0.833507\pi$$
0.866298 0.499527i $$-0.166493\pi$$
$$882$$ 0 0
$$883$$ 160327. 0.205630 0.102815 0.994701i $$-0.467215\pi$$
0.102815 + 0.994701i $$0.467215\pi$$
$$884$$ − 749485.i − 0.959089i
$$885$$ 0 0
$$886$$ −368519. −0.469454
$$887$$ − 183914.i − 0.233759i −0.993146 0.116879i $$-0.962711\pi$$
0.993146 0.116879i $$-0.0372891\pi$$
$$888$$ 0 0
$$889$$ −537958. −0.680682
$$890$$ 121888.i 0.153880i
$$891$$ 0 0
$$892$$ −19516.3 −0.0245283
$$893$$ − 1.52468e6i − 1.91195i
$$894$$ 0 0
$$895$$ −10347.0 −0.0129172
$$896$$ 26820.2i 0.0334077i
$$897$$ 0 0
$$898$$ 263737. 0.327053
$$899$$ 356518.i 0.441126i
$$900$$ 0 0
$$901$$ −837982. −1.03225
$$902$$ 29433.2i 0.0361763i
$$903$$ 0 0
$$904$$ 365745. 0.447550
$$905$$ 121303.i 0.148107i
$$906$$ 0 0
$$907$$ 184492. 0.224266 0.112133 0.993693i $$-0.464232\pi$$
0.112133 + 0.993693i $$0.464232\pi$$
$$908$$ − 27271.1i − 0.0330774i
$$909$$ 0 0
$$910$$ 38894.4 0.0469682
$$911$$ 979147.i 1.17981i 0.807474 + 0.589904i $$0.200834\pi$$
−0.807474 + 0.589904i $$0.799166\pi$$
$$912$$ 0 0
$$913$$ 340486. 0.408467
$$914$$ 390020.i 0.466868i
$$915$$ 0 0
$$916$$ 86304.9 0.102860
$$917$$ 617692.i 0.734570i
$$918$$ 0 0
$$919$$ 1.32373e6 1.56736 0.783679 0.621166i $$-0.213341\pi$$
0.783679 + 0.621166i $$0.213341\pi$$
$$920$$ − 34493.2i − 0.0407529i
$$921$$ 0 0
$$922$$ −917818. −1.07968
$$923$$ 2.04371e6i 2.39892i
$$924$$ 0 0
$$925$$ −234391. −0.273941
$$926$$ 756417.i 0.882143i
$$927$$ 0 0
$$928$$ 217543. 0.252609
$$929$$ 1.15956e6i 1.34358i 0.740743 + 0.671789i $$0.234474\pi$$
−0.740743 + 0.671789i $$0.765526\pi$$
$$930$$ 0 0
$$931$$ 183811. 0.212066
$$932$$ 505638.i 0.582114i
$$933$$ 0 0
$$934$$ 218733. 0.250738
$$935$$ − 174031.i − 0.199069i
$$936$$ 0 0
$$937$$ −1.49483e6 −1.70260 −0.851300 0.524679i $$-0.824185\pi$$
−0.851300 + 0.524679i $$0.824185\pi$$
$$938$$ − 384629.i − 0.437156i
$$939$$ 0 0
$$940$$ 71121.0 0.0804900
$$941$$ − 1.18590e6i − 1.33927i −0.742689 0.669637i $$-0.766450\pi$$
0.742689 0.669637i $$-0.233550\pi$$
$$942$$ 0 0
$$943$$ 35937.5 0.0404133
$$944$$ − 351114.i − 0.394007i
$$945$$ 0 0
$$946$$ 1.16362e6 1.30026
$$947$$ − 865824.i − 0.965450i −0.875772 0.482725i $$-0.839647\pi$$
0.875772 0.482725i $$-0.160353\pi$$
$$948$$ 0 0
$$949$$ −514521. −0.571309
$$950$$ − 932532.i − 1.03328i
$$951$$ 0 0
$$952$$ 165221. 0.182302
$$953$$ − 388687.i − 0.427971i −0.976837 0.213986i $$-0.931355\pi$$
0.976837 0.213986i $$-0.0686446\pi$$
$$954$$ 0 0
$$955$$ 127708. 0.140026
$$956$$ − 343917.i − 0.376304i
$$957$$ 0 0
$$958$$ −824682. −0.898577
$$959$$ − 67175.3i − 0.0730419i
$$960$$ 0 0
$$961$$ −835513. −0.904704
$$962$$ − 256055.i − 0.276683i
$$963$$ 0 0
$$964$$ −301678. −0.324631
$$965$$ − 119399.i − 0.128218i
$$966$$ 0 0
$$967$$ 192495. 0.205858 0.102929 0.994689i $$-0.467179\pi$$
0.102929 + 0.994689i $$0.467179\pi$$
$$968$$ − 120267.i − 0.128350i
$$969$$ 0 0
$$970$$ −93797.3 −0.0996889
$$971$$ − 453950.i − 0.481470i −0.970591 0.240735i $$-0.922612\pi$$
0.970591 0.240735i $$-0.0773885\pi$$
$$972$$ 0 0
$$973$$ 168668. 0.178159
$$974$$ 279183.i 0.294287i
$$975$$ 0 0
$$976$$ −322073. −0.338107
$$977$$ − 727881.i − 0.762555i −0.924461 0.381278i $$-0.875484\pi$$
0.924461 0.381278i $$-0.124516\pi$$
$$978$$ 0 0
$$979$$ 1.94827e6 2.03275
$$980$$ 8574.12i 0.00892765i
$$981$$ 0 0
$$982$$ 194311. 0.201500
$$983$$ − 662759.i − 0.685881i −0.939357 0.342941i $$-0.888577\pi$$
0.939357 0.342941i $$-0.111423\pi$$
$$984$$ 0 0
$$985$$ 66322.0 0.0683573
$$986$$ − 1.34014e6i − 1.37846i
$$987$$ 0 0
$$988$$ 1.01872e6 1.04362
$$989$$ − 1.42077e6i − 1.45255i
$$990$$ 0 0
$$991$$ −1.16764e6 −1.18894 −0.594472 0.804116i $$-0.702639\pi$$
−0.594472 + 0.804116i $$0.702639\pi$$
$$992$$ − 53701.5i − 0.0545711i
$$993$$ 0 0
$$994$$ −450529. −0.455984
$$995$$ − 138887.i − 0.140286i
$$996$$ 0 0
$$997$$ 1.12634e6 1.13313 0.566563 0.824019i $$-0.308273\pi$$
0.566563 + 0.824019i $$0.308273\pi$$
$$998$$ 510487.i 0.512535i
$$999$$ 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 378.5.b.a.323.3 8
3.2 odd 2 inner 378.5.b.a.323.6 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.a.323.3 8 1.1 even 1 trivial
378.5.b.a.323.6 yes 8 3.2 odd 2 inner