# Properties

 Label 378.5.b.a.323.5 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 323.5 Root $$2.39656i$$ of defining polynomial Character $$\chi$$ $$=$$ 378.323 Dual form 378.5.b.a.323.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.82843i q^{2} -8.00000 q^{4} -14.9735i q^{5} -18.5203 q^{7} -22.6274i q^{8} +O(q^{10})$$ $$q+2.82843i q^{2} -8.00000 q^{4} -14.9735i q^{5} -18.5203 q^{7} -22.6274i q^{8} +42.3515 q^{10} +207.672i q^{11} -62.8805 q^{13} -52.3832i q^{14} +64.0000 q^{16} -187.297i q^{17} +462.879 q^{19} +119.788i q^{20} -587.385 q^{22} +455.668i q^{23} +400.793 q^{25} -177.853i q^{26} +148.162 q^{28} -1181.31i q^{29} -1516.38 q^{31} +181.019i q^{32} +529.755 q^{34} +277.314i q^{35} -2092.12 q^{37} +1309.22i q^{38} -338.812 q^{40} -2995.00i q^{41} -3082.43 q^{43} -1661.38i q^{44} -1288.82 q^{46} -1652.15i q^{47} +343.000 q^{49} +1133.62i q^{50} +503.044 q^{52} -2863.40i q^{53} +3109.58 q^{55} +419.066i q^{56} +3341.26 q^{58} -3120.03i q^{59} +3220.68 q^{61} -4288.97i q^{62} -512.000 q^{64} +941.542i q^{65} -5231.48 q^{67} +1498.37i q^{68} -784.361 q^{70} +2620.11i q^{71} -6236.52 q^{73} -5917.41i q^{74} -3703.04 q^{76} -3846.14i q^{77} +3408.06 q^{79} -958.306i q^{80} +8471.13 q^{82} +6087.85i q^{83} -2804.49 q^{85} -8718.43i q^{86} +4699.08 q^{88} -13322.1i q^{89} +1164.56 q^{91} -3645.34i q^{92} +4672.99 q^{94} -6930.94i q^{95} -6073.09 q^{97} +970.151i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 64 q^{4}+O(q^{10})$$ 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
$$5$$ − 14.9735i − 0.598941i −0.954106 0.299471i $$-0.903190\pi$$
0.954106 0.299471i $$-0.0968100\pi$$
$$6$$ 0 0
$$7$$ −18.5203 −0.377964
$$8$$ − 22.6274i − 0.353553i
$$9$$ 0 0
$$10$$ 42.3515 0.423515
$$11$$ 207.672i 1.71630i 0.513401 + 0.858149i $$0.328385\pi$$
−0.513401 + 0.858149i $$0.671615\pi$$
$$12$$ 0 0
$$13$$ −62.8805 −0.372074 −0.186037 0.982543i $$-0.559564\pi$$
−0.186037 + 0.982543i $$0.559564\pi$$
$$14$$ − 52.3832i − 0.267261i
$$15$$ 0 0
$$16$$ 64.0000 0.250000
$$17$$ − 187.297i − 0.648085i −0.946042 0.324043i $$-0.894958\pi$$
0.946042 0.324043i $$-0.105042\pi$$
$$18$$ 0 0
$$19$$ 462.879 1.28221 0.641107 0.767451i $$-0.278475\pi$$
0.641107 + 0.767451i $$0.278475\pi$$
$$20$$ 119.788i 0.299471i
$$21$$ 0 0
$$22$$ −587.385 −1.21361
$$23$$ 455.668i 0.861376i 0.902501 + 0.430688i $$0.141729\pi$$
−0.902501 + 0.430688i $$0.858271\pi$$
$$24$$ 0 0
$$25$$ 400.793 0.641270
$$26$$ − 177.853i − 0.263096i
$$27$$ 0 0
$$28$$ 148.162 0.188982
$$29$$ − 1181.31i − 1.40465i −0.711855 0.702326i $$-0.752145\pi$$
0.711855 0.702326i $$-0.247855\pi$$
$$30$$ 0 0
$$31$$ −1516.38 −1.57792 −0.788960 0.614445i $$-0.789380\pi$$
−0.788960 + 0.614445i $$0.789380\pi$$
$$32$$ 181.019i 0.176777i
$$33$$ 0 0
$$34$$ 529.755 0.458265
$$35$$ 277.314i 0.226378i
$$36$$ 0 0
$$37$$ −2092.12 −1.52821 −0.764105 0.645092i $$-0.776819\pi$$
−0.764105 + 0.645092i $$0.776819\pi$$
$$38$$ 1309.22i 0.906663i
$$39$$ 0 0
$$40$$ −338.812 −0.211758
$$41$$ − 2995.00i − 1.78168i −0.454321 0.890838i $$-0.650118\pi$$
0.454321 0.890838i $$-0.349882\pi$$
$$42$$ 0 0
$$43$$ −3082.43 −1.66708 −0.833540 0.552459i $$-0.813690\pi$$
−0.833540 + 0.552459i $$0.813690\pi$$
$$44$$ − 1661.38i − 0.858149i
$$45$$ 0 0
$$46$$ −1288.82 −0.609085
$$47$$ − 1652.15i − 0.747918i −0.927445 0.373959i $$-0.878000\pi$$
0.927445 0.373959i $$-0.122000\pi$$
$$48$$ 0 0
$$49$$ 343.000 0.142857
$$50$$ 1133.62i 0.453446i
$$51$$ 0 0
$$52$$ 503.044 0.186037
$$53$$ − 2863.40i − 1.01936i −0.860363 0.509682i $$-0.829763\pi$$
0.860363 0.509682i $$-0.170237\pi$$
$$54$$ 0 0
$$55$$ 3109.58 1.02796
$$56$$ 419.066i 0.133631i
$$57$$ 0 0
$$58$$ 3341.26 0.993239
$$59$$ − 3120.03i − 0.896302i −0.893958 0.448151i $$-0.852083\pi$$
0.893958 0.448151i $$-0.147917\pi$$
$$60$$ 0 0
$$61$$ 3220.68 0.865540 0.432770 0.901504i $$-0.357536\pi$$
0.432770 + 0.901504i $$0.357536\pi$$
$$62$$ − 4288.97i − 1.11576i
$$63$$ 0 0
$$64$$ −512.000 −0.125000
$$65$$ 941.542i 0.222850i
$$66$$ 0 0
$$67$$ −5231.48 −1.16540 −0.582700 0.812688i $$-0.698004\pi$$
−0.582700 + 0.812688i $$0.698004\pi$$
$$68$$ 1498.37i 0.324043i
$$69$$ 0 0
$$70$$ −784.361 −0.160074
$$71$$ 2620.11i 0.519759i 0.965641 + 0.259880i $$0.0836828\pi$$
−0.965641 + 0.259880i $$0.916317\pi$$
$$72$$ 0 0
$$73$$ −6236.52 −1.17030 −0.585149 0.810926i $$-0.698964\pi$$
−0.585149 + 0.810926i $$0.698964\pi$$
$$74$$ − 5917.41i − 1.08061i
$$75$$ 0 0
$$76$$ −3703.04 −0.641107
$$77$$ − 3846.14i − 0.648700i
$$78$$ 0 0
$$79$$ 3408.06 0.546076 0.273038 0.962003i $$-0.411971\pi$$
0.273038 + 0.962003i $$0.411971\pi$$
$$80$$ − 958.306i − 0.149735i
$$81$$ 0 0
$$82$$ 8471.13 1.25984
$$83$$ 6087.85i 0.883706i 0.897087 + 0.441853i $$0.145679\pi$$
−0.897087 + 0.441853i $$0.854321\pi$$
$$84$$ 0 0
$$85$$ −2804.49 −0.388165
$$86$$ − 8718.43i − 1.17880i
$$87$$ 0 0
$$88$$ 4699.08 0.606803
$$89$$ − 13322.1i − 1.68187i −0.541134 0.840936i $$-0.682005\pi$$
0.541134 0.840936i $$-0.317995\pi$$
$$90$$ 0 0
$$91$$ 1164.56 0.140631
$$92$$ − 3645.34i − 0.430688i
$$93$$ 0 0
$$94$$ 4672.99 0.528858
$$95$$ − 6930.94i − 0.767971i
$$96$$ 0 0
$$97$$ −6073.09 −0.645455 −0.322728 0.946492i $$-0.604600\pi$$
−0.322728 + 0.946492i $$0.604600\pi$$
$$98$$ 970.151i 0.101015i
$$99$$ 0 0
$$100$$ −3206.35 −0.320635
$$101$$ − 5450.61i − 0.534321i −0.963652 0.267160i $$-0.913915\pi$$
0.963652 0.267160i $$-0.0860854\pi$$
$$102$$ 0 0
$$103$$ 4836.88 0.455922 0.227961 0.973670i $$-0.426794\pi$$
0.227961 + 0.973670i $$0.426794\pi$$
$$104$$ 1422.82i 0.131548i
$$105$$ 0 0
$$106$$ 8098.91 0.720800
$$107$$ − 7429.22i − 0.648897i −0.945903 0.324448i $$-0.894821\pi$$
0.945903 0.324448i $$-0.105179\pi$$
$$108$$ 0 0
$$109$$ 15978.9 1.34492 0.672458 0.740136i $$-0.265239\pi$$
0.672458 + 0.740136i $$0.265239\pi$$
$$110$$ 8795.23i 0.726878i
$$111$$ 0 0
$$112$$ −1185.30 −0.0944911
$$113$$ 8283.62i 0.648729i 0.945932 + 0.324364i $$0.105150\pi$$
−0.945932 + 0.324364i $$0.894850\pi$$
$$114$$ 0 0
$$115$$ 6822.96 0.515914
$$116$$ 9450.50i 0.702326i
$$117$$ 0 0
$$118$$ 8824.77 0.633781
$$119$$ 3468.78i 0.244953i
$$120$$ 0 0
$$121$$ −28486.7 −1.94568
$$122$$ 9109.45i 0.612029i
$$123$$ 0 0
$$124$$ 12131.0 0.788960
$$125$$ − 15359.7i − 0.983024i
$$126$$ 0 0
$$127$$ 25189.7 1.56176 0.780882 0.624679i $$-0.214770\pi$$
0.780882 + 0.624679i $$0.214770\pi$$
$$128$$ − 1448.15i − 0.0883883i
$$129$$ 0 0
$$130$$ −2663.08 −0.157579
$$131$$ − 10668.0i − 0.621644i −0.950468 0.310822i $$-0.899396\pi$$
0.950468 0.310822i $$-0.100604\pi$$
$$132$$ 0 0
$$133$$ −8572.65 −0.484632
$$134$$ − 14796.9i − 0.824062i
$$135$$ 0 0
$$136$$ −4238.04 −0.229133
$$137$$ 9682.43i 0.515873i 0.966162 + 0.257937i $$0.0830426\pi$$
−0.966162 + 0.257937i $$0.916957\pi$$
$$138$$ 0 0
$$139$$ −17916.1 −0.927285 −0.463643 0.886022i $$-0.653458\pi$$
−0.463643 + 0.886022i $$0.653458\pi$$
$$140$$ − 2218.51i − 0.113189i
$$141$$ 0 0
$$142$$ −7410.78 −0.367525
$$143$$ − 13058.5i − 0.638589i
$$144$$ 0 0
$$145$$ −17688.4 −0.841304
$$146$$ − 17639.5i − 0.827526i
$$147$$ 0 0
$$148$$ 16737.0 0.764105
$$149$$ 17588.5i 0.792240i 0.918199 + 0.396120i $$0.129644\pi$$
−0.918199 + 0.396120i $$0.870356\pi$$
$$150$$ 0 0
$$151$$ −35597.0 −1.56120 −0.780602 0.625028i $$-0.785088\pi$$
−0.780602 + 0.625028i $$0.785088\pi$$
$$152$$ − 10473.8i − 0.453331i
$$153$$ 0 0
$$154$$ 10878.5 0.458700
$$155$$ 22705.6i 0.945081i
$$156$$ 0 0
$$157$$ 32966.9 1.33745 0.668726 0.743509i $$-0.266840\pi$$
0.668726 + 0.743509i $$0.266840\pi$$
$$158$$ 9639.46i 0.386134i
$$159$$ 0 0
$$160$$ 2710.50 0.105879
$$161$$ − 8439.09i − 0.325570i
$$162$$ 0 0
$$163$$ −20460.9 −0.770102 −0.385051 0.922895i $$-0.625816\pi$$
−0.385051 + 0.922895i $$0.625816\pi$$
$$164$$ 23960.0i 0.890838i
$$165$$ 0 0
$$166$$ −17219.1 −0.624875
$$167$$ 19104.1i 0.685006i 0.939517 + 0.342503i $$0.111275\pi$$
−0.939517 + 0.342503i $$0.888725\pi$$
$$168$$ 0 0
$$169$$ −24607.0 −0.861561
$$170$$ − 7932.30i − 0.274474i
$$171$$ 0 0
$$172$$ 24659.5 0.833540
$$173$$ − 27932.8i − 0.933302i −0.884442 0.466651i $$-0.845460\pi$$
0.884442 0.466651i $$-0.154540\pi$$
$$174$$ 0 0
$$175$$ −7422.80 −0.242377
$$176$$ 13291.0i 0.429074i
$$177$$ 0 0
$$178$$ 37680.6 1.18926
$$179$$ − 26824.0i − 0.837179i −0.908176 0.418589i $$-0.862525\pi$$
0.908176 0.418589i $$-0.137475\pi$$
$$180$$ 0 0
$$181$$ −8214.15 −0.250729 −0.125365 0.992111i $$-0.540010\pi$$
−0.125365 + 0.992111i $$0.540010\pi$$
$$182$$ 3293.88i 0.0994409i
$$183$$ 0 0
$$184$$ 10310.6 0.304543
$$185$$ 31326.4i 0.915308i
$$186$$ 0 0
$$187$$ 38896.3 1.11231
$$188$$ 13217.2i 0.373959i
$$189$$ 0 0
$$190$$ 19603.7 0.543038
$$191$$ − 19528.1i − 0.535296i −0.963517 0.267648i $$-0.913754\pi$$
0.963517 0.267648i $$-0.0862464\pi$$
$$192$$ 0 0
$$193$$ 1684.59 0.0452252 0.0226126 0.999744i $$-0.492802\pi$$
0.0226126 + 0.999744i $$0.492802\pi$$
$$194$$ − 17177.3i − 0.456406i
$$195$$ 0 0
$$196$$ −2744.00 −0.0714286
$$197$$ 25114.6i 0.647134i 0.946205 + 0.323567i $$0.104882\pi$$
−0.946205 + 0.323567i $$0.895118\pi$$
$$198$$ 0 0
$$199$$ −48183.6 −1.21673 −0.608363 0.793659i $$-0.708173\pi$$
−0.608363 + 0.793659i $$0.708173\pi$$
$$200$$ − 9068.92i − 0.226723i
$$201$$ 0 0
$$202$$ 15416.6 0.377822
$$203$$ 21878.2i 0.530909i
$$204$$ 0 0
$$205$$ −44845.7 −1.06712
$$206$$ 13680.8i 0.322386i
$$207$$ 0 0
$$208$$ −4024.35 −0.0930184
$$209$$ 96127.1i 2.20066i
$$210$$ 0 0
$$211$$ −37374.5 −0.839480 −0.419740 0.907644i $$-0.637879\pi$$
−0.419740 + 0.907644i $$0.637879\pi$$
$$212$$ 22907.2i 0.509682i
$$213$$ 0 0
$$214$$ 21013.0 0.458839
$$215$$ 46154.9i 0.998483i
$$216$$ 0 0
$$217$$ 28083.8 0.596397
$$218$$ 45195.3i 0.950999i
$$219$$ 0 0
$$220$$ −24876.7 −0.513981
$$221$$ 11777.3i 0.241135i
$$222$$ 0 0
$$223$$ −14929.9 −0.300225 −0.150113 0.988669i $$-0.547964\pi$$
−0.150113 + 0.988669i $$0.547964\pi$$
$$224$$ − 3352.53i − 0.0668153i
$$225$$ 0 0
$$226$$ −23429.6 −0.458720
$$227$$ 81222.3i 1.57625i 0.615518 + 0.788123i $$0.288947\pi$$
−0.615518 + 0.788123i $$0.711053\pi$$
$$228$$ 0 0
$$229$$ −26288.7 −0.501300 −0.250650 0.968078i $$-0.580644\pi$$
−0.250650 + 0.968078i $$0.580644\pi$$
$$230$$ 19298.2i 0.364806i
$$231$$ 0 0
$$232$$ −26730.1 −0.496620
$$233$$ − 77377.7i − 1.42529i −0.701523 0.712647i $$-0.747496\pi$$
0.701523 0.712647i $$-0.252504\pi$$
$$234$$ 0 0
$$235$$ −24738.5 −0.447959
$$236$$ 24960.2i 0.448151i
$$237$$ 0 0
$$238$$ −9811.20 −0.173208
$$239$$ 214.251i 0.00375083i 0.999998 + 0.00187541i $$0.000596963\pi$$
−0.999998 + 0.00187541i $$0.999403\pi$$
$$240$$ 0 0
$$241$$ 3165.30 0.0544980 0.0272490 0.999629i $$-0.491325\pi$$
0.0272490 + 0.999629i $$0.491325\pi$$
$$242$$ − 80572.4i − 1.37580i
$$243$$ 0 0
$$244$$ −25765.4 −0.432770
$$245$$ − 5135.92i − 0.0855630i
$$246$$ 0 0
$$247$$ −29106.1 −0.477078
$$248$$ 34311.8i 0.557879i
$$249$$ 0 0
$$250$$ 43443.9 0.695103
$$251$$ − 31889.6i − 0.506177i −0.967443 0.253088i $$-0.918554\pi$$
0.967443 0.253088i $$-0.0814464\pi$$
$$252$$ 0 0
$$253$$ −94629.5 −1.47838
$$254$$ 71247.2i 1.10433i
$$255$$ 0 0
$$256$$ 4096.00 0.0625000
$$257$$ 89826.0i 1.35999i 0.733217 + 0.679995i $$0.238018\pi$$
−0.733217 + 0.679995i $$0.761982\pi$$
$$258$$ 0 0
$$259$$ 38746.6 0.577609
$$260$$ − 7532.34i − 0.111425i
$$261$$ 0 0
$$262$$ 30173.8 0.439569
$$263$$ 23175.4i 0.335055i 0.985867 + 0.167528i $$0.0535783\pi$$
−0.985867 + 0.167528i $$0.946422\pi$$
$$264$$ 0 0
$$265$$ −42875.1 −0.610539
$$266$$ − 24247.1i − 0.342686i
$$267$$ 0 0
$$268$$ 41851.8 0.582700
$$269$$ − 56937.5i − 0.786853i −0.919356 0.393427i $$-0.871290\pi$$
0.919356 0.393427i $$-0.128710\pi$$
$$270$$ 0 0
$$271$$ 123059. 1.67561 0.837807 0.545967i $$-0.183838\pi$$
0.837807 + 0.545967i $$0.183838\pi$$
$$272$$ − 11987.0i − 0.162021i
$$273$$ 0 0
$$274$$ −27386.0 −0.364778
$$275$$ 83233.6i 1.10061i
$$276$$ 0 0
$$277$$ −10073.2 −0.131283 −0.0656415 0.997843i $$-0.520909\pi$$
−0.0656415 + 0.997843i $$0.520909\pi$$
$$278$$ − 50674.3i − 0.655690i
$$279$$ 0 0
$$280$$ 6274.89 0.0800369
$$281$$ 123794.i 1.56778i 0.620898 + 0.783892i $$0.286768\pi$$
−0.620898 + 0.783892i $$0.713232\pi$$
$$282$$ 0 0
$$283$$ 112317. 1.40240 0.701201 0.712964i $$-0.252648\pi$$
0.701201 + 0.712964i $$0.252648\pi$$
$$284$$ − 20960.8i − 0.259880i
$$285$$ 0 0
$$286$$ 36935.0 0.451551
$$287$$ 55468.1i 0.673410i
$$288$$ 0 0
$$289$$ 48441.0 0.579986
$$290$$ − 50030.4i − 0.594892i
$$291$$ 0 0
$$292$$ 49892.2 0.585149
$$293$$ − 64511.2i − 0.751449i −0.926731 0.375725i $$-0.877394\pi$$
0.926731 0.375725i $$-0.122606\pi$$
$$294$$ 0 0
$$295$$ −46717.8 −0.536832
$$296$$ 47339.3i 0.540304i
$$297$$ 0 0
$$298$$ −49747.9 −0.560198
$$299$$ − 28652.6i − 0.320496i
$$300$$ 0 0
$$301$$ 57087.4 0.630097
$$302$$ − 100684.i − 1.10394i
$$303$$ 0 0
$$304$$ 29624.3 0.320554
$$305$$ − 48224.9i − 0.518408i
$$306$$ 0 0
$$307$$ 11483.3 0.121840 0.0609199 0.998143i $$-0.480597\pi$$
0.0609199 + 0.998143i $$0.480597\pi$$
$$308$$ 30769.1i 0.324350i
$$309$$ 0 0
$$310$$ −64221.0 −0.668273
$$311$$ 11256.9i 0.116385i 0.998305 + 0.0581926i $$0.0185337\pi$$
−0.998305 + 0.0581926i $$0.981466\pi$$
$$312$$ 0 0
$$313$$ −112068. −1.14392 −0.571959 0.820282i $$-0.693816\pi$$
−0.571959 + 0.820282i $$0.693816\pi$$
$$314$$ 93244.4i 0.945722i
$$315$$ 0 0
$$316$$ −27264.5 −0.273038
$$317$$ − 47792.1i − 0.475595i −0.971315 0.237798i $$-0.923574\pi$$
0.971315 0.237798i $$-0.0764255\pi$$
$$318$$ 0 0
$$319$$ 245326. 2.41080
$$320$$ 7666.45i 0.0748676i
$$321$$ 0 0
$$322$$ 23869.4 0.230213
$$323$$ − 86695.8i − 0.830984i
$$324$$ 0 0
$$325$$ −25202.1 −0.238600
$$326$$ − 57872.0i − 0.544545i
$$327$$ 0 0
$$328$$ −67769.0 −0.629918
$$329$$ 30598.3i 0.282686i
$$330$$ 0 0
$$331$$ −108343. −0.988885 −0.494443 0.869210i $$-0.664628\pi$$
−0.494443 + 0.869210i $$0.664628\pi$$
$$332$$ − 48702.8i − 0.441853i
$$333$$ 0 0
$$334$$ −54034.7 −0.484373
$$335$$ 78333.7i 0.698006i
$$336$$ 0 0
$$337$$ 163814. 1.44241 0.721207 0.692719i $$-0.243588\pi$$
0.721207 + 0.692719i $$0.243588\pi$$
$$338$$ − 69599.2i − 0.609216i
$$339$$ 0 0
$$340$$ 22435.9 0.194082
$$341$$ − 314910.i − 2.70818i
$$342$$ 0 0
$$343$$ −6352.45 −0.0539949
$$344$$ 69747.5i 0.589402i
$$345$$ 0 0
$$346$$ 79005.8 0.659944
$$347$$ − 199534.i − 1.65713i −0.559892 0.828566i $$-0.689157\pi$$
0.559892 0.828566i $$-0.310843\pi$$
$$348$$ 0 0
$$349$$ 28305.5 0.232391 0.116196 0.993226i $$-0.462930\pi$$
0.116196 + 0.993226i $$0.462930\pi$$
$$350$$ − 20994.8i − 0.171386i
$$351$$ 0 0
$$352$$ −37592.6 −0.303401
$$353$$ 134579.i 1.08001i 0.841661 + 0.540007i $$0.181578\pi$$
−0.841661 + 0.540007i $$0.818422\pi$$
$$354$$ 0 0
$$355$$ 39232.2 0.311305
$$356$$ 106577.i 0.840936i
$$357$$ 0 0
$$358$$ 75869.8 0.591975
$$359$$ − 125745.i − 0.975666i −0.872937 0.487833i $$-0.837787\pi$$
0.872937 0.487833i $$-0.162213\pi$$
$$360$$ 0 0
$$361$$ 83936.4 0.644074
$$362$$ − 23233.1i − 0.177292i
$$363$$ 0 0
$$364$$ −9316.50 −0.0703153
$$365$$ 93382.7i 0.700940i
$$366$$ 0 0
$$367$$ −22221.8 −0.164986 −0.0824931 0.996592i $$-0.526288\pi$$
−0.0824931 + 0.996592i $$0.526288\pi$$
$$368$$ 29162.8i 0.215344i
$$369$$ 0 0
$$370$$ −88604.5 −0.647221
$$371$$ 53030.8i 0.385284i
$$372$$ 0 0
$$373$$ 34876.0 0.250674 0.125337 0.992114i $$-0.459999\pi$$
0.125337 + 0.992114i $$0.459999\pi$$
$$374$$ 110015.i 0.786520i
$$375$$ 0 0
$$376$$ −37383.9 −0.264429
$$377$$ 74281.5i 0.522634i
$$378$$ 0 0
$$379$$ −102131. −0.711016 −0.355508 0.934673i $$-0.615692\pi$$
−0.355508 + 0.934673i $$0.615692\pi$$
$$380$$ 55447.5i 0.383986i
$$381$$ 0 0
$$382$$ 55233.9 0.378511
$$383$$ − 12013.5i − 0.0818981i −0.999161 0.0409490i $$-0.986962\pi$$
0.999161 0.0409490i $$-0.0130381\pi$$
$$384$$ 0 0
$$385$$ −57590.3 −0.388533
$$386$$ 4764.75i 0.0319791i
$$387$$ 0 0
$$388$$ 48584.7 0.322728
$$389$$ − 102479.i − 0.677227i −0.940926 0.338614i $$-0.890042\pi$$
0.940926 0.338614i $$-0.109958\pi$$
$$390$$ 0 0
$$391$$ 85345.1 0.558245
$$392$$ − 7761.20i − 0.0505076i
$$393$$ 0 0
$$394$$ −71034.9 −0.457593
$$395$$ − 51030.7i − 0.327068i
$$396$$ 0 0
$$397$$ 102844. 0.652528 0.326264 0.945279i $$-0.394210\pi$$
0.326264 + 0.945279i $$0.394210\pi$$
$$398$$ − 136284.i − 0.860355i
$$399$$ 0 0
$$400$$ 25650.8 0.160317
$$401$$ − 252453.i − 1.56997i −0.619514 0.784986i $$-0.712670\pi$$
0.619514 0.784986i $$-0.287330\pi$$
$$402$$ 0 0
$$403$$ 95350.7 0.587102
$$404$$ 43604.9i 0.267160i
$$405$$ 0 0
$$406$$ −61881.0 −0.375409
$$407$$ − 434475.i − 2.62286i
$$408$$ 0 0
$$409$$ −42140.0 −0.251911 −0.125956 0.992036i $$-0.540200\pi$$
−0.125956 + 0.992036i $$0.540200\pi$$
$$410$$ − 126843.i − 0.754567i
$$411$$ 0 0
$$412$$ −38695.0 −0.227961
$$413$$ 57783.7i 0.338770i
$$414$$ 0 0
$$415$$ 91156.7 0.529288
$$416$$ − 11382.6i − 0.0657740i
$$417$$ 0 0
$$418$$ −271889. −1.55610
$$419$$ 219896.i 1.25254i 0.779608 + 0.626268i $$0.215418\pi$$
−0.779608 + 0.626268i $$0.784582\pi$$
$$420$$ 0 0
$$421$$ −148584. −0.838316 −0.419158 0.907913i $$-0.637675\pi$$
−0.419158 + 0.907913i $$0.637675\pi$$
$$422$$ − 105711.i − 0.593602i
$$423$$ 0 0
$$424$$ −64791.2 −0.360400
$$425$$ − 75067.3i − 0.415597i
$$426$$ 0 0
$$427$$ −59647.7 −0.327143
$$428$$ 59433.7i 0.324448i
$$429$$ 0 0
$$430$$ −130546. −0.706034
$$431$$ − 340490.i − 1.83295i −0.400097 0.916473i $$-0.631024\pi$$
0.400097 0.916473i $$-0.368976\pi$$
$$432$$ 0 0
$$433$$ −341800. −1.82304 −0.911519 0.411257i $$-0.865090\pi$$
−0.911519 + 0.411257i $$0.865090\pi$$
$$434$$ 79432.9i 0.421717i
$$435$$ 0 0
$$436$$ −127831. −0.672458
$$437$$ 210919.i 1.10447i
$$438$$ 0 0
$$439$$ 34198.2 0.177449 0.0887246 0.996056i $$-0.471721\pi$$
0.0887246 + 0.996056i $$0.471721\pi$$
$$440$$ − 70361.8i − 0.363439i
$$441$$ 0 0
$$442$$ −33311.2 −0.170509
$$443$$ 127446.i 0.649409i 0.945815 + 0.324705i $$0.105265\pi$$
−0.945815 + 0.324705i $$0.894735\pi$$
$$444$$ 0 0
$$445$$ −199479. −1.00734
$$446$$ − 42228.1i − 0.212291i
$$447$$ 0 0
$$448$$ 9482.37 0.0472456
$$449$$ 250976.i 1.24491i 0.782655 + 0.622456i $$0.213865\pi$$
−0.782655 + 0.622456i $$0.786135\pi$$
$$450$$ 0 0
$$451$$ 621977. 3.05789
$$452$$ − 66268.9i − 0.324364i
$$453$$ 0 0
$$454$$ −229731. −1.11457
$$455$$ − 17437.6i − 0.0842295i
$$456$$ 0 0
$$457$$ −311362. −1.49085 −0.745425 0.666590i $$-0.767753\pi$$
−0.745425 + 0.666590i $$0.767753\pi$$
$$458$$ − 74355.6i − 0.354473i
$$459$$ 0 0
$$460$$ −54583.7 −0.257957
$$461$$ 299717.i 1.41029i 0.709061 + 0.705147i $$0.249119\pi$$
−0.709061 + 0.705147i $$0.750881\pi$$
$$462$$ 0 0
$$463$$ 350647. 1.63572 0.817859 0.575418i $$-0.195161\pi$$
0.817859 + 0.575418i $$0.195161\pi$$
$$464$$ − 75604.0i − 0.351163i
$$465$$ 0 0
$$466$$ 218857. 1.00783
$$467$$ 187935.i 0.861734i 0.902416 + 0.430867i $$0.141792\pi$$
−0.902416 + 0.430867i $$0.858208\pi$$
$$468$$ 0 0
$$469$$ 96888.3 0.440479
$$470$$ − 69971.1i − 0.316755i
$$471$$ 0 0
$$472$$ −70598.1 −0.316891
$$473$$ − 640135.i − 2.86121i
$$474$$ 0 0
$$475$$ 185519. 0.822245
$$476$$ − 27750.3i − 0.122477i
$$477$$ 0 0
$$478$$ −605.993 −0.00265223
$$479$$ − 440865.i − 1.92148i −0.277459 0.960738i $$-0.589492\pi$$
0.277459 0.960738i $$-0.410508\pi$$
$$480$$ 0 0
$$481$$ 131553. 0.568607
$$482$$ 8952.81i 0.0385359i
$$483$$ 0 0
$$484$$ 227893. 0.972839
$$485$$ 90935.6i 0.386590i
$$486$$ 0 0
$$487$$ −309791. −1.30620 −0.653101 0.757270i $$-0.726532\pi$$
−0.653101 + 0.757270i $$0.726532\pi$$
$$488$$ − 72875.6i − 0.306015i
$$489$$ 0 0
$$490$$ 14526.6 0.0605022
$$491$$ 43693.9i 0.181241i 0.995885 + 0.0906207i $$0.0288851\pi$$
−0.995885 + 0.0906207i $$0.971115\pi$$
$$492$$ 0 0
$$493$$ −221256. −0.910335
$$494$$ − 82324.4i − 0.337345i
$$495$$ 0 0
$$496$$ −97048.4 −0.394480
$$497$$ − 48525.0i − 0.196450i
$$498$$ 0 0
$$499$$ −18463.9 −0.0741521 −0.0370760 0.999312i $$-0.511804\pi$$
−0.0370760 + 0.999312i $$0.511804\pi$$
$$500$$ 122878.i 0.491512i
$$501$$ 0 0
$$502$$ 90197.5 0.357921
$$503$$ 186650.i 0.737722i 0.929484 + 0.368861i $$0.120252\pi$$
−0.929484 + 0.368861i $$0.879748\pi$$
$$504$$ 0 0
$$505$$ −81614.8 −0.320027
$$506$$ − 267653.i − 1.04537i
$$507$$ 0 0
$$508$$ −201518. −0.780882
$$509$$ − 158194.i − 0.610595i −0.952257 0.305298i $$-0.901244\pi$$
0.952257 0.305298i $$-0.0987559\pi$$
$$510$$ 0 0
$$511$$ 115502. 0.442331
$$512$$ 11585.2i 0.0441942i
$$513$$ 0 0
$$514$$ −254066. −0.961658
$$515$$ − 72425.1i − 0.273070i
$$516$$ 0 0
$$517$$ 343106. 1.28365
$$518$$ 109592.i 0.408431i
$$519$$ 0 0
$$520$$ 21304.7 0.0787895
$$521$$ 352048.i 1.29696i 0.761232 + 0.648479i $$0.224595\pi$$
−0.761232 + 0.648479i $$0.775405\pi$$
$$522$$ 0 0
$$523$$ 182994. 0.669013 0.334506 0.942393i $$-0.391430\pi$$
0.334506 + 0.942393i $$0.391430\pi$$
$$524$$ 85344.3i 0.310822i
$$525$$ 0 0
$$526$$ −65550.0 −0.236920
$$527$$ 284013.i 1.02263i
$$528$$ 0 0
$$529$$ 72207.6 0.258031
$$530$$ − 121269.i − 0.431717i
$$531$$ 0 0
$$532$$ 68581.2 0.242316
$$533$$ 188327.i 0.662915i
$$534$$ 0 0
$$535$$ −111242. −0.388651
$$536$$ 118375.i 0.412031i
$$537$$ 0 0
$$538$$ 161044. 0.556389
$$539$$ 71231.5i 0.245185i
$$540$$ 0 0
$$541$$ 216902. 0.741087 0.370543 0.928815i $$-0.379172\pi$$
0.370543 + 0.928815i $$0.379172\pi$$
$$542$$ 348063.i 1.18484i
$$543$$ 0 0
$$544$$ 33904.3 0.114566
$$545$$ − 239261.i − 0.805525i
$$546$$ 0 0
$$547$$ −307754. −1.02856 −0.514279 0.857623i $$-0.671940\pi$$
−0.514279 + 0.857623i $$0.671940\pi$$
$$548$$ − 77459.4i − 0.257937i
$$549$$ 0 0
$$550$$ −235420. −0.778248
$$551$$ − 546805.i − 1.80107i
$$552$$ 0 0
$$553$$ −63118.2 −0.206397
$$554$$ − 28491.4i − 0.0928311i
$$555$$ 0 0
$$556$$ 143329. 0.463643
$$557$$ 74238.2i 0.239286i 0.992817 + 0.119643i $$0.0381750\pi$$
−0.992817 + 0.119643i $$0.961825\pi$$
$$558$$ 0 0
$$559$$ 193825. 0.620277
$$560$$ 17748.1i 0.0565946i
$$561$$ 0 0
$$562$$ −350142. −1.10859
$$563$$ 279024.i 0.880286i 0.897928 + 0.440143i $$0.145072\pi$$
−0.897928 + 0.440143i $$0.854928\pi$$
$$564$$ 0 0
$$565$$ 124035. 0.388550
$$566$$ 317680.i 0.991648i
$$567$$ 0 0
$$568$$ 59286.2 0.183763
$$569$$ − 154079.i − 0.475902i −0.971277 0.237951i $$-0.923524\pi$$
0.971277 0.237951i $$-0.0764758\pi$$
$$570$$ 0 0
$$571$$ 202500. 0.621087 0.310543 0.950559i $$-0.399489\pi$$
0.310543 + 0.950559i $$0.399489\pi$$
$$572$$ 104468.i 0.319295i
$$573$$ 0 0
$$574$$ −156888. −0.476173
$$575$$ 182629.i 0.552374i
$$576$$ 0 0
$$577$$ −404034. −1.21358 −0.606788 0.794864i $$-0.707542\pi$$
−0.606788 + 0.794864i $$0.707542\pi$$
$$578$$ 137012.i 0.410112i
$$579$$ 0 0
$$580$$ 141507. 0.420652
$$581$$ − 112749.i − 0.334010i
$$582$$ 0 0
$$583$$ 594647. 1.74953
$$584$$ 141116.i 0.413763i
$$585$$ 0 0
$$586$$ 182465. 0.531355
$$587$$ 204467.i 0.593399i 0.954971 + 0.296700i $$0.0958860\pi$$
−0.954971 + 0.296700i $$0.904114\pi$$
$$588$$ 0 0
$$589$$ −701901. −2.02323
$$590$$ − 132138.i − 0.379598i
$$591$$ 0 0
$$592$$ −133896. −0.382053
$$593$$ 226013.i 0.642723i 0.946957 + 0.321361i $$0.104140\pi$$
−0.946957 + 0.321361i $$0.895860\pi$$
$$594$$ 0 0
$$595$$ 51939.9 0.146713
$$596$$ − 140708.i − 0.396120i
$$597$$ 0 0
$$598$$ 81041.8 0.226625
$$599$$ − 45246.9i − 0.126106i −0.998010 0.0630530i $$-0.979916\pi$$
0.998010 0.0630530i $$-0.0200837\pi$$
$$600$$ 0 0
$$601$$ 24718.9 0.0684352 0.0342176 0.999414i $$-0.489106\pi$$
0.0342176 + 0.999414i $$0.489106\pi$$
$$602$$ 161468.i 0.445546i
$$603$$ 0 0
$$604$$ 284776. 0.780602
$$605$$ 426546.i 1.16535i
$$606$$ 0 0
$$607$$ 34864.1 0.0946241 0.0473120 0.998880i $$-0.484934\pi$$
0.0473120 + 0.998880i $$0.484934\pi$$
$$608$$ 83790.1i 0.226666i
$$609$$ 0 0
$$610$$ 136401. 0.366570
$$611$$ 103888.i 0.278281i
$$612$$ 0 0
$$613$$ −175378. −0.466717 −0.233359 0.972391i $$-0.574972\pi$$
−0.233359 + 0.972391i $$0.574972\pi$$
$$614$$ 32479.6i 0.0861537i
$$615$$ 0 0
$$616$$ −87028.2 −0.229350
$$617$$ 192394.i 0.505383i 0.967547 + 0.252691i $$0.0813158\pi$$
−0.967547 + 0.252691i $$0.918684\pi$$
$$618$$ 0 0
$$619$$ −235328. −0.614174 −0.307087 0.951681i $$-0.599354\pi$$
−0.307087 + 0.951681i $$0.599354\pi$$
$$620$$ − 181645.i − 0.472540i
$$621$$ 0 0
$$622$$ −31839.3 −0.0822967
$$623$$ 246729.i 0.635688i
$$624$$ 0 0
$$625$$ 20506.3 0.0524961
$$626$$ − 316977.i − 0.808872i
$$627$$ 0 0
$$628$$ −263735. −0.668726
$$629$$ 391847.i 0.990411i
$$630$$ 0 0
$$631$$ 534962. 1.34358 0.671791 0.740741i $$-0.265525\pi$$
0.671791 + 0.740741i $$0.265525\pi$$
$$632$$ − 77115.7i − 0.193067i
$$633$$ 0 0
$$634$$ 135176. 0.336297
$$635$$ − 377179.i − 0.935405i
$$636$$ 0 0
$$637$$ −21568.0 −0.0531534
$$638$$ 693886.i 1.70469i
$$639$$ 0 0
$$640$$ −21684.0 −0.0529394
$$641$$ − 715669.i − 1.74179i −0.491469 0.870895i $$-0.663540\pi$$
0.491469 0.870895i $$-0.336460\pi$$
$$642$$ 0 0
$$643$$ 401264. 0.970528 0.485264 0.874368i $$-0.338723\pi$$
0.485264 + 0.874368i $$0.338723\pi$$
$$644$$ 67512.7i 0.162785i
$$645$$ 0 0
$$646$$ 245213. 0.587595
$$647$$ 761674.i 1.81953i 0.415119 + 0.909767i $$0.363740\pi$$
−0.415119 + 0.909767i $$0.636260\pi$$
$$648$$ 0 0
$$649$$ 647942. 1.53832
$$650$$ − 71282.2i − 0.168715i
$$651$$ 0 0
$$652$$ 163687. 0.385051
$$653$$ 225493.i 0.528819i 0.964411 + 0.264409i $$0.0851770\pi$$
−0.964411 + 0.264409i $$0.914823\pi$$
$$654$$ 0 0
$$655$$ −159738. −0.372328
$$656$$ − 191680.i − 0.445419i
$$657$$ 0 0
$$658$$ −86545.0 −0.199890
$$659$$ 844001.i 1.94344i 0.236127 + 0.971722i $$0.424122\pi$$
−0.236127 + 0.971722i $$0.575878\pi$$
$$660$$ 0 0
$$661$$ −233508. −0.534441 −0.267220 0.963635i $$-0.586105\pi$$
−0.267220 + 0.963635i $$0.586105\pi$$
$$662$$ − 306441.i − 0.699247i
$$663$$ 0 0
$$664$$ 137752. 0.312437
$$665$$ 128363.i 0.290266i
$$666$$ 0 0
$$667$$ 538287. 1.20993
$$668$$ − 152833.i − 0.342503i
$$669$$ 0 0
$$670$$ −221561. −0.493564
$$671$$ 668844.i 1.48552i
$$672$$ 0 0
$$673$$ −380168. −0.839355 −0.419677 0.907673i $$-0.637857\pi$$
−0.419677 + 0.907673i $$0.637857\pi$$
$$674$$ 463335.i 1.01994i
$$675$$ 0 0
$$676$$ 196856. 0.430781
$$677$$ − 390571.i − 0.852163i −0.904685 0.426082i $$-0.859894\pi$$
0.904685 0.426082i $$-0.140106\pi$$
$$678$$ 0 0
$$679$$ 112475. 0.243959
$$680$$ 63458.4i 0.137237i
$$681$$ 0 0
$$682$$ 890699. 1.91497
$$683$$ − 483186.i − 1.03579i −0.855443 0.517896i $$-0.826715\pi$$
0.855443 0.517896i $$-0.173285\pi$$
$$684$$ 0 0
$$685$$ 144980. 0.308978
$$686$$ − 17967.4i − 0.0381802i
$$687$$ 0 0
$$688$$ −197276. −0.416770
$$689$$ 180052.i 0.379279i
$$690$$ 0 0
$$691$$ −145791. −0.305334 −0.152667 0.988278i $$-0.548786\pi$$
−0.152667 + 0.988278i $$0.548786\pi$$
$$692$$ 223462.i 0.466651i
$$693$$ 0 0
$$694$$ 564366. 1.17177
$$695$$ 268267.i 0.555389i
$$696$$ 0 0
$$697$$ −560953. −1.15468
$$698$$ 80060.0i 0.164325i
$$699$$ 0 0
$$700$$ 59382.4 0.121189
$$701$$ 427557.i 0.870078i 0.900412 + 0.435039i $$0.143265\pi$$
−0.900412 + 0.435039i $$0.856735\pi$$
$$702$$ 0 0
$$703$$ −968400. −1.95949
$$704$$ − 106328.i − 0.214537i
$$705$$ 0 0
$$706$$ −380648. −0.763685
$$707$$ 100947.i 0.201954i
$$708$$ 0 0
$$709$$ −568967. −1.13186 −0.565932 0.824452i $$-0.691484\pi$$
−0.565932 + 0.824452i $$0.691484\pi$$
$$710$$ 110965.i 0.220126i
$$711$$ 0 0
$$712$$ −301445. −0.594632
$$713$$ − 690966.i − 1.35918i
$$714$$ 0 0
$$715$$ −195532. −0.382477
$$716$$ 214592.i 0.418589i
$$717$$ 0 0
$$718$$ 355660. 0.689900
$$719$$ 460546.i 0.890873i 0.895314 + 0.445436i $$0.146951\pi$$
−0.895314 + 0.445436i $$0.853049\pi$$
$$720$$ 0 0
$$721$$ −89580.2 −0.172322
$$722$$ 237408.i 0.455429i
$$723$$ 0 0
$$724$$ 65713.2 0.125365
$$725$$ − 473462.i − 0.900761i
$$726$$ 0 0
$$727$$ 551066. 1.04264 0.521321 0.853361i $$-0.325439\pi$$
0.521321 + 0.853361i $$0.325439\pi$$
$$728$$ − 26351.0i − 0.0497204i
$$729$$ 0 0
$$730$$ −264126. −0.495639
$$731$$ 577329.i 1.08041i
$$732$$ 0 0
$$733$$ −280002. −0.521138 −0.260569 0.965455i $$-0.583910\pi$$
−0.260569 + 0.965455i $$0.583910\pi$$
$$734$$ − 62852.8i − 0.116663i
$$735$$ 0 0
$$736$$ −82484.7 −0.152271
$$737$$ − 1.08643e6i − 2.00017i
$$738$$ 0 0
$$739$$ 8667.21 0.0158705 0.00793525 0.999969i $$-0.497474\pi$$
0.00793525 + 0.999969i $$0.497474\pi$$
$$740$$ − 250611.i − 0.457654i
$$741$$ 0 0
$$742$$ −149994. −0.272437
$$743$$ − 370104.i − 0.670419i −0.942144 0.335210i $$-0.891193\pi$$
0.942144 0.335210i $$-0.108807\pi$$
$$744$$ 0 0
$$745$$ 263362. 0.474505
$$746$$ 98644.2i 0.177253i
$$747$$ 0 0
$$748$$ −311170. −0.556154
$$749$$ 137591.i 0.245260i
$$750$$ 0 0
$$751$$ 508370. 0.901363 0.450681 0.892685i $$-0.351181\pi$$
0.450681 + 0.892685i $$0.351181\pi$$
$$752$$ − 105738.i − 0.186980i
$$753$$ 0 0
$$754$$ −210100. −0.369558
$$755$$ 533013.i 0.935070i
$$756$$ 0 0
$$757$$ 832192. 1.45222 0.726109 0.687580i $$-0.241327\pi$$
0.726109 + 0.687580i $$0.241327\pi$$
$$758$$ − 288870.i − 0.502765i
$$759$$ 0 0
$$760$$ −156829. −0.271519
$$761$$ 599649.i 1.03545i 0.855548 + 0.517724i $$0.173220\pi$$
−0.855548 + 0.517724i $$0.826780\pi$$
$$762$$ 0 0
$$763$$ −295934. −0.508330
$$764$$ 156225.i 0.267648i
$$765$$ 0 0
$$766$$ 33979.4 0.0579107
$$767$$ 196189.i 0.333490i
$$768$$ 0 0
$$769$$ 122085. 0.206447 0.103224 0.994658i $$-0.467084\pi$$
0.103224 + 0.994658i $$0.467084\pi$$
$$770$$ − 162890.i − 0.274734i
$$771$$ 0 0
$$772$$ −13476.8 −0.0226126
$$773$$ − 655751.i − 1.09744i −0.836007 0.548719i $$-0.815116\pi$$
0.836007 0.548719i $$-0.184884\pi$$
$$774$$ 0 0
$$775$$ −607755. −1.01187
$$776$$ 137418.i 0.228203i
$$777$$ 0 0
$$778$$ 289854. 0.478872
$$779$$ − 1.38632e6i − 2.28449i
$$780$$ 0 0
$$781$$ −544123. −0.892061
$$782$$ 241392.i 0.394739i
$$783$$ 0 0
$$784$$ 21952.0 0.0357143
$$785$$ − 493630.i − 0.801055i
$$786$$ 0 0
$$787$$ 208274. 0.336268 0.168134 0.985764i $$-0.446226\pi$$
0.168134 + 0.985764i $$0.446226\pi$$
$$788$$ − 200917.i − 0.323567i
$$789$$ 0 0
$$790$$ 144337. 0.231272
$$791$$ − 153415.i − 0.245196i
$$792$$ 0 0
$$793$$ −202518. −0.322045
$$794$$ 290888.i 0.461407i
$$795$$ 0 0
$$796$$ 385468. 0.608363
$$797$$ 464263.i 0.730883i 0.930834 + 0.365442i $$0.119082\pi$$
−0.930834 + 0.365442i $$0.880918\pi$$
$$798$$ 0 0
$$799$$ −309442. −0.484715
$$800$$ 72551.4i 0.113362i
$$801$$ 0 0
$$802$$ 714045. 1.11014
$$803$$ − 1.29515e6i − 2.00858i
$$804$$ 0 0
$$805$$ −126363. −0.194997
$$806$$ 269693.i 0.415144i
$$807$$ 0 0
$$808$$ −123333. −0.188911
$$809$$ 653300.i 0.998195i 0.866546 + 0.499097i $$0.166335\pi$$
−0.866546 + 0.499097i $$0.833665\pi$$
$$810$$ 0 0
$$811$$ −1.15300e6 −1.75302 −0.876510 0.481384i $$-0.840134\pi$$
−0.876510 + 0.481384i $$0.840134\pi$$
$$812$$ − 175026.i − 0.265454i
$$813$$ 0 0
$$814$$ 1.22888e6 1.85464
$$815$$ 306371.i 0.461246i
$$816$$ 0 0
$$817$$ −1.42679e6 −2.13755
$$818$$ − 119190.i − 0.178128i
$$819$$ 0 0
$$820$$ 358765. 0.533559
$$821$$ 440108.i 0.652939i 0.945208 + 0.326470i $$0.105859\pi$$
−0.945208 + 0.326470i $$0.894141\pi$$
$$822$$ 0 0
$$823$$ 628536. 0.927963 0.463981 0.885845i $$-0.346420\pi$$
0.463981 + 0.885845i $$0.346420\pi$$
$$824$$ − 109446.i − 0.161193i
$$825$$ 0 0
$$826$$ −163437. −0.239547
$$827$$ − 92273.2i − 0.134916i −0.997722 0.0674582i $$-0.978511\pi$$
0.997722 0.0674582i $$-0.0214889\pi$$
$$828$$ 0 0
$$829$$ −745793. −1.08520 −0.542599 0.839992i $$-0.682560\pi$$
−0.542599 + 0.839992i $$0.682560\pi$$
$$830$$ 257830.i 0.374263i
$$831$$ 0 0
$$832$$ 32194.8 0.0465092
$$833$$ − 64242.7i − 0.0925836i
$$834$$ 0 0
$$835$$ 286056. 0.410278
$$836$$ − 769017.i − 1.10033i
$$837$$ 0 0
$$838$$ −621961. −0.885677
$$839$$ − 957819.i − 1.36069i −0.732891 0.680346i $$-0.761830\pi$$
0.732891 0.680346i $$-0.238170\pi$$
$$840$$ 0 0
$$841$$ −688219. −0.973049
$$842$$ − 420259.i − 0.592779i
$$843$$ 0 0
$$844$$ 298996. 0.419740
$$845$$ 368454.i 0.516024i
$$846$$ 0 0
$$847$$ 527580. 0.735397
$$848$$ − 183257.i − 0.254841i
$$849$$ 0 0
$$850$$ 212322. 0.293872
$$851$$ − 953312.i − 1.31636i
$$852$$ 0 0
$$853$$ −943000. −1.29603 −0.648013 0.761629i $$-0.724400\pi$$
−0.648013 + 0.761629i $$0.724400\pi$$
$$854$$ − 168709.i − 0.231325i
$$855$$ 0 0
$$856$$ −168104. −0.229420
$$857$$ 601580.i 0.819090i 0.912290 + 0.409545i $$0.134313\pi$$
−0.912290 + 0.409545i $$0.865687\pi$$
$$858$$ 0 0
$$859$$ 456174. 0.618221 0.309111 0.951026i $$-0.399969\pi$$
0.309111 + 0.951026i $$0.399969\pi$$
$$860$$ − 369239.i − 0.499241i
$$861$$ 0 0
$$862$$ 963051. 1.29609
$$863$$ 387585.i 0.520410i 0.965553 + 0.260205i $$0.0837901\pi$$
−0.965553 + 0.260205i $$0.916210\pi$$
$$864$$ 0 0
$$865$$ −418252. −0.558993
$$866$$ − 966756.i − 1.28908i
$$867$$ 0 0
$$868$$ −224670. −0.298199
$$869$$ 707759.i 0.937230i
$$870$$ 0 0
$$871$$ 328958. 0.433614
$$872$$ − 361562.i − 0.475499i
$$873$$ 0 0
$$874$$ −596570. −0.780978
$$875$$ 284467.i 0.371548i
$$876$$ 0 0
$$877$$ 382540. 0.497368 0.248684 0.968585i $$-0.420002\pi$$
0.248684 + 0.968585i $$0.420002\pi$$
$$878$$ 96727.1i 0.125476i
$$879$$ 0 0
$$880$$ 199013. 0.256990
$$881$$ 166044.i 0.213930i 0.994263 + 0.106965i $$0.0341132\pi$$
−0.994263 + 0.106965i $$0.965887\pi$$
$$882$$ 0 0
$$883$$ −882718. −1.13214 −0.566071 0.824357i $$-0.691537\pi$$
−0.566071 + 0.824357i $$0.691537\pi$$
$$884$$ − 94218.4i − 0.120568i
$$885$$ 0 0
$$886$$ −360472. −0.459202
$$887$$ − 233588.i − 0.296896i −0.988920 0.148448i $$-0.952572\pi$$
0.988920 0.148448i $$-0.0474277\pi$$
$$888$$ 0 0
$$889$$ −466520. −0.590291
$$890$$ − 564212.i − 0.712299i
$$891$$ 0 0
$$892$$ 119439. 0.150113
$$893$$ − 764747.i − 0.958992i
$$894$$ 0 0
$$895$$ −401651. −0.501421
$$896$$ 26820.2i 0.0334077i
$$897$$ 0 0
$$898$$ −709866. −0.880286
$$899$$ 1.79132e6i 2.21643i
$$900$$ 0 0
$$901$$ −536304. −0.660635
$$902$$ 1.75922e6i 2.16225i
$$903$$ 0 0
$$904$$ 187437. 0.229360
$$905$$ 122995.i 0.150172i
$$906$$ 0 0
$$907$$ −192790. −0.234353 −0.117176 0.993111i $$-0.537384\pi$$
−0.117176 + 0.993111i $$0.537384\pi$$
$$908$$ − 649779.i − 0.788123i
$$909$$ 0 0
$$910$$ 49321.0 0.0595592
$$911$$ − 1.15573e6i − 1.39258i −0.717759 0.696292i $$-0.754832\pi$$
0.717759 0.696292i $$-0.245168\pi$$
$$912$$ 0 0
$$913$$ −1.26428e6 −1.51670
$$914$$ − 880666.i − 1.05419i
$$915$$ 0 0
$$916$$ 210309. 0.250650
$$917$$ 197575.i 0.234959i
$$918$$ 0 0
$$919$$ 1.17909e6 1.39610 0.698049 0.716050i $$-0.254052\pi$$
0.698049 + 0.716050i $$0.254052\pi$$
$$920$$ − 154386.i − 0.182403i
$$921$$ 0 0
$$922$$ −847728. −0.997229
$$923$$ − 164753.i − 0.193389i
$$924$$ 0 0
$$925$$ −838508. −0.979995
$$926$$ 991780.i 1.15663i
$$927$$ 0 0
$$928$$ 213840. 0.248310
$$929$$ − 630809.i − 0.730915i −0.930828 0.365457i $$-0.880913\pi$$
0.930828 0.365457i $$-0.119087\pi$$
$$930$$ 0 0
$$931$$ 158768. 0.183174
$$932$$ 619022.i 0.712647i
$$933$$ 0 0
$$934$$ −531559. −0.609338
$$935$$ − 582414.i − 0.666206i
$$936$$ 0 0
$$937$$ 936497. 1.06666 0.533331 0.845906i $$-0.320940\pi$$
0.533331 + 0.845906i $$0.320940\pi$$
$$938$$ 274042.i 0.311466i
$$939$$ 0 0
$$940$$ 197908. 0.223979
$$941$$ − 1.58769e6i − 1.79303i −0.443016 0.896514i $$-0.646092\pi$$
0.443016 0.896514i $$-0.353908\pi$$
$$942$$ 0 0
$$943$$ 1.36472e6 1.53469
$$944$$ − 199682.i − 0.224075i
$$945$$ 0 0
$$946$$ 1.81057e6 2.02318
$$947$$ − 75297.7i − 0.0839618i −0.999118 0.0419809i $$-0.986633\pi$$
0.999118 0.0419809i $$-0.0133669\pi$$
$$948$$ 0 0
$$949$$ 392155. 0.435437
$$950$$ 524727.i 0.581415i
$$951$$ 0 0
$$952$$ 78489.6 0.0866040
$$953$$ 20401.2i 0.0224632i 0.999937 + 0.0112316i $$0.00357520\pi$$
−0.999937 + 0.0112316i $$0.996425\pi$$
$$954$$ 0 0
$$955$$ −292405. −0.320611
$$956$$ − 1714.01i − 0.00187541i
$$957$$ 0 0
$$958$$ 1.24696e6 1.35869
$$959$$ − 179321.i − 0.194982i
$$960$$ 0 0
$$961$$ 1.37589e6 1.48983
$$962$$ 372089.i 0.402066i
$$963$$ 0 0
$$964$$ −25322.4 −0.0272490
$$965$$ − 25224.3i − 0.0270873i
$$966$$ 0 0
$$967$$ −619443. −0.662443 −0.331222 0.943553i $$-0.607461\pi$$
−0.331222 + 0.943553i $$0.607461\pi$$
$$968$$ 644580.i 0.687901i
$$969$$ 0 0
$$970$$ −257205. −0.273360
$$971$$ − 1.61783e6i − 1.71591i −0.513725 0.857955i $$-0.671735\pi$$
0.513725 0.857955i $$-0.328265\pi$$
$$972$$ 0 0
$$973$$ 331810. 0.350481
$$974$$ − 876221.i − 0.923625i
$$975$$ 0 0
$$976$$ 206123. 0.216385
$$977$$ 320209.i 0.335463i 0.985833 + 0.167732i $$0.0536442\pi$$
−0.985833 + 0.167732i $$0.946356\pi$$
$$978$$ 0 0
$$979$$ 2.76663e6 2.88659
$$980$$ 41087.4i 0.0427815i
$$981$$ 0 0
$$982$$ −123585. −0.128157
$$983$$ − 947603.i − 0.980662i −0.871536 0.490331i $$-0.836876\pi$$
0.871536 0.490331i $$-0.163124\pi$$
$$984$$ 0 0
$$985$$ 376055. 0.387595
$$986$$ − 625806.i − 0.643704i
$$987$$ 0 0
$$988$$ 232849. 0.238539
$$989$$ − 1.40457e6i − 1.43598i
$$990$$ 0 0
$$991$$ −309271. −0.314914 −0.157457 0.987526i $$-0.550330\pi$$
−0.157457 + 0.987526i $$0.550330\pi$$
$$992$$ − 274494.i − 0.278939i
$$993$$ 0 0
$$994$$ 137250. 0.138911
$$995$$ 721478.i 0.728747i
$$996$$ 0 0
$$997$$ 149332. 0.150232 0.0751159 0.997175i $$-0.476067\pi$$
0.0751159 + 0.997175i $$0.476067\pi$$
$$998$$ − 52223.9i − 0.0524334i
$$999$$ 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.323.5 yes 8
3.2 odd 2 inner 378.5.b.a.323.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.a.323.4 8 3.2 odd 2 inner
378.5.b.a.323.5 yes 8 1.1 even 1 trivial