# Properties

 Label 384.5.b.c.319.1 Level $384$ Weight $5$ Character 384.319 Analytic conductor $39.694$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 384.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.6940658242$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ x^8 + 3*x^6 + 5*x^4 + 12*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{28}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 319.1 Root $$1.09445 + 0.895644i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.319 Dual form 384.5.b.c.319.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.19615 q^{3} -39.7490i q^{5} -46.0431i q^{7} +27.0000 q^{9} +O(q^{10})$$ $$q-5.19615 q^{3} -39.7490i q^{5} -46.0431i q^{7} +27.0000 q^{9} +181.283 q^{11} +183.498i q^{13} +206.542i q^{15} +427.992 q^{17} +668.558 q^{19} +239.247i q^{21} -882.463i q^{23} -954.984 q^{25} -140.296 q^{27} +807.247i q^{29} +391.276i q^{31} -941.976 q^{33} -1830.17 q^{35} -466.980i q^{37} -953.484i q^{39} +2159.93 q^{41} -509.182 q^{43} -1073.22i q^{45} -2056.83i q^{47} +281.031 q^{49} -2223.91 q^{51} +753.725i q^{53} -7205.84i q^{55} -3473.93 q^{57} +1309.43 q^{59} +801.913i q^{61} -1243.16i q^{63} +7293.87 q^{65} -505.813 q^{67} +4585.41i q^{69} +2170.94i q^{71} -2297.97 q^{73} +4962.24 q^{75} -8346.85i q^{77} -10705.3i q^{79} +729.000 q^{81} +2977.81 q^{83} -17012.3i q^{85} -4194.58i q^{87} -6493.92 q^{89} +8448.82 q^{91} -2033.13i q^{93} -26574.5i q^{95} -8189.92 q^{97} +4894.65 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 216 q^{9}+O(q^{10})$$ 8 * q + 216 * q^9 $$8 q + 216 q^{9} + 1392 q^{17} - 3576 q^{25} - 1440 q^{33} - 1008 q^{41} + 10376 q^{49} - 9504 q^{57} + 23808 q^{65} - 10256 q^{73} + 5832 q^{81} - 31632 q^{89} - 45200 q^{97}+O(q^{100})$$ 8 * q + 216 * q^9 + 1392 * q^17 - 3576 * q^25 - 1440 * q^33 - 1008 * q^41 + 10376 * q^49 - 9504 * q^57 + 23808 * q^65 - 10256 * q^73 + 5832 * q^81 - 31632 * q^89 - 45200 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −5.19615 −0.577350
$$4$$ 0 0
$$5$$ − 39.7490i − 1.58996i −0.606635 0.794980i $$-0.707481\pi$$
0.606635 0.794980i $$-0.292519\pi$$
$$6$$ 0 0
$$7$$ − 46.0431i − 0.939655i −0.882758 0.469828i $$-0.844316\pi$$
0.882758 0.469828i $$-0.155684\pi$$
$$8$$ 0 0
$$9$$ 27.0000 0.333333
$$10$$ 0 0
$$11$$ 181.283 1.49821 0.749105 0.662451i $$-0.230484\pi$$
0.749105 + 0.662451i $$0.230484\pi$$
$$12$$ 0 0
$$13$$ 183.498i 1.08579i 0.839801 + 0.542894i $$0.182671\pi$$
−0.839801 + 0.542894i $$0.817329\pi$$
$$14$$ 0 0
$$15$$ 206.542i 0.917964i
$$16$$ 0 0
$$17$$ 427.992 1.48094 0.740471 0.672089i $$-0.234603\pi$$
0.740471 + 0.672089i $$0.234603\pi$$
$$18$$ 0 0
$$19$$ 668.558 1.85196 0.925981 0.377571i $$-0.123241\pi$$
0.925981 + 0.377571i $$0.123241\pi$$
$$20$$ 0 0
$$21$$ 239.247i 0.542510i
$$22$$ 0 0
$$23$$ − 882.463i − 1.66817i −0.551635 0.834086i $$-0.685996\pi$$
0.551635 0.834086i $$-0.314004\pi$$
$$24$$ 0 0
$$25$$ −954.984 −1.52797
$$26$$ 0 0
$$27$$ −140.296 −0.192450
$$28$$ 0 0
$$29$$ 807.247i 0.959866i 0.877305 + 0.479933i $$0.159339\pi$$
−0.877305 + 0.479933i $$0.840661\pi$$
$$30$$ 0 0
$$31$$ 391.276i 0.407155i 0.979059 + 0.203577i $$0.0652569\pi$$
−0.979059 + 0.203577i $$0.934743\pi$$
$$32$$ 0 0
$$33$$ −941.976 −0.864992
$$34$$ 0 0
$$35$$ −1830.17 −1.49402
$$36$$ 0 0
$$37$$ − 466.980i − 0.341111i −0.985348 0.170555i $$-0.945444\pi$$
0.985348 0.170555i $$-0.0545561\pi$$
$$38$$ 0 0
$$39$$ − 953.484i − 0.626880i
$$40$$ 0 0
$$41$$ 2159.93 1.28491 0.642454 0.766325i $$-0.277917\pi$$
0.642454 + 0.766325i $$0.277917\pi$$
$$42$$ 0 0
$$43$$ −509.182 −0.275382 −0.137691 0.990475i $$-0.543968\pi$$
−0.137691 + 0.990475i $$0.543968\pi$$
$$44$$ 0 0
$$45$$ − 1073.22i − 0.529987i
$$46$$ 0 0
$$47$$ − 2056.83i − 0.931116i −0.885017 0.465558i $$-0.845854\pi$$
0.885017 0.465558i $$-0.154146\pi$$
$$48$$ 0 0
$$49$$ 281.031 0.117048
$$50$$ 0 0
$$51$$ −2223.91 −0.855022
$$52$$ 0 0
$$53$$ 753.725i 0.268325i 0.990959 + 0.134163i $$0.0428344\pi$$
−0.990959 + 0.134163i $$0.957166\pi$$
$$54$$ 0 0
$$55$$ − 7205.84i − 2.38210i
$$56$$ 0 0
$$57$$ −3473.93 −1.06923
$$58$$ 0 0
$$59$$ 1309.43 0.376165 0.188083 0.982153i $$-0.439773\pi$$
0.188083 + 0.982153i $$0.439773\pi$$
$$60$$ 0 0
$$61$$ 801.913i 0.215510i 0.994177 + 0.107755i $$0.0343662\pi$$
−0.994177 + 0.107755i $$0.965634\pi$$
$$62$$ 0 0
$$63$$ − 1243.16i − 0.313218i
$$64$$ 0 0
$$65$$ 7293.87 1.72636
$$66$$ 0 0
$$67$$ −505.813 −0.112678 −0.0563392 0.998412i $$-0.517943\pi$$
−0.0563392 + 0.998412i $$0.517943\pi$$
$$68$$ 0 0
$$69$$ 4585.41i 0.963119i
$$70$$ 0 0
$$71$$ 2170.94i 0.430658i 0.976542 + 0.215329i $$0.0690823\pi$$
−0.976542 + 0.215329i $$0.930918\pi$$
$$72$$ 0 0
$$73$$ −2297.97 −0.431219 −0.215610 0.976480i $$-0.569174\pi$$
−0.215610 + 0.976480i $$0.569174\pi$$
$$74$$ 0 0
$$75$$ 4962.24 0.882177
$$76$$ 0 0
$$77$$ − 8346.85i − 1.40780i
$$78$$ 0 0
$$79$$ − 10705.3i − 1.71532i −0.514219 0.857659i $$-0.671918\pi$$
0.514219 0.857659i $$-0.328082\pi$$
$$80$$ 0 0
$$81$$ 729.000 0.111111
$$82$$ 0 0
$$83$$ 2977.81 0.432256 0.216128 0.976365i $$-0.430657\pi$$
0.216128 + 0.976365i $$0.430657\pi$$
$$84$$ 0 0
$$85$$ − 17012.3i − 2.35464i
$$86$$ 0 0
$$87$$ − 4194.58i − 0.554179i
$$88$$ 0 0
$$89$$ −6493.92 −0.819836 −0.409918 0.912122i $$-0.634443\pi$$
−0.409918 + 0.912122i $$0.634443\pi$$
$$90$$ 0 0
$$91$$ 8448.82 1.02027
$$92$$ 0 0
$$93$$ − 2033.13i − 0.235071i
$$94$$ 0 0
$$95$$ − 26574.5i − 2.94455i
$$96$$ 0 0
$$97$$ −8189.92 −0.870435 −0.435217 0.900325i $$-0.643328\pi$$
−0.435217 + 0.900325i $$0.643328\pi$$
$$98$$ 0 0
$$99$$ 4894.65 0.499403
$$100$$ 0 0
$$101$$ 15576.5i 1.52696i 0.645833 + 0.763479i $$0.276510\pi$$
−0.645833 + 0.763479i $$0.723490\pi$$
$$102$$ 0 0
$$103$$ 6628.58i 0.624807i 0.949950 + 0.312403i $$0.101134\pi$$
−0.949950 + 0.312403i $$0.898866\pi$$
$$104$$ 0 0
$$105$$ 9509.83 0.862570
$$106$$ 0 0
$$107$$ 11796.4 1.03035 0.515173 0.857086i $$-0.327728\pi$$
0.515173 + 0.857086i $$0.327728\pi$$
$$108$$ 0 0
$$109$$ 13286.1i 1.11826i 0.829080 + 0.559130i $$0.188865\pi$$
−0.829080 + 0.559130i $$0.811135\pi$$
$$110$$ 0 0
$$111$$ 2426.50i 0.196940i
$$112$$ 0 0
$$113$$ −11713.6 −0.917350 −0.458675 0.888604i $$-0.651676\pi$$
−0.458675 + 0.888604i $$0.651676\pi$$
$$114$$ 0 0
$$115$$ −35077.0 −2.65233
$$116$$ 0 0
$$117$$ 4954.45i 0.361929i
$$118$$ 0 0
$$119$$ − 19706.1i − 1.39157i
$$120$$ 0 0
$$121$$ 18222.7 1.24463
$$122$$ 0 0
$$123$$ −11223.3 −0.741842
$$124$$ 0 0
$$125$$ 13116.5i 0.839459i
$$126$$ 0 0
$$127$$ − 5640.55i − 0.349715i −0.984594 0.174858i $$-0.944054\pi$$
0.984594 0.174858i $$-0.0559465\pi$$
$$128$$ 0 0
$$129$$ 2645.79 0.158992
$$130$$ 0 0
$$131$$ −31922.5 −1.86018 −0.930089 0.367335i $$-0.880270\pi$$
−0.930089 + 0.367335i $$0.880270\pi$$
$$132$$ 0 0
$$133$$ − 30782.5i − 1.74021i
$$134$$ 0 0
$$135$$ 5576.63i 0.305988i
$$136$$ 0 0
$$137$$ −10515.6 −0.560263 −0.280132 0.959962i $$-0.590378\pi$$
−0.280132 + 0.959962i $$0.590378\pi$$
$$138$$ 0 0
$$139$$ 14985.9 0.775626 0.387813 0.921738i $$-0.373231\pi$$
0.387813 + 0.921738i $$0.373231\pi$$
$$140$$ 0 0
$$141$$ 10687.6i 0.537580i
$$142$$ 0 0
$$143$$ 33265.2i 1.62674i
$$144$$ 0 0
$$145$$ 32087.3 1.52615
$$146$$ 0 0
$$147$$ −1460.28 −0.0675775
$$148$$ 0 0
$$149$$ 11795.4i 0.531298i 0.964070 + 0.265649i $$0.0855863\pi$$
−0.964070 + 0.265649i $$0.914414\pi$$
$$150$$ 0 0
$$151$$ − 33792.9i − 1.48208i −0.671460 0.741040i $$-0.734333\pi$$
0.671460 0.741040i $$-0.265667\pi$$
$$152$$ 0 0
$$153$$ 11555.8 0.493647
$$154$$ 0 0
$$155$$ 15552.8 0.647360
$$156$$ 0 0
$$157$$ − 28788.9i − 1.16796i −0.811770 0.583978i $$-0.801496\pi$$
0.811770 0.583978i $$-0.198504\pi$$
$$158$$ 0 0
$$159$$ − 3916.47i − 0.154918i
$$160$$ 0 0
$$161$$ −40631.3 −1.56751
$$162$$ 0 0
$$163$$ 13409.0 0.504684 0.252342 0.967638i $$-0.418799\pi$$
0.252342 + 0.967638i $$0.418799\pi$$
$$164$$ 0 0
$$165$$ 37442.6i 1.37530i
$$166$$ 0 0
$$167$$ 13206.7i 0.473546i 0.971565 + 0.236773i $$0.0760898\pi$$
−0.971565 + 0.236773i $$0.923910\pi$$
$$168$$ 0 0
$$169$$ −5110.53 −0.178934
$$170$$ 0 0
$$171$$ 18051.1 0.617320
$$172$$ 0 0
$$173$$ − 364.956i − 0.0121940i −0.999981 0.00609702i $$-0.998059\pi$$
0.999981 0.00609702i $$-0.00194076\pi$$
$$174$$ 0 0
$$175$$ 43970.5i 1.43577i
$$176$$ 0 0
$$177$$ −6804.00 −0.217179
$$178$$ 0 0
$$179$$ −14372.0 −0.448549 −0.224275 0.974526i $$-0.572001\pi$$
−0.224275 + 0.974526i $$0.572001\pi$$
$$180$$ 0 0
$$181$$ 4050.54i 0.123639i 0.998087 + 0.0618195i $$0.0196903\pi$$
−0.998087 + 0.0618195i $$0.980310\pi$$
$$182$$ 0 0
$$183$$ − 4166.86i − 0.124425i
$$184$$ 0 0
$$185$$ −18562.0 −0.542352
$$186$$ 0 0
$$187$$ 77587.9 2.21876
$$188$$ 0 0
$$189$$ 6459.67i 0.180837i
$$190$$ 0 0
$$191$$ 49880.9i 1.36731i 0.729805 + 0.683656i $$0.239611\pi$$
−0.729805 + 0.683656i $$0.760389\pi$$
$$192$$ 0 0
$$193$$ −48425.7 −1.30005 −0.650026 0.759912i $$-0.725242\pi$$
−0.650026 + 0.759912i $$0.725242\pi$$
$$194$$ 0 0
$$195$$ −37900.0 −0.996714
$$196$$ 0 0
$$197$$ 5556.74i 0.143182i 0.997434 + 0.0715908i $$0.0228076\pi$$
−0.997434 + 0.0715908i $$0.977192\pi$$
$$198$$ 0 0
$$199$$ 60594.7i 1.53013i 0.643952 + 0.765066i $$0.277294\pi$$
−0.643952 + 0.765066i $$0.722706\pi$$
$$200$$ 0 0
$$201$$ 2628.28 0.0650549
$$202$$ 0 0
$$203$$ 37168.2 0.901943
$$204$$ 0 0
$$205$$ − 85855.1i − 2.04295i
$$206$$ 0 0
$$207$$ − 23826.5i − 0.556057i
$$208$$ 0 0
$$209$$ 121198. 2.77463
$$210$$ 0 0
$$211$$ −27539.9 −0.618583 −0.309292 0.950967i $$-0.600092\pi$$
−0.309292 + 0.950967i $$0.600092\pi$$
$$212$$ 0 0
$$213$$ − 11280.6i − 0.248640i
$$214$$ 0 0
$$215$$ 20239.5i 0.437847i
$$216$$ 0 0
$$217$$ 18015.6 0.382585
$$218$$ 0 0
$$219$$ 11940.6 0.248965
$$220$$ 0 0
$$221$$ 78535.7i 1.60799i
$$222$$ 0 0
$$223$$ 3021.35i 0.0607564i 0.999538 + 0.0303782i $$0.00967116\pi$$
−0.999538 + 0.0303782i $$0.990329\pi$$
$$224$$ 0 0
$$225$$ −25784.6 −0.509325
$$226$$ 0 0
$$227$$ 7077.28 0.137346 0.0686728 0.997639i $$-0.478124\pi$$
0.0686728 + 0.997639i $$0.478124\pi$$
$$228$$ 0 0
$$229$$ − 102761.i − 1.95956i −0.200088 0.979778i $$-0.564123\pi$$
0.200088 0.979778i $$-0.435877\pi$$
$$230$$ 0 0
$$231$$ 43371.5i 0.812795i
$$232$$ 0 0
$$233$$ 22976.6 0.423227 0.211613 0.977353i $$-0.432128\pi$$
0.211613 + 0.977353i $$0.432128\pi$$
$$234$$ 0 0
$$235$$ −81757.1 −1.48044
$$236$$ 0 0
$$237$$ 55626.4i 0.990339i
$$238$$ 0 0
$$239$$ 35566.7i 0.622656i 0.950303 + 0.311328i $$0.100774\pi$$
−0.950303 + 0.311328i $$0.899226\pi$$
$$240$$ 0 0
$$241$$ −92683.4 −1.59576 −0.797881 0.602816i $$-0.794045\pi$$
−0.797881 + 0.602816i $$0.794045\pi$$
$$242$$ 0 0
$$243$$ −3788.00 −0.0641500
$$244$$ 0 0
$$245$$ − 11170.7i − 0.186101i
$$246$$ 0 0
$$247$$ 122679.i 2.01084i
$$248$$ 0 0
$$249$$ −15473.2 −0.249563
$$250$$ 0 0
$$251$$ −27590.5 −0.437937 −0.218968 0.975732i $$-0.570269\pi$$
−0.218968 + 0.975732i $$0.570269\pi$$
$$252$$ 0 0
$$253$$ − 159976.i − 2.49927i
$$254$$ 0 0
$$255$$ 88398.3i 1.35945i
$$256$$ 0 0
$$257$$ 92304.6 1.39752 0.698758 0.715358i $$-0.253736\pi$$
0.698758 + 0.715358i $$0.253736\pi$$
$$258$$ 0 0
$$259$$ −21501.2 −0.320526
$$260$$ 0 0
$$261$$ 21795.7i 0.319955i
$$262$$ 0 0
$$263$$ 13884.4i 0.200732i 0.994951 + 0.100366i $$0.0320014\pi$$
−0.994951 + 0.100366i $$0.967999\pi$$
$$264$$ 0 0
$$265$$ 29959.8 0.426626
$$266$$ 0 0
$$267$$ 33743.4 0.473333
$$268$$ 0 0
$$269$$ 52698.1i 0.728266i 0.931347 + 0.364133i $$0.118635\pi$$
−0.931347 + 0.364133i $$0.881365\pi$$
$$270$$ 0 0
$$271$$ − 47028.8i − 0.640361i −0.947356 0.320181i $$-0.896256\pi$$
0.947356 0.320181i $$-0.103744\pi$$
$$272$$ 0 0
$$273$$ −43901.4 −0.589051
$$274$$ 0 0
$$275$$ −173123. −2.28923
$$276$$ 0 0
$$277$$ 108393.i 1.41268i 0.707873 + 0.706340i $$0.249655\pi$$
−0.707873 + 0.706340i $$0.750345\pi$$
$$278$$ 0 0
$$279$$ 10564.4i 0.135718i
$$280$$ 0 0
$$281$$ −73090.7 −0.925656 −0.462828 0.886448i $$-0.653165\pi$$
−0.462828 + 0.886448i $$0.653165\pi$$
$$282$$ 0 0
$$283$$ −44763.1 −0.558917 −0.279458 0.960158i $$-0.590155\pi$$
−0.279458 + 0.960158i $$0.590155\pi$$
$$284$$ 0 0
$$285$$ 138085.i 1.70003i
$$286$$ 0 0
$$287$$ − 99449.9i − 1.20737i
$$288$$ 0 0
$$289$$ 99656.3 1.19319
$$290$$ 0 0
$$291$$ 42556.1 0.502546
$$292$$ 0 0
$$293$$ 42030.2i 0.489583i 0.969576 + 0.244792i $$0.0787195\pi$$
−0.969576 + 0.244792i $$0.921280\pi$$
$$294$$ 0 0
$$295$$ − 52048.6i − 0.598088i
$$296$$ 0 0
$$297$$ −25433.4 −0.288331
$$298$$ 0 0
$$299$$ 161930. 1.81128
$$300$$ 0 0
$$301$$ 23444.3i 0.258765i
$$302$$ 0 0
$$303$$ − 80937.9i − 0.881590i
$$304$$ 0 0
$$305$$ 31875.3 0.342653
$$306$$ 0 0
$$307$$ −8820.31 −0.0935852 −0.0467926 0.998905i $$-0.514900\pi$$
−0.0467926 + 0.998905i $$0.514900\pi$$
$$308$$ 0 0
$$309$$ − 34443.1i − 0.360732i
$$310$$ 0 0
$$311$$ 155059.i 1.60315i 0.597893 + 0.801576i $$0.296005\pi$$
−0.597893 + 0.801576i $$0.703995\pi$$
$$312$$ 0 0
$$313$$ −179666. −1.83390 −0.916951 0.398999i $$-0.869358\pi$$
−0.916951 + 0.398999i $$0.869358\pi$$
$$314$$ 0 0
$$315$$ −49414.6 −0.498005
$$316$$ 0 0
$$317$$ 6188.77i 0.0615865i 0.999526 + 0.0307933i $$0.00980335\pi$$
−0.999526 + 0.0307933i $$0.990197\pi$$
$$318$$ 0 0
$$319$$ 146341.i 1.43808i
$$320$$ 0 0
$$321$$ −61296.0 −0.594870
$$322$$ 0 0
$$323$$ 286138. 2.74265
$$324$$ 0 0
$$325$$ − 175238.i − 1.65906i
$$326$$ 0 0
$$327$$ − 69036.4i − 0.645628i
$$328$$ 0 0
$$329$$ −94703.1 −0.874928
$$330$$ 0 0
$$331$$ 132159. 1.20626 0.603130 0.797643i $$-0.293920\pi$$
0.603130 + 0.797643i $$0.293920\pi$$
$$332$$ 0 0
$$333$$ − 12608.5i − 0.113704i
$$334$$ 0 0
$$335$$ 20105.6i 0.179154i
$$336$$ 0 0
$$337$$ 70040.0 0.616718 0.308359 0.951270i $$-0.400220\pi$$
0.308359 + 0.951270i $$0.400220\pi$$
$$338$$ 0 0
$$339$$ 60865.8 0.529632
$$340$$ 0 0
$$341$$ 70931.8i 0.610004i
$$342$$ 0 0
$$343$$ − 123489.i − 1.04964i
$$344$$ 0 0
$$345$$ 182266. 1.53132
$$346$$ 0 0
$$347$$ −157722. −1.30988 −0.654942 0.755680i $$-0.727307\pi$$
−0.654942 + 0.755680i $$0.727307\pi$$
$$348$$ 0 0
$$349$$ − 198747.i − 1.63174i −0.578238 0.815868i $$-0.696259\pi$$
0.578238 0.815868i $$-0.303741\pi$$
$$350$$ 0 0
$$351$$ − 25744.1i − 0.208960i
$$352$$ 0 0
$$353$$ 13376.1 0.107345 0.0536724 0.998559i $$-0.482907\pi$$
0.0536724 + 0.998559i $$0.482907\pi$$
$$354$$ 0 0
$$355$$ 86292.9 0.684729
$$356$$ 0 0
$$357$$ 102396.i 0.803426i
$$358$$ 0 0
$$359$$ − 190011.i − 1.47432i −0.675721 0.737158i $$-0.736168\pi$$
0.675721 0.737158i $$-0.263832\pi$$
$$360$$ 0 0
$$361$$ 316649. 2.42976
$$362$$ 0 0
$$363$$ −94687.8 −0.718590
$$364$$ 0 0
$$365$$ 91342.0i 0.685622i
$$366$$ 0 0
$$367$$ − 94140.5i − 0.698947i −0.936946 0.349474i $$-0.886360\pi$$
0.936946 0.349474i $$-0.113640\pi$$
$$368$$ 0 0
$$369$$ 58318.1 0.428302
$$370$$ 0 0
$$371$$ 34703.9 0.252133
$$372$$ 0 0
$$373$$ 66152.4i 0.475475i 0.971329 + 0.237738i $$0.0764058\pi$$
−0.971329 + 0.237738i $$0.923594\pi$$
$$374$$ 0 0
$$375$$ − 68155.6i − 0.484662i
$$376$$ 0 0
$$377$$ −148128. −1.04221
$$378$$ 0 0
$$379$$ −64395.2 −0.448306 −0.224153 0.974554i $$-0.571962\pi$$
−0.224153 + 0.974554i $$0.571962\pi$$
$$380$$ 0 0
$$381$$ 29309.2i 0.201908i
$$382$$ 0 0
$$383$$ − 30714.5i − 0.209385i −0.994505 0.104693i $$-0.966614\pi$$
0.994505 0.104693i $$-0.0333859\pi$$
$$384$$ 0 0
$$385$$ −331779. −2.23835
$$386$$ 0 0
$$387$$ −13747.9 −0.0917941
$$388$$ 0 0
$$389$$ 199122.i 1.31589i 0.753066 + 0.657945i $$0.228574\pi$$
−0.753066 + 0.657945i $$0.771426\pi$$
$$390$$ 0 0
$$391$$ − 377687.i − 2.47046i
$$392$$ 0 0
$$393$$ 165874. 1.07397
$$394$$ 0 0
$$395$$ −425525. −2.72729
$$396$$ 0 0
$$397$$ − 68565.8i − 0.435037i −0.976056 0.217519i $$-0.930204\pi$$
0.976056 0.217519i $$-0.0697963\pi$$
$$398$$ 0 0
$$399$$ 159951.i 1.00471i
$$400$$ 0 0
$$401$$ −21797.1 −0.135553 −0.0677765 0.997701i $$-0.521590\pi$$
−0.0677765 + 0.997701i $$0.521590\pi$$
$$402$$ 0 0
$$403$$ −71798.3 −0.442084
$$404$$ 0 0
$$405$$ − 28977.0i − 0.176662i
$$406$$ 0 0
$$407$$ − 84655.8i − 0.511055i
$$408$$ 0 0
$$409$$ 230211. 1.37620 0.688098 0.725618i $$-0.258446\pi$$
0.688098 + 0.725618i $$0.258446\pi$$
$$410$$ 0 0
$$411$$ 54640.6 0.323468
$$412$$ 0 0
$$413$$ − 60290.3i − 0.353465i
$$414$$ 0 0
$$415$$ − 118365.i − 0.687270i
$$416$$ 0 0
$$417$$ −77868.9 −0.447808
$$418$$ 0 0
$$419$$ −17360.5 −0.0988859 −0.0494430 0.998777i $$-0.515745\pi$$
−0.0494430 + 0.998777i $$0.515745\pi$$
$$420$$ 0 0
$$421$$ − 186829.i − 1.05410i −0.849836 0.527048i $$-0.823299\pi$$
0.849836 0.527048i $$-0.176701\pi$$
$$422$$ 0 0
$$423$$ − 55534.5i − 0.310372i
$$424$$ 0 0
$$425$$ −408726. −2.26284
$$426$$ 0 0
$$427$$ 36922.6 0.202505
$$428$$ 0 0
$$429$$ − 172851.i − 0.939197i
$$430$$ 0 0
$$431$$ − 153753.i − 0.827693i −0.910347 0.413846i $$-0.864185\pi$$
0.910347 0.413846i $$-0.135815\pi$$
$$432$$ 0 0
$$433$$ 9168.87 0.0489035 0.0244517 0.999701i $$-0.492216\pi$$
0.0244517 + 0.999701i $$0.492216\pi$$
$$434$$ 0 0
$$435$$ −166730. −0.881122
$$436$$ 0 0
$$437$$ − 589978.i − 3.08939i
$$438$$ 0 0
$$439$$ 178760.i 0.927561i 0.885950 + 0.463780i $$0.153507\pi$$
−0.885950 + 0.463780i $$0.846493\pi$$
$$440$$ 0 0
$$441$$ 7587.85 0.0390159
$$442$$ 0 0
$$443$$ 17798.2 0.0906920 0.0453460 0.998971i $$-0.485561\pi$$
0.0453460 + 0.998971i $$0.485561\pi$$
$$444$$ 0 0
$$445$$ 258127.i 1.30351i
$$446$$ 0 0
$$447$$ − 61290.5i − 0.306745i
$$448$$ 0 0
$$449$$ 242915. 1.20493 0.602464 0.798146i $$-0.294186\pi$$
0.602464 + 0.798146i $$0.294186\pi$$
$$450$$ 0 0
$$451$$ 391559. 1.92506
$$452$$ 0 0
$$453$$ 175593.i 0.855680i
$$454$$ 0 0
$$455$$ − 335832.i − 1.62218i
$$456$$ 0 0
$$457$$ −190762. −0.913396 −0.456698 0.889622i $$-0.650968\pi$$
−0.456698 + 0.889622i $$0.650968\pi$$
$$458$$ 0 0
$$459$$ −60045.6 −0.285007
$$460$$ 0 0
$$461$$ − 155121.i − 0.729909i −0.931025 0.364955i $$-0.881084\pi$$
0.931025 0.364955i $$-0.118916\pi$$
$$462$$ 0 0
$$463$$ − 230925.i − 1.07723i −0.842551 0.538616i $$-0.818947\pi$$
0.842551 0.538616i $$-0.181053\pi$$
$$464$$ 0 0
$$465$$ −80814.9 −0.373754
$$466$$ 0 0
$$467$$ 154112. 0.706647 0.353323 0.935501i $$-0.385052\pi$$
0.353323 + 0.935501i $$0.385052\pi$$
$$468$$ 0 0
$$469$$ 23289.2i 0.105879i
$$470$$ 0 0
$$471$$ 149592.i 0.674319i
$$472$$ 0 0
$$473$$ −92306.3 −0.412581
$$474$$ 0 0
$$475$$ −638462. −2.82975
$$476$$ 0 0
$$477$$ 20350.6i 0.0894417i
$$478$$ 0 0
$$479$$ − 405668.i − 1.76807i −0.467420 0.884035i $$-0.654816\pi$$
0.467420 0.884035i $$-0.345184\pi$$
$$480$$ 0 0
$$481$$ 85690.0 0.370373
$$482$$ 0 0
$$483$$ 211127. 0.905000
$$484$$ 0 0
$$485$$ 325541.i 1.38396i
$$486$$ 0 0
$$487$$ 426114.i 1.79667i 0.439314 + 0.898334i $$0.355222\pi$$
−0.439314 + 0.898334i $$0.644778\pi$$
$$488$$ 0 0
$$489$$ −69675.0 −0.291379
$$490$$ 0 0
$$491$$ 273125. 1.13292 0.566459 0.824090i $$-0.308313\pi$$
0.566459 + 0.824090i $$0.308313\pi$$
$$492$$ 0 0
$$493$$ 345495.i 1.42151i
$$494$$ 0 0
$$495$$ − 194558.i − 0.794032i
$$496$$ 0 0
$$497$$ 99957.1 0.404670
$$498$$ 0 0
$$499$$ 103548. 0.415855 0.207928 0.978144i $$-0.433328\pi$$
0.207928 + 0.978144i $$0.433328\pi$$
$$500$$ 0 0
$$501$$ − 68624.2i − 0.273402i
$$502$$ 0 0
$$503$$ 217063.i 0.857925i 0.903322 + 0.428963i $$0.141121\pi$$
−0.903322 + 0.428963i $$0.858879\pi$$
$$504$$ 0 0
$$505$$ 619151. 2.42780
$$506$$ 0 0
$$507$$ 26555.1 0.103307
$$508$$ 0 0
$$509$$ 115098.i 0.444256i 0.975018 + 0.222128i $$0.0713003\pi$$
−0.975018 + 0.222128i $$0.928700\pi$$
$$510$$ 0 0
$$511$$ 105806.i 0.405198i
$$512$$ 0 0
$$513$$ −93796.1 −0.356410
$$514$$ 0 0
$$515$$ 263479. 0.993418
$$516$$ 0 0
$$517$$ − 372870.i − 1.39501i
$$518$$ 0 0
$$519$$ 1896.37i 0.00704024i
$$520$$ 0 0
$$521$$ 288543. 1.06300 0.531502 0.847057i $$-0.321628\pi$$
0.531502 + 0.847057i $$0.321628\pi$$
$$522$$ 0 0
$$523$$ 176826. 0.646461 0.323231 0.946320i $$-0.395231\pi$$
0.323231 + 0.946320i $$0.395231\pi$$
$$524$$ 0 0
$$525$$ − 228477.i − 0.828942i
$$526$$ 0 0
$$527$$ 167463.i 0.602973i
$$528$$ 0 0
$$529$$ −498900. −1.78280
$$530$$ 0 0
$$531$$ 35354.6 0.125388
$$532$$ 0 0
$$533$$ 396343.i 1.39514i
$$534$$ 0 0
$$535$$ − 468896.i − 1.63821i
$$536$$ 0 0
$$537$$ 74678.9 0.258970
$$538$$ 0 0
$$539$$ 50946.4 0.175362
$$540$$ 0 0
$$541$$ 425220.i 1.45284i 0.687249 + 0.726422i $$0.258818\pi$$
−0.687249 + 0.726422i $$0.741182\pi$$
$$542$$ 0 0
$$543$$ − 21047.2i − 0.0713830i
$$544$$ 0 0
$$545$$ 528108. 1.77799
$$546$$ 0 0
$$547$$ 495505. 1.65605 0.828024 0.560692i $$-0.189465\pi$$
0.828024 + 0.560692i $$0.189465\pi$$
$$548$$ 0 0
$$549$$ 21651.7i 0.0718367i
$$550$$ 0 0
$$551$$ 539691.i 1.77763i
$$552$$ 0 0
$$553$$ −492905. −1.61181
$$554$$ 0 0
$$555$$ 96451.0 0.313127
$$556$$ 0 0
$$557$$ − 257895.i − 0.831251i −0.909536 0.415626i $$-0.863563\pi$$
0.909536 0.415626i $$-0.136437\pi$$
$$558$$ 0 0
$$559$$ − 93433.9i − 0.299007i
$$560$$ 0 0
$$561$$ −403158. −1.28100
$$562$$ 0 0
$$563$$ −417799. −1.31811 −0.659054 0.752096i $$-0.729043\pi$$
−0.659054 + 0.752096i $$0.729043\pi$$
$$564$$ 0 0
$$565$$ 465606.i 1.45855i
$$566$$ 0 0
$$567$$ − 33565.4i − 0.104406i
$$568$$ 0 0
$$569$$ −100935. −0.311756 −0.155878 0.987776i $$-0.549821\pi$$
−0.155878 + 0.987776i $$0.549821\pi$$
$$570$$ 0 0
$$571$$ −453320. −1.39038 −0.695188 0.718828i $$-0.744679\pi$$
−0.695188 + 0.718828i $$0.744679\pi$$
$$572$$ 0 0
$$573$$ − 259189.i − 0.789418i
$$574$$ 0 0
$$575$$ 842738.i 2.54892i
$$576$$ 0 0
$$577$$ 12119.6 0.0364031 0.0182015 0.999834i $$-0.494206\pi$$
0.0182015 + 0.999834i $$0.494206\pi$$
$$578$$ 0 0
$$579$$ 251627. 0.750586
$$580$$ 0 0
$$581$$ − 137108.i − 0.406172i
$$582$$ 0 0
$$583$$ 136638.i 0.402008i
$$584$$ 0 0
$$585$$ 196934. 0.575453
$$586$$ 0 0
$$587$$ −327468. −0.950370 −0.475185 0.879886i $$-0.657619\pi$$
−0.475185 + 0.879886i $$0.657619\pi$$
$$588$$ 0 0
$$589$$ 261591.i 0.754035i
$$590$$ 0 0
$$591$$ − 28873.7i − 0.0826660i
$$592$$ 0 0
$$593$$ 610182. 1.73520 0.867601 0.497260i $$-0.165661\pi$$
0.867601 + 0.497260i $$0.165661\pi$$
$$594$$ 0 0
$$595$$ −783298. −2.21255
$$596$$ 0 0
$$597$$ − 314859.i − 0.883422i
$$598$$ 0 0
$$599$$ − 499941.i − 1.39337i −0.717379 0.696683i $$-0.754658\pi$$
0.717379 0.696683i $$-0.245342\pi$$
$$600$$ 0 0
$$601$$ −486965. −1.34818 −0.674091 0.738649i $$-0.735464\pi$$
−0.674091 + 0.738649i $$0.735464\pi$$
$$602$$ 0 0
$$603$$ −13657.0 −0.0375595
$$604$$ 0 0
$$605$$ − 724334.i − 1.97892i
$$606$$ 0 0
$$607$$ − 41693.4i − 0.113159i −0.998398 0.0565796i $$-0.981981\pi$$
0.998398 0.0565796i $$-0.0180195\pi$$
$$608$$ 0 0
$$609$$ −193131. −0.520737
$$610$$ 0 0
$$611$$ 377425. 1.01099
$$612$$ 0 0
$$613$$ 542042.i 1.44249i 0.692681 + 0.721244i $$0.256429\pi$$
−0.692681 + 0.721244i $$0.743571\pi$$
$$614$$ 0 0
$$615$$ 446116.i 1.17950i
$$616$$ 0 0
$$617$$ −146814. −0.385653 −0.192827 0.981233i $$-0.561765\pi$$
−0.192827 + 0.981233i $$0.561765\pi$$
$$618$$ 0 0
$$619$$ −72720.6 −0.189791 −0.0948956 0.995487i $$-0.530252\pi$$
−0.0948956 + 0.995487i $$0.530252\pi$$
$$620$$ 0 0
$$621$$ 123806.i 0.321040i
$$622$$ 0 0
$$623$$ 299000.i 0.770363i
$$624$$ 0 0
$$625$$ −75495.2 −0.193268
$$626$$ 0 0
$$627$$ −629766. −1.60193
$$628$$ 0 0
$$629$$ − 199864.i − 0.505165i
$$630$$ 0 0
$$631$$ 568570.i 1.42799i 0.700151 + 0.713995i $$0.253116\pi$$
−0.700151 + 0.713995i $$0.746884\pi$$
$$632$$ 0 0
$$633$$ 143102. 0.357139
$$634$$ 0 0
$$635$$ −224206. −0.556033
$$636$$ 0 0
$$637$$ 51568.7i 0.127089i
$$638$$ 0 0
$$639$$ 58615.5i 0.143553i
$$640$$ 0 0
$$641$$ 292494. 0.711870 0.355935 0.934511i $$-0.384162\pi$$
0.355935 + 0.934511i $$0.384162\pi$$
$$642$$ 0 0
$$643$$ −103161. −0.249514 −0.124757 0.992187i $$-0.539815\pi$$
−0.124757 + 0.992187i $$0.539815\pi$$
$$644$$ 0 0
$$645$$ − 105167.i − 0.252791i
$$646$$ 0 0
$$647$$ 372569.i 0.890017i 0.895526 + 0.445009i $$0.146799\pi$$
−0.895526 + 0.445009i $$0.853201\pi$$
$$648$$ 0 0
$$649$$ 237378. 0.563574
$$650$$ 0 0
$$651$$ −93611.6 −0.220886
$$652$$ 0 0
$$653$$ 85619.0i 0.200791i 0.994948 + 0.100395i $$0.0320108\pi$$
−0.994948 + 0.100395i $$0.967989\pi$$
$$654$$ 0 0
$$655$$ 1.26889e6i 2.95761i
$$656$$ 0 0
$$657$$ −62045.1 −0.143740
$$658$$ 0 0
$$659$$ −258812. −0.595954 −0.297977 0.954573i $$-0.596312\pi$$
−0.297977 + 0.954573i $$0.596312\pi$$
$$660$$ 0 0
$$661$$ 327660.i 0.749928i 0.927039 + 0.374964i $$0.122345\pi$$
−0.927039 + 0.374964i $$0.877655\pi$$
$$662$$ 0 0
$$663$$ − 408084.i − 0.928372i
$$664$$ 0 0
$$665$$ −1.22357e6 −2.76686
$$666$$ 0 0
$$667$$ 712366. 1.60122
$$668$$ 0 0
$$669$$ − 15699.4i − 0.0350777i
$$670$$ 0 0
$$671$$ 145374.i 0.322880i
$$672$$ 0 0
$$673$$ 137121. 0.302742 0.151371 0.988477i $$-0.451631\pi$$
0.151371 + 0.988477i $$0.451631\pi$$
$$674$$ 0 0
$$675$$ 133981. 0.294059
$$676$$ 0 0
$$677$$ − 573451.i − 1.25118i −0.780153 0.625589i $$-0.784859\pi$$
0.780153 0.625589i $$-0.215141\pi$$
$$678$$ 0 0
$$679$$ 377089.i 0.817909i
$$680$$ 0 0
$$681$$ −36774.6 −0.0792965
$$682$$ 0 0
$$683$$ 175872. 0.377012 0.188506 0.982072i $$-0.439636\pi$$
0.188506 + 0.982072i $$0.439636\pi$$
$$684$$ 0 0
$$685$$ 417984.i 0.890797i
$$686$$ 0 0
$$687$$ 533962.i 1.13135i
$$688$$ 0 0
$$689$$ −138307. −0.291344
$$690$$ 0 0
$$691$$ 7216.66 0.0151140 0.00755702 0.999971i $$-0.497595\pi$$
0.00755702 + 0.999971i $$0.497595\pi$$
$$692$$ 0 0
$$693$$ − 225365.i − 0.469267i
$$694$$ 0 0
$$695$$ − 595673.i − 1.23321i
$$696$$ 0 0
$$697$$ 924433. 1.90287
$$698$$ 0 0
$$699$$ −119390. −0.244350
$$700$$ 0 0
$$701$$ 502543.i 1.02267i 0.859380 + 0.511337i $$0.170850\pi$$
−0.859380 + 0.511337i $$0.829150\pi$$
$$702$$ 0 0
$$703$$ − 312203.i − 0.631723i
$$704$$ 0 0
$$705$$ 424823. 0.854731
$$706$$ 0 0
$$707$$ 717191. 1.43481
$$708$$ 0 0
$$709$$ 95591.1i 0.190163i 0.995470 + 0.0950813i $$0.0303111\pi$$
−0.995470 + 0.0950813i $$0.969689\pi$$
$$710$$ 0 0
$$711$$ − 289043.i − 0.571772i
$$712$$ 0 0
$$713$$ 345286. 0.679204
$$714$$ 0 0
$$715$$ 1.32226e6 2.58645
$$716$$ 0 0
$$717$$ − 184810.i − 0.359491i
$$718$$ 0 0
$$719$$ − 475789.i − 0.920357i −0.887826 0.460178i $$-0.847785\pi$$
0.887826 0.460178i $$-0.152215\pi$$
$$720$$ 0 0
$$721$$ 305200. 0.587103
$$722$$ 0 0
$$723$$ 481597. 0.921313
$$724$$ 0 0
$$725$$ − 770908.i − 1.46665i
$$726$$ 0 0
$$727$$ − 193262.i − 0.365661i −0.983144 0.182830i $$-0.941474\pi$$
0.983144 0.182830i $$-0.0585259\pi$$
$$728$$ 0 0
$$729$$ 19683.0 0.0370370
$$730$$ 0 0
$$731$$ −217926. −0.407825
$$732$$ 0 0
$$733$$ − 650799.i − 1.21126i −0.795745 0.605632i $$-0.792920\pi$$
0.795745 0.605632i $$-0.207080\pi$$
$$734$$ 0 0
$$735$$ 58044.8i 0.107446i
$$736$$ 0 0
$$737$$ −91695.6 −0.168816
$$738$$ 0 0
$$739$$ −99472.6 −0.182144 −0.0910719 0.995844i $$-0.529029\pi$$
−0.0910719 + 0.995844i $$0.529029\pi$$
$$740$$ 0 0
$$741$$ − 637459.i − 1.16096i
$$742$$ 0 0
$$743$$ 715681.i 1.29641i 0.761466 + 0.648205i $$0.224480\pi$$
−0.761466 + 0.648205i $$0.775520\pi$$
$$744$$ 0 0
$$745$$ 468854. 0.844744
$$746$$ 0 0
$$747$$ 80401.0 0.144085
$$748$$ 0 0
$$749$$ − 543144.i − 0.968170i
$$750$$ 0 0
$$751$$ 603532.i 1.07009i 0.844824 + 0.535045i $$0.179705\pi$$
−0.844824 + 0.535045i $$0.820295\pi$$
$$752$$ 0 0
$$753$$ 143364. 0.252843
$$754$$ 0 0
$$755$$ −1.34324e6 −2.35645
$$756$$ 0 0
$$757$$ 784192.i 1.36846i 0.729268 + 0.684228i $$0.239861\pi$$
−0.729268 + 0.684228i $$0.760139\pi$$
$$758$$ 0 0
$$759$$ 831259.i 1.44296i
$$760$$ 0 0
$$761$$ −568076. −0.980927 −0.490464 0.871462i $$-0.663173\pi$$
−0.490464 + 0.871462i $$0.663173\pi$$
$$762$$ 0 0
$$763$$ 611731. 1.05078
$$764$$ 0 0
$$765$$ − 459331.i − 0.784880i
$$766$$ 0 0
$$767$$ 240278.i 0.408435i
$$768$$ 0 0
$$769$$ −531757. −0.899209 −0.449605 0.893228i $$-0.648435\pi$$
−0.449605 + 0.893228i $$0.648435\pi$$
$$770$$ 0 0
$$771$$ −479629. −0.806856
$$772$$ 0 0
$$773$$ − 260161.i − 0.435394i −0.976016 0.217697i $$-0.930145\pi$$
0.976016 0.217697i $$-0.0698545\pi$$
$$774$$ 0 0
$$775$$ − 373662.i − 0.622122i
$$776$$ 0 0
$$777$$ 111724. 0.185056
$$778$$ 0 0
$$779$$ 1.44404e6 2.37960
$$780$$ 0 0
$$781$$ 393556.i 0.645216i
$$782$$ 0 0
$$783$$ − 113254.i − 0.184726i
$$784$$ 0 0
$$785$$ −1.14433e6 −1.85700
$$786$$ 0 0
$$787$$ −724342. −1.16948 −0.584742 0.811219i $$-0.698804\pi$$
−0.584742 + 0.811219i $$0.698804\pi$$
$$788$$ 0 0
$$789$$ − 72145.7i − 0.115893i
$$790$$ 0 0
$$791$$ 539332.i 0.861993i
$$792$$ 0 0
$$793$$ −147150. −0.233998
$$794$$ 0 0
$$795$$ −155676. −0.246313
$$796$$ 0 0
$$797$$ 1.04249e6i 1.64117i 0.571524 + 0.820586i $$0.306353\pi$$
−0.571524 + 0.820586i $$0.693647\pi$$
$$798$$ 0 0
$$799$$ − 880309.i − 1.37893i
$$800$$ 0 0
$$801$$ −175336. −0.273279
$$802$$ 0 0
$$803$$ −416584. −0.646057
$$804$$ 0 0
$$805$$ 1.61506e6i 2.49227i
$$806$$ 0 0
$$807$$ − 273827.i − 0.420465i
$$808$$ 0 0
$$809$$ −647222. −0.988909 −0.494455 0.869204i $$-0.664632\pi$$
−0.494455 + 0.869204i $$0.664632\pi$$
$$810$$ 0 0
$$811$$ −1.24087e6 −1.88662 −0.943309 0.331916i $$-0.892305\pi$$
−0.943309 + 0.331916i $$0.892305\pi$$
$$812$$ 0 0
$$813$$ 244369.i 0.369713i
$$814$$ 0 0
$$815$$ − 532993.i − 0.802428i
$$816$$ 0 0
$$817$$ −340418. −0.509997
$$818$$ 0 0
$$819$$ 228118. 0.340089
$$820$$ 0 0
$$821$$ 819276.i 1.21547i 0.794141 + 0.607734i $$0.207921\pi$$
−0.794141 + 0.607734i $$0.792079\pi$$
$$822$$ 0 0
$$823$$ − 515929.i − 0.761710i −0.924635 0.380855i $$-0.875630\pi$$
0.924635 0.380855i $$-0.124370\pi$$
$$824$$ 0 0
$$825$$ 899573. 1.32169
$$826$$ 0 0
$$827$$ 140674. 0.205685 0.102842 0.994698i $$-0.467206\pi$$
0.102842 + 0.994698i $$0.467206\pi$$
$$828$$ 0 0
$$829$$ 137443.i 0.199993i 0.994988 + 0.0999964i $$0.0318831\pi$$
−0.994988 + 0.0999964i $$0.968117\pi$$
$$830$$ 0 0
$$831$$ − 563229.i − 0.815611i
$$832$$ 0 0
$$833$$ 120279. 0.173341
$$834$$ 0 0
$$835$$ 524955. 0.752920
$$836$$ 0 0
$$837$$ − 54894.5i − 0.0783570i
$$838$$ 0 0
$$839$$ 591422.i 0.840182i 0.907482 + 0.420091i $$0.138002\pi$$
−0.907482 + 0.420091i $$0.861998\pi$$
$$840$$ 0 0
$$841$$ 55633.2 0.0786579
$$842$$ 0 0
$$843$$ 379790. 0.534428
$$844$$ 0 0
$$845$$ 203138.i 0.284498i
$$846$$ 0 0
$$847$$ − 839029.i − 1.16953i
$$848$$ 0 0
$$849$$ 232596. 0.322691
$$850$$ 0 0
$$851$$ −412093. −0.569031
$$852$$ 0 0
$$853$$ − 169773.i − 0.233331i −0.993171 0.116665i $$-0.962780\pi$$
0.993171 0.116665i $$-0.0372205\pi$$
$$854$$ 0 0
$$855$$ − 717512.i − 0.981515i
$$856$$ 0 0
$$857$$ 1.05083e6 1.43077 0.715384 0.698732i $$-0.246252\pi$$
0.715384 + 0.698732i $$0.246252\pi$$
$$858$$ 0 0
$$859$$ −1.05896e6 −1.43514 −0.717569 0.696487i $$-0.754745\pi$$
−0.717569 + 0.696487i $$0.754745\pi$$
$$860$$ 0 0
$$861$$ 516757.i 0.697075i
$$862$$ 0 0
$$863$$ − 1.22011e6i − 1.63824i −0.573621 0.819121i $$-0.694462\pi$$
0.573621 0.819121i $$-0.305538\pi$$
$$864$$ 0 0
$$865$$ −14506.6 −0.0193881
$$866$$ 0 0
$$867$$ −517829. −0.688887
$$868$$ 0 0
$$869$$ − 1.94069e6i − 2.56991i
$$870$$ 0 0
$$871$$ − 92815.8i − 0.122345i
$$872$$ 0 0
$$873$$ −221128. −0.290145
$$874$$ 0 0
$$875$$ 603927. 0.788802
$$876$$ 0 0
$$877$$ − 707060.i − 0.919299i −0.888100 0.459650i $$-0.847975\pi$$
0.888100 0.459650i $$-0.152025\pi$$
$$878$$ 0 0
$$879$$ − 218395.i − 0.282661i
$$880$$ 0 0
$$881$$ 430965. 0.555252 0.277626 0.960689i $$-0.410452\pi$$
0.277626 + 0.960689i $$0.410452\pi$$
$$882$$ 0 0
$$883$$ −1.08382e6 −1.39007 −0.695037 0.718974i $$-0.744612\pi$$
−0.695037 + 0.718974i $$0.744612\pi$$
$$884$$ 0 0
$$885$$ 270452.i 0.345306i
$$886$$ 0 0
$$887$$ 620229.i 0.788324i 0.919041 + 0.394162i $$0.128965\pi$$
−0.919041 + 0.394162i $$0.871035\pi$$
$$888$$ 0 0
$$889$$ −259709. −0.328612
$$890$$ 0 0
$$891$$ 132156. 0.166468
$$892$$ 0 0
$$893$$ − 1.37511e6i − 1.72439i
$$894$$ 0 0
$$895$$ 571272.i 0.713176i
$$896$$ 0 0
$$897$$ −841414. −1.04574
$$898$$ 0 0
$$899$$ −315856. −0.390814
$$900$$ 0 0
$$901$$ 322589.i 0.397374i
$$902$$ 0 0
$$903$$ − 121820.i − 0.149398i
$$904$$ 0 0
$$905$$ 161005. 0.196581
$$906$$ 0 0
$$907$$ 155091. 0.188526 0.0942629 0.995547i $$-0.469951\pi$$
0.0942629 + 0.995547i $$0.469951\pi$$
$$908$$ 0 0
$$909$$ 420566.i 0.508986i
$$910$$ 0 0
$$911$$ − 670134.i − 0.807467i −0.914877 0.403734i $$-0.867712\pi$$
0.914877 0.403734i $$-0.132288\pi$$
$$912$$ 0 0
$$913$$ 539828. 0.647611
$$914$$ 0 0
$$915$$ −165629. −0.197831
$$916$$ 0 0
$$917$$ 1.46981e6i 1.74793i
$$918$$ 0 0
$$919$$ 631844.i 0.748134i 0.927402 + 0.374067i $$0.122037\pi$$
−0.927402 + 0.374067i $$0.877963\pi$$
$$920$$ 0 0
$$921$$ 45831.7 0.0540314
$$922$$ 0 0
$$923$$ −398364. −0.467602
$$924$$ 0 0
$$925$$ 445959.i 0.521208i
$$926$$ 0 0
$$927$$ 178972.i 0.208269i
$$928$$ 0 0
$$929$$ 1.16204e6 1.34645 0.673225 0.739437i $$-0.264908\pi$$
0.673225 + 0.739437i $$0.264908\pi$$
$$930$$ 0 0
$$931$$ 187886. 0.216768
$$932$$ 0 0
$$933$$ − 805708.i − 0.925580i
$$934$$ 0 0
$$935$$ − 3.08404e6i − 3.52774i
$$936$$ 0 0
$$937$$ −1.18305e6 −1.34748 −0.673741 0.738967i $$-0.735314\pi$$
−0.673741 + 0.738967i $$0.735314\pi$$
$$938$$ 0 0
$$939$$ 933570. 1.05880
$$940$$ 0 0
$$941$$ − 159241.i − 0.179836i −0.995949 0.0899179i $$-0.971340\pi$$
0.995949 0.0899179i $$-0.0286605\pi$$
$$942$$ 0 0
$$943$$ − 1.90606e6i − 2.14345i
$$944$$ 0 0
$$945$$ 256766. 0.287523
$$946$$ 0 0
$$947$$ −940316. −1.04851 −0.524257 0.851560i $$-0.675657\pi$$
−0.524257 + 0.851560i $$0.675657\pi$$
$$948$$ 0 0
$$949$$ − 421673.i − 0.468213i
$$950$$ 0 0
$$951$$ − 32157.8i − 0.0355570i
$$952$$ 0 0
$$953$$ 577950. 0.636362 0.318181 0.948030i $$-0.396928\pi$$
0.318181 + 0.948030i $$0.396928\pi$$
$$954$$ 0 0
$$955$$ 1.98272e6 2.17397
$$956$$ 0 0
$$957$$ − 760408.i − 0.830276i
$$958$$ 0 0
$$959$$ 484170.i 0.526454i
$$960$$ 0 0
$$961$$ 770424. 0.834225
$$962$$ 0 0
$$963$$ 318504. 0.343449
$$964$$ 0 0
$$965$$ 1.92487e6i 2.06703i
$$966$$ 0 0
$$967$$ − 785047.i − 0.839542i −0.907630 0.419771i $$-0.862110\pi$$
0.907630 0.419771i $$-0.137890\pi$$
$$968$$ 0 0
$$969$$ −1.48681e6 −1.58347
$$970$$ 0 0
$$971$$ 1.05236e6 1.11616 0.558082 0.829786i $$-0.311538\pi$$
0.558082 + 0.829786i $$0.311538\pi$$
$$972$$ 0 0
$$973$$ − 689996.i − 0.728821i
$$974$$ 0 0
$$975$$ 910562.i 0.957856i
$$976$$ 0 0
$$977$$ 20197.7 0.0211598 0.0105799 0.999944i $$-0.496632\pi$$
0.0105799 + 0.999944i $$0.496632\pi$$
$$978$$ 0 0
$$979$$ −1.17724e6 −1.22829
$$980$$ 0 0
$$981$$ 358723.i 0.372754i
$$982$$ 0 0
$$983$$ − 1.43076e6i − 1.48068i −0.672234 0.740338i $$-0.734665\pi$$
0.672234 0.740338i $$-0.265335\pi$$
$$984$$ 0 0
$$985$$ 220875. 0.227653
$$986$$ 0 0
$$987$$ 492092. 0.505140
$$988$$ 0 0
$$989$$ 449334.i 0.459385i
$$990$$ 0 0
$$991$$ 787866.i 0.802241i 0.916025 + 0.401120i $$0.131379\pi$$
−0.916025 + 0.401120i $$0.868621\pi$$
$$992$$ 0 0
$$993$$ −686718. −0.696434
$$994$$ 0 0
$$995$$ 2.40858e6 2.43285
$$996$$ 0 0
$$997$$ − 453709.i − 0.456444i −0.973609 0.228222i $$-0.926709\pi$$
0.973609 0.228222i $$-0.0732912\pi$$
$$998$$ 0 0
$$999$$ 65515.5i 0.0656468i
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 384.5.b.c.319.1 8
3.2 odd 2 1152.5.b.k.703.7 8
4.3 odd 2 inner 384.5.b.c.319.5 yes 8
8.3 odd 2 inner 384.5.b.c.319.4 yes 8
8.5 even 2 inner 384.5.b.c.319.8 yes 8
12.11 even 2 1152.5.b.k.703.8 8
16.3 odd 4 768.5.g.c.511.3 4
16.5 even 4 768.5.g.g.511.4 4
16.11 odd 4 768.5.g.g.511.2 4
16.13 even 4 768.5.g.c.511.1 4
24.5 odd 2 1152.5.b.k.703.1 8
24.11 even 2 1152.5.b.k.703.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.b.c.319.1 8 1.1 even 1 trivial
384.5.b.c.319.4 yes 8 8.3 odd 2 inner
384.5.b.c.319.5 yes 8 4.3 odd 2 inner
384.5.b.c.319.8 yes 8 8.5 even 2 inner
768.5.g.c.511.1 4 16.13 even 4
768.5.g.c.511.3 4 16.3 odd 4
768.5.g.g.511.2 4 16.11 odd 4
768.5.g.g.511.4 4 16.5 even 4
1152.5.b.k.703.1 8 24.5 odd 2
1152.5.b.k.703.2 8 24.11 even 2
1152.5.b.k.703.7 8 3.2 odd 2
1152.5.b.k.703.8 8 12.11 even 2