# Properties

 Label 378.5.b.b.323.6 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: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884$$ x^8 + 92*x^6 + 2949*x^4 + 37548*x^2 + 142884 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 323.6 Root $$2.54824i$$ of defining polynomial Character $$\chi$$ $$=$$ 378.323 Dual form 378.5.b.b.323.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.82843i q^{2} -8.00000 q^{4} -17.8674i q^{5} -18.5203 q^{7} -22.6274i q^{8} +O(q^{10})$$ $$q+2.82843i q^{2} -8.00000 q^{4} -17.8674i q^{5} -18.5203 q^{7} -22.6274i q^{8} +50.5366 q^{10} +45.4134i q^{11} -155.432 q^{13} -52.3832i q^{14} +64.0000 q^{16} -114.905i q^{17} -17.2296 q^{19} +142.939i q^{20} -128.449 q^{22} -55.0943i q^{23} +305.756 q^{25} -439.629i q^{26} +148.162 q^{28} +578.935i q^{29} +514.064 q^{31} +181.019i q^{32} +325.001 q^{34} +330.909i q^{35} +1287.01 q^{37} -48.7326i q^{38} -404.293 q^{40} +2151.69i q^{41} +2349.59 q^{43} -363.307i q^{44} +155.830 q^{46} +1965.66i q^{47} +343.000 q^{49} +864.809i q^{50} +1243.46 q^{52} +3259.56i q^{53} +811.419 q^{55} +419.066i q^{56} -1637.48 q^{58} -276.141i q^{59} -438.948 q^{61} +1453.99i q^{62} -512.000 q^{64} +2777.17i q^{65} -1761.95 q^{67} +919.241i q^{68} -935.951 q^{70} -1378.26i q^{71} +1873.02 q^{73} +3640.20i q^{74} +137.837 q^{76} -841.068i q^{77} +4585.41 q^{79} -1143.51i q^{80} -6085.89 q^{82} +7058.33i q^{83} -2053.05 q^{85} +6645.64i q^{86} +1027.59 q^{88} +7138.01i q^{89} +2878.64 q^{91} +440.755i q^{92} -5559.72 q^{94} +307.848i q^{95} -1109.01 q^{97} +970.151i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 64 q^{4}+O(q^{10})$$ 8 * q - 64 * q^4 $$8 q - 64 q^{4} - 32 q^{10} + 296 q^{13} + 512 q^{16} - 80 q^{19} + 800 q^{22} - 1464 q^{25} - 1024 q^{31} + 896 q^{34} + 904 q^{37} + 256 q^{40} - 3224 q^{43} - 4384 q^{46} + 2744 q^{49} - 2368 q^{52} + 8312 q^{55} + 2944 q^{58} + 128 q^{61} - 4096 q^{64} - 10152 q^{67} + 1568 q^{70} + 10632 q^{73} + 640 q^{76} - 4072 q^{79} - 3552 q^{82} + 13448 q^{85} - 6400 q^{88} - 6664 q^{91} - 384 q^{94} + 35616 q^{97}+O(q^{100})$$ 8 * q - 64 * q^4 - 32 * q^10 + 296 * q^13 + 512 * q^16 - 80 * q^19 + 800 * q^22 - 1464 * q^25 - 1024 * q^31 + 896 * q^34 + 904 * q^37 + 256 * q^40 - 3224 * q^43 - 4384 * q^46 + 2744 * q^49 - 2368 * q^52 + 8312 * q^55 + 2944 * q^58 + 128 * q^61 - 4096 * q^64 - 10152 * q^67 + 1568 * q^70 + 10632 * q^73 + 640 * q^76 - 4072 * q^79 - 3552 * q^82 + 13448 * q^85 - 6400 * q^88 - 6664 * q^91 - 384 * q^94 + 35616 * 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
$$5$$ − 17.8674i − 0.714696i −0.933971 0.357348i $$-0.883681\pi$$
0.933971 0.357348i $$-0.116319\pi$$
$$6$$ 0 0
$$7$$ −18.5203 −0.377964
$$8$$ − 22.6274i − 0.353553i
$$9$$ 0 0
$$10$$ 50.5366 0.505366
$$11$$ 45.4134i 0.375317i 0.982234 + 0.187659i $$0.0600899\pi$$
−0.982234 + 0.187659i $$0.939910\pi$$
$$12$$ 0 0
$$13$$ −155.432 −0.919717 −0.459858 0.887992i $$-0.652100\pi$$
−0.459858 + 0.887992i $$0.652100\pi$$
$$14$$ − 52.3832i − 0.267261i
$$15$$ 0 0
$$16$$ 64.0000 0.250000
$$17$$ − 114.905i − 0.397596i −0.980041 0.198798i $$-0.936296\pi$$
0.980041 0.198798i $$-0.0637037\pi$$
$$18$$ 0 0
$$19$$ −17.2296 −0.0477274 −0.0238637 0.999715i $$-0.507597\pi$$
−0.0238637 + 0.999715i $$0.507597\pi$$
$$20$$ 142.939i 0.357348i
$$21$$ 0 0
$$22$$ −128.449 −0.265390
$$23$$ − 55.0943i − 0.104148i −0.998643 0.0520740i $$-0.983417\pi$$
0.998643 0.0520740i $$-0.0165832\pi$$
$$24$$ 0 0
$$25$$ 305.756 0.489210
$$26$$ − 439.629i − 0.650338i
$$27$$ 0 0
$$28$$ 148.162 0.188982
$$29$$ 578.935i 0.688389i 0.938898 + 0.344195i $$0.111848\pi$$
−0.938898 + 0.344195i $$0.888152\pi$$
$$30$$ 0 0
$$31$$ 514.064 0.534926 0.267463 0.963568i $$-0.413815\pi$$
0.267463 + 0.963568i $$0.413815\pi$$
$$32$$ 181.019i 0.176777i
$$33$$ 0 0
$$34$$ 325.001 0.281143
$$35$$ 330.909i 0.270130i
$$36$$ 0 0
$$37$$ 1287.01 0.940107 0.470053 0.882638i $$-0.344235\pi$$
0.470053 + 0.882638i $$0.344235\pi$$
$$38$$ − 48.7326i − 0.0337483i
$$39$$ 0 0
$$40$$ −404.293 −0.252683
$$41$$ 2151.69i 1.28001i 0.768373 + 0.640003i $$0.221067\pi$$
−0.768373 + 0.640003i $$0.778933\pi$$
$$42$$ 0 0
$$43$$ 2349.59 1.27073 0.635367 0.772210i $$-0.280849\pi$$
0.635367 + 0.772210i $$0.280849\pi$$
$$44$$ − 363.307i − 0.187659i
$$45$$ 0 0
$$46$$ 155.830 0.0736438
$$47$$ 1965.66i 0.889841i 0.895570 + 0.444920i $$0.146768\pi$$
−0.895570 + 0.444920i $$0.853232\pi$$
$$48$$ 0 0
$$49$$ 343.000 0.142857
$$50$$ 864.809i 0.345924i
$$51$$ 0 0
$$52$$ 1243.46 0.459858
$$53$$ 3259.56i 1.16040i 0.814474 + 0.580200i $$0.197026\pi$$
−0.814474 + 0.580200i $$0.802974\pi$$
$$54$$ 0 0
$$55$$ 811.419 0.268238
$$56$$ 419.066i 0.133631i
$$57$$ 0 0
$$58$$ −1637.48 −0.486765
$$59$$ − 276.141i − 0.0793281i −0.999213 0.0396641i $$-0.987371\pi$$
0.999213 0.0396641i $$-0.0126288\pi$$
$$60$$ 0 0
$$61$$ −438.948 −0.117965 −0.0589826 0.998259i $$-0.518786\pi$$
−0.0589826 + 0.998259i $$0.518786\pi$$
$$62$$ 1453.99i 0.378250i
$$63$$ 0 0
$$64$$ −512.000 −0.125000
$$65$$ 2777.17i 0.657318i
$$66$$ 0 0
$$67$$ −1761.95 −0.392503 −0.196251 0.980554i $$-0.562877\pi$$
−0.196251 + 0.980554i $$0.562877\pi$$
$$68$$ 919.241i 0.198798i
$$69$$ 0 0
$$70$$ −935.951 −0.191010
$$71$$ − 1378.26i − 0.273409i −0.990612 0.136705i $$-0.956349\pi$$
0.990612 0.136705i $$-0.0436511\pi$$
$$72$$ 0 0
$$73$$ 1873.02 0.351477 0.175738 0.984437i $$-0.443769\pi$$
0.175738 + 0.984437i $$0.443769\pi$$
$$74$$ 3640.20i 0.664756i
$$75$$ 0 0
$$76$$ 137.837 0.0238637
$$77$$ − 841.068i − 0.141857i
$$78$$ 0 0
$$79$$ 4585.41 0.734724 0.367362 0.930078i $$-0.380261\pi$$
0.367362 + 0.930078i $$0.380261\pi$$
$$80$$ − 1143.51i − 0.178674i
$$81$$ 0 0
$$82$$ −6085.89 −0.905100
$$83$$ 7058.33i 1.02458i 0.858813 + 0.512290i $$0.171203\pi$$
−0.858813 + 0.512290i $$0.828797\pi$$
$$84$$ 0 0
$$85$$ −2053.05 −0.284160
$$86$$ 6645.64i 0.898545i
$$87$$ 0 0
$$88$$ 1027.59 0.132695
$$89$$ 7138.01i 0.901150i 0.892739 + 0.450575i $$0.148781\pi$$
−0.892739 + 0.450575i $$0.851219\pi$$
$$90$$ 0 0
$$91$$ 2878.64 0.347620
$$92$$ 440.755i 0.0520740i
$$93$$ 0 0
$$94$$ −5559.72 −0.629213
$$95$$ 307.848i 0.0341105i
$$96$$ 0 0
$$97$$ −1109.01 −0.117867 −0.0589336 0.998262i $$-0.518770\pi$$
−0.0589336 + 0.998262i $$0.518770\pi$$
$$98$$ 970.151i 0.101015i
$$99$$ 0 0
$$100$$ −2446.05 −0.244605
$$101$$ 15526.5i 1.52205i 0.648721 + 0.761027i $$0.275304\pi$$
−0.648721 + 0.761027i $$0.724696\pi$$
$$102$$ 0 0
$$103$$ −17118.9 −1.61362 −0.806809 0.590813i $$-0.798807\pi$$
−0.806809 + 0.590813i $$0.798807\pi$$
$$104$$ 3517.03i 0.325169i
$$105$$ 0 0
$$106$$ −9219.44 −0.820526
$$107$$ − 4086.42i − 0.356924i −0.983947 0.178462i $$-0.942888\pi$$
0.983947 0.178462i $$-0.0571121\pi$$
$$108$$ 0 0
$$109$$ 9630.93 0.810616 0.405308 0.914180i $$-0.367164\pi$$
0.405308 + 0.914180i $$0.367164\pi$$
$$110$$ 2295.04i 0.189673i
$$111$$ 0 0
$$112$$ −1185.30 −0.0944911
$$113$$ − 5446.96i − 0.426577i −0.976989 0.213288i $$-0.931583\pi$$
0.976989 0.213288i $$-0.0684174\pi$$
$$114$$ 0 0
$$115$$ −984.392 −0.0744342
$$116$$ − 4631.48i − 0.344195i
$$117$$ 0 0
$$118$$ 781.045 0.0560935
$$119$$ 2128.07i 0.150277i
$$120$$ 0 0
$$121$$ 12578.6 0.859137
$$122$$ − 1241.53i − 0.0834140i
$$123$$ 0 0
$$124$$ −4112.51 −0.267463
$$125$$ − 16630.2i − 1.06433i
$$126$$ 0 0
$$127$$ 3615.47 0.224160 0.112080 0.993699i $$-0.464249\pi$$
0.112080 + 0.993699i $$0.464249\pi$$
$$128$$ − 1448.15i − 0.0883883i
$$129$$ 0 0
$$130$$ −7855.02 −0.464794
$$131$$ 13601.5i 0.792584i 0.918125 + 0.396292i $$0.129703\pi$$
−0.918125 + 0.396292i $$0.870297\pi$$
$$132$$ 0 0
$$133$$ 319.096 0.0180392
$$134$$ − 4983.53i − 0.277541i
$$135$$ 0 0
$$136$$ −2600.01 −0.140571
$$137$$ 1456.98i 0.0776268i 0.999246 + 0.0388134i $$0.0123578\pi$$
−0.999246 + 0.0388134i $$0.987642\pi$$
$$138$$ 0 0
$$139$$ −2883.53 −0.149243 −0.0746217 0.997212i $$-0.523775\pi$$
−0.0746217 + 0.997212i $$0.523775\pi$$
$$140$$ − 2647.27i − 0.135065i
$$141$$ 0 0
$$142$$ 3898.29 0.193329
$$143$$ − 7058.71i − 0.345186i
$$144$$ 0 0
$$145$$ 10344.1 0.491989
$$146$$ 5297.70i 0.248531i
$$147$$ 0 0
$$148$$ −10296.0 −0.470053
$$149$$ 40158.5i 1.80886i 0.426623 + 0.904430i $$0.359703\pi$$
−0.426623 + 0.904430i $$0.640297\pi$$
$$150$$ 0 0
$$151$$ −3953.92 −0.173410 −0.0867051 0.996234i $$-0.527634\pi$$
−0.0867051 + 0.996234i $$0.527634\pi$$
$$152$$ 389.861i 0.0168742i
$$153$$ 0 0
$$154$$ 2378.90 0.100308
$$155$$ − 9184.98i − 0.382309i
$$156$$ 0 0
$$157$$ 15472.7 0.627722 0.313861 0.949469i $$-0.398377\pi$$
0.313861 + 0.949469i $$0.398377\pi$$
$$158$$ 12969.5i 0.519528i
$$159$$ 0 0
$$160$$ 3234.34 0.126342
$$161$$ 1020.36i 0.0393643i
$$162$$ 0 0
$$163$$ 37395.2 1.40748 0.703738 0.710460i $$-0.251513\pi$$
0.703738 + 0.710460i $$0.251513\pi$$
$$164$$ − 17213.5i − 0.640003i
$$165$$ 0 0
$$166$$ −19964.0 −0.724487
$$167$$ − 5416.39i − 0.194212i −0.995274 0.0971061i $$-0.969041\pi$$
0.995274 0.0971061i $$-0.0309586\pi$$
$$168$$ 0 0
$$169$$ −4401.84 −0.154121
$$170$$ − 5806.92i − 0.200931i
$$171$$ 0 0
$$172$$ −18796.7 −0.635367
$$173$$ 14162.2i 0.473194i 0.971608 + 0.236597i $$0.0760321\pi$$
−0.971608 + 0.236597i $$0.923968\pi$$
$$174$$ 0 0
$$175$$ −5662.69 −0.184904
$$176$$ 2906.46i 0.0938294i
$$177$$ 0 0
$$178$$ −20189.3 −0.637209
$$179$$ − 16147.2i − 0.503955i −0.967733 0.251977i $$-0.918919\pi$$
0.967733 0.251977i $$-0.0810809\pi$$
$$180$$ 0 0
$$181$$ 26540.9 0.810138 0.405069 0.914286i $$-0.367248\pi$$
0.405069 + 0.914286i $$0.367248\pi$$
$$182$$ 8142.03i 0.245805i
$$183$$ 0 0
$$184$$ −1246.64 −0.0368219
$$185$$ − 22995.4i − 0.671890i
$$186$$ 0 0
$$187$$ 5218.23 0.149225
$$188$$ − 15725.3i − 0.444920i
$$189$$ 0 0
$$190$$ −870.725 −0.0241198
$$191$$ − 2149.99i − 0.0589344i −0.999566 0.0294672i $$-0.990619\pi$$
0.999566 0.0294672i $$-0.00938106\pi$$
$$192$$ 0 0
$$193$$ 26344.9 0.707265 0.353633 0.935384i $$-0.384946\pi$$
0.353633 + 0.935384i $$0.384946\pi$$
$$194$$ − 3136.76i − 0.0833448i
$$195$$ 0 0
$$196$$ −2744.00 −0.0714286
$$197$$ − 62745.0i − 1.61676i −0.588658 0.808382i $$-0.700344\pi$$
0.588658 0.808382i $$-0.299656\pi$$
$$198$$ 0 0
$$199$$ −41364.4 −1.04453 −0.522264 0.852784i $$-0.674912\pi$$
−0.522264 + 0.852784i $$0.674912\pi$$
$$200$$ − 6918.48i − 0.172962i
$$201$$ 0 0
$$202$$ −43915.5 −1.07625
$$203$$ − 10722.0i − 0.260187i
$$204$$ 0 0
$$205$$ 38445.1 0.914814
$$206$$ − 48419.5i − 1.14100i
$$207$$ 0 0
$$208$$ −9947.66 −0.229929
$$209$$ − 782.454i − 0.0179129i
$$210$$ 0 0
$$211$$ 60192.2 1.35200 0.675998 0.736904i $$-0.263713\pi$$
0.675998 + 0.736904i $$0.263713\pi$$
$$212$$ − 26076.5i − 0.580200i
$$213$$ 0 0
$$214$$ 11558.1 0.252383
$$215$$ − 41981.0i − 0.908188i
$$216$$ 0 0
$$217$$ −9520.60 −0.202183
$$218$$ 27240.4i 0.573192i
$$219$$ 0 0
$$220$$ −6491.35 −0.134119
$$221$$ 17860.0i 0.365675i
$$222$$ 0 0
$$223$$ 54766.3 1.10129 0.550647 0.834738i $$-0.314381\pi$$
0.550647 + 0.834738i $$0.314381\pi$$
$$224$$ − 3352.53i − 0.0668153i
$$225$$ 0 0
$$226$$ 15406.3 0.301635
$$227$$ − 23659.5i − 0.459149i −0.973291 0.229575i $$-0.926267\pi$$
0.973291 0.229575i $$-0.0737335\pi$$
$$228$$ 0 0
$$229$$ −18364.7 −0.350198 −0.175099 0.984551i $$-0.556025\pi$$
−0.175099 + 0.984551i $$0.556025\pi$$
$$230$$ − 2784.28i − 0.0526329i
$$231$$ 0 0
$$232$$ 13099.8 0.243382
$$233$$ − 71340.7i − 1.31409i −0.753851 0.657046i $$-0.771806\pi$$
0.753851 0.657046i $$-0.228194\pi$$
$$234$$ 0 0
$$235$$ 35121.2 0.635965
$$236$$ 2209.13i 0.0396641i
$$237$$ 0 0
$$238$$ −6019.10 −0.106262
$$239$$ − 43626.2i − 0.763751i −0.924214 0.381876i $$-0.875278\pi$$
0.924214 0.381876i $$-0.124722\pi$$
$$240$$ 0 0
$$241$$ 12447.2 0.214308 0.107154 0.994242i $$-0.465826\pi$$
0.107154 + 0.994242i $$0.465826\pi$$
$$242$$ 35577.7i 0.607501i
$$243$$ 0 0
$$244$$ 3511.59 0.0589826
$$245$$ − 6128.52i − 0.102099i
$$246$$ 0 0
$$247$$ 2678.03 0.0438957
$$248$$ − 11631.9i − 0.189125i
$$249$$ 0 0
$$250$$ 47037.3 0.752596
$$251$$ − 78794.7i − 1.25069i −0.780348 0.625345i $$-0.784958\pi$$
0.780348 0.625345i $$-0.215042\pi$$
$$252$$ 0 0
$$253$$ 2502.02 0.0390886
$$254$$ 10226.1i 0.158505i
$$255$$ 0 0
$$256$$ 4096.00 0.0625000
$$257$$ 28911.2i 0.437723i 0.975756 + 0.218861i $$0.0702343\pi$$
−0.975756 + 0.218861i $$0.929766\pi$$
$$258$$ 0 0
$$259$$ −23835.7 −0.355327
$$260$$ − 22217.3i − 0.328659i
$$261$$ 0 0
$$262$$ −38470.9 −0.560442
$$263$$ 65442.0i 0.946118i 0.881031 + 0.473059i $$0.156850\pi$$
−0.881031 + 0.473059i $$0.843150\pi$$
$$264$$ 0 0
$$265$$ 58239.9 0.829333
$$266$$ 902.540i 0.0127557i
$$267$$ 0 0
$$268$$ 14095.6 0.196251
$$269$$ − 104321.i − 1.44167i −0.693107 0.720835i $$-0.743759\pi$$
0.693107 0.720835i $$-0.256241\pi$$
$$270$$ 0 0
$$271$$ −31635.6 −0.430762 −0.215381 0.976530i $$-0.569099\pi$$
−0.215381 + 0.976530i $$0.569099\pi$$
$$272$$ − 7353.93i − 0.0993989i
$$273$$ 0 0
$$274$$ −4120.96 −0.0548905
$$275$$ 13885.4i 0.183609i
$$276$$ 0 0
$$277$$ −116481. −1.51809 −0.759045 0.651039i $$-0.774334\pi$$
−0.759045 + 0.651039i $$0.774334\pi$$
$$278$$ − 8155.86i − 0.105531i
$$279$$ 0 0
$$280$$ 7487.61 0.0955052
$$281$$ 17406.9i 0.220449i 0.993907 + 0.110224i $$0.0351570\pi$$
−0.993907 + 0.110224i $$0.964843\pi$$
$$282$$ 0 0
$$283$$ −10431.0 −0.130242 −0.0651211 0.997877i $$-0.520743\pi$$
−0.0651211 + 0.997877i $$0.520743\pi$$
$$284$$ 11026.0i 0.136705i
$$285$$ 0 0
$$286$$ 19965.0 0.244083
$$287$$ − 39849.8i − 0.483796i
$$288$$ 0 0
$$289$$ 70317.8 0.841918
$$290$$ 29257.4i 0.347888i
$$291$$ 0 0
$$292$$ −14984.1 −0.175738
$$293$$ − 64199.7i − 0.747822i −0.927465 0.373911i $$-0.878017\pi$$
0.927465 0.373911i $$-0.121983\pi$$
$$294$$ 0 0
$$295$$ −4933.92 −0.0566955
$$296$$ − 29121.6i − 0.332378i
$$297$$ 0 0
$$298$$ −113585. −1.27906
$$299$$ 8563.43i 0.0957867i
$$300$$ 0 0
$$301$$ −43515.0 −0.480292
$$302$$ − 11183.4i − 0.122619i
$$303$$ 0 0
$$304$$ −1102.69 −0.0119318
$$305$$ 7842.86i 0.0843092i
$$306$$ 0 0
$$307$$ −65196.6 −0.691749 −0.345874 0.938281i $$-0.612418\pi$$
−0.345874 + 0.938281i $$0.612418\pi$$
$$308$$ 6728.55i 0.0709283i
$$309$$ 0 0
$$310$$ 25979.1 0.270334
$$311$$ 151896.i 1.57046i 0.619206 + 0.785229i $$0.287455\pi$$
−0.619206 + 0.785229i $$0.712545\pi$$
$$312$$ 0 0
$$313$$ −54154.6 −0.552773 −0.276387 0.961047i $$-0.589137\pi$$
−0.276387 + 0.961047i $$0.589137\pi$$
$$314$$ 43763.5i 0.443867i
$$315$$ 0 0
$$316$$ −36683.3 −0.367362
$$317$$ 137384.i 1.36715i 0.729878 + 0.683577i $$0.239577\pi$$
−0.729878 + 0.683577i $$0.760423\pi$$
$$318$$ 0 0
$$319$$ −26291.4 −0.258364
$$320$$ 9148.10i 0.0893370i
$$321$$ 0 0
$$322$$ −2886.02 −0.0278347
$$323$$ 1979.77i 0.0189762i
$$324$$ 0 0
$$325$$ −47524.4 −0.449935
$$326$$ 105770.i 0.995236i
$$327$$ 0 0
$$328$$ 48687.2 0.452550
$$329$$ − 36404.5i − 0.336328i
$$330$$ 0 0
$$331$$ −163805. −1.49510 −0.747550 0.664205i $$-0.768770\pi$$
−0.747550 + 0.664205i $$0.768770\pi$$
$$332$$ − 56466.6i − 0.512290i
$$333$$ 0 0
$$334$$ 15319.9 0.137329
$$335$$ 31481.4i 0.280520i
$$336$$ 0 0
$$337$$ 164077. 1.44473 0.722367 0.691510i $$-0.243054\pi$$
0.722367 + 0.691510i $$0.243054\pi$$
$$338$$ − 12450.3i − 0.108980i
$$339$$ 0 0
$$340$$ 16424.4 0.142080
$$341$$ 23345.4i 0.200767i
$$342$$ 0 0
$$343$$ −6352.45 −0.0539949
$$344$$ − 53165.1i − 0.449272i
$$345$$ 0 0
$$346$$ −40056.8 −0.334599
$$347$$ 72809.2i 0.604683i 0.953200 + 0.302341i $$0.0977682\pi$$
−0.953200 + 0.302341i $$0.902232\pi$$
$$348$$ 0 0
$$349$$ 15848.6 0.130118 0.0650592 0.997881i $$-0.479276\pi$$
0.0650592 + 0.997881i $$0.479276\pi$$
$$350$$ − 16016.5i − 0.130747i
$$351$$ 0 0
$$352$$ −8220.71 −0.0663474
$$353$$ 40491.9i 0.324951i 0.986713 + 0.162476i $$0.0519479\pi$$
−0.986713 + 0.162476i $$0.948052\pi$$
$$354$$ 0 0
$$355$$ −24625.8 −0.195404
$$356$$ − 57104.1i − 0.450575i
$$357$$ 0 0
$$358$$ 45671.2 0.356350
$$359$$ 30479.4i 0.236492i 0.992984 + 0.118246i $$0.0377272\pi$$
−0.992984 + 0.118246i $$0.962273\pi$$
$$360$$ 0 0
$$361$$ −130024. −0.997722
$$362$$ 75069.1i 0.572854i
$$363$$ 0 0
$$364$$ −23029.2 −0.173810
$$365$$ − 33466.0i − 0.251199i
$$366$$ 0 0
$$367$$ −252948. −1.87801 −0.939007 0.343898i $$-0.888253\pi$$
−0.939007 + 0.343898i $$0.888253\pi$$
$$368$$ − 3526.04i − 0.0260370i
$$369$$ 0 0
$$370$$ 65040.9 0.475098
$$371$$ − 60367.9i − 0.438590i
$$372$$ 0 0
$$373$$ 131647. 0.946221 0.473110 0.881003i $$-0.343131\pi$$
0.473110 + 0.881003i $$0.343131\pi$$
$$374$$ 14759.4i 0.105518i
$$375$$ 0 0
$$376$$ 44477.8 0.314606
$$377$$ − 89985.1i − 0.633123i
$$378$$ 0 0
$$379$$ −64033.9 −0.445791 −0.222896 0.974842i $$-0.571551\pi$$
−0.222896 + 0.974842i $$0.571551\pi$$
$$380$$ − 2462.78i − 0.0170553i
$$381$$ 0 0
$$382$$ 6081.08 0.0416729
$$383$$ 272341.i 1.85659i 0.371849 + 0.928293i $$0.378724\pi$$
−0.371849 + 0.928293i $$0.621276\pi$$
$$384$$ 0 0
$$385$$ −15027.7 −0.101384
$$386$$ 74514.7i 0.500112i
$$387$$ 0 0
$$388$$ 8872.11 0.0589336
$$389$$ 285818.i 1.88882i 0.328775 + 0.944408i $$0.393364\pi$$
−0.328775 + 0.944408i $$0.606636\pi$$
$$390$$ 0 0
$$391$$ −6330.62 −0.0414088
$$392$$ − 7761.20i − 0.0505076i
$$393$$ 0 0
$$394$$ 177470. 1.14323
$$395$$ − 81929.4i − 0.525104i
$$396$$ 0 0
$$397$$ 201945. 1.28130 0.640652 0.767831i $$-0.278664\pi$$
0.640652 + 0.767831i $$0.278664\pi$$
$$398$$ − 116996.i − 0.738593i
$$399$$ 0 0
$$400$$ 19568.4 0.122303
$$401$$ 58597.3i 0.364409i 0.983261 + 0.182204i $$0.0583232\pi$$
−0.983261 + 0.182204i $$0.941677\pi$$
$$402$$ 0 0
$$403$$ −79902.1 −0.491981
$$404$$ − 124212.i − 0.761027i
$$405$$ 0 0
$$406$$ 30326.5 0.183980
$$407$$ 58447.3i 0.352838i
$$408$$ 0 0
$$409$$ 179325. 1.07200 0.535999 0.844219i $$-0.319935\pi$$
0.535999 + 0.844219i $$0.319935\pi$$
$$410$$ 108739.i 0.646871i
$$411$$ 0 0
$$412$$ 136951. 0.806809
$$413$$ 5114.21i 0.0299832i
$$414$$ 0 0
$$415$$ 126114. 0.732263
$$416$$ − 28136.2i − 0.162585i
$$417$$ 0 0
$$418$$ 2213.11 0.0126663
$$419$$ − 153806.i − 0.876085i −0.898954 0.438042i $$-0.855672\pi$$
0.898954 0.438042i $$-0.144328\pi$$
$$420$$ 0 0
$$421$$ −91254.8 −0.514863 −0.257432 0.966297i $$-0.582876\pi$$
−0.257432 + 0.966297i $$0.582876\pi$$
$$422$$ 170249.i 0.956005i
$$423$$ 0 0
$$424$$ 73755.5 0.410263
$$425$$ − 35133.0i − 0.194508i
$$426$$ 0 0
$$427$$ 8129.44 0.0445866
$$428$$ 32691.4i 0.178462i
$$429$$ 0 0
$$430$$ 118740. 0.642186
$$431$$ 348521.i 1.87618i 0.346394 + 0.938089i $$0.387406\pi$$
−0.346394 + 0.938089i $$0.612594\pi$$
$$432$$ 0 0
$$433$$ −86162.8 −0.459562 −0.229781 0.973242i $$-0.573801\pi$$
−0.229781 + 0.973242i $$0.573801\pi$$
$$434$$ − 26928.3i − 0.142965i
$$435$$ 0 0
$$436$$ −77047.4 −0.405308
$$437$$ 949.252i 0.00497071i
$$438$$ 0 0
$$439$$ −284089. −1.47409 −0.737047 0.675842i $$-0.763780\pi$$
−0.737047 + 0.675842i $$0.763780\pi$$
$$440$$ − 18360.3i − 0.0948364i
$$441$$ 0 0
$$442$$ −50515.6 −0.258572
$$443$$ 77877.1i 0.396828i 0.980118 + 0.198414i $$0.0635790\pi$$
−0.980118 + 0.198414i $$0.936421\pi$$
$$444$$ 0 0
$$445$$ 127538. 0.644048
$$446$$ 154902.i 0.778733i
$$447$$ 0 0
$$448$$ 9482.37 0.0472456
$$449$$ 87226.0i 0.432667i 0.976320 + 0.216333i $$0.0694098\pi$$
−0.976320 + 0.216333i $$0.930590\pi$$
$$450$$ 0 0
$$451$$ −97715.5 −0.480408
$$452$$ 43575.7i 0.213288i
$$453$$ 0 0
$$454$$ 66919.2 0.324667
$$455$$ − 51433.9i − 0.248443i
$$456$$ 0 0
$$457$$ 60354.0 0.288984 0.144492 0.989506i $$-0.453845\pi$$
0.144492 + 0.989506i $$0.453845\pi$$
$$458$$ − 51943.3i − 0.247627i
$$459$$ 0 0
$$460$$ 7875.14 0.0372171
$$461$$ − 136850.i − 0.643938i −0.946750 0.321969i $$-0.895655\pi$$
0.946750 0.321969i $$-0.104345\pi$$
$$462$$ 0 0
$$463$$ 167651. 0.782068 0.391034 0.920376i $$-0.372118\pi$$
0.391034 + 0.920376i $$0.372118\pi$$
$$464$$ 37051.8i 0.172097i
$$465$$ 0 0
$$466$$ 201782. 0.929203
$$467$$ 188881.i 0.866074i 0.901376 + 0.433037i $$0.142558\pi$$
−0.901376 + 0.433037i $$0.857442\pi$$
$$468$$ 0 0
$$469$$ 32631.7 0.148352
$$470$$ 99337.7i 0.449696i
$$471$$ 0 0
$$472$$ −6248.36 −0.0280467
$$473$$ 106703.i 0.476929i
$$474$$ 0 0
$$475$$ −5268.05 −0.0233487
$$476$$ − 17024.6i − 0.0751385i
$$477$$ 0 0
$$478$$ 123394. 0.540054
$$479$$ − 396256.i − 1.72705i −0.504305 0.863526i $$-0.668251\pi$$
0.504305 0.863526i $$-0.331749\pi$$
$$480$$ 0 0
$$481$$ −200042. −0.864632
$$482$$ 35206.1i 0.151539i
$$483$$ 0 0
$$484$$ −100629. −0.429568
$$485$$ 19815.2i 0.0842392i
$$486$$ 0 0
$$487$$ −205617. −0.866962 −0.433481 0.901163i $$-0.642715\pi$$
−0.433481 + 0.901163i $$0.642715\pi$$
$$488$$ 9932.27i 0.0417070i
$$489$$ 0 0
$$490$$ 17334.1 0.0721952
$$491$$ 146443.i 0.607445i 0.952761 + 0.303723i $$0.0982296\pi$$
−0.952761 + 0.303723i $$0.901770\pi$$
$$492$$ 0 0
$$493$$ 66522.6 0.273700
$$494$$ 7574.61i 0.0310389i
$$495$$ 0 0
$$496$$ 32900.1 0.133732
$$497$$ 25525.6i 0.103339i
$$498$$ 0 0
$$499$$ −266336. −1.06962 −0.534810 0.844973i $$-0.679617\pi$$
−0.534810 + 0.844973i $$0.679617\pi$$
$$500$$ 133042.i 0.532166i
$$501$$ 0 0
$$502$$ 222865. 0.884371
$$503$$ − 8405.60i − 0.0332225i −0.999862 0.0166113i $$-0.994712\pi$$
0.999862 0.0166113i $$-0.00528777\pi$$
$$504$$ 0 0
$$505$$ 277417. 1.08780
$$506$$ 7076.79i 0.0276398i
$$507$$ 0 0
$$508$$ −28923.8 −0.112080
$$509$$ − 106126.i − 0.409624i −0.978801 0.204812i $$-0.934342\pi$$
0.978801 0.204812i $$-0.0656584\pi$$
$$510$$ 0 0
$$511$$ −34688.8 −0.132846
$$512$$ 11585.2i 0.0441942i
$$513$$ 0 0
$$514$$ −81773.1 −0.309517
$$515$$ 305870.i 1.15325i
$$516$$ 0 0
$$517$$ −89267.3 −0.333973
$$518$$ − 67417.5i − 0.251254i
$$519$$ 0 0
$$520$$ 62840.1 0.232397
$$521$$ − 68150.9i − 0.251071i −0.992089 0.125535i $$-0.959935\pi$$
0.992089 0.125535i $$-0.0400648\pi$$
$$522$$ 0 0
$$523$$ −117480. −0.429496 −0.214748 0.976669i $$-0.568893\pi$$
−0.214748 + 0.976669i $$0.568893\pi$$
$$524$$ − 108812.i − 0.396292i
$$525$$ 0 0
$$526$$ −185098. −0.669006
$$527$$ − 59068.6i − 0.212684i
$$528$$ 0 0
$$529$$ 276806. 0.989153
$$530$$ 164727.i 0.586427i
$$531$$ 0 0
$$532$$ −2552.77 −0.00901962
$$533$$ − 334442.i − 1.17724i
$$534$$ 0 0
$$535$$ −73013.7 −0.255092
$$536$$ 39868.3i 0.138771i
$$537$$ 0 0
$$538$$ 295063. 1.01941
$$539$$ 15576.8i 0.0536168i
$$540$$ 0 0
$$541$$ 78291.9 0.267499 0.133750 0.991015i $$-0.457298\pi$$
0.133750 + 0.991015i $$0.457298\pi$$
$$542$$ − 89478.9i − 0.304594i
$$543$$ 0 0
$$544$$ 20800.0 0.0702856
$$545$$ − 172080.i − 0.579344i
$$546$$ 0 0
$$547$$ −294769. −0.985160 −0.492580 0.870267i $$-0.663946\pi$$
−0.492580 + 0.870267i $$0.663946\pi$$
$$548$$ − 11655.8i − 0.0388134i
$$549$$ 0 0
$$550$$ −39273.9 −0.129831
$$551$$ − 9974.81i − 0.0328550i
$$552$$ 0 0
$$553$$ −84923.0 −0.277700
$$554$$ − 329459.i − 1.07345i
$$555$$ 0 0
$$556$$ 23068.3 0.0746217
$$557$$ 477782.i 1.54000i 0.638047 + 0.769998i $$0.279743\pi$$
−0.638047 + 0.769998i $$0.720257\pi$$
$$558$$ 0 0
$$559$$ −365202. −1.16872
$$560$$ 21178.2i 0.0675324i
$$561$$ 0 0
$$562$$ −49234.1 −0.155881
$$563$$ − 450077.i − 1.41994i −0.704232 0.709970i $$-0.748708\pi$$
0.704232 0.709970i $$-0.251292\pi$$
$$564$$ 0 0
$$565$$ −97323.0 −0.304873
$$566$$ − 29503.2i − 0.0920951i
$$567$$ 0 0
$$568$$ −31186.4 −0.0966647
$$569$$ − 2913.20i − 0.00899800i −0.999990 0.00449900i $$-0.998568\pi$$
0.999990 0.00449900i $$-0.00143208\pi$$
$$570$$ 0 0
$$571$$ 108683. 0.333342 0.166671 0.986013i $$-0.446698\pi$$
0.166671 + 0.986013i $$0.446698\pi$$
$$572$$ 56469.6i 0.172593i
$$573$$ 0 0
$$574$$ 112712. 0.342096
$$575$$ − 16845.4i − 0.0509503i
$$576$$ 0 0
$$577$$ 388049. 1.16556 0.582780 0.812630i $$-0.301965\pi$$
0.582780 + 0.812630i $$0.301965\pi$$
$$578$$ 198889.i 0.595326i
$$579$$ 0 0
$$580$$ −82752.5 −0.245994
$$581$$ − 130722.i − 0.387255i
$$582$$ 0 0
$$583$$ −148028. −0.435518
$$584$$ − 42381.6i − 0.124266i
$$585$$ 0 0
$$586$$ 181584. 0.528790
$$587$$ − 541195.i − 1.57064i −0.619087 0.785322i $$-0.712497\pi$$
0.619087 0.785322i $$-0.287503\pi$$
$$588$$ 0 0
$$589$$ −8857.11 −0.0255306
$$590$$ − 13955.2i − 0.0400897i
$$591$$ 0 0
$$592$$ 82368.4 0.235027
$$593$$ 545815.i 1.55216i 0.630636 + 0.776079i $$0.282794\pi$$
−0.630636 + 0.776079i $$0.717206\pi$$
$$594$$ 0 0
$$595$$ 38023.1 0.107402
$$596$$ − 321268.i − 0.904430i
$$597$$ 0 0
$$598$$ −24221.0 −0.0677315
$$599$$ 30038.9i 0.0837202i 0.999123 + 0.0418601i $$0.0133284\pi$$
−0.999123 + 0.0418601i $$0.986672\pi$$
$$600$$ 0 0
$$601$$ −250122. −0.692473 −0.346237 0.938147i $$-0.612541\pi$$
−0.346237 + 0.938147i $$0.612541\pi$$
$$602$$ − 123079.i − 0.339618i
$$603$$ 0 0
$$604$$ 31631.4 0.0867051
$$605$$ − 224747.i − 0.614021i
$$606$$ 0 0
$$607$$ −270171. −0.733265 −0.366632 0.930366i $$-0.619489\pi$$
−0.366632 + 0.930366i $$0.619489\pi$$
$$608$$ − 3118.89i − 0.00843709i
$$609$$ 0 0
$$610$$ −22183.0 −0.0596156
$$611$$ − 305527.i − 0.818402i
$$612$$ 0 0
$$613$$ 378425. 1.00707 0.503535 0.863975i $$-0.332033\pi$$
0.503535 + 0.863975i $$0.332033\pi$$
$$614$$ − 184404.i − 0.489140i
$$615$$ 0 0
$$616$$ −19031.2 −0.0501539
$$617$$ − 580490.i − 1.52484i −0.647083 0.762420i $$-0.724011\pi$$
0.647083 0.762420i $$-0.275989\pi$$
$$618$$ 0 0
$$619$$ 196149. 0.511922 0.255961 0.966687i $$-0.417608\pi$$
0.255961 + 0.966687i $$0.417608\pi$$
$$620$$ 73479.9i 0.191155i
$$621$$ 0 0
$$622$$ −429627. −1.11048
$$623$$ − 132198.i − 0.340603i
$$624$$ 0 0
$$625$$ −106040. −0.271463
$$626$$ − 153172.i − 0.390870i
$$627$$ 0 0
$$628$$ −123782. −0.313861
$$629$$ − 147884.i − 0.373782i
$$630$$ 0 0
$$631$$ 69136.2 0.173639 0.0868194 0.996224i $$-0.472330\pi$$
0.0868194 + 0.996224i $$0.472330\pi$$
$$632$$ − 103756.i − 0.259764i
$$633$$ 0 0
$$634$$ −388581. −0.966724
$$635$$ − 64599.1i − 0.160206i
$$636$$ 0 0
$$637$$ −53313.2 −0.131388
$$638$$ − 74363.4i − 0.182691i
$$639$$ 0 0
$$640$$ −25874.7 −0.0631708
$$641$$ 426696.i 1.03849i 0.854625 + 0.519245i $$0.173787\pi$$
−0.854625 + 0.519245i $$0.826213\pi$$
$$642$$ 0 0
$$643$$ −129308. −0.312755 −0.156377 0.987697i $$-0.549982\pi$$
−0.156377 + 0.987697i $$0.549982\pi$$
$$644$$ − 8162.89i − 0.0196821i
$$645$$ 0 0
$$646$$ −5599.63 −0.0134182
$$647$$ − 71367.5i − 0.170487i −0.996360 0.0852437i $$-0.972833\pi$$
0.996360 0.0852437i $$-0.0271669\pi$$
$$648$$ 0 0
$$649$$ 12540.5 0.0297732
$$650$$ − 134419.i − 0.318152i
$$651$$ 0 0
$$652$$ −299162. −0.703738
$$653$$ 308085.i 0.722511i 0.932467 + 0.361256i $$0.117652\pi$$
−0.932467 + 0.361256i $$0.882348\pi$$
$$654$$ 0 0
$$655$$ 243024. 0.566456
$$656$$ 137708.i 0.320001i
$$657$$ 0 0
$$658$$ 102967. 0.237820
$$659$$ − 576984.i − 1.32860i −0.747468 0.664298i $$-0.768731\pi$$
0.747468 0.664298i $$-0.231269\pi$$
$$660$$ 0 0
$$661$$ 13885.9 0.0317813 0.0158907 0.999874i $$-0.494942\pi$$
0.0158907 + 0.999874i $$0.494942\pi$$
$$662$$ − 463310.i − 1.05720i
$$663$$ 0 0
$$664$$ 159712. 0.362244
$$665$$ − 5701.42i − 0.0128926i
$$666$$ 0 0
$$667$$ 31896.0 0.0716944
$$668$$ 43331.1i 0.0971061i
$$669$$ 0 0
$$670$$ −89042.8 −0.198358
$$671$$ − 19934.1i − 0.0442744i
$$672$$ 0 0
$$673$$ 61470.0 0.135717 0.0678583 0.997695i $$-0.478383\pi$$
0.0678583 + 0.997695i $$0.478383\pi$$
$$674$$ 464080.i 1.02158i
$$675$$ 0 0
$$676$$ 35214.7 0.0770603
$$677$$ − 553915.i − 1.20855i −0.796774 0.604277i $$-0.793462\pi$$
0.796774 0.604277i $$-0.206538\pi$$
$$678$$ 0 0
$$679$$ 20539.2 0.0445497
$$680$$ 46455.3i 0.100466i
$$681$$ 0 0
$$682$$ −66030.8 −0.141964
$$683$$ 111142.i 0.238253i 0.992879 + 0.119127i $$0.0380094\pi$$
−0.992879 + 0.119127i $$0.961991\pi$$
$$684$$ 0 0
$$685$$ 26032.4 0.0554796
$$686$$ − 17967.4i − 0.0381802i
$$687$$ 0 0
$$688$$ 150374. 0.317684
$$689$$ − 506641.i − 1.06724i
$$690$$ 0 0
$$691$$ −284092. −0.594981 −0.297491 0.954725i $$-0.596150\pi$$
−0.297491 + 0.954725i $$0.596150\pi$$
$$692$$ − 113298.i − 0.236597i
$$693$$ 0 0
$$694$$ −205936. −0.427575
$$695$$ 51521.2i 0.106664i
$$696$$ 0 0
$$697$$ 247240. 0.508924
$$698$$ 44826.5i 0.0920076i
$$699$$ 0 0
$$700$$ 45301.5 0.0924520
$$701$$ − 125819.i − 0.256041i −0.991772 0.128020i $$-0.959138\pi$$
0.991772 0.128020i $$-0.0408623\pi$$
$$702$$ 0 0
$$703$$ −22174.6 −0.0448688
$$704$$ − 23251.7i − 0.0469147i
$$705$$ 0 0
$$706$$ −114528. −0.229775
$$707$$ − 287554.i − 0.575282i
$$708$$ 0 0
$$709$$ −864378. −1.71954 −0.859768 0.510686i $$-0.829392\pi$$
−0.859768 + 0.510686i $$0.829392\pi$$
$$710$$ − 69652.3i − 0.138172i
$$711$$ 0 0
$$712$$ 161515. 0.318605
$$713$$ − 28322.0i − 0.0557115i
$$714$$ 0 0
$$715$$ −126121. −0.246703
$$716$$ 129178.i 0.251977i
$$717$$ 0 0
$$718$$ −86208.6 −0.167225
$$719$$ 537857.i 1.04042i 0.854038 + 0.520210i $$0.174146\pi$$
−0.854038 + 0.520210i $$0.825854\pi$$
$$720$$ 0 0
$$721$$ 317046. 0.609890
$$722$$ − 367764.i − 0.705496i
$$723$$ 0 0
$$724$$ −212327. −0.405069
$$725$$ 177013.i 0.336767i
$$726$$ 0 0
$$727$$ −182656. −0.345592 −0.172796 0.984958i $$-0.555280\pi$$
−0.172796 + 0.984958i $$0.555280\pi$$
$$728$$ − 65136.3i − 0.122902i
$$729$$ 0 0
$$730$$ 94656.0 0.177624
$$731$$ − 269980.i − 0.505238i
$$732$$ 0 0
$$733$$ 690415. 1.28500 0.642499 0.766287i $$-0.277898\pi$$
0.642499 + 0.766287i $$0.277898\pi$$
$$734$$ − 715445.i − 1.32796i
$$735$$ 0 0
$$736$$ 9973.14 0.0184110
$$737$$ − 80015.9i − 0.147313i
$$738$$ 0 0
$$739$$ −447899. −0.820146 −0.410073 0.912053i $$-0.634497\pi$$
−0.410073 + 0.912053i $$0.634497\pi$$
$$740$$ 183964.i 0.335945i
$$741$$ 0 0
$$742$$ 170746. 0.310130
$$743$$ 156899.i 0.284212i 0.989851 + 0.142106i $$0.0453873\pi$$
−0.989851 + 0.142106i $$0.954613\pi$$
$$744$$ 0 0
$$745$$ 717527. 1.29278
$$746$$ 372353.i 0.669079i
$$747$$ 0 0
$$748$$ −41745.9 −0.0746123
$$749$$ 75681.6i 0.134905i
$$750$$ 0 0
$$751$$ −733687. −1.30086 −0.650431 0.759565i $$-0.725412\pi$$
−0.650431 + 0.759565i $$0.725412\pi$$
$$752$$ 125802.i 0.222460i
$$753$$ 0 0
$$754$$ 254516. 0.447686
$$755$$ 70646.3i 0.123935i
$$756$$ 0 0
$$757$$ 1.11424e6 1.94440 0.972202 0.234142i $$-0.0752281\pi$$
0.972202 + 0.234142i $$0.0752281\pi$$
$$758$$ − 181115.i − 0.315222i
$$759$$ 0 0
$$760$$ 6965.80 0.0120599
$$761$$ 747308.i 1.29042i 0.764006 + 0.645209i $$0.223230\pi$$
−0.764006 + 0.645209i $$0.776770\pi$$
$$762$$ 0 0
$$763$$ −178367. −0.306384
$$764$$ 17199.9i 0.0294672i
$$765$$ 0 0
$$766$$ −770296. −1.31280
$$767$$ 42921.2i 0.0729594i
$$768$$ 0 0
$$769$$ 965130. 1.63205 0.816024 0.578017i $$-0.196173\pi$$
0.816024 + 0.578017i $$0.196173\pi$$
$$770$$ − 42504.7i − 0.0716896i
$$771$$ 0 0
$$772$$ −210759. −0.353633
$$773$$ − 195019.i − 0.326376i −0.986595 0.163188i $$-0.947822\pi$$
0.986595 0.163188i $$-0.0521776\pi$$
$$774$$ 0 0
$$775$$ 157178. 0.261691
$$776$$ 25094.1i 0.0416724i
$$777$$ 0 0
$$778$$ −808414. −1.33560
$$779$$ − 37072.7i − 0.0610913i
$$780$$ 0 0
$$781$$ 62591.3 0.102615
$$782$$ − 17905.7i − 0.0292804i
$$783$$ 0 0
$$784$$ 21952.0 0.0357143
$$785$$ − 276457.i − 0.448630i
$$786$$ 0 0
$$787$$ 700518. 1.13102 0.565510 0.824742i $$-0.308679\pi$$
0.565510 + 0.824742i $$0.308679\pi$$
$$788$$ 501960.i 0.808382i
$$789$$ 0 0
$$790$$ 231731. 0.371305
$$791$$ 100879.i 0.161231i
$$792$$ 0 0
$$793$$ 68226.7 0.108495
$$794$$ 571187.i 0.906019i
$$795$$ 0 0
$$796$$ 330915. 0.522264
$$797$$ 6535.45i 0.0102887i 0.999987 + 0.00514433i $$0.00163750\pi$$
−0.999987 + 0.00514433i $$0.998363\pi$$
$$798$$ 0 0
$$799$$ 225864. 0.353797
$$800$$ 55347.8i 0.0864809i
$$801$$ 0 0
$$802$$ −165738. −0.257676
$$803$$ 85060.2i 0.131915i
$$804$$ 0 0
$$805$$ 18231.2 0.0281335
$$806$$ − 225997.i − 0.347883i
$$807$$ 0 0
$$808$$ 351324. 0.538127
$$809$$ 1.10789e6i 1.69278i 0.532566 + 0.846388i $$0.321228\pi$$
−0.532566 + 0.846388i $$0.678772\pi$$
$$810$$ 0 0
$$811$$ −961969. −1.46258 −0.731289 0.682067i $$-0.761081\pi$$
−0.731289 + 0.682067i $$0.761081\pi$$
$$812$$ 85776.2i 0.130093i
$$813$$ 0 0
$$814$$ −165314. −0.249494
$$815$$ − 668155.i − 1.00592i
$$816$$ 0 0
$$817$$ −40482.4 −0.0606488
$$818$$ 507208.i 0.758017i
$$819$$ 0 0
$$820$$ −307560. −0.457407
$$821$$ 926435.i 1.37445i 0.726445 + 0.687224i $$0.241171\pi$$
−0.726445 + 0.687224i $$0.758829\pi$$
$$822$$ 0 0
$$823$$ 608329. 0.898129 0.449064 0.893499i $$-0.351757\pi$$
0.449064 + 0.893499i $$0.351757\pi$$
$$824$$ 387356.i 0.570500i
$$825$$ 0 0
$$826$$ −14465.2 −0.0212013
$$827$$ − 26569.8i − 0.0388487i −0.999811 0.0194244i $$-0.993817\pi$$
0.999811 0.0194244i $$-0.00618335\pi$$
$$828$$ 0 0
$$829$$ 785642. 1.14318 0.571591 0.820538i $$-0.306326\pi$$
0.571591 + 0.820538i $$0.306326\pi$$
$$830$$ 356704.i 0.517788i
$$831$$ 0 0
$$832$$ 79581.3 0.114965
$$833$$ − 39412.5i − 0.0567994i
$$834$$ 0 0
$$835$$ −96776.7 −0.138803
$$836$$ 6259.63i 0.00895646i
$$837$$ 0 0
$$838$$ 435030. 0.619485
$$839$$ − 338072.i − 0.480270i −0.970739 0.240135i $$-0.922808\pi$$
0.970739 0.240135i $$-0.0771918\pi$$
$$840$$ 0 0
$$841$$ 372115. 0.526121
$$842$$ − 258108.i − 0.364063i
$$843$$ 0 0
$$844$$ −481538. −0.675998
$$845$$ 78649.4i 0.110149i
$$846$$ 0 0
$$847$$ −232959. −0.324723
$$848$$ 208612.i 0.290100i
$$849$$ 0 0
$$850$$ 99371.0 0.137538
$$851$$ − 70906.7i − 0.0979103i
$$852$$ 0 0
$$853$$ 742542. 1.02052 0.510262 0.860019i $$-0.329548\pi$$
0.510262 + 0.860019i $$0.329548\pi$$
$$854$$ 22993.5i 0.0315275i
$$855$$ 0 0
$$856$$ −92465.2 −0.126192
$$857$$ − 419964.i − 0.571808i −0.958258 0.285904i $$-0.907706\pi$$
0.958258 0.285904i $$-0.0922938\pi$$
$$858$$ 0 0
$$859$$ 843636. 1.14332 0.571662 0.820490i $$-0.306299\pi$$
0.571662 + 0.820490i $$0.306299\pi$$
$$860$$ 335848.i 0.454094i
$$861$$ 0 0
$$862$$ −985766. −1.32666
$$863$$ 346514.i 0.465264i 0.972565 + 0.232632i $$0.0747338\pi$$
−0.972565 + 0.232632i $$0.925266\pi$$
$$864$$ 0 0
$$865$$ 253042. 0.338190
$$866$$ − 243705.i − 0.324959i
$$867$$ 0 0
$$868$$ 76164.8 0.101092
$$869$$ 208239.i 0.275755i
$$870$$ 0 0
$$871$$ 273863. 0.360992
$$872$$ − 217923.i − 0.286596i
$$873$$ 0 0
$$874$$ −2684.89 −0.00351482
$$875$$ 307995.i 0.402280i
$$876$$ 0 0
$$877$$ 316397. 0.411370 0.205685 0.978618i $$-0.434058\pi$$
0.205685 + 0.978618i $$0.434058\pi$$
$$878$$ − 803524.i − 1.04234i
$$879$$ 0 0
$$880$$ 51930.8 0.0670594
$$881$$ − 67739.5i − 0.0872751i −0.999047 0.0436375i $$-0.986105\pi$$
0.999047 0.0436375i $$-0.0138947\pi$$
$$882$$ 0 0
$$883$$ 1.13315e6 1.45333 0.726666 0.686991i $$-0.241069\pi$$
0.726666 + 0.686991i $$0.241069\pi$$
$$884$$ − 142880.i − 0.182838i
$$885$$ 0 0
$$886$$ −220270. −0.280600
$$887$$ 925863.i 1.17679i 0.808573 + 0.588396i $$0.200240\pi$$
−0.808573 + 0.588396i $$0.799760\pi$$
$$888$$ 0 0
$$889$$ −66959.5 −0.0847244
$$890$$ 360731.i 0.455411i
$$891$$ 0 0
$$892$$ −438130. −0.550647
$$893$$ − 33867.5i − 0.0424698i
$$894$$ 0 0
$$895$$ −288509. −0.360174
$$896$$ 26820.2i 0.0334077i
$$897$$ 0 0
$$898$$ −246712. −0.305942
$$899$$ 297610.i 0.368237i
$$900$$ 0 0
$$901$$ 374540. 0.461370
$$902$$ − 276381.i − 0.339700i
$$903$$ 0 0
$$904$$ −123251. −0.150818
$$905$$ − 474217.i − 0.579002i
$$906$$ 0 0
$$907$$ 478626. 0.581811 0.290905 0.956752i $$-0.406043\pi$$
0.290905 + 0.956752i $$0.406043\pi$$
$$908$$ 189276.i 0.229575i
$$909$$ 0 0
$$910$$ 145477. 0.175676
$$911$$ 388418.i 0.468018i 0.972234 + 0.234009i $$0.0751845\pi$$
−0.972234 + 0.234009i $$0.924815\pi$$
$$912$$ 0 0
$$913$$ −320543. −0.384543
$$914$$ 170707.i 0.204342i
$$915$$ 0 0
$$916$$ 146918. 0.175099
$$917$$ − 251904.i − 0.299569i
$$918$$ 0 0
$$919$$ −807823. −0.956501 −0.478250 0.878223i $$-0.658729\pi$$
−0.478250 + 0.878223i $$0.658729\pi$$
$$920$$ 22274.2i 0.0263165i
$$921$$ 0 0
$$922$$ 387071. 0.455333
$$923$$ 214225.i 0.251459i
$$924$$ 0 0
$$925$$ 393510. 0.459910
$$926$$ 474189.i 0.553005i
$$927$$ 0 0
$$928$$ −104798. −0.121691
$$929$$ − 1.62506e6i − 1.88294i −0.337094 0.941471i $$-0.609444\pi$$
0.337094 0.941471i $$-0.390556\pi$$
$$930$$ 0 0
$$931$$ −5909.75 −0.00681819
$$932$$ 570726.i 0.657046i
$$933$$ 0 0
$$934$$ −534237. −0.612407
$$935$$ − 93236.2i − 0.106650i
$$936$$ 0 0
$$937$$ −1.17872e6 −1.34255 −0.671275 0.741208i $$-0.734253\pi$$
−0.671275 + 0.741208i $$0.734253\pi$$
$$938$$ 92296.3i 0.104901i
$$939$$ 0 0
$$940$$ −280970. −0.317983
$$941$$ 418331.i 0.472433i 0.971700 + 0.236217i $$0.0759075\pi$$
−0.971700 + 0.236217i $$0.924093\pi$$
$$942$$ 0 0
$$943$$ 118546. 0.133310
$$944$$ − 17673.0i − 0.0198320i
$$945$$ 0 0
$$946$$ −301801. −0.337240
$$947$$ 860451.i 0.959458i 0.877417 + 0.479729i $$0.159265\pi$$
−0.877417 + 0.479729i $$0.840735\pi$$
$$948$$ 0 0
$$949$$ −291127. −0.323259
$$950$$ − 14900.3i − 0.0165100i
$$951$$ 0 0
$$952$$ 48152.8 0.0531309
$$953$$ 555766.i 0.611937i 0.952042 + 0.305968i $$0.0989801\pi$$
−0.952042 + 0.305968i $$0.901020\pi$$
$$954$$ 0 0
$$955$$ −38414.7 −0.0421202
$$956$$ 349010.i 0.381876i
$$957$$ 0 0
$$958$$ 1.12078e6 1.22121
$$959$$ − 26983.6i − 0.0293402i
$$960$$ 0 0
$$961$$ −659259. −0.713854
$$962$$ − 565805.i − 0.611387i
$$963$$ 0 0
$$964$$ −99577.9 −0.107154
$$965$$ − 470715.i − 0.505479i
$$966$$ 0 0
$$967$$ −862383. −0.922247 −0.461124 0.887336i $$-0.652554\pi$$
−0.461124 + 0.887336i $$0.652554\pi$$
$$968$$ − 284622.i − 0.303751i
$$969$$ 0 0
$$970$$ −56045.8 −0.0595661
$$971$$ 1.69499e6i 1.79775i 0.438209 + 0.898873i $$0.355613\pi$$
−0.438209 + 0.898873i $$0.644387\pi$$
$$972$$ 0 0
$$973$$ 53403.8 0.0564087
$$974$$ − 581572.i − 0.613035i
$$975$$ 0 0
$$976$$ −28092.7 −0.0294913
$$977$$ 1.20992e6i 1.26755i 0.773516 + 0.633777i $$0.218496\pi$$
−0.773516 + 0.633777i $$0.781504\pi$$
$$978$$ 0 0
$$979$$ −324161. −0.338217
$$980$$ 49028.1i 0.0510497i
$$981$$ 0 0
$$982$$ −414205. −0.429529
$$983$$ 1.28838e6i 1.33332i 0.745360 + 0.666662i $$0.232277\pi$$
−0.745360 + 0.666662i $$0.767723\pi$$
$$984$$ 0 0
$$985$$ −1.12109e6 −1.15549
$$986$$ 188154.i 0.193535i
$$987$$ 0 0
$$988$$ −21424.2 −0.0219478
$$989$$ − 129449.i − 0.132345i
$$990$$ 0 0
$$991$$ −1.23803e6 −1.26062 −0.630312 0.776342i $$-0.717073\pi$$
−0.630312 + 0.776342i $$0.717073\pi$$
$$992$$ 93055.5i 0.0945625i
$$993$$ 0 0
$$994$$ −72197.4 −0.0730716
$$995$$ 739073.i 0.746520i
$$996$$ 0 0
$$997$$ −288690. −0.290430 −0.145215 0.989400i $$-0.546387\pi$$
−0.145215 + 0.989400i $$0.546387\pi$$
$$998$$ − 753313.i − 0.756335i
$$999$$ 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.b.323.6 yes 8
3.2 odd 2 inner 378.5.b.b.323.3 8

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