# Properties

 Label 441.6.a.be.1.8 Level $441$ Weight $6$ Character 441.1 Self dual yes Analytic conductor $70.729$ Analytic rank $1$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 146x^{6} + 5453x^{4} - 40868x^{2} + 3844$$ x^8 - 146*x^6 + 5453*x^4 - 40868*x^2 + 3844 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}\cdot 7^{4}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.8 Root $$-0.308653$$ of defining polynomial Character $$\chi$$ $$=$$ 441.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+8.12599 q^{2} +34.0317 q^{4} +17.4201 q^{5} +16.5098 q^{8} +O(q^{10})$$ $$q+8.12599 q^{2} +34.0317 q^{4} +17.4201 q^{5} +16.5098 q^{8} +141.556 q^{10} -114.280 q^{11} -205.240 q^{13} -954.857 q^{16} -757.277 q^{17} -1013.65 q^{19} +592.838 q^{20} -928.635 q^{22} -916.299 q^{23} -2821.54 q^{25} -1667.78 q^{26} -1095.47 q^{29} +8233.32 q^{31} -8287.47 q^{32} -6153.62 q^{34} -10716.4 q^{37} -8236.94 q^{38} +287.604 q^{40} +18758.1 q^{41} -4643.49 q^{43} -3889.13 q^{44} -7445.84 q^{46} -13969.4 q^{47} -22927.8 q^{50} -6984.69 q^{52} -29306.0 q^{53} -1990.77 q^{55} -8901.78 q^{58} -30378.0 q^{59} +18658.2 q^{61} +66903.9 q^{62} -36788.5 q^{64} -3575.32 q^{65} +19933.4 q^{67} -25771.4 q^{68} -57338.5 q^{71} -60194.4 q^{73} -87081.2 q^{74} -34496.4 q^{76} +35715.5 q^{79} -16633.7 q^{80} +152428. q^{82} +86641.2 q^{83} -13191.9 q^{85} -37733.0 q^{86} -1886.73 q^{88} -42941.7 q^{89} -31183.2 q^{92} -113515. q^{94} -17658.0 q^{95} +20619.4 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 20 q^{4}+O(q^{10})$$ 8 * q + 20 * q^4 $$8 q + 20 q^{4} - 828 q^{16} - 2384 q^{22} - 2392 q^{25} - 19136 q^{37} - 41184 q^{43} - 13152 q^{46} - 88872 q^{58} - 210812 q^{64} - 42336 q^{67} - 251072 q^{79} - 567664 q^{85} - 88752 q^{88}+O(q^{100})$$ 8 * q + 20 * q^4 - 828 * q^16 - 2384 * q^22 - 2392 * q^25 - 19136 * q^37 - 41184 * q^43 - 13152 * q^46 - 88872 * q^58 - 210812 * q^64 - 42336 * q^67 - 251072 * q^79 - 567664 * q^85 - 88752 * q^88

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 8.12599 1.43649 0.718243 0.695792i $$-0.244947\pi$$
0.718243 + 0.695792i $$0.244947\pi$$
$$3$$ 0 0
$$4$$ 34.0317 1.06349
$$5$$ 17.4201 0.311621 0.155811 0.987787i $$-0.450201\pi$$
0.155811 + 0.987787i $$0.450201\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 16.5098 0.0912047
$$9$$ 0 0
$$10$$ 141.556 0.447639
$$11$$ −114.280 −0.284765 −0.142383 0.989812i $$-0.545476\pi$$
−0.142383 + 0.989812i $$0.545476\pi$$
$$12$$ 0 0
$$13$$ −205.240 −0.336825 −0.168413 0.985717i $$-0.553864\pi$$
−0.168413 + 0.985717i $$0.553864\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −954.857 −0.932477
$$17$$ −757.277 −0.635524 −0.317762 0.948170i $$-0.602931\pi$$
−0.317762 + 0.948170i $$0.602931\pi$$
$$18$$ 0 0
$$19$$ −1013.65 −0.644178 −0.322089 0.946709i $$-0.604385\pi$$
−0.322089 + 0.946709i $$0.604385\pi$$
$$20$$ 592.838 0.331406
$$21$$ 0 0
$$22$$ −928.635 −0.409061
$$23$$ −916.299 −0.361175 −0.180587 0.983559i $$-0.557800\pi$$
−0.180587 + 0.983559i $$0.557800\pi$$
$$24$$ 0 0
$$25$$ −2821.54 −0.902892
$$26$$ −1667.78 −0.483845
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1095.47 −0.241883 −0.120942 0.992660i $$-0.538591\pi$$
−0.120942 + 0.992660i $$0.538591\pi$$
$$30$$ 0 0
$$31$$ 8233.32 1.53876 0.769380 0.638791i $$-0.220565\pi$$
0.769380 + 0.638791i $$0.220565\pi$$
$$32$$ −8287.47 −1.43070
$$33$$ 0 0
$$34$$ −6153.62 −0.912922
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10716.4 −1.28690 −0.643448 0.765490i $$-0.722497\pi$$
−0.643448 + 0.765490i $$0.722497\pi$$
$$38$$ −8236.94 −0.925352
$$39$$ 0 0
$$40$$ 287.604 0.0284213
$$41$$ 18758.1 1.74273 0.871365 0.490636i $$-0.163235\pi$$
0.871365 + 0.490636i $$0.163235\pi$$
$$42$$ 0 0
$$43$$ −4643.49 −0.382978 −0.191489 0.981495i $$-0.561332\pi$$
−0.191489 + 0.981495i $$0.561332\pi$$
$$44$$ −3889.13 −0.302845
$$45$$ 0 0
$$46$$ −7445.84 −0.518823
$$47$$ −13969.4 −0.922428 −0.461214 0.887289i $$-0.652586\pi$$
−0.461214 + 0.887289i $$0.652586\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −22927.8 −1.29699
$$51$$ 0 0
$$52$$ −6984.69 −0.358211
$$53$$ −29306.0 −1.43307 −0.716533 0.697553i $$-0.754272\pi$$
−0.716533 + 0.697553i $$0.754272\pi$$
$$54$$ 0 0
$$55$$ −1990.77 −0.0887388
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −8901.78 −0.347462
$$59$$ −30378.0 −1.13613 −0.568067 0.822983i $$-0.692308\pi$$
−0.568067 + 0.822983i $$0.692308\pi$$
$$60$$ 0 0
$$61$$ 18658.2 0.642014 0.321007 0.947077i $$-0.395979\pi$$
0.321007 + 0.947077i $$0.395979\pi$$
$$62$$ 66903.9 2.21041
$$63$$ 0 0
$$64$$ −36788.5 −1.12270
$$65$$ −3575.32 −0.104962
$$66$$ 0 0
$$67$$ 19933.4 0.542493 0.271246 0.962510i $$-0.412564\pi$$
0.271246 + 0.962510i $$0.412564\pi$$
$$68$$ −25771.4 −0.675875
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −57338.5 −1.34990 −0.674948 0.737866i $$-0.735834\pi$$
−0.674948 + 0.737866i $$0.735834\pi$$
$$72$$ 0 0
$$73$$ −60194.4 −1.32205 −0.661027 0.750362i $$-0.729879\pi$$
−0.661027 + 0.750362i $$0.729879\pi$$
$$74$$ −87081.2 −1.84861
$$75$$ 0 0
$$76$$ −34496.4 −0.685078
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 35715.5 0.643857 0.321928 0.946764i $$-0.395669\pi$$
0.321928 + 0.946764i $$0.395669\pi$$
$$80$$ −16633.7 −0.290580
$$81$$ 0 0
$$82$$ 152428. 2.50341
$$83$$ 86641.2 1.38048 0.690239 0.723582i $$-0.257506\pi$$
0.690239 + 0.723582i $$0.257506\pi$$
$$84$$ 0 0
$$85$$ −13191.9 −0.198043
$$86$$ −37733.0 −0.550142
$$87$$ 0 0
$$88$$ −1886.73 −0.0259719
$$89$$ −42941.7 −0.574651 −0.287326 0.957833i $$-0.592766\pi$$
−0.287326 + 0.957833i $$0.592766\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −31183.2 −0.384107
$$93$$ 0 0
$$94$$ −113515. −1.32506
$$95$$ −17658.0 −0.200739
$$96$$ 0 0
$$97$$ 20619.4 0.222508 0.111254 0.993792i $$-0.464513\pi$$
0.111254 + 0.993792i $$0.464513\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −96021.8 −0.960218
$$101$$ −106588. −1.03970 −0.519848 0.854259i $$-0.674012\pi$$
−0.519848 + 0.854259i $$0.674012\pi$$
$$102$$ 0 0
$$103$$ 87392.9 0.811677 0.405839 0.913945i $$-0.366980\pi$$
0.405839 + 0.913945i $$0.366980\pi$$
$$104$$ −3388.48 −0.0307201
$$105$$ 0 0
$$106$$ −238140. −2.05858
$$107$$ 39306.9 0.331902 0.165951 0.986134i $$-0.446931\pi$$
0.165951 + 0.986134i $$0.446931\pi$$
$$108$$ 0 0
$$109$$ −44341.3 −0.357472 −0.178736 0.983897i $$-0.557201\pi$$
−0.178736 + 0.983897i $$0.557201\pi$$
$$110$$ −16177.0 −0.127472
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 64665.1 0.476402 0.238201 0.971216i $$-0.423442\pi$$
0.238201 + 0.971216i $$0.423442\pi$$
$$114$$ 0 0
$$115$$ −15962.1 −0.112550
$$116$$ −37280.7 −0.257241
$$117$$ 0 0
$$118$$ −246851. −1.63204
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −147991. −0.918909
$$122$$ 151616. 0.922244
$$123$$ 0 0
$$124$$ 280194. 1.63646
$$125$$ −103590. −0.592981
$$126$$ 0 0
$$127$$ −155448. −0.855215 −0.427607 0.903965i $$-0.640643\pi$$
−0.427607 + 0.903965i $$0.640643\pi$$
$$128$$ −33744.0 −0.182042
$$129$$ 0 0
$$130$$ −29053.0 −0.150776
$$131$$ 189150. 0.963002 0.481501 0.876446i $$-0.340092\pi$$
0.481501 + 0.876446i $$0.340092\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 161978. 0.779283
$$135$$ 0 0
$$136$$ −12502.5 −0.0579628
$$137$$ 344445. 1.56790 0.783950 0.620824i $$-0.213202\pi$$
0.783950 + 0.620824i $$0.213202\pi$$
$$138$$ 0 0
$$139$$ −270646. −1.18813 −0.594065 0.804417i $$-0.702478\pi$$
−0.594065 + 0.804417i $$0.702478\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −465932. −1.93911
$$143$$ 23454.8 0.0959161
$$144$$ 0 0
$$145$$ −19083.2 −0.0753759
$$146$$ −489139. −1.89911
$$147$$ 0 0
$$148$$ −364697. −1.36860
$$149$$ 131911. 0.486759 0.243379 0.969931i $$-0.421744\pi$$
0.243379 + 0.969931i $$0.421744\pi$$
$$150$$ 0 0
$$151$$ −322961. −1.15268 −0.576338 0.817211i $$-0.695519\pi$$
−0.576338 + 0.817211i $$0.695519\pi$$
$$152$$ −16735.2 −0.0587521
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 143426. 0.479510
$$156$$ 0 0
$$157$$ 181881. 0.588894 0.294447 0.955668i $$-0.404865\pi$$
0.294447 + 0.955668i $$0.404865\pi$$
$$158$$ 290224. 0.924891
$$159$$ 0 0
$$160$$ −144369. −0.445835
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −196117. −0.578157 −0.289078 0.957305i $$-0.593349\pi$$
−0.289078 + 0.957305i $$0.593349\pi$$
$$164$$ 638372. 1.85338
$$165$$ 0 0
$$166$$ 704046. 1.98304
$$167$$ −293276. −0.813741 −0.406870 0.913486i $$-0.633380\pi$$
−0.406870 + 0.913486i $$0.633380\pi$$
$$168$$ 0 0
$$169$$ −329169. −0.886549
$$170$$ −107197. −0.284486
$$171$$ 0 0
$$172$$ −158026. −0.407294
$$173$$ 442828. 1.12492 0.562458 0.826826i $$-0.309856\pi$$
0.562458 + 0.826826i $$0.309856\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 109121. 0.265537
$$177$$ 0 0
$$178$$ −348944. −0.825479
$$179$$ 661446. 1.54298 0.771492 0.636239i $$-0.219511\pi$$
0.771492 + 0.636239i $$0.219511\pi$$
$$180$$ 0 0
$$181$$ 139700. 0.316956 0.158478 0.987363i $$-0.449341\pi$$
0.158478 + 0.987363i $$0.449341\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −15127.9 −0.0329409
$$185$$ −186681. −0.401024
$$186$$ 0 0
$$187$$ 86541.2 0.180975
$$188$$ −475402. −0.980995
$$189$$ 0 0
$$190$$ −143489. −0.288359
$$191$$ 751907. 1.49135 0.745677 0.666308i $$-0.232126\pi$$
0.745677 + 0.666308i $$0.232126\pi$$
$$192$$ 0 0
$$193$$ 464929. 0.898449 0.449224 0.893419i $$-0.351700\pi$$
0.449224 + 0.893419i $$0.351700\pi$$
$$194$$ 167553. 0.319630
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 518493. 0.951870 0.475935 0.879480i $$-0.342110\pi$$
0.475935 + 0.879480i $$0.342110\pi$$
$$198$$ 0 0
$$199$$ −590287. −1.05665 −0.528324 0.849043i $$-0.677179\pi$$
−0.528324 + 0.849043i $$0.677179\pi$$
$$200$$ −46583.1 −0.0823480
$$201$$ 0 0
$$202$$ −866137. −1.49351
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 326770. 0.543071
$$206$$ 710154. 1.16596
$$207$$ 0 0
$$208$$ 195975. 0.314082
$$209$$ 115840. 0.183439
$$210$$ 0 0
$$211$$ 207037. 0.320142 0.160071 0.987106i $$-0.448828\pi$$
0.160071 + 0.987106i $$0.448828\pi$$
$$212$$ −997333. −1.52405
$$213$$ 0 0
$$214$$ 319408. 0.476772
$$215$$ −80890.3 −0.119344
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −360317. −0.513503
$$219$$ 0 0
$$220$$ −67749.2 −0.0943730
$$221$$ 155424. 0.214061
$$222$$ 0 0
$$223$$ 1.23347e6 1.66098 0.830491 0.557033i $$-0.188060\pi$$
0.830491 + 0.557033i $$0.188060\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 525468. 0.684345
$$227$$ −1.12039e6 −1.44312 −0.721562 0.692350i $$-0.756576\pi$$
−0.721562 + 0.692350i $$0.756576\pi$$
$$228$$ 0 0
$$229$$ 412271. 0.519510 0.259755 0.965675i $$-0.416358\pi$$
0.259755 + 0.965675i $$0.416358\pi$$
$$230$$ −129708. −0.161676
$$231$$ 0 0
$$232$$ −18086.0 −0.0220609
$$233$$ 836448. 1.00937 0.504684 0.863304i $$-0.331609\pi$$
0.504684 + 0.863304i $$0.331609\pi$$
$$234$$ 0 0
$$235$$ −243349. −0.287448
$$236$$ −1.03382e6 −1.20827
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −319694. −0.362026 −0.181013 0.983481i $$-0.557938\pi$$
−0.181013 + 0.983481i $$0.557938\pi$$
$$240$$ 0 0
$$241$$ −965937. −1.07129 −0.535644 0.844444i $$-0.679931\pi$$
−0.535644 + 0.844444i $$0.679931\pi$$
$$242$$ −1.20258e6 −1.32000
$$243$$ 0 0
$$244$$ 634970. 0.682776
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 208043. 0.216975
$$248$$ 135931. 0.140342
$$249$$ 0 0
$$250$$ −841768. −0.851809
$$251$$ 1.60171e6 1.60472 0.802361 0.596839i $$-0.203577\pi$$
0.802361 + 0.596839i $$0.203577\pi$$
$$252$$ 0 0
$$253$$ 104714. 0.102850
$$254$$ −1.26317e6 −1.22850
$$255$$ 0 0
$$256$$ 903029. 0.861196
$$257$$ 1.08451e6 1.02424 0.512120 0.858914i $$-0.328860\pi$$
0.512120 + 0.858914i $$0.328860\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −121674. −0.111626
$$261$$ 0 0
$$262$$ 1.53703e6 1.38334
$$263$$ −1.04604e6 −0.932517 −0.466259 0.884648i $$-0.654398\pi$$
−0.466259 + 0.884648i $$0.654398\pi$$
$$264$$ 0 0
$$265$$ −510514. −0.446574
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 678368. 0.576937
$$269$$ −1.41567e6 −1.19284 −0.596419 0.802673i $$-0.703410\pi$$
−0.596419 + 0.802673i $$0.703410\pi$$
$$270$$ 0 0
$$271$$ −884078. −0.731252 −0.365626 0.930762i $$-0.619145\pi$$
−0.365626 + 0.930762i $$0.619145\pi$$
$$272$$ 723091. 0.592612
$$273$$ 0 0
$$274$$ 2.79896e6 2.25227
$$275$$ 322444. 0.257112
$$276$$ 0 0
$$277$$ −1.77062e6 −1.38652 −0.693258 0.720690i $$-0.743825\pi$$
−0.693258 + 0.720690i $$0.743825\pi$$
$$278$$ −2.19926e6 −1.70673
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −802195. −0.606058 −0.303029 0.952981i $$-0.597998\pi$$
−0.303029 + 0.952981i $$0.597998\pi$$
$$282$$ 0 0
$$283$$ 1.81693e6 1.34856 0.674282 0.738474i $$-0.264453\pi$$
0.674282 + 0.738474i $$0.264453\pi$$
$$284$$ −1.95133e6 −1.43560
$$285$$ 0 0
$$286$$ 190593. 0.137782
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −846389. −0.596109
$$290$$ −155070. −0.108276
$$291$$ 0 0
$$292$$ −2.04852e6 −1.40599
$$293$$ 2.34122e6 1.59321 0.796605 0.604500i $$-0.206627\pi$$
0.796605 + 0.604500i $$0.206627\pi$$
$$294$$ 0 0
$$295$$ −529189. −0.354043
$$296$$ −176925. −0.117371
$$297$$ 0 0
$$298$$ 1.07190e6 0.699222
$$299$$ 188062. 0.121653
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −2.62438e6 −1.65580
$$303$$ 0 0
$$304$$ 967894. 0.600681
$$305$$ 325028. 0.200065
$$306$$ 0 0
$$307$$ 855540. 0.518077 0.259039 0.965867i $$-0.416594\pi$$
0.259039 + 0.965867i $$0.416594\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 1.16548e6 0.688809
$$311$$ −2.58071e6 −1.51300 −0.756498 0.653996i $$-0.773091\pi$$
−0.756498 + 0.653996i $$0.773091\pi$$
$$312$$ 0 0
$$313$$ −3.03363e6 −1.75026 −0.875129 0.483889i $$-0.839224\pi$$
−0.875129 + 0.483889i $$0.839224\pi$$
$$314$$ 1.47796e6 0.845938
$$315$$ 0 0
$$316$$ 1.21546e6 0.684736
$$317$$ 751440. 0.419997 0.209999 0.977702i $$-0.432654\pi$$
0.209999 + 0.977702i $$0.432654\pi$$
$$318$$ 0 0
$$319$$ 125190. 0.0688799
$$320$$ −640861. −0.349856
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 767616. 0.409391
$$324$$ 0 0
$$325$$ 579094. 0.304117
$$326$$ −1.59364e6 −0.830514
$$327$$ 0 0
$$328$$ 309693. 0.158945
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −726862. −0.364655 −0.182327 0.983238i $$-0.558363\pi$$
−0.182327 + 0.983238i $$0.558363\pi$$
$$332$$ 2.94855e6 1.46813
$$333$$ 0 0
$$334$$ −2.38316e6 −1.16893
$$335$$ 347243. 0.169052
$$336$$ 0 0
$$337$$ 650213. 0.311875 0.155938 0.987767i $$-0.450160\pi$$
0.155938 + 0.987767i $$0.450160\pi$$
$$338$$ −2.67483e6 −1.27351
$$339$$ 0 0
$$340$$ −448942. −0.210617
$$341$$ −940900. −0.438185
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −76663.2 −0.0349294
$$345$$ 0 0
$$346$$ 3.59842e6 1.61593
$$347$$ 60293.3 0.0268810 0.0134405 0.999910i $$-0.495722\pi$$
0.0134405 + 0.999910i $$0.495722\pi$$
$$348$$ 0 0
$$349$$ 3.61218e6 1.58747 0.793735 0.608263i $$-0.208134\pi$$
0.793735 + 0.608263i $$0.208134\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 947088. 0.407412
$$353$$ −681893. −0.291259 −0.145629 0.989339i $$-0.546521\pi$$
−0.145629 + 0.989339i $$0.546521\pi$$
$$354$$ 0 0
$$355$$ −998844. −0.420656
$$356$$ −1.46138e6 −0.611137
$$357$$ 0 0
$$358$$ 5.37490e6 2.21648
$$359$$ 3.14620e6 1.28840 0.644200 0.764857i $$-0.277190\pi$$
0.644200 + 0.764857i $$0.277190\pi$$
$$360$$ 0 0
$$361$$ −1.44860e6 −0.585035
$$362$$ 1.13520e6 0.455303
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1.04860e6 −0.411980
$$366$$ 0 0
$$367$$ −2.97099e6 −1.15143 −0.575714 0.817651i $$-0.695276\pi$$
−0.575714 + 0.817651i $$0.695276\pi$$
$$368$$ 874934. 0.336787
$$369$$ 0 0
$$370$$ −1.51697e6 −0.576066
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3.00412e6 1.11801 0.559005 0.829164i $$-0.311183\pi$$
0.559005 + 0.829164i $$0.311183\pi$$
$$374$$ 703233. 0.259968
$$375$$ 0 0
$$376$$ −230632. −0.0841298
$$377$$ 224835. 0.0814723
$$378$$ 0 0
$$379$$ −754946. −0.269972 −0.134986 0.990848i $$-0.543099\pi$$
−0.134986 + 0.990848i $$0.543099\pi$$
$$380$$ −600932. −0.213485
$$381$$ 0 0
$$382$$ 6.10999e6 2.14231
$$383$$ 3.66829e6 1.27781 0.638905 0.769286i $$-0.279388\pi$$
0.638905 + 0.769286i $$0.279388\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 3.77801e6 1.29061
$$387$$ 0 0
$$388$$ 701712. 0.236635
$$389$$ 3.00960e6 1.00841 0.504203 0.863585i $$-0.331786\pi$$
0.504203 + 0.863585i $$0.331786\pi$$
$$390$$ 0 0
$$391$$ 693892. 0.229536
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 4.21327e6 1.36735
$$395$$ 622170. 0.200639
$$396$$ 0 0
$$397$$ 287204. 0.0914564 0.0457282 0.998954i $$-0.485439\pi$$
0.0457282 + 0.998954i $$0.485439\pi$$
$$398$$ −4.79666e6 −1.51786
$$399$$ 0 0
$$400$$ 2.69416e6 0.841927
$$401$$ −863171. −0.268063 −0.134031 0.990977i $$-0.542792\pi$$
−0.134031 + 0.990977i $$0.542792\pi$$
$$402$$ 0 0
$$403$$ −1.68981e6 −0.518293
$$404$$ −3.62739e6 −1.10571
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.22466e6 0.366463
$$408$$ 0 0
$$409$$ −2.82932e6 −0.836321 −0.418161 0.908373i $$-0.637325\pi$$
−0.418161 + 0.908373i $$0.637325\pi$$
$$410$$ 2.65533e6 0.780114
$$411$$ 0 0
$$412$$ 2.97413e6 0.863212
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.50930e6 0.430186
$$416$$ 1.70092e6 0.481894
$$417$$ 0 0
$$418$$ 941314. 0.263508
$$419$$ −7.08132e6 −1.97051 −0.985256 0.171089i $$-0.945272\pi$$
−0.985256 + 0.171089i $$0.945272\pi$$
$$420$$ 0 0
$$421$$ −2.95296e6 −0.811994 −0.405997 0.913874i $$-0.633076\pi$$
−0.405997 + 0.913874i $$0.633076\pi$$
$$422$$ 1.68238e6 0.459879
$$423$$ 0 0
$$424$$ −483836. −0.130702
$$425$$ 2.13668e6 0.573810
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 1.33768e6 0.352975
$$429$$ 0 0
$$430$$ −657314. −0.171436
$$431$$ 4.47340e6 1.15996 0.579981 0.814630i $$-0.303060\pi$$
0.579981 + 0.814630i $$0.303060\pi$$
$$432$$ 0 0
$$433$$ 5.11279e6 1.31050 0.655252 0.755411i $$-0.272563\pi$$
0.655252 + 0.755411i $$0.272563\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −1.50901e6 −0.380168
$$437$$ 928810. 0.232661
$$438$$ 0 0
$$439$$ −590666. −0.146279 −0.0731393 0.997322i $$-0.523302\pi$$
−0.0731393 + 0.997322i $$0.523302\pi$$
$$440$$ −32867.2 −0.00809340
$$441$$ 0 0
$$442$$ 1.26297e6 0.307495
$$443$$ 160765. 0.0389209 0.0194605 0.999811i $$-0.493805\pi$$
0.0194605 + 0.999811i $$0.493805\pi$$
$$444$$ 0 0
$$445$$ −748051. −0.179073
$$446$$ 1.00231e7 2.38598
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 7.98274e6 1.86869 0.934343 0.356376i $$-0.115988\pi$$
0.934343 + 0.356376i $$0.115988\pi$$
$$450$$ 0 0
$$451$$ −2.14367e6 −0.496269
$$452$$ 2.20067e6 0.506650
$$453$$ 0 0
$$454$$ −9.10426e6 −2.07303
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.22463e6 −0.274292 −0.137146 0.990551i $$-0.543793\pi$$
−0.137146 + 0.990551i $$0.543793\pi$$
$$458$$ 3.35011e6 0.746269
$$459$$ 0 0
$$460$$ −543217. −0.119696
$$461$$ −1.99446e6 −0.437091 −0.218546 0.975827i $$-0.570131\pi$$
−0.218546 + 0.975827i $$0.570131\pi$$
$$462$$ 0 0
$$463$$ −125144. −0.0271304 −0.0135652 0.999908i $$-0.504318\pi$$
−0.0135652 + 0.999908i $$0.504318\pi$$
$$464$$ 1.04602e6 0.225550
$$465$$ 0 0
$$466$$ 6.79697e6 1.44994
$$467$$ −7.06372e6 −1.49879 −0.749396 0.662122i $$-0.769656\pi$$
−0.749396 + 0.662122i $$0.769656\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −1.97745e6 −0.412915
$$471$$ 0 0
$$472$$ −501535. −0.103621
$$473$$ 530656. 0.109059
$$474$$ 0 0
$$475$$ 2.86006e6 0.581623
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −2.59783e6 −0.520045
$$479$$ 4.64810e6 0.925629 0.462814 0.886455i $$-0.346840\pi$$
0.462814 + 0.886455i $$0.346840\pi$$
$$480$$ 0 0
$$481$$ 2.19943e6 0.433459
$$482$$ −7.84919e6 −1.53889
$$483$$ 0 0
$$484$$ −5.03640e6 −0.977252
$$485$$ 359192. 0.0693382
$$486$$ 0 0
$$487$$ −5.57906e6 −1.06595 −0.532977 0.846130i $$-0.678927\pi$$
−0.532977 + 0.846130i $$0.678927\pi$$
$$488$$ 308043. 0.0585547
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −5.81454e6 −1.08846 −0.544229 0.838937i $$-0.683178\pi$$
−0.544229 + 0.838937i $$0.683178\pi$$
$$492$$ 0 0
$$493$$ 829574. 0.153723
$$494$$ 1.69055e6 0.311682
$$495$$ 0 0
$$496$$ −7.86164e6 −1.43486
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −7.17797e6 −1.29048 −0.645238 0.763982i $$-0.723242\pi$$
−0.645238 + 0.763982i $$0.723242\pi$$
$$500$$ −3.52533e6 −0.630631
$$501$$ 0 0
$$502$$ 1.30155e7 2.30516
$$503$$ 3.00238e6 0.529109 0.264555 0.964371i $$-0.414775\pi$$
0.264555 + 0.964371i $$0.414775\pi$$
$$504$$ 0 0
$$505$$ −1.85679e6 −0.323991
$$506$$ 850907. 0.147743
$$507$$ 0 0
$$508$$ −5.29016e6 −0.909514
$$509$$ 3.52386e6 0.602870 0.301435 0.953487i $$-0.402534\pi$$
0.301435 + 0.953487i $$0.402534\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 8.41781e6 1.41914
$$513$$ 0 0
$$514$$ 8.81274e6 1.47131
$$515$$ 1.52240e6 0.252936
$$516$$ 0 0
$$517$$ 1.59641e6 0.262675
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −59027.9 −0.00957302
$$521$$ 3.20451e6 0.517210 0.258605 0.965983i $$-0.416737\pi$$
0.258605 + 0.965983i $$0.416737\pi$$
$$522$$ 0 0
$$523$$ −8.14584e6 −1.30221 −0.651106 0.758987i $$-0.725695\pi$$
−0.651106 + 0.758987i $$0.725695\pi$$
$$524$$ 6.43709e6 1.02414
$$525$$ 0 0
$$526$$ −8.50007e6 −1.33955
$$527$$ −6.23490e6 −0.977920
$$528$$ 0 0
$$529$$ −5.59674e6 −0.869553
$$530$$ −4.14843e6 −0.641497
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3.84993e6 −0.586995
$$534$$ 0 0
$$535$$ 684733. 0.103428
$$536$$ 329097. 0.0494779
$$537$$ 0 0
$$538$$ −1.15037e7 −1.71349
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −9.33721e6 −1.37159 −0.685794 0.727795i $$-0.740545\pi$$
−0.685794 + 0.727795i $$0.740545\pi$$
$$542$$ −7.18401e6 −1.05043
$$543$$ 0 0
$$544$$ 6.27591e6 0.909242
$$545$$ −772432. −0.111396
$$546$$ 0 0
$$547$$ −7.27605e6 −1.03975 −0.519873 0.854244i $$-0.674021\pi$$
−0.519873 + 0.854244i $$0.674021\pi$$
$$548$$ 1.17221e7 1.66745
$$549$$ 0 0
$$550$$ 2.62018e6 0.369338
$$551$$ 1.11043e6 0.155816
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −1.43880e7 −1.99171
$$555$$ 0 0
$$556$$ −9.21054e6 −1.26357
$$557$$ −9.10577e6 −1.24359 −0.621797 0.783178i $$-0.713597\pi$$
−0.621797 + 0.783178i $$0.713597\pi$$
$$558$$ 0 0
$$559$$ 953032. 0.128997
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.51863e6 −0.870594
$$563$$ −1.93485e6 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$564$$ 0 0
$$565$$ 1.12648e6 0.148457
$$566$$ 1.47643e7 1.93719
$$567$$ 0 0
$$568$$ −946648. −0.123117
$$569$$ −8.73948e6 −1.13163 −0.565816 0.824532i $$-0.691439\pi$$
−0.565816 + 0.824532i $$0.691439\pi$$
$$570$$ 0 0
$$571$$ −1.03137e7 −1.32380 −0.661902 0.749591i $$-0.730250\pi$$
−0.661902 + 0.749591i $$0.730250\pi$$
$$572$$ 798207. 0.102006
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 2.58537e6 0.326102
$$576$$ 0 0
$$577$$ −5.56614e6 −0.696009 −0.348004 0.937493i $$-0.613141\pi$$
−0.348004 + 0.937493i $$0.613141\pi$$
$$578$$ −6.87775e6 −0.856302
$$579$$ 0 0
$$580$$ −649436. −0.0801616
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 3.34907e6 0.408087
$$584$$ −993799. −0.120578
$$585$$ 0 0
$$586$$ 1.90247e7 2.28862
$$587$$ 1.57310e6 0.188435 0.0942173 0.995552i $$-0.469965\pi$$
0.0942173 + 0.995552i $$0.469965\pi$$
$$588$$ 0 0
$$589$$ −8.34574e6 −0.991235
$$590$$ −4.30019e6 −0.508578
$$591$$ 0 0
$$592$$ 1.02326e7 1.20000
$$593$$ −1.62461e7 −1.89720 −0.948599 0.316481i $$-0.897499\pi$$
−0.948599 + 0.316481i $$0.897499\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4.48914e6 0.517664
$$597$$ 0 0
$$598$$ 1.52819e6 0.174753
$$599$$ −1.69388e7 −1.92893 −0.964465 0.264210i $$-0.914889\pi$$
−0.964465 + 0.264210i $$0.914889\pi$$
$$600$$ 0 0
$$601$$ −700155. −0.0790693 −0.0395347 0.999218i $$-0.512588\pi$$
−0.0395347 + 0.999218i $$0.512588\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −1.09909e7 −1.22586
$$605$$ −2.57803e6 −0.286351
$$606$$ 0 0
$$607$$ 3.13306e6 0.345141 0.172570 0.984997i $$-0.444793\pi$$
0.172570 + 0.984997i $$0.444793\pi$$
$$608$$ 8.40063e6 0.921622
$$609$$ 0 0
$$610$$ 2.64118e6 0.287391
$$611$$ 2.86708e6 0.310697
$$612$$ 0 0
$$613$$ 2.65140e6 0.284986 0.142493 0.989796i $$-0.454488\pi$$
0.142493 + 0.989796i $$0.454488\pi$$
$$614$$ 6.95211e6 0.744210
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1.47657e7 −1.56149 −0.780747 0.624848i $$-0.785161\pi$$
−0.780747 + 0.624848i $$0.785161\pi$$
$$618$$ 0 0
$$619$$ −8.33398e6 −0.874230 −0.437115 0.899406i $$-0.644000\pi$$
−0.437115 + 0.899406i $$0.644000\pi$$
$$620$$ 4.88103e6 0.509955
$$621$$ 0 0
$$622$$ −2.09708e7 −2.17340
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 7.01276e6 0.718107
$$626$$ −2.46513e7 −2.51422
$$627$$ 0 0
$$628$$ 6.18971e6 0.626284
$$629$$ 8.11526e6 0.817854
$$630$$ 0 0
$$631$$ −1.17243e7 −1.17223 −0.586116 0.810227i $$-0.699344\pi$$
−0.586116 + 0.810227i $$0.699344\pi$$
$$632$$ 589657. 0.0587228
$$633$$ 0 0
$$634$$ 6.10620e6 0.603320
$$635$$ −2.70792e6 −0.266503
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 1.01729e6 0.0989449
$$639$$ 0 0
$$640$$ −587825. −0.0567281
$$641$$ −8.58605e6 −0.825369 −0.412685 0.910874i $$-0.635409\pi$$
−0.412685 + 0.910874i $$0.635409\pi$$
$$642$$ 0 0
$$643$$ 1.87633e7 1.78970 0.894852 0.446364i $$-0.147281\pi$$
0.894852 + 0.446364i $$0.147281\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 6.23764e6 0.588084
$$647$$ 1.59559e7 1.49851 0.749255 0.662282i $$-0.230412\pi$$
0.749255 + 0.662282i $$0.230412\pi$$
$$648$$ 0 0
$$649$$ 3.47159e6 0.323531
$$650$$ 4.70571e6 0.436860
$$651$$ 0 0
$$652$$ −6.67419e6 −0.614865
$$653$$ −1.36524e7 −1.25293 −0.626464 0.779451i $$-0.715498\pi$$
−0.626464 + 0.779451i $$0.715498\pi$$
$$654$$ 0 0
$$655$$ 3.29501e6 0.300092
$$656$$ −1.79113e7 −1.62506
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −4.32613e6 −0.388049 −0.194024 0.980997i $$-0.562154\pi$$
−0.194024 + 0.980997i $$0.562154\pi$$
$$660$$ 0 0
$$661$$ −1.97317e7 −1.75655 −0.878275 0.478157i $$-0.841305\pi$$
−0.878275 + 0.478157i $$0.841305\pi$$
$$662$$ −5.90647e6 −0.523821
$$663$$ 0 0
$$664$$ 1.43043e6 0.125906
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.00378e6 0.0873621
$$668$$ −9.98070e6 −0.865406
$$669$$ 0 0
$$670$$ 2.82169e6 0.242841
$$671$$ −2.13225e6 −0.182823
$$672$$ 0 0
$$673$$ 9.88089e6 0.840927 0.420464 0.907309i $$-0.361867\pi$$
0.420464 + 0.907309i $$0.361867\pi$$
$$674$$ 5.28362e6 0.448004
$$675$$ 0 0
$$676$$ −1.12022e7 −0.942837
$$677$$ −2.22205e6 −0.186329 −0.0931647 0.995651i $$-0.529698\pi$$
−0.0931647 + 0.995651i $$0.529698\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −217795. −0.0180624
$$681$$ 0 0
$$682$$ −7.64575e6 −0.629447
$$683$$ −5.54335e6 −0.454695 −0.227347 0.973814i $$-0.573005\pi$$
−0.227347 + 0.973814i $$0.573005\pi$$
$$684$$ 0 0
$$685$$ 6.00028e6 0.488591
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.43387e6 0.357118
$$689$$ 6.01477e6 0.482693
$$690$$ 0 0
$$691$$ 1.43508e7 1.14335 0.571677 0.820478i $$-0.306293\pi$$
0.571677 + 0.820478i $$0.306293\pi$$
$$692$$ 1.50702e7 1.19634
$$693$$ 0 0
$$694$$ 489943. 0.0386142
$$695$$ −4.71469e6 −0.370246
$$696$$ 0 0
$$697$$ −1.42051e7 −1.10755
$$698$$ 2.93525e7 2.28038
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.50894e7 1.15978 0.579891 0.814694i $$-0.303095\pi$$
0.579891 + 0.814694i $$0.303095\pi$$
$$702$$ 0 0
$$703$$ 1.08627e7 0.828990
$$704$$ 4.20417e6 0.319705
$$705$$ 0 0
$$706$$ −5.54105e6 −0.418389
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.04016e7 0.777115 0.388557 0.921425i $$-0.372974\pi$$
0.388557 + 0.921425i $$0.372974\pi$$
$$710$$ −8.11660e6 −0.604266
$$711$$ 0 0
$$712$$ −708960. −0.0524109
$$713$$ −7.54419e6 −0.555762
$$714$$ 0 0
$$715$$ 408586. 0.0298895
$$716$$ 2.25101e7 1.64095
$$717$$ 0 0
$$718$$ 2.55660e7 1.85077
$$719$$ −1.11331e7 −0.803145 −0.401572 0.915827i $$-0.631536\pi$$
−0.401572 + 0.915827i $$0.631536\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −1.17713e7 −0.840395
$$723$$ 0 0
$$724$$ 4.75422e6 0.337080
$$725$$ 3.09091e6 0.218394
$$726$$ 0 0
$$727$$ −8.95327e6 −0.628269 −0.314135 0.949378i $$-0.601714\pi$$
−0.314135 + 0.949378i $$0.601714\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −8.52088e6 −0.591803
$$731$$ 3.51641e6 0.243392
$$732$$ 0 0
$$733$$ −2.60536e7 −1.79105 −0.895524 0.445014i $$-0.853199\pi$$
−0.895524 + 0.445014i $$0.853199\pi$$
$$734$$ −2.41423e7 −1.65401
$$735$$ 0 0
$$736$$ 7.59380e6 0.516731
$$737$$ −2.27798e6 −0.154483
$$738$$ 0 0
$$739$$ −5.97111e6 −0.402202 −0.201101 0.979571i $$-0.564452\pi$$
−0.201101 + 0.979571i $$0.564452\pi$$
$$740$$ −6.35307e6 −0.426486
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −6.80330e6 −0.452114 −0.226057 0.974114i $$-0.572583\pi$$
−0.226057 + 0.974114i $$0.572583\pi$$
$$744$$ 0 0
$$745$$ 2.29790e6 0.151684
$$746$$ 2.44115e7 1.60601
$$747$$ 0 0
$$748$$ 2.94515e6 0.192466
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −2.12648e7 −1.37582 −0.687911 0.725795i $$-0.741472\pi$$
−0.687911 + 0.725795i $$0.741472\pi$$
$$752$$ 1.33388e7 0.860143
$$753$$ 0 0
$$754$$ 1.82701e6 0.117034
$$755$$ −5.62602e6 −0.359198
$$756$$ 0 0
$$757$$ −9.49238e6 −0.602054 −0.301027 0.953616i $$-0.597329\pi$$
−0.301027 + 0.953616i $$0.597329\pi$$
$$758$$ −6.13469e6 −0.387810
$$759$$ 0 0
$$760$$ −291530. −0.0183084
$$761$$ 1.71806e7 1.07542 0.537710 0.843130i $$-0.319290\pi$$
0.537710 + 0.843130i $$0.319290\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 2.55887e7 1.58604
$$765$$ 0 0
$$766$$ 2.98085e7 1.83556
$$767$$ 6.23480e6 0.382678
$$768$$ 0 0
$$769$$ 1.39968e7 0.853517 0.426758 0.904366i $$-0.359655\pi$$
0.426758 + 0.904366i $$0.359655\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.58223e7 0.955493
$$773$$ 2.54449e7 1.53162 0.765812 0.643065i $$-0.222337\pi$$
0.765812 + 0.643065i $$0.222337\pi$$
$$774$$ 0 0
$$775$$ −2.32306e7 −1.38933
$$776$$ 340422. 0.0202938
$$777$$ 0 0
$$778$$ 2.44560e7 1.44856
$$779$$ −1.90143e7 −1.12263
$$780$$ 0 0
$$781$$ 6.55261e6 0.384403
$$782$$ 5.63856e6 0.329725
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 3.16839e6 0.183512
$$786$$ 0 0
$$787$$ 9.95632e6 0.573010 0.286505 0.958079i $$-0.407507\pi$$
0.286505 + 0.958079i $$0.407507\pi$$
$$788$$ 1.76452e7 1.01231
$$789$$ 0 0
$$790$$ 5.05575e6 0.288216
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −3.82941e6 −0.216246
$$794$$ 2.33382e6 0.131376
$$795$$ 0 0
$$796$$ −2.00885e7 −1.12374
$$797$$ −2.19815e7 −1.22578 −0.612888 0.790170i $$-0.709992\pi$$
−0.612888 + 0.790170i $$0.709992\pi$$
$$798$$ 0 0
$$799$$ 1.05787e7 0.586226
$$800$$ 2.33834e7 1.29176
$$801$$ 0 0
$$802$$ −7.01412e6 −0.385068
$$803$$ 6.87899e6 0.376475
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −1.37314e7 −0.744521
$$807$$ 0 0
$$808$$ −1.75976e6 −0.0948253
$$809$$ 1.01820e7 0.546969 0.273485 0.961876i $$-0.411824\pi$$
0.273485 + 0.961876i $$0.411824\pi$$
$$810$$ 0 0
$$811$$ −1.06750e7 −0.569923 −0.284961 0.958539i $$-0.591981\pi$$
−0.284961 + 0.958539i $$0.591981\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 9.95160e6 0.526419
$$815$$ −3.41638e6 −0.180166
$$816$$ 0 0
$$817$$ 4.70689e6 0.246706
$$818$$ −2.29910e7 −1.20136
$$819$$ 0 0
$$820$$ 1.11205e7 0.577552
$$821$$ 2.22260e7 1.15081 0.575404 0.817869i $$-0.304845\pi$$
0.575404 + 0.817869i $$0.304845\pi$$
$$822$$ 0 0
$$823$$ −2.42908e7 −1.25009 −0.625046 0.780588i $$-0.714920\pi$$
−0.625046 + 0.780588i $$0.714920\pi$$
$$824$$ 1.44284e6 0.0740288
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.66722e7 1.35611 0.678055 0.735011i $$-0.262823\pi$$
0.678055 + 0.735011i $$0.262823\pi$$
$$828$$ 0 0
$$829$$ 6.78413e6 0.342853 0.171427 0.985197i $$-0.445162\pi$$
0.171427 + 0.985197i $$0.445162\pi$$
$$830$$ 1.22646e7 0.617956
$$831$$ 0 0
$$832$$ 7.55049e6 0.378152
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −5.10892e6 −0.253579
$$836$$ 3.94223e6 0.195086
$$837$$ 0 0
$$838$$ −5.75427e7 −2.83061
$$839$$ −1.88975e7 −0.926830 −0.463415 0.886141i $$-0.653376\pi$$
−0.463415 + 0.886141i $$0.653376\pi$$
$$840$$ 0 0
$$841$$ −1.93111e7 −0.941493
$$842$$ −2.39958e7 −1.16642
$$843$$ 0 0
$$844$$ 7.04584e6 0.340468
$$845$$ −5.73418e6 −0.276267
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.79830e7 1.33630
$$849$$ 0 0
$$850$$ 1.73627e7 0.824270
$$851$$ 9.81941e6 0.464795
$$852$$ 0 0
$$853$$ 1.69205e7 0.796235 0.398118 0.917334i $$-0.369664\pi$$
0.398118 + 0.917334i $$0.369664\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 648950. 0.0302710
$$857$$ 2.87564e7 1.33747 0.668733 0.743502i $$-0.266837\pi$$
0.668733 + 0.743502i $$0.266837\pi$$
$$858$$ 0 0
$$859$$ −8.92517e6 −0.412699 −0.206350 0.978478i $$-0.566158\pi$$
−0.206350 + 0.978478i $$0.566158\pi$$
$$860$$ −2.75284e6 −0.126921
$$861$$ 0 0
$$862$$ 3.63508e7 1.66627
$$863$$ 7.18205e6 0.328262 0.164131 0.986439i $$-0.447518\pi$$
0.164131 + 0.986439i $$0.447518\pi$$
$$864$$ 0 0
$$865$$ 7.71413e6 0.350547
$$866$$ 4.15465e7 1.88252
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −4.08155e6 −0.183348
$$870$$ 0 0
$$871$$ −4.09114e6 −0.182725
$$872$$ −732066. −0.0326031
$$873$$ 0 0
$$874$$ 7.54750e6 0.334214
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 5.05341e6 0.221863 0.110932 0.993828i $$-0.464617\pi$$
0.110932 + 0.993828i $$0.464617\pi$$
$$878$$ −4.79975e6 −0.210127
$$879$$ 0 0
$$880$$ 1.90090e6 0.0827469
$$881$$ −3.15002e6 −0.136733 −0.0683665 0.997660i $$-0.521779\pi$$
−0.0683665 + 0.997660i $$0.521779\pi$$
$$882$$ 0 0
$$883$$ 3.33141e7 1.43789 0.718946 0.695066i $$-0.244625\pi$$
0.718946 + 0.695066i $$0.244625\pi$$
$$884$$ 5.28934e6 0.227652
$$885$$ 0 0
$$886$$ 1.30638e6 0.0559094
$$887$$ −2.04549e6 −0.0872946 −0.0436473 0.999047i $$-0.513898\pi$$
−0.0436473 + 0.999047i $$0.513898\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −6.07866e6 −0.257237
$$891$$ 0 0
$$892$$ 4.19770e7 1.76644
$$893$$ 1.41601e7 0.594208
$$894$$ 0 0
$$895$$ 1.15225e7 0.480827
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 6.48677e7 2.68434
$$899$$ −9.01936e6 −0.372200
$$900$$ 0 0
$$901$$ 2.21927e7 0.910749
$$902$$ −1.74195e7 −0.712883
$$903$$ 0 0
$$904$$ 1.06761e6 0.0434502
$$905$$ 2.43359e6 0.0987702
$$906$$ 0 0
$$907$$ 2.69774e7 1.08888 0.544442 0.838798i $$-0.316741\pi$$
0.544442 + 0.838798i $$0.316741\pi$$
$$908$$ −3.81287e7 −1.53475
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −5.42821e6 −0.216701 −0.108350 0.994113i $$-0.534557\pi$$
−0.108350 + 0.994113i $$0.534557\pi$$
$$912$$ 0 0
$$913$$ −9.90132e6 −0.393112
$$914$$ −9.95129e6 −0.394016
$$915$$ 0 0
$$916$$ 1.40303e7 0.552495
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 9.41696e6 0.367809 0.183904 0.982944i $$-0.441126\pi$$
0.183904 + 0.982944i $$0.441126\pi$$
$$920$$ −263531. −0.0102651
$$921$$ 0 0
$$922$$ −1.62069e7 −0.627876
$$923$$ 1.17682e7 0.454679
$$924$$ 0 0
$$925$$ 3.02367e7 1.16193
$$926$$ −1.01692e6 −0.0389724
$$927$$ 0 0
$$928$$ 9.07868e6 0.346061
$$929$$ −8.43871e6 −0.320802 −0.160401 0.987052i $$-0.551279\pi$$
−0.160401 + 0.987052i $$0.551279\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 2.84658e7 1.07345
$$933$$ 0 0
$$934$$ −5.73997e7 −2.15299
$$935$$ 1.50756e6 0.0563957
$$936$$ 0 0
$$937$$ −683755. −0.0254420 −0.0127210 0.999919i $$-0.504049\pi$$
−0.0127210 + 0.999919i $$0.504049\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −8.28158e6 −0.305699
$$941$$ −4.97436e7 −1.83132 −0.915658 0.401958i $$-0.868330\pi$$
−0.915658 + 0.401958i $$0.868330\pi$$
$$942$$ 0 0
$$943$$ −1.71881e7 −0.629430
$$944$$ 2.90066e7 1.05942
$$945$$ 0 0
$$946$$ 4.31211e6 0.156661
$$947$$ −2.21967e7 −0.804292 −0.402146 0.915575i $$-0.631736\pi$$
−0.402146 + 0.915575i $$0.631736\pi$$
$$948$$ 0 0
$$949$$ 1.23543e7 0.445301
$$950$$ 2.32408e7 0.835493
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −1.34934e7 −0.481270 −0.240635 0.970616i $$-0.577356\pi$$
−0.240635 + 0.970616i $$0.577356\pi$$
$$954$$ 0 0
$$955$$ 1.30983e7 0.464737
$$956$$ −1.08797e7 −0.385012
$$957$$ 0 0
$$958$$ 3.77704e7 1.32965
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 3.91585e7 1.36778
$$962$$ 1.78726e7 0.622658
$$963$$ 0 0
$$964$$ −3.28725e7 −1.13931
$$965$$ 8.09913e6 0.279976
$$966$$ 0 0
$$967$$ −1.59641e7 −0.549008 −0.274504 0.961586i $$-0.588514\pi$$
−0.274504 + 0.961586i $$0.588514\pi$$
$$968$$ −2.44331e6 −0.0838088
$$969$$ 0 0
$$970$$ 2.91879e6 0.0996033
$$971$$ 2.03921e7 0.694087 0.347043 0.937849i $$-0.387186\pi$$
0.347043 + 0.937849i $$0.387186\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −4.53354e7 −1.53123
$$975$$ 0 0
$$976$$ −1.78159e7 −0.598663
$$977$$ −2.62595e7 −0.880136 −0.440068 0.897964i $$-0.645046\pi$$
−0.440068 + 0.897964i $$0.645046\pi$$
$$978$$ 0 0
$$979$$ 4.90736e6 0.163641
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −4.72489e7 −1.56355
$$983$$ −8.01076e6 −0.264417 −0.132209 0.991222i $$-0.542207\pi$$
−0.132209 + 0.991222i $$0.542207\pi$$
$$984$$ 0 0
$$985$$ 9.03223e6 0.296623
$$986$$ 6.74111e6 0.220820
$$987$$ 0 0
$$988$$ 7.08006e6 0.230751
$$989$$ 4.25483e6 0.138322
$$990$$ 0 0
$$991$$ −1.60083e7 −0.517798 −0.258899 0.965904i $$-0.583360\pi$$
−0.258899 + 0.965904i $$0.583360\pi$$
$$992$$ −6.82334e7 −2.20150
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −1.02829e7 −0.329274
$$996$$ 0 0
$$997$$ −2.97034e7 −0.946386 −0.473193 0.880959i $$-0.656899\pi$$
−0.473193 + 0.880959i $$0.656899\pi$$
$$998$$ −5.83281e7 −1.85375
$$999$$ 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 441.6.a.be.1.8 yes 8
3.2 odd 2 inner 441.6.a.be.1.1 8
7.6 odd 2 inner 441.6.a.be.1.7 yes 8
21.20 even 2 inner 441.6.a.be.1.2 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
441.6.a.be.1.1 8 3.2 odd 2 inner
441.6.a.be.1.2 yes 8 21.20 even 2 inner
441.6.a.be.1.7 yes 8 7.6 odd 2 inner
441.6.a.be.1.8 yes 8 1.1 even 1 trivial