# Properties

 Label 256.8.a.r.1.2 Level $256$ Weight $8$ Character 256.1 Self dual yes Analytic conductor $79.971$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 256.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.9705665239$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 163x^{4} + 4820x^{2} - 15296$$ x^6 - 163*x^4 + 4820*x^2 - 15296 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{27}$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-5.81430$$ of defining polynomial Character $$\chi$$ $$=$$ 256.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-40.2163 q^{3} +324.492 q^{5} +956.960 q^{7} -569.651 q^{9} +O(q^{10})$$ $$q-40.2163 q^{3} +324.492 q^{5} +956.960 q^{7} -569.651 q^{9} +5452.20 q^{11} +6289.38 q^{13} -13049.8 q^{15} +34587.3 q^{17} +14595.6 q^{19} -38485.4 q^{21} +24667.5 q^{23} +27169.8 q^{25} +110862. q^{27} -171116. q^{29} +111688. q^{31} -219267. q^{33} +310526. q^{35} -103636. q^{37} -252935. q^{39} -71691.3 q^{41} -328419. q^{43} -184847. q^{45} +119043. q^{47} +92230.3 q^{49} -1.39097e6 q^{51} -1.04011e6 q^{53} +1.76919e6 q^{55} -586982. q^{57} -225984. q^{59} -1.55268e6 q^{61} -545133. q^{63} +2.04085e6 q^{65} -316375. q^{67} -992033. q^{69} -538965. q^{71} +2.68512e6 q^{73} -1.09267e6 q^{75} +5.21754e6 q^{77} +8.22632e6 q^{79} -3.21264e6 q^{81} +5.89510e6 q^{83} +1.12233e7 q^{85} +6.88164e6 q^{87} -437005. q^{89} +6.01868e6 q^{91} -4.49169e6 q^{93} +4.73616e6 q^{95} -7.84322e6 q^{97} -3.10585e6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 688 q^{7} + 2918 q^{9}+O(q^{10})$$ 6 * q + 688 * q^7 + 2918 * q^9 $$6 q + 688 q^{7} + 2918 q^{9} + 17872 q^{15} + 1452 q^{17} + 1296 q^{23} + 39314 q^{25} - 89280 q^{31} + 53880 q^{33} + 328208 q^{39} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} + 1889896 q^{57} + 5776816 q^{63} + 1416480 q^{65} + 7597104 q^{71} - 2089564 q^{73} + 16015904 q^{79} - 723058 q^{81} + 37453776 q^{87} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100})$$ 6 * q + 688 * q^7 + 2918 * q^9 + 17872 * q^15 + 1452 * q^17 + 1296 * q^23 + 39314 * q^25 - 89280 * q^31 + 53880 * q^33 + 328208 * q^39 - 521244 * q^41 + 1566432 * q^47 - 511050 * q^49 + 3270256 * q^55 + 1889896 * q^57 + 5776816 * q^63 + 1416480 * q^65 + 7597104 * q^71 - 2089564 * q^73 + 16015904 * q^79 - 723058 * q^81 + 37453776 * q^87 - 2169084 * q^89 + 48537936 * q^95 - 1088308 * q^97

## 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
$$3$$ −40.2163 −0.859959 −0.429979 0.902839i $$-0.641479\pi$$
−0.429979 + 0.902839i $$0.641479\pi$$
$$4$$ 0 0
$$5$$ 324.492 1.16094 0.580468 0.814283i $$-0.302870\pi$$
0.580468 + 0.814283i $$0.302870\pi$$
$$6$$ 0 0
$$7$$ 956.960 1.05451 0.527255 0.849707i $$-0.323221\pi$$
0.527255 + 0.849707i $$0.323221\pi$$
$$8$$ 0 0
$$9$$ −569.651 −0.260471
$$10$$ 0 0
$$11$$ 5452.20 1.23509 0.617544 0.786537i $$-0.288128\pi$$
0.617544 + 0.786537i $$0.288128\pi$$
$$12$$ 0 0
$$13$$ 6289.38 0.793973 0.396987 0.917824i $$-0.370056\pi$$
0.396987 + 0.917824i $$0.370056\pi$$
$$14$$ 0 0
$$15$$ −13049.8 −0.998357
$$16$$ 0 0
$$17$$ 34587.3 1.70744 0.853720 0.520733i $$-0.174341\pi$$
0.853720 + 0.520733i $$0.174341\pi$$
$$18$$ 0 0
$$19$$ 14595.6 0.488186 0.244093 0.969752i $$-0.421510\pi$$
0.244093 + 0.969752i $$0.421510\pi$$
$$20$$ 0 0
$$21$$ −38485.4 −0.906835
$$22$$ 0 0
$$23$$ 24667.5 0.422743 0.211372 0.977406i $$-0.432207\pi$$
0.211372 + 0.977406i $$0.432207\pi$$
$$24$$ 0 0
$$25$$ 27169.8 0.347774
$$26$$ 0 0
$$27$$ 110862. 1.08395
$$28$$ 0 0
$$29$$ −171116. −1.30286 −0.651429 0.758710i $$-0.725830\pi$$
−0.651429 + 0.758710i $$0.725830\pi$$
$$30$$ 0 0
$$31$$ 111688. 0.673352 0.336676 0.941620i $$-0.390697\pi$$
0.336676 + 0.941620i $$0.390697\pi$$
$$32$$ 0 0
$$33$$ −219267. −1.06212
$$34$$ 0 0
$$35$$ 310526. 1.22422
$$36$$ 0 0
$$37$$ −103636. −0.336360 −0.168180 0.985756i $$-0.553789\pi$$
−0.168180 + 0.985756i $$0.553789\pi$$
$$38$$ 0 0
$$39$$ −252935. −0.682784
$$40$$ 0 0
$$41$$ −71691.3 −0.162451 −0.0812256 0.996696i $$-0.525883\pi$$
−0.0812256 + 0.996696i $$0.525883\pi$$
$$42$$ 0 0
$$43$$ −328419. −0.629925 −0.314962 0.949104i $$-0.601992\pi$$
−0.314962 + 0.949104i $$0.601992\pi$$
$$44$$ 0 0
$$45$$ −184847. −0.302391
$$46$$ 0 0
$$47$$ 119043. 0.167248 0.0836241 0.996497i $$-0.473350\pi$$
0.0836241 + 0.996497i $$0.473350\pi$$
$$48$$ 0 0
$$49$$ 92230.3 0.111992
$$50$$ 0 0
$$51$$ −1.39097e6 −1.46833
$$52$$ 0 0
$$53$$ −1.04011e6 −0.959648 −0.479824 0.877365i $$-0.659300\pi$$
−0.479824 + 0.877365i $$0.659300\pi$$
$$54$$ 0 0
$$55$$ 1.76919e6 1.43386
$$56$$ 0 0
$$57$$ −586982. −0.419820
$$58$$ 0 0
$$59$$ −225984. −0.143250 −0.0716250 0.997432i $$-0.522818\pi$$
−0.0716250 + 0.997432i $$0.522818\pi$$
$$60$$ 0 0
$$61$$ −1.55268e6 −0.875843 −0.437922 0.899013i $$-0.644285\pi$$
−0.437922 + 0.899013i $$0.644285\pi$$
$$62$$ 0 0
$$63$$ −545133. −0.274670
$$64$$ 0 0
$$65$$ 2.04085e6 0.921753
$$66$$ 0 0
$$67$$ −316375. −0.128511 −0.0642555 0.997933i $$-0.520467\pi$$
−0.0642555 + 0.997933i $$0.520467\pi$$
$$68$$ 0 0
$$69$$ −992033. −0.363542
$$70$$ 0 0
$$71$$ −538965. −0.178713 −0.0893566 0.996000i $$-0.528481\pi$$
−0.0893566 + 0.996000i $$0.528481\pi$$
$$72$$ 0 0
$$73$$ 2.68512e6 0.807856 0.403928 0.914791i $$-0.367645\pi$$
0.403928 + 0.914791i $$0.367645\pi$$
$$74$$ 0 0
$$75$$ −1.09267e6 −0.299071
$$76$$ 0 0
$$77$$ 5.21754e6 1.30241
$$78$$ 0 0
$$79$$ 8.22632e6 1.87720 0.938600 0.345007i $$-0.112123\pi$$
0.938600 + 0.345007i $$0.112123\pi$$
$$80$$ 0 0
$$81$$ −3.21264e6 −0.671684
$$82$$ 0 0
$$83$$ 5.89510e6 1.13167 0.565833 0.824520i $$-0.308555\pi$$
0.565833 + 0.824520i $$0.308555\pi$$
$$84$$ 0 0
$$85$$ 1.12233e7 1.98223
$$86$$ 0 0
$$87$$ 6.88164e6 1.12040
$$88$$ 0 0
$$89$$ −437005. −0.0657085 −0.0328542 0.999460i $$-0.510460\pi$$
−0.0328542 + 0.999460i $$0.510460\pi$$
$$90$$ 0 0
$$91$$ 6.01868e6 0.837253
$$92$$ 0 0
$$93$$ −4.49169e6 −0.579055
$$94$$ 0 0
$$95$$ 4.73616e6 0.566753
$$96$$ 0 0
$$97$$ −7.84322e6 −0.872556 −0.436278 0.899812i $$-0.643704\pi$$
−0.436278 + 0.899812i $$0.643704\pi$$
$$98$$ 0 0
$$99$$ −3.10585e6 −0.321705
$$100$$ 0 0
$$101$$ −6.19757e6 −0.598545 −0.299272 0.954168i $$-0.596744\pi$$
−0.299272 + 0.954168i $$0.596744\pi$$
$$102$$ 0 0
$$103$$ 6.59816e6 0.594966 0.297483 0.954727i $$-0.403853\pi$$
0.297483 + 0.954727i $$0.403853\pi$$
$$104$$ 0 0
$$105$$ −1.24882e7 −1.05278
$$106$$ 0 0
$$107$$ 512845. 0.0404709 0.0202354 0.999795i $$-0.493558\pi$$
0.0202354 + 0.999795i $$0.493558\pi$$
$$108$$ 0 0
$$109$$ 1.95882e7 1.44878 0.724388 0.689393i $$-0.242123\pi$$
0.724388 + 0.689393i $$0.242123\pi$$
$$110$$ 0 0
$$111$$ 4.16785e6 0.289256
$$112$$ 0 0
$$113$$ 1.88876e7 1.23141 0.615705 0.787977i $$-0.288871\pi$$
0.615705 + 0.787977i $$0.288871\pi$$
$$114$$ 0 0
$$115$$ 8.00438e6 0.490778
$$116$$ 0 0
$$117$$ −3.58275e6 −0.206807
$$118$$ 0 0
$$119$$ 3.30987e7 1.80051
$$120$$ 0 0
$$121$$ 1.02394e7 0.525441
$$122$$ 0 0
$$123$$ 2.88316e6 0.139701
$$124$$ 0 0
$$125$$ −1.65345e7 −0.757193
$$126$$ 0 0
$$127$$ 3.96314e7 1.71683 0.858413 0.512959i $$-0.171451\pi$$
0.858413 + 0.512959i $$0.171451\pi$$
$$128$$ 0 0
$$129$$ 1.32078e7 0.541709
$$130$$ 0 0
$$131$$ −3.65337e7 −1.41986 −0.709928 0.704274i $$-0.751273\pi$$
−0.709928 + 0.704274i $$0.751273\pi$$
$$132$$ 0 0
$$133$$ 1.39675e7 0.514797
$$134$$ 0 0
$$135$$ 3.59739e7 1.25840
$$136$$ 0 0
$$137$$ 2.56967e7 0.853799 0.426899 0.904299i $$-0.359606\pi$$
0.426899 + 0.904299i $$0.359606\pi$$
$$138$$ 0 0
$$139$$ −5.23001e7 −1.65177 −0.825886 0.563836i $$-0.809325\pi$$
−0.825886 + 0.563836i $$0.809325\pi$$
$$140$$ 0 0
$$141$$ −4.78747e6 −0.143827
$$142$$ 0 0
$$143$$ 3.42910e7 0.980626
$$144$$ 0 0
$$145$$ −5.55256e7 −1.51254
$$146$$ 0 0
$$147$$ −3.70916e6 −0.0963085
$$148$$ 0 0
$$149$$ −1.80406e7 −0.446785 −0.223392 0.974729i $$-0.571713\pi$$
−0.223392 + 0.974729i $$0.571713\pi$$
$$150$$ 0 0
$$151$$ −3.87385e7 −0.915637 −0.457818 0.889046i $$-0.651369\pi$$
−0.457818 + 0.889046i $$0.651369\pi$$
$$152$$ 0 0
$$153$$ −1.97027e7 −0.444739
$$154$$ 0 0
$$155$$ 3.62420e7 0.781719
$$156$$ 0 0
$$157$$ −5.12341e7 −1.05660 −0.528300 0.849058i $$-0.677170\pi$$
−0.528300 + 0.849058i $$0.677170\pi$$
$$158$$ 0 0
$$159$$ 4.18292e7 0.825258
$$160$$ 0 0
$$161$$ 2.36058e7 0.445787
$$162$$ 0 0
$$163$$ 8.57572e7 1.55101 0.775504 0.631343i $$-0.217496\pi$$
0.775504 + 0.631343i $$0.217496\pi$$
$$164$$ 0 0
$$165$$ −7.11504e7 −1.23306
$$166$$ 0 0
$$167$$ 1.05871e8 1.75901 0.879503 0.475893i $$-0.157875\pi$$
0.879503 + 0.475893i $$0.157875\pi$$
$$168$$ 0 0
$$169$$ −2.31923e7 −0.369606
$$170$$ 0 0
$$171$$ −8.31441e6 −0.127158
$$172$$ 0 0
$$173$$ −1.98148e7 −0.290956 −0.145478 0.989361i $$-0.546472\pi$$
−0.145478 + 0.989361i $$0.546472\pi$$
$$174$$ 0 0
$$175$$ 2.60005e7 0.366731
$$176$$ 0 0
$$177$$ 9.08822e6 0.123189
$$178$$ 0 0
$$179$$ −2.97800e7 −0.388096 −0.194048 0.980992i $$-0.562162\pi$$
−0.194048 + 0.980992i $$0.562162\pi$$
$$180$$ 0 0
$$181$$ 3.96227e6 0.0496671 0.0248335 0.999692i $$-0.492094\pi$$
0.0248335 + 0.999692i $$0.492094\pi$$
$$182$$ 0 0
$$183$$ 6.24429e7 0.753189
$$184$$ 0 0
$$185$$ −3.36290e7 −0.390493
$$186$$ 0 0
$$187$$ 1.88577e8 2.10884
$$188$$ 0 0
$$189$$ 1.06091e8 1.14304
$$190$$ 0 0
$$191$$ −4.80105e7 −0.498562 −0.249281 0.968431i $$-0.580194\pi$$
−0.249281 + 0.968431i $$0.580194\pi$$
$$192$$ 0 0
$$193$$ −4.72502e6 −0.0473100 −0.0236550 0.999720i $$-0.507530\pi$$
−0.0236550 + 0.999720i $$0.507530\pi$$
$$194$$ 0 0
$$195$$ −8.20754e7 −0.792669
$$196$$ 0 0
$$197$$ 1.14882e8 1.07058 0.535290 0.844668i $$-0.320202\pi$$
0.535290 + 0.844668i $$0.320202\pi$$
$$198$$ 0 0
$$199$$ −1.20933e7 −0.108782 −0.0543911 0.998520i $$-0.517322\pi$$
−0.0543911 + 0.998520i $$0.517322\pi$$
$$200$$ 0 0
$$201$$ 1.27234e7 0.110514
$$202$$ 0 0
$$203$$ −1.63751e8 −1.37388
$$204$$ 0 0
$$205$$ −2.32632e7 −0.188596
$$206$$ 0 0
$$207$$ −1.40518e7 −0.110112
$$208$$ 0 0
$$209$$ 7.95784e7 0.602953
$$210$$ 0 0
$$211$$ 1.95850e8 1.43527 0.717636 0.696418i $$-0.245224\pi$$
0.717636 + 0.696418i $$0.245224\pi$$
$$212$$ 0 0
$$213$$ 2.16752e7 0.153686
$$214$$ 0 0
$$215$$ −1.06569e8 −0.731302
$$216$$ 0 0
$$217$$ 1.06881e8 0.710057
$$218$$ 0 0
$$219$$ −1.07986e8 −0.694723
$$220$$ 0 0
$$221$$ 2.17532e8 1.35566
$$222$$ 0 0
$$223$$ 1.08024e8 0.652311 0.326156 0.945316i $$-0.394247\pi$$
0.326156 + 0.945316i $$0.394247\pi$$
$$224$$ 0 0
$$225$$ −1.54773e7 −0.0905851
$$226$$ 0 0
$$227$$ −1.61144e8 −0.914374 −0.457187 0.889371i $$-0.651143\pi$$
−0.457187 + 0.889371i $$0.651143\pi$$
$$228$$ 0 0
$$229$$ 5.27173e7 0.290088 0.145044 0.989425i $$-0.453668\pi$$
0.145044 + 0.989425i $$0.453668\pi$$
$$230$$ 0 0
$$231$$ −2.09830e8 −1.12002
$$232$$ 0 0
$$233$$ 1.79423e8 0.929249 0.464625 0.885508i $$-0.346189\pi$$
0.464625 + 0.885508i $$0.346189\pi$$
$$234$$ 0 0
$$235$$ 3.86285e7 0.194165
$$236$$ 0 0
$$237$$ −3.30832e8 −1.61431
$$238$$ 0 0
$$239$$ −8.42441e7 −0.399160 −0.199580 0.979882i $$-0.563958\pi$$
−0.199580 + 0.979882i $$0.563958\pi$$
$$240$$ 0 0
$$241$$ −2.12302e8 −0.977000 −0.488500 0.872564i $$-0.662456\pi$$
−0.488500 + 0.872564i $$0.662456\pi$$
$$242$$ 0 0
$$243$$ −1.13255e8 −0.506333
$$244$$ 0 0
$$245$$ 2.99280e7 0.130016
$$246$$ 0 0
$$247$$ 9.17975e7 0.387607
$$248$$ 0 0
$$249$$ −2.37079e8 −0.973186
$$250$$ 0 0
$$251$$ −1.18102e8 −0.471411 −0.235706 0.971825i $$-0.575740\pi$$
−0.235706 + 0.971825i $$0.575740\pi$$
$$252$$ 0 0
$$253$$ 1.34492e8 0.522125
$$254$$ 0 0
$$255$$ −4.51359e8 −1.70463
$$256$$ 0 0
$$257$$ 1.27463e8 0.468402 0.234201 0.972188i $$-0.424753\pi$$
0.234201 + 0.972188i $$0.424753\pi$$
$$258$$ 0 0
$$259$$ −9.91755e7 −0.354695
$$260$$ 0 0
$$261$$ 9.74762e7 0.339357
$$262$$ 0 0
$$263$$ −4.33125e8 −1.46814 −0.734071 0.679073i $$-0.762382\pi$$
−0.734071 + 0.679073i $$0.762382\pi$$
$$264$$ 0 0
$$265$$ −3.37506e8 −1.11409
$$266$$ 0 0
$$267$$ 1.75747e7 0.0565066
$$268$$ 0 0
$$269$$ 3.44748e8 1.07986 0.539931 0.841709i $$-0.318450\pi$$
0.539931 + 0.841709i $$0.318450\pi$$
$$270$$ 0 0
$$271$$ −4.42513e8 −1.35062 −0.675311 0.737533i $$-0.735990\pi$$
−0.675311 + 0.737533i $$0.735990\pi$$
$$272$$ 0 0
$$273$$ −2.42049e8 −0.720003
$$274$$ 0 0
$$275$$ 1.48135e8 0.429531
$$276$$ 0 0
$$277$$ −3.18148e8 −0.899395 −0.449697 0.893181i $$-0.648468\pi$$
−0.449697 + 0.893181i $$0.648468\pi$$
$$278$$ 0 0
$$279$$ −6.36234e7 −0.175389
$$280$$ 0 0
$$281$$ −1.28497e8 −0.345478 −0.172739 0.984968i $$-0.555262\pi$$
−0.172739 + 0.984968i $$0.555262\pi$$
$$282$$ 0 0
$$283$$ 3.98970e8 1.04637 0.523187 0.852218i $$-0.324743\pi$$
0.523187 + 0.852218i $$0.324743\pi$$
$$284$$ 0 0
$$285$$ −1.90471e8 −0.487384
$$286$$ 0 0
$$287$$ −6.86057e7 −0.171306
$$288$$ 0 0
$$289$$ 7.85942e8 1.91535
$$290$$ 0 0
$$291$$ 3.15425e8 0.750362
$$292$$ 0 0
$$293$$ −2.00958e8 −0.466732 −0.233366 0.972389i $$-0.574974\pi$$
−0.233366 + 0.972389i $$0.574974\pi$$
$$294$$ 0 0
$$295$$ −7.33298e7 −0.166304
$$296$$ 0 0
$$297$$ 6.04444e8 1.33878
$$298$$ 0 0
$$299$$ 1.55143e8 0.335647
$$300$$ 0 0
$$301$$ −3.14284e8 −0.664262
$$302$$ 0 0
$$303$$ 2.49243e8 0.514724
$$304$$ 0 0
$$305$$ −5.03830e8 −1.01680
$$306$$ 0 0
$$307$$ 1.58918e7 0.0313465 0.0156733 0.999877i $$-0.495011\pi$$
0.0156733 + 0.999877i $$0.495011\pi$$
$$308$$ 0 0
$$309$$ −2.65353e8 −0.511646
$$310$$ 0 0
$$311$$ −4.87710e8 −0.919391 −0.459695 0.888077i $$-0.652041\pi$$
−0.459695 + 0.888077i $$0.652041\pi$$
$$312$$ 0 0
$$313$$ 3.24731e8 0.598576 0.299288 0.954163i $$-0.403251\pi$$
0.299288 + 0.954163i $$0.403251\pi$$
$$314$$ 0 0
$$315$$ −1.76891e8 −0.318874
$$316$$ 0 0
$$317$$ −1.06084e9 −1.87043 −0.935213 0.354086i $$-0.884792\pi$$
−0.935213 + 0.354086i $$0.884792\pi$$
$$318$$ 0 0
$$319$$ −9.32958e8 −1.60914
$$320$$ 0 0
$$321$$ −2.06247e7 −0.0348033
$$322$$ 0 0
$$323$$ 5.04824e8 0.833548
$$324$$ 0 0
$$325$$ 1.70881e8 0.276123
$$326$$ 0 0
$$327$$ −7.87763e8 −1.24589
$$328$$ 0 0
$$329$$ 1.13919e8 0.176365
$$330$$ 0 0
$$331$$ 2.88487e8 0.437249 0.218624 0.975809i $$-0.429843\pi$$
0.218624 + 0.975809i $$0.429843\pi$$
$$332$$ 0 0
$$333$$ 5.90363e7 0.0876121
$$334$$ 0 0
$$335$$ −1.02661e8 −0.149193
$$336$$ 0 0
$$337$$ −1.10595e8 −0.157410 −0.0787051 0.996898i $$-0.525079\pi$$
−0.0787051 + 0.996898i $$0.525079\pi$$
$$338$$ 0 0
$$339$$ −7.59590e8 −1.05896
$$340$$ 0 0
$$341$$ 6.08948e8 0.831649
$$342$$ 0 0
$$343$$ −6.99837e8 −0.936414
$$344$$ 0 0
$$345$$ −3.21907e8 −0.422049
$$346$$ 0 0
$$347$$ −1.10651e9 −1.42168 −0.710841 0.703352i $$-0.751686\pi$$
−0.710841 + 0.703352i $$0.751686\pi$$
$$348$$ 0 0
$$349$$ 1.38337e9 1.74201 0.871003 0.491278i $$-0.163470\pi$$
0.871003 + 0.491278i $$0.163470\pi$$
$$350$$ 0 0
$$351$$ 6.97254e8 0.860630
$$352$$ 0 0
$$353$$ −2.47617e8 −0.299618 −0.149809 0.988715i $$-0.547866\pi$$
−0.149809 + 0.988715i $$0.547866\pi$$
$$354$$ 0 0
$$355$$ −1.74890e8 −0.207475
$$356$$ 0 0
$$357$$ −1.33111e9 −1.54837
$$358$$ 0 0
$$359$$ 1.38641e9 1.58148 0.790738 0.612155i $$-0.209697\pi$$
0.790738 + 0.612155i $$0.209697\pi$$
$$360$$ 0 0
$$361$$ −6.80839e8 −0.761674
$$362$$ 0 0
$$363$$ −4.11789e8 −0.451858
$$364$$ 0 0
$$365$$ 8.71299e8 0.937869
$$366$$ 0 0
$$367$$ −7.49367e8 −0.791341 −0.395670 0.918393i $$-0.629488\pi$$
−0.395670 + 0.918393i $$0.629488\pi$$
$$368$$ 0 0
$$369$$ 4.08390e7 0.0423139
$$370$$ 0 0
$$371$$ −9.95340e8 −1.01196
$$372$$ 0 0
$$373$$ 1.49519e9 1.49181 0.745906 0.666051i $$-0.232017\pi$$
0.745906 + 0.666051i $$0.232017\pi$$
$$374$$ 0 0
$$375$$ 6.64957e8 0.651155
$$376$$ 0 0
$$377$$ −1.07621e9 −1.03443
$$378$$ 0 0
$$379$$ −7.92096e7 −0.0747379 −0.0373689 0.999302i $$-0.511898\pi$$
−0.0373689 + 0.999302i $$0.511898\pi$$
$$380$$ 0 0
$$381$$ −1.59383e9 −1.47640
$$382$$ 0 0
$$383$$ 4.80285e8 0.436820 0.218410 0.975857i $$-0.429913\pi$$
0.218410 + 0.975857i $$0.429913\pi$$
$$384$$ 0 0
$$385$$ 1.69305e9 1.51202
$$386$$ 0 0
$$387$$ 1.87084e8 0.164077
$$388$$ 0 0
$$389$$ 1.07150e9 0.922928 0.461464 0.887159i $$-0.347324\pi$$
0.461464 + 0.887159i $$0.347324\pi$$
$$390$$ 0 0
$$391$$ 8.53180e8 0.721809
$$392$$ 0 0
$$393$$ 1.46925e9 1.22102
$$394$$ 0 0
$$395$$ 2.66937e9 2.17931
$$396$$ 0 0
$$397$$ 2.03185e9 1.62976 0.814882 0.579627i $$-0.196802\pi$$
0.814882 + 0.579627i $$0.196802\pi$$
$$398$$ 0 0
$$399$$ −5.61719e8 −0.442705
$$400$$ 0 0
$$401$$ −2.57759e9 −1.99622 −0.998111 0.0614301i $$-0.980434\pi$$
−0.998111 + 0.0614301i $$0.980434\pi$$
$$402$$ 0 0
$$403$$ 7.02451e8 0.534624
$$404$$ 0 0
$$405$$ −1.04248e9 −0.779782
$$406$$ 0 0
$$407$$ −5.65044e8 −0.415434
$$408$$ 0 0
$$409$$ −3.30242e8 −0.238672 −0.119336 0.992854i $$-0.538077\pi$$
−0.119336 + 0.992854i $$0.538077\pi$$
$$410$$ 0 0
$$411$$ −1.03343e9 −0.734232
$$412$$ 0 0
$$413$$ −2.16257e8 −0.151059
$$414$$ 0 0
$$415$$ 1.91291e9 1.31379
$$416$$ 0 0
$$417$$ 2.10331e9 1.42046
$$418$$ 0 0
$$419$$ 5.80021e7 0.0385207 0.0192604 0.999815i $$-0.493869\pi$$
0.0192604 + 0.999815i $$0.493869\pi$$
$$420$$ 0 0
$$421$$ −1.90609e8 −0.124496 −0.0622480 0.998061i $$-0.519827\pi$$
−0.0622480 + 0.998061i $$0.519827\pi$$
$$422$$ 0 0
$$423$$ −6.78129e7 −0.0435633
$$424$$ 0 0
$$425$$ 9.39731e8 0.593803
$$426$$ 0 0
$$427$$ −1.48585e9 −0.923586
$$428$$ 0 0
$$429$$ −1.37906e9 −0.843298
$$430$$ 0 0
$$431$$ 2.42923e9 1.46150 0.730749 0.682646i $$-0.239171\pi$$
0.730749 + 0.682646i $$0.239171\pi$$
$$432$$ 0 0
$$433$$ −2.37902e9 −1.40828 −0.704141 0.710060i $$-0.748668\pi$$
−0.704141 + 0.710060i $$0.748668\pi$$
$$434$$ 0 0
$$435$$ 2.23303e9 1.30072
$$436$$ 0 0
$$437$$ 3.60037e8 0.206378
$$438$$ 0 0
$$439$$ 1.33161e9 0.751194 0.375597 0.926783i $$-0.377438\pi$$
0.375597 + 0.926783i $$0.377438\pi$$
$$440$$ 0 0
$$441$$ −5.25390e7 −0.0291707
$$442$$ 0 0
$$443$$ −5.02643e8 −0.274692 −0.137346 0.990523i $$-0.543857\pi$$
−0.137346 + 0.990523i $$0.543857\pi$$
$$444$$ 0 0
$$445$$ −1.41805e8 −0.0762834
$$446$$ 0 0
$$447$$ 7.25525e8 0.384217
$$448$$ 0 0
$$449$$ 3.14785e9 1.64116 0.820580 0.571531i $$-0.193650\pi$$
0.820580 + 0.571531i $$0.193650\pi$$
$$450$$ 0 0
$$451$$ −3.90876e8 −0.200641
$$452$$ 0 0
$$453$$ 1.55792e9 0.787410
$$454$$ 0 0
$$455$$ 1.95301e9 0.971998
$$456$$ 0 0
$$457$$ 2.68422e9 1.31556 0.657782 0.753209i $$-0.271495\pi$$
0.657782 + 0.753209i $$0.271495\pi$$
$$458$$ 0 0
$$459$$ 3.83442e9 1.85078
$$460$$ 0 0
$$461$$ 1.30434e9 0.620065 0.310033 0.950726i $$-0.399660\pi$$
0.310033 + 0.950726i $$0.399660\pi$$
$$462$$ 0 0
$$463$$ 2.86853e9 1.34315 0.671577 0.740934i $$-0.265617\pi$$
0.671577 + 0.740934i $$0.265617\pi$$
$$464$$ 0 0
$$465$$ −1.45752e9 −0.672246
$$466$$ 0 0
$$467$$ 5.96519e8 0.271029 0.135514 0.990775i $$-0.456731\pi$$
0.135514 + 0.990775i $$0.456731\pi$$
$$468$$ 0 0
$$469$$ −3.02758e8 −0.135516
$$470$$ 0 0
$$471$$ 2.06045e9 0.908632
$$472$$ 0 0
$$473$$ −1.79061e9 −0.778012
$$474$$ 0 0
$$475$$ 3.96561e8 0.169778
$$476$$ 0 0
$$477$$ 5.92497e8 0.249961
$$478$$ 0 0
$$479$$ −2.16068e9 −0.898289 −0.449144 0.893459i $$-0.648271\pi$$
−0.449144 + 0.893459i $$0.648271\pi$$
$$480$$ 0 0
$$481$$ −6.51805e8 −0.267061
$$482$$ 0 0
$$483$$ −9.49337e8 −0.383359
$$484$$ 0 0
$$485$$ −2.54506e9 −1.01298
$$486$$ 0 0
$$487$$ −1.41934e8 −0.0556847 −0.0278424 0.999612i $$-0.508864\pi$$
−0.0278424 + 0.999612i $$0.508864\pi$$
$$488$$ 0 0
$$489$$ −3.44884e9 −1.33380
$$490$$ 0 0
$$491$$ −2.38677e9 −0.909966 −0.454983 0.890500i $$-0.650355\pi$$
−0.454983 + 0.890500i $$0.650355\pi$$
$$492$$ 0 0
$$493$$ −5.91843e9 −2.22455
$$494$$ 0 0
$$495$$ −1.00782e9 −0.373479
$$496$$ 0 0
$$497$$ −5.15769e8 −0.188455
$$498$$ 0 0
$$499$$ −5.23900e9 −1.88754 −0.943771 0.330601i $$-0.892749\pi$$
−0.943771 + 0.330601i $$0.892749\pi$$
$$500$$ 0 0
$$501$$ −4.25772e9 −1.51267
$$502$$ 0 0
$$503$$ 3.63292e9 1.27282 0.636411 0.771350i $$-0.280418\pi$$
0.636411 + 0.771350i $$0.280418\pi$$
$$504$$ 0 0
$$505$$ −2.01106e9 −0.694872
$$506$$ 0 0
$$507$$ 9.32706e8 0.317846
$$508$$ 0 0
$$509$$ 2.58693e9 0.869505 0.434753 0.900550i $$-0.356836\pi$$
0.434753 + 0.900550i $$0.356836\pi$$
$$510$$ 0 0
$$511$$ 2.56955e9 0.851892
$$512$$ 0 0
$$513$$ 1.61811e9 0.529171
$$514$$ 0 0
$$515$$ 2.14105e9 0.690718
$$516$$ 0 0
$$517$$ 6.49047e8 0.206566
$$518$$ 0 0
$$519$$ 7.96876e8 0.250210
$$520$$ 0 0
$$521$$ −1.08542e8 −0.0336253 −0.0168127 0.999859i $$-0.505352\pi$$
−0.0168127 + 0.999859i $$0.505352\pi$$
$$522$$ 0 0
$$523$$ −6.10725e9 −1.86676 −0.933382 0.358884i $$-0.883157\pi$$
−0.933382 + 0.358884i $$0.883157\pi$$
$$524$$ 0 0
$$525$$ −1.04564e9 −0.315374
$$526$$ 0 0
$$527$$ 3.86300e9 1.14971
$$528$$ 0 0
$$529$$ −2.79634e9 −0.821288
$$530$$ 0 0
$$531$$ 1.28732e8 0.0373125
$$532$$ 0 0
$$533$$ −4.50894e8 −0.128982
$$534$$ 0 0
$$535$$ 1.66414e8 0.0469841
$$536$$ 0 0
$$537$$ 1.19764e9 0.333747
$$538$$ 0 0
$$539$$ 5.02858e8 0.138320
$$540$$ 0 0
$$541$$ 5.39345e8 0.146445 0.0732227 0.997316i $$-0.476672\pi$$
0.0732227 + 0.997316i $$0.476672\pi$$
$$542$$ 0 0
$$543$$ −1.59348e8 −0.0427116
$$544$$ 0 0
$$545$$ 6.35620e9 1.68194
$$546$$ 0 0
$$547$$ 8.82287e7 0.0230491 0.0115246 0.999934i $$-0.496332\pi$$
0.0115246 + 0.999934i $$0.496332\pi$$
$$548$$ 0 0
$$549$$ 8.84483e8 0.228132
$$550$$ 0 0
$$551$$ −2.49754e9 −0.636037
$$552$$ 0 0
$$553$$ 7.87226e9 1.97953
$$554$$ 0 0
$$555$$ 1.35243e9 0.335807
$$556$$ 0 0
$$557$$ −5.57233e8 −0.136629 −0.0683147 0.997664i $$-0.521762\pi$$
−0.0683147 + 0.997664i $$0.521762\pi$$
$$558$$ 0 0
$$559$$ −2.06555e9 −0.500143
$$560$$ 0 0
$$561$$ −7.58386e9 −1.81351
$$562$$ 0 0
$$563$$ −1.17012e9 −0.276344 −0.138172 0.990408i $$-0.544123\pi$$
−0.138172 + 0.990408i $$0.544123\pi$$
$$564$$ 0 0
$$565$$ 6.12887e9 1.42959
$$566$$ 0 0
$$567$$ −3.07437e9 −0.708297
$$568$$ 0 0
$$569$$ 2.39181e9 0.544295 0.272147 0.962256i $$-0.412266\pi$$
0.272147 + 0.962256i $$0.412266\pi$$
$$570$$ 0 0
$$571$$ 3.15823e9 0.709933 0.354966 0.934879i $$-0.384492\pi$$
0.354966 + 0.934879i $$0.384492\pi$$
$$572$$ 0 0
$$573$$ 1.93080e9 0.428743
$$574$$ 0 0
$$575$$ 6.70211e8 0.147019
$$576$$ 0 0
$$577$$ 4.03435e9 0.874296 0.437148 0.899390i $$-0.355989\pi$$
0.437148 + 0.899390i $$0.355989\pi$$
$$578$$ 0 0
$$579$$ 1.90023e8 0.0406846
$$580$$ 0 0
$$581$$ 5.64138e9 1.19335
$$582$$ 0 0
$$583$$ −5.67087e9 −1.18525
$$584$$ 0 0
$$585$$ −1.16257e9 −0.240090
$$586$$ 0 0
$$587$$ 2.72240e9 0.555544 0.277772 0.960647i $$-0.410404\pi$$
0.277772 + 0.960647i $$0.410404\pi$$
$$588$$ 0 0
$$589$$ 1.63016e9 0.328721
$$590$$ 0 0
$$591$$ −4.62012e9 −0.920655
$$592$$ 0 0
$$593$$ −2.29251e9 −0.451460 −0.225730 0.974190i $$-0.572477\pi$$
−0.225730 + 0.974190i $$0.572477\pi$$
$$594$$ 0 0
$$595$$ 1.07402e10 2.09028
$$596$$ 0 0
$$597$$ 4.86346e8 0.0935482
$$598$$ 0 0
$$599$$ 3.58734e9 0.681991 0.340995 0.940065i $$-0.389236\pi$$
0.340995 + 0.940065i $$0.389236\pi$$
$$600$$ 0 0
$$601$$ −8.20369e9 −1.54152 −0.770759 0.637127i $$-0.780123\pi$$
−0.770759 + 0.637127i $$0.780123\pi$$
$$602$$ 0 0
$$603$$ 1.80223e8 0.0334734
$$604$$ 0 0
$$605$$ 3.32259e9 0.610004
$$606$$ 0 0
$$607$$ −4.60087e9 −0.834986 −0.417493 0.908680i $$-0.637091\pi$$
−0.417493 + 0.908680i $$0.637091\pi$$
$$608$$ 0 0
$$609$$ 6.58546e9 1.18148
$$610$$ 0 0
$$611$$ 7.48707e8 0.132791
$$612$$ 0 0
$$613$$ −8.55728e9 −1.50046 −0.750229 0.661178i $$-0.770057\pi$$
−0.750229 + 0.661178i $$0.770057\pi$$
$$614$$ 0 0
$$615$$ 9.35561e8 0.162184
$$616$$ 0 0
$$617$$ 2.58089e9 0.442355 0.221178 0.975234i $$-0.429010\pi$$
0.221178 + 0.975234i $$0.429010\pi$$
$$618$$ 0 0
$$619$$ −5.26641e9 −0.892478 −0.446239 0.894914i $$-0.647237\pi$$
−0.446239 + 0.894914i $$0.647237\pi$$
$$620$$ 0 0
$$621$$ 2.73469e9 0.458234
$$622$$ 0 0
$$623$$ −4.18197e8 −0.0692903
$$624$$ 0 0
$$625$$ −7.48796e9 −1.22683
$$626$$ 0 0
$$627$$ −3.20035e9 −0.518514
$$628$$ 0 0
$$629$$ −3.58449e9 −0.574314
$$630$$ 0 0
$$631$$ −8.32515e9 −1.31914 −0.659568 0.751645i $$-0.729261\pi$$
−0.659568 + 0.751645i $$0.729261\pi$$
$$632$$ 0 0
$$633$$ −7.87635e9 −1.23428
$$634$$ 0 0
$$635$$ 1.28601e10 1.99313
$$636$$ 0 0
$$637$$ 5.80071e8 0.0889187
$$638$$ 0 0
$$639$$ 3.07022e8 0.0465496
$$640$$ 0 0
$$641$$ 4.26190e9 0.639146 0.319573 0.947562i $$-0.396460\pi$$
0.319573 + 0.947562i $$0.396460\pi$$
$$642$$ 0 0
$$643$$ −1.26588e10 −1.87782 −0.938908 0.344167i $$-0.888161\pi$$
−0.938908 + 0.344167i $$0.888161\pi$$
$$644$$ 0 0
$$645$$ 4.28582e9 0.628890
$$646$$ 0 0
$$647$$ 7.38061e9 1.07134 0.535670 0.844427i $$-0.320059\pi$$
0.535670 + 0.844427i $$0.320059\pi$$
$$648$$ 0 0
$$649$$ −1.23211e9 −0.176926
$$650$$ 0 0
$$651$$ −4.29837e9 −0.610620
$$652$$ 0 0
$$653$$ −3.21579e9 −0.451951 −0.225975 0.974133i $$-0.572557\pi$$
−0.225975 + 0.974133i $$0.572557\pi$$
$$654$$ 0 0
$$655$$ −1.18549e10 −1.64836
$$656$$ 0 0
$$657$$ −1.52958e9 −0.210423
$$658$$ 0 0
$$659$$ 5.17004e9 0.703711 0.351856 0.936054i $$-0.385551\pi$$
0.351856 + 0.936054i $$0.385551\pi$$
$$660$$ 0 0
$$661$$ 1.95604e9 0.263435 0.131717 0.991287i $$-0.457951\pi$$
0.131717 + 0.991287i $$0.457951\pi$$
$$662$$ 0 0
$$663$$ −8.74835e9 −1.16581
$$664$$ 0 0
$$665$$ 4.53232e9 0.597647
$$666$$ 0 0
$$667$$ −4.22099e9 −0.550775
$$668$$ 0 0
$$669$$ −4.34434e9 −0.560961
$$670$$ 0 0
$$671$$ −8.46551e9 −1.08174
$$672$$ 0 0
$$673$$ 1.54679e9 0.195605 0.0978024 0.995206i $$-0.468819\pi$$
0.0978024 + 0.995206i $$0.468819\pi$$
$$674$$ 0 0
$$675$$ 3.01211e9 0.376971
$$676$$ 0 0
$$677$$ 8.55209e9 1.05928 0.529642 0.848222i $$-0.322326\pi$$
0.529642 + 0.848222i $$0.322326\pi$$
$$678$$ 0 0
$$679$$ −7.50565e9 −0.920119
$$680$$ 0 0
$$681$$ 6.48061e9 0.786323
$$682$$ 0 0
$$683$$ −7.26976e9 −0.873067 −0.436534 0.899688i $$-0.643794\pi$$
−0.436534 + 0.899688i $$0.643794\pi$$
$$684$$ 0 0
$$685$$ 8.33837e9 0.991206
$$686$$ 0 0
$$687$$ −2.12010e9 −0.249463
$$688$$ 0 0
$$689$$ −6.54162e9 −0.761935
$$690$$ 0 0
$$691$$ −5.19893e9 −0.599434 −0.299717 0.954028i $$-0.596892\pi$$
−0.299717 + 0.954028i $$0.596892\pi$$
$$692$$ 0 0
$$693$$ −2.97218e9 −0.339241
$$694$$ 0 0
$$695$$ −1.69709e10 −1.91760
$$696$$ 0 0
$$697$$ −2.47961e9 −0.277376
$$698$$ 0 0
$$699$$ −7.21572e9 −0.799116
$$700$$ 0 0
$$701$$ −1.35221e10 −1.48262 −0.741310 0.671163i $$-0.765795\pi$$
−0.741310 + 0.671163i $$0.765795\pi$$
$$702$$ 0 0
$$703$$ −1.51263e9 −0.164206
$$704$$ 0 0
$$705$$ −1.55349e9 −0.166974
$$706$$ 0 0
$$707$$ −5.93083e9 −0.631172
$$708$$ 0 0
$$709$$ −1.05901e10 −1.11594 −0.557969 0.829862i $$-0.688419\pi$$
−0.557969 + 0.829862i $$0.688419\pi$$
$$710$$ 0 0
$$711$$ −4.68613e9 −0.488957
$$712$$ 0 0
$$713$$ 2.75507e9 0.284655
$$714$$ 0 0
$$715$$ 1.11271e10 1.13845
$$716$$ 0 0
$$717$$ 3.38799e9 0.343261
$$718$$ 0 0
$$719$$ 1.49161e10 1.49659 0.748297 0.663363i $$-0.230872\pi$$
0.748297 + 0.663363i $$0.230872\pi$$
$$720$$ 0 0
$$721$$ 6.31417e9 0.627398
$$722$$ 0 0
$$723$$ 8.53800e9 0.840180
$$724$$ 0 0
$$725$$ −4.64919e9 −0.453100
$$726$$ 0 0
$$727$$ 8.90159e9 0.859206 0.429603 0.903018i $$-0.358654\pi$$
0.429603 + 0.903018i $$0.358654\pi$$
$$728$$ 0 0
$$729$$ 1.15808e10 1.10711
$$730$$ 0 0
$$731$$ −1.13591e10 −1.07556
$$732$$ 0 0
$$733$$ −7.99792e9 −0.750090 −0.375045 0.927007i $$-0.622373\pi$$
−0.375045 + 0.927007i $$0.622373\pi$$
$$734$$ 0 0
$$735$$ −1.20359e9 −0.111808
$$736$$ 0 0
$$737$$ −1.72494e9 −0.158722
$$738$$ 0 0
$$739$$ 1.03852e10 0.946588 0.473294 0.880905i $$-0.343065\pi$$
0.473294 + 0.880905i $$0.343065\pi$$
$$740$$ 0 0
$$741$$ −3.69175e9 −0.333326
$$742$$ 0 0
$$743$$ 3.73477e9 0.334044 0.167022 0.985953i $$-0.446585\pi$$
0.167022 + 0.985953i $$0.446585\pi$$
$$744$$ 0 0
$$745$$ −5.85402e9 −0.518689
$$746$$ 0 0
$$747$$ −3.35815e9 −0.294766
$$748$$ 0 0
$$749$$ 4.90772e8 0.0426770
$$750$$ 0 0
$$751$$ −1.34330e10 −1.15726 −0.578631 0.815589i $$-0.696413\pi$$
−0.578631 + 0.815589i $$0.696413\pi$$
$$752$$ 0 0
$$753$$ 4.74963e9 0.405394
$$754$$ 0 0
$$755$$ −1.25703e10 −1.06300
$$756$$ 0 0
$$757$$ 6.78007e9 0.568065 0.284033 0.958815i $$-0.408328\pi$$
0.284033 + 0.958815i $$0.408328\pi$$
$$758$$ 0 0
$$759$$ −5.40877e9 −0.449006
$$760$$ 0 0
$$761$$ −8.01137e9 −0.658962 −0.329481 0.944162i $$-0.606874\pi$$
−0.329481 + 0.944162i $$0.606874\pi$$
$$762$$ 0 0
$$763$$ 1.87451e10 1.52775
$$764$$ 0 0
$$765$$ −6.39335e9 −0.516314
$$766$$ 0 0
$$767$$ −1.42130e9 −0.113737
$$768$$ 0 0
$$769$$ 1.46553e10 1.16213 0.581063 0.813858i $$-0.302637\pi$$
0.581063 + 0.813858i $$0.302637\pi$$
$$770$$ 0 0
$$771$$ −5.12609e9 −0.402806
$$772$$ 0 0
$$773$$ −2.33296e10 −1.81668 −0.908340 0.418233i $$-0.862650\pi$$
−0.908340 + 0.418233i $$0.862650\pi$$
$$774$$ 0 0
$$775$$ 3.03456e9 0.234174
$$776$$ 0 0
$$777$$ 3.98847e9 0.305023
$$778$$ 0 0
$$779$$ −1.04638e9 −0.0793064
$$780$$ 0 0
$$781$$ −2.93855e9 −0.220726
$$782$$ 0 0
$$783$$ −1.89703e10 −1.41224
$$784$$ 0 0
$$785$$ −1.66250e10 −1.22664
$$786$$ 0 0
$$787$$ −5.27740e8 −0.0385930 −0.0192965 0.999814i $$-0.506143\pi$$
−0.0192965 + 0.999814i $$0.506143\pi$$
$$788$$ 0 0
$$789$$ 1.74187e10 1.26254
$$790$$ 0 0
$$791$$ 1.80747e10 1.29853
$$792$$ 0 0
$$793$$ −9.76536e9 −0.695396
$$794$$ 0 0
$$795$$ 1.35732e10 0.958072
$$796$$ 0 0
$$797$$ 2.41514e9 0.168981 0.0844907 0.996424i $$-0.473074\pi$$
0.0844907 + 0.996424i $$0.473074\pi$$
$$798$$ 0 0
$$799$$ 4.11738e9 0.285566
$$800$$ 0 0
$$801$$ 2.48940e8 0.0171152
$$802$$ 0 0
$$803$$ 1.46398e10 0.997773
$$804$$ 0 0
$$805$$ 7.65988e9 0.517531
$$806$$ 0 0
$$807$$ −1.38645e10 −0.928636
$$808$$ 0 0
$$809$$ −1.16364e10 −0.772679 −0.386340 0.922357i $$-0.626261\pi$$
−0.386340 + 0.922357i $$0.626261\pi$$
$$810$$ 0 0
$$811$$ 1.44359e10 0.950320 0.475160 0.879899i $$-0.342390\pi$$
0.475160 + 0.879899i $$0.342390\pi$$
$$812$$ 0 0
$$813$$ 1.77962e10 1.16148
$$814$$ 0 0
$$815$$ 2.78275e10 1.80062
$$816$$ 0 0
$$817$$ −4.79348e9 −0.307521
$$818$$ 0 0
$$819$$ −3.42855e9 −0.218080
$$820$$ 0 0
$$821$$ −7.63805e9 −0.481706 −0.240853 0.970562i $$-0.577427\pi$$
−0.240853 + 0.970562i $$0.577427\pi$$
$$822$$ 0 0
$$823$$ 2.16446e10 1.35348 0.676738 0.736224i $$-0.263393\pi$$
0.676738 + 0.736224i $$0.263393\pi$$
$$824$$ 0 0
$$825$$ −5.95746e9 −0.369379
$$826$$ 0 0
$$827$$ −1.57823e10 −0.970288 −0.485144 0.874434i $$-0.661233\pi$$
−0.485144 + 0.874434i $$0.661233\pi$$
$$828$$ 0 0
$$829$$ 2.63296e10 1.60510 0.802552 0.596582i $$-0.203475\pi$$
0.802552 + 0.596582i $$0.203475\pi$$
$$830$$ 0 0
$$831$$ 1.27947e10 0.773442
$$832$$ 0 0
$$833$$ 3.19000e9 0.191220
$$834$$ 0 0
$$835$$ 3.43541e10 2.04210
$$836$$ 0 0
$$837$$ 1.23820e10 0.729882
$$838$$ 0 0
$$839$$ 1.84995e9 0.108142 0.0540710 0.998537i $$-0.482780\pi$$
0.0540710 + 0.998537i $$0.482780\pi$$
$$840$$ 0 0
$$841$$ 1.20307e10 0.697438
$$842$$ 0 0
$$843$$ 5.16767e9 0.297097
$$844$$ 0 0
$$845$$ −7.52569e9 −0.429090
$$846$$ 0 0
$$847$$ 9.79866e9 0.554083
$$848$$ 0 0
$$849$$ −1.60451e10 −0.899839
$$850$$ 0 0
$$851$$ −2.55643e9 −0.142194
$$852$$ 0 0
$$853$$ −6.28088e7 −0.00346496 −0.00173248 0.999998i $$-0.500551\pi$$
−0.00173248 + 0.999998i $$0.500551\pi$$
$$854$$ 0 0
$$855$$ −2.69796e9 −0.147623
$$856$$ 0 0
$$857$$ 6.25595e9 0.339516 0.169758 0.985486i $$-0.445701\pi$$
0.169758 + 0.985486i $$0.445701\pi$$
$$858$$ 0 0
$$859$$ −2.48119e10 −1.33562 −0.667811 0.744331i $$-0.732769\pi$$
−0.667811 + 0.744331i $$0.732769\pi$$
$$860$$ 0 0
$$861$$ 2.75907e9 0.147316
$$862$$ 0 0
$$863$$ 1.28944e10 0.682909 0.341455 0.939898i $$-0.389080\pi$$
0.341455 + 0.939898i $$0.389080\pi$$
$$864$$ 0 0
$$865$$ −6.42973e9 −0.337782
$$866$$ 0 0
$$867$$ −3.16077e10 −1.64712
$$868$$ 0 0
$$869$$ 4.48516e10 2.31851
$$870$$ 0 0
$$871$$ −1.98980e9 −0.102034
$$872$$ 0 0
$$873$$ 4.46789e9 0.227276
$$874$$ 0 0
$$875$$ −1.58229e10 −0.798468
$$876$$ 0 0
$$877$$ −2.75670e10 −1.38004 −0.690018 0.723792i $$-0.742397\pi$$
−0.690018 + 0.723792i $$0.742397\pi$$
$$878$$ 0 0
$$879$$ 8.08177e9 0.401371
$$880$$ 0 0
$$881$$ 2.74343e10 1.35169 0.675847 0.737042i $$-0.263778\pi$$
0.675847 + 0.737042i $$0.263778\pi$$
$$882$$ 0 0
$$883$$ −2.09906e10 −1.02604 −0.513018 0.858378i $$-0.671473\pi$$
−0.513018 + 0.858378i $$0.671473\pi$$
$$884$$ 0 0
$$885$$ 2.94905e9 0.143015
$$886$$ 0 0
$$887$$ 2.39134e10 1.15056 0.575280 0.817957i $$-0.304893\pi$$
0.575280 + 0.817957i $$0.304893\pi$$
$$888$$ 0 0
$$889$$ 3.79257e10 1.81041
$$890$$ 0 0
$$891$$ −1.75160e10 −0.829588
$$892$$ 0 0
$$893$$ 1.73751e9 0.0816483
$$894$$ 0 0
$$895$$ −9.66337e9 −0.450555
$$896$$ 0 0
$$897$$ −6.23927e9 −0.288643
$$898$$ 0 0
$$899$$ −1.91117e10 −0.877282
$$900$$ 0 0
$$901$$ −3.59744e10 −1.63854
$$902$$ 0 0
$$903$$ 1.26393e10 0.571238
$$904$$ 0 0
$$905$$ 1.28572e9 0.0576603
$$906$$ 0 0
$$907$$ 3.84425e10 1.71075 0.855373 0.518012i $$-0.173328\pi$$
0.855373 + 0.518012i $$0.173328\pi$$
$$908$$ 0 0
$$909$$ 3.53045e9 0.155904
$$910$$ 0 0
$$911$$ 3.64507e10 1.59732 0.798660 0.601783i $$-0.205543\pi$$
0.798660 + 0.601783i $$0.205543\pi$$
$$912$$ 0 0
$$913$$ 3.21413e10 1.39771
$$914$$ 0 0
$$915$$ 2.02622e10 0.874405
$$916$$ 0 0
$$917$$ −3.49613e10 −1.49725
$$918$$ 0 0
$$919$$ −2.37343e10 −1.00872 −0.504361 0.863493i $$-0.668272\pi$$
−0.504361 + 0.863493i $$0.668272\pi$$
$$920$$ 0 0
$$921$$ −6.39111e8 −0.0269567
$$922$$ 0 0
$$923$$ −3.38976e9 −0.141894
$$924$$ 0 0
$$925$$ −2.81577e9 −0.116977
$$926$$ 0 0
$$927$$ −3.75864e9 −0.154972
$$928$$ 0 0
$$929$$ −4.23952e10 −1.73485 −0.867424 0.497570i $$-0.834226\pi$$
−0.867424 + 0.497570i $$0.834226\pi$$
$$930$$ 0 0
$$931$$ 1.34616e9 0.0546730
$$932$$ 0 0
$$933$$ 1.96139e10 0.790638
$$934$$ 0 0
$$935$$ 6.11916e10 2.44823
$$936$$ 0 0
$$937$$ −3.70014e10 −1.46937 −0.734683 0.678411i $$-0.762669\pi$$
−0.734683 + 0.678411i $$0.762669\pi$$
$$938$$ 0 0
$$939$$ −1.30595e10 −0.514750
$$940$$ 0 0
$$941$$ 2.10617e10 0.824005 0.412003 0.911183i $$-0.364829\pi$$
0.412003 + 0.911183i $$0.364829\pi$$
$$942$$ 0 0
$$943$$ −1.76844e9 −0.0686752
$$944$$ 0 0
$$945$$ 3.44256e10 1.32700
$$946$$ 0 0
$$947$$ 1.35146e10 0.517104 0.258552 0.965997i $$-0.416755\pi$$
0.258552 + 0.965997i $$0.416755\pi$$
$$948$$ 0 0
$$949$$ 1.68877e10 0.641416
$$950$$ 0 0
$$951$$ 4.26628e10 1.60849
$$952$$ 0 0
$$953$$ −4.26831e10 −1.59746 −0.798731 0.601688i $$-0.794495\pi$$
−0.798731 + 0.601688i $$0.794495\pi$$
$$954$$ 0 0
$$955$$ −1.55790e10 −0.578799
$$956$$ 0 0
$$957$$ 3.75201e10 1.38380
$$958$$ 0 0
$$959$$ 2.45907e10 0.900340
$$960$$ 0 0
$$961$$ −1.50383e10 −0.546597
$$962$$ 0 0
$$963$$ −2.92142e8 −0.0105415
$$964$$ 0 0
$$965$$ −1.53323e9 −0.0549239
$$966$$ 0 0
$$967$$ −2.32274e10 −0.826053 −0.413026 0.910719i $$-0.635528\pi$$
−0.413026 + 0.910719i $$0.635528\pi$$
$$968$$ 0 0
$$969$$ −2.03021e10 −0.716817
$$970$$ 0 0
$$971$$ 4.85678e10 1.70248 0.851238 0.524780i $$-0.175852\pi$$
0.851238 + 0.524780i $$0.175852\pi$$
$$972$$ 0 0
$$973$$ −5.00491e10 −1.74181
$$974$$ 0 0
$$975$$ −6.87221e9 −0.237454
$$976$$ 0 0
$$977$$ 1.54549e10 0.530193 0.265096 0.964222i $$-0.414596\pi$$
0.265096 + 0.964222i $$0.414596\pi$$
$$978$$ 0 0
$$979$$ −2.38264e9 −0.0811557
$$980$$ 0 0
$$981$$ −1.11584e10 −0.377364
$$982$$ 0 0
$$983$$ 1.59192e10 0.534546 0.267273 0.963621i $$-0.413877\pi$$
0.267273 + 0.963621i $$0.413877\pi$$
$$984$$ 0 0
$$985$$ 3.72782e10 1.24288
$$986$$ 0 0
$$987$$ −4.58142e9 −0.151667
$$988$$ 0 0
$$989$$ −8.10126e9 −0.266296
$$990$$ 0 0
$$991$$ 1.49883e10 0.489208 0.244604 0.969623i $$-0.421342\pi$$
0.244604 + 0.969623i $$0.421342\pi$$
$$992$$ 0 0
$$993$$ −1.16019e10 −0.376016
$$994$$ 0 0
$$995$$ −3.92416e9 −0.126289
$$996$$ 0 0
$$997$$ −3.44101e10 −1.09965 −0.549824 0.835281i $$-0.685305\pi$$
−0.549824 + 0.835281i $$0.685305\pi$$
$$998$$ 0 0
$$999$$ −1.14893e10 −0.364598
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 256.8.a.r.1.2 6
4.3 odd 2 256.8.a.q.1.5 6
8.3 odd 2 256.8.a.q.1.2 6
8.5 even 2 inner 256.8.a.r.1.5 6
16.3 odd 4 32.8.b.a.17.5 6
16.5 even 4 8.8.b.a.5.1 6
16.11 odd 4 32.8.b.a.17.2 6
16.13 even 4 8.8.b.a.5.2 yes 6
48.5 odd 4 72.8.d.b.37.6 6
48.11 even 4 288.8.d.b.145.2 6
48.29 odd 4 72.8.d.b.37.5 6
48.35 even 4 288.8.d.b.145.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.b.a.5.1 6 16.5 even 4
8.8.b.a.5.2 yes 6 16.13 even 4
32.8.b.a.17.2 6 16.11 odd 4
32.8.b.a.17.5 6 16.3 odd 4
72.8.d.b.37.5 6 48.29 odd 4
72.8.d.b.37.6 6 48.5 odd 4
256.8.a.q.1.2 6 8.3 odd 2
256.8.a.q.1.5 6 4.3 odd 2
256.8.a.r.1.2 6 1.1 even 1 trivial
256.8.a.r.1.5 6 8.5 even 2 inner
288.8.d.b.145.2 6 48.11 even 4
288.8.d.b.145.5 6 48.35 even 4