# Properties

 Label 825.6.a.t.1.6 Level $825$ Weight $6$ Character 825.1 Self dual yes Analytic conductor $132.317$ Analytic rank $0$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + 7443872 x^{2} - 12856224 x - 7036608$$ x^10 - x^9 - 271*x^8 + 309*x^7 + 24456*x^6 - 33410*x^5 - 822204*x^4 + 1367872*x^3 + 7443872*x^2 - 12856224*x - 7036608 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 5^{4}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$-0.444648$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.444648 q^{2} +9.00000 q^{3} -31.8023 q^{4} +4.00183 q^{6} +205.804 q^{7} -28.3695 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+0.444648 q^{2} +9.00000 q^{3} -31.8023 q^{4} +4.00183 q^{6} +205.804 q^{7} -28.3695 q^{8} +81.0000 q^{9} +121.000 q^{11} -286.221 q^{12} -893.005 q^{13} +91.5105 q^{14} +1005.06 q^{16} +836.291 q^{17} +36.0165 q^{18} +816.231 q^{19} +1852.24 q^{21} +53.8024 q^{22} +3125.01 q^{23} -255.326 q^{24} -397.073 q^{26} +729.000 q^{27} -6545.05 q^{28} -7307.37 q^{29} +152.387 q^{31} +1354.72 q^{32} +1089.00 q^{33} +371.855 q^{34} -2575.99 q^{36} +8734.38 q^{37} +362.935 q^{38} -8037.05 q^{39} -9608.76 q^{41} +823.594 q^{42} +9208.31 q^{43} -3848.08 q^{44} +1389.53 q^{46} +11806.9 q^{47} +9045.53 q^{48} +25548.5 q^{49} +7526.62 q^{51} +28399.6 q^{52} +12879.4 q^{53} +324.148 q^{54} -5838.58 q^{56} +7346.08 q^{57} -3249.21 q^{58} -517.608 q^{59} -50201.8 q^{61} +67.7587 q^{62} +16670.2 q^{63} -31559.5 q^{64} +484.221 q^{66} +4050.14 q^{67} -26596.0 q^{68} +28125.1 q^{69} +16467.7 q^{71} -2297.93 q^{72} -55131.2 q^{73} +3883.72 q^{74} -25958.0 q^{76} +24902.3 q^{77} -3573.66 q^{78} +25009.6 q^{79} +6561.00 q^{81} -4272.51 q^{82} -76147.7 q^{83} -58905.5 q^{84} +4094.45 q^{86} -65766.3 q^{87} -3432.72 q^{88} +104461. q^{89} -183784. q^{91} -99382.6 q^{92} +1371.49 q^{93} +5249.89 q^{94} +12192.5 q^{96} +39347.8 q^{97} +11360.1 q^{98} +9801.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10})$$ 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9 $$10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9} + 1210 q^{11} + 2007 q^{12} - 1102 q^{13} + 684 q^{14} + 7019 q^{16} + 1106 q^{17} - 81 q^{18} + 2586 q^{19} + 1692 q^{21} - 121 q^{22} + 2206 q^{23} + 1593 q^{24} + 12001 q^{26} + 7290 q^{27} + 14452 q^{28} + 5824 q^{29} - 4586 q^{31} - 10627 q^{32} + 10890 q^{33} + 7426 q^{34} + 18063 q^{36} - 18362 q^{37} + 44001 q^{38} - 9918 q^{39} - 5474 q^{41} + 6156 q^{42} + 20496 q^{43} + 26983 q^{44} + 19981 q^{46} - 14970 q^{47} + 63171 q^{48} + 68582 q^{49} + 9954 q^{51} - 58603 q^{52} + 61980 q^{53} - 729 q^{54} + 7132 q^{56} + 23274 q^{57} + 7161 q^{58} + 61190 q^{59} + 8230 q^{61} - 4509 q^{62} + 15228 q^{63} + 152223 q^{64} - 1089 q^{66} - 11930 q^{67} + 105598 q^{68} + 19854 q^{69} + 59822 q^{71} + 14337 q^{72} - 20680 q^{73} + 132564 q^{74} + 68165 q^{76} + 22748 q^{77} + 108009 q^{78} + 234494 q^{79} + 65610 q^{81} + 151948 q^{82} + 185478 q^{83} + 130068 q^{84} - 17825 q^{86} + 52416 q^{87} + 21417 q^{88} + 181834 q^{89} + 206274 q^{91} + 98373 q^{92} - 41274 q^{93} - 64998 q^{94} - 95643 q^{96} + 304358 q^{97} + 153453 q^{98} + 98010 q^{99}+O(q^{100})$$ 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9 + 1210 * q^11 + 2007 * q^12 - 1102 * q^13 + 684 * q^14 + 7019 * q^16 + 1106 * q^17 - 81 * q^18 + 2586 * q^19 + 1692 * q^21 - 121 * q^22 + 2206 * q^23 + 1593 * q^24 + 12001 * q^26 + 7290 * q^27 + 14452 * q^28 + 5824 * q^29 - 4586 * q^31 - 10627 * q^32 + 10890 * q^33 + 7426 * q^34 + 18063 * q^36 - 18362 * q^37 + 44001 * q^38 - 9918 * q^39 - 5474 * q^41 + 6156 * q^42 + 20496 * q^43 + 26983 * q^44 + 19981 * q^46 - 14970 * q^47 + 63171 * q^48 + 68582 * q^49 + 9954 * q^51 - 58603 * q^52 + 61980 * q^53 - 729 * q^54 + 7132 * q^56 + 23274 * q^57 + 7161 * q^58 + 61190 * q^59 + 8230 * q^61 - 4509 * q^62 + 15228 * q^63 + 152223 * q^64 - 1089 * q^66 - 11930 * q^67 + 105598 * q^68 + 19854 * q^69 + 59822 * q^71 + 14337 * q^72 - 20680 * q^73 + 132564 * q^74 + 68165 * q^76 + 22748 * q^77 + 108009 * q^78 + 234494 * q^79 + 65610 * q^81 + 151948 * q^82 + 185478 * q^83 + 130068 * q^84 - 17825 * q^86 + 52416 * q^87 + 21417 * q^88 + 181834 * q^89 + 206274 * q^91 + 98373 * q^92 - 41274 * q^93 - 64998 * q^94 - 95643 * q^96 + 304358 * q^97 + 153453 * q^98 + 98010 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.444648 0.0786034 0.0393017 0.999227i $$-0.487487\pi$$
0.0393017 + 0.999227i $$0.487487\pi$$
$$3$$ 9.00000 0.577350
$$4$$ −31.8023 −0.993822
$$5$$ 0 0
$$6$$ 4.00183 0.0453817
$$7$$ 205.804 1.58748 0.793742 0.608254i $$-0.208130\pi$$
0.793742 + 0.608254i $$0.208130\pi$$
$$8$$ −28.3695 −0.156721
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ 121.000 0.301511
$$12$$ −286.221 −0.573783
$$13$$ −893.005 −1.46553 −0.732767 0.680480i $$-0.761771\pi$$
−0.732767 + 0.680480i $$0.761771\pi$$
$$14$$ 91.5105 0.124782
$$15$$ 0 0
$$16$$ 1005.06 0.981503
$$17$$ 836.291 0.701835 0.350918 0.936406i $$-0.385870\pi$$
0.350918 + 0.936406i $$0.385870\pi$$
$$18$$ 36.0165 0.0262011
$$19$$ 816.231 0.518715 0.259358 0.965781i $$-0.416489\pi$$
0.259358 + 0.965781i $$0.416489\pi$$
$$20$$ 0 0
$$21$$ 1852.24 0.916535
$$22$$ 53.8024 0.0236998
$$23$$ 3125.01 1.23178 0.615889 0.787833i $$-0.288797\pi$$
0.615889 + 0.787833i $$0.288797\pi$$
$$24$$ −255.326 −0.0904830
$$25$$ 0 0
$$26$$ −397.073 −0.115196
$$27$$ 729.000 0.192450
$$28$$ −6545.05 −1.57768
$$29$$ −7307.37 −1.61349 −0.806745 0.590900i $$-0.798773\pi$$
−0.806745 + 0.590900i $$0.798773\pi$$
$$30$$ 0 0
$$31$$ 152.387 0.0284803 0.0142401 0.999899i $$-0.495467\pi$$
0.0142401 + 0.999899i $$0.495467\pi$$
$$32$$ 1354.72 0.233871
$$33$$ 1089.00 0.174078
$$34$$ 371.855 0.0551666
$$35$$ 0 0
$$36$$ −2575.99 −0.331274
$$37$$ 8734.38 1.04888 0.524442 0.851446i $$-0.324274\pi$$
0.524442 + 0.851446i $$0.324274\pi$$
$$38$$ 362.935 0.0407728
$$39$$ −8037.05 −0.846126
$$40$$ 0 0
$$41$$ −9608.76 −0.892704 −0.446352 0.894857i $$-0.647277\pi$$
−0.446352 + 0.894857i $$0.647277\pi$$
$$42$$ 823.594 0.0720427
$$43$$ 9208.31 0.759467 0.379733 0.925096i $$-0.376016\pi$$
0.379733 + 0.925096i $$0.376016\pi$$
$$44$$ −3848.08 −0.299648
$$45$$ 0 0
$$46$$ 1389.53 0.0968219
$$47$$ 11806.9 0.779632 0.389816 0.920893i $$-0.372539\pi$$
0.389816 + 0.920893i $$0.372539\pi$$
$$48$$ 9045.53 0.566671
$$49$$ 25548.5 1.52011
$$50$$ 0 0
$$51$$ 7526.62 0.405205
$$52$$ 28399.6 1.45648
$$53$$ 12879.4 0.629807 0.314904 0.949124i $$-0.398028\pi$$
0.314904 + 0.949124i $$0.398028\pi$$
$$54$$ 324.148 0.0151272
$$55$$ 0 0
$$56$$ −5838.58 −0.248792
$$57$$ 7346.08 0.299480
$$58$$ −3249.21 −0.126826
$$59$$ −517.608 −0.0193585 −0.00967924 0.999953i $$-0.503081\pi$$
−0.00967924 + 0.999953i $$0.503081\pi$$
$$60$$ 0 0
$$61$$ −50201.8 −1.72741 −0.863704 0.503999i $$-0.831861\pi$$
−0.863704 + 0.503999i $$0.831861\pi$$
$$62$$ 67.7587 0.00223865
$$63$$ 16670.2 0.529162
$$64$$ −31559.5 −0.963120
$$65$$ 0 0
$$66$$ 484.221 0.0136831
$$67$$ 4050.14 0.110226 0.0551129 0.998480i $$-0.482448\pi$$
0.0551129 + 0.998480i $$0.482448\pi$$
$$68$$ −26596.0 −0.697499
$$69$$ 28125.1 0.711167
$$70$$ 0 0
$$71$$ 16467.7 0.387692 0.193846 0.981032i $$-0.437904\pi$$
0.193846 + 0.981032i $$0.437904\pi$$
$$72$$ −2297.93 −0.0522404
$$73$$ −55131.2 −1.21085 −0.605425 0.795903i $$-0.706997\pi$$
−0.605425 + 0.795903i $$0.706997\pi$$
$$74$$ 3883.72 0.0824458
$$75$$ 0 0
$$76$$ −25958.0 −0.515510
$$77$$ 24902.3 0.478645
$$78$$ −3573.66 −0.0665084
$$79$$ 25009.6 0.450858 0.225429 0.974260i $$-0.427622\pi$$
0.225429 + 0.974260i $$0.427622\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ −4272.51 −0.0701696
$$83$$ −76147.7 −1.21328 −0.606641 0.794976i $$-0.707483\pi$$
−0.606641 + 0.794976i $$0.707483\pi$$
$$84$$ −58905.5 −0.910872
$$85$$ 0 0
$$86$$ 4094.45 0.0596966
$$87$$ −65766.3 −0.931548
$$88$$ −3432.72 −0.0472532
$$89$$ 104461. 1.39790 0.698952 0.715169i $$-0.253650\pi$$
0.698952 + 0.715169i $$0.253650\pi$$
$$90$$ 0 0
$$91$$ −183784. −2.32651
$$92$$ −99382.6 −1.22417
$$93$$ 1371.49 0.0164431
$$94$$ 5249.89 0.0612817
$$95$$ 0 0
$$96$$ 12192.5 0.135025
$$97$$ 39347.8 0.424611 0.212306 0.977203i $$-0.431903\pi$$
0.212306 + 0.977203i $$0.431903\pi$$
$$98$$ 11360.1 0.119486
$$99$$ 9801.00 0.100504
$$100$$ 0 0
$$101$$ −96213.1 −0.938493 −0.469246 0.883067i $$-0.655474\pi$$
−0.469246 + 0.883067i $$0.655474\pi$$
$$102$$ 3346.70 0.0318505
$$103$$ 54269.8 0.504040 0.252020 0.967722i $$-0.418905\pi$$
0.252020 + 0.967722i $$0.418905\pi$$
$$104$$ 25334.2 0.229680
$$105$$ 0 0
$$106$$ 5726.82 0.0495050
$$107$$ −46180.6 −0.389942 −0.194971 0.980809i $$-0.562461\pi$$
−0.194971 + 0.980809i $$0.562461\pi$$
$$108$$ −23183.9 −0.191261
$$109$$ 162090. 1.30675 0.653373 0.757036i $$-0.273353\pi$$
0.653373 + 0.757036i $$0.273353\pi$$
$$110$$ 0 0
$$111$$ 78609.4 0.605574
$$112$$ 206846. 1.55812
$$113$$ 252815. 1.86255 0.931274 0.364321i $$-0.118699\pi$$
0.931274 + 0.364321i $$0.118699\pi$$
$$114$$ 3266.42 0.0235402
$$115$$ 0 0
$$116$$ 232391. 1.60352
$$117$$ −72333.4 −0.488511
$$118$$ −230.153 −0.00152164
$$119$$ 172112. 1.11415
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ −22322.1 −0.135780
$$123$$ −86478.8 −0.515403
$$124$$ −4846.26 −0.0283043
$$125$$ 0 0
$$126$$ 7412.35 0.0415939
$$127$$ 171792. 0.945135 0.472568 0.881294i $$-0.343327\pi$$
0.472568 + 0.881294i $$0.343327\pi$$
$$128$$ −57384.0 −0.309575
$$129$$ 82874.8 0.438478
$$130$$ 0 0
$$131$$ 140420. 0.714911 0.357455 0.933930i $$-0.383644\pi$$
0.357455 + 0.933930i $$0.383644\pi$$
$$132$$ −34632.7 −0.173002
$$133$$ 167984. 0.823453
$$134$$ 1800.89 0.00866411
$$135$$ 0 0
$$136$$ −23725.2 −0.109992
$$137$$ 194429. 0.885034 0.442517 0.896760i $$-0.354086\pi$$
0.442517 + 0.896760i $$0.354086\pi$$
$$138$$ 12505.8 0.0559001
$$139$$ −246396. −1.08168 −0.540838 0.841127i $$-0.681893\pi$$
−0.540838 + 0.841127i $$0.681893\pi$$
$$140$$ 0 0
$$141$$ 106262. 0.450121
$$142$$ 7322.32 0.0304739
$$143$$ −108054. −0.441875
$$144$$ 81409.8 0.327168
$$145$$ 0 0
$$146$$ −24514.0 −0.0951768
$$147$$ 229936. 0.877635
$$148$$ −277773. −1.04240
$$149$$ −329064. −1.21427 −0.607135 0.794599i $$-0.707681\pi$$
−0.607135 + 0.794599i $$0.707681\pi$$
$$150$$ 0 0
$$151$$ 125794. 0.448971 0.224486 0.974477i $$-0.427930\pi$$
0.224486 + 0.974477i $$0.427930\pi$$
$$152$$ −23156.1 −0.0812936
$$153$$ 67739.6 0.233945
$$154$$ 11072.8 0.0376231
$$155$$ 0 0
$$156$$ 255597. 0.840898
$$157$$ 21521.1 0.0696810 0.0348405 0.999393i $$-0.488908\pi$$
0.0348405 + 0.999393i $$0.488908\pi$$
$$158$$ 11120.5 0.0354389
$$159$$ 115915. 0.363619
$$160$$ 0 0
$$161$$ 643142. 1.95543
$$162$$ 2917.33 0.00873371
$$163$$ 383784. 1.13140 0.565702 0.824610i $$-0.308605\pi$$
0.565702 + 0.824610i $$0.308605\pi$$
$$164$$ 305580. 0.887189
$$165$$ 0 0
$$166$$ −33858.9 −0.0953681
$$167$$ 512066. 1.42081 0.710403 0.703795i $$-0.248513\pi$$
0.710403 + 0.703795i $$0.248513\pi$$
$$168$$ −52547.2 −0.143640
$$169$$ 426166. 1.14779
$$170$$ 0 0
$$171$$ 66114.7 0.172905
$$172$$ −292845. −0.754774
$$173$$ −260098. −0.660726 −0.330363 0.943854i $$-0.607171\pi$$
−0.330363 + 0.943854i $$0.607171\pi$$
$$174$$ −29242.8 −0.0732228
$$175$$ 0 0
$$176$$ 121612. 0.295934
$$177$$ −4658.47 −0.0111766
$$178$$ 46448.2 0.109880
$$179$$ 112495. 0.262421 0.131211 0.991354i $$-0.458114\pi$$
0.131211 + 0.991354i $$0.458114\pi$$
$$180$$ 0 0
$$181$$ 56459.6 0.128098 0.0640488 0.997947i $$-0.479599\pi$$
0.0640488 + 0.997947i $$0.479599\pi$$
$$182$$ −81719.4 −0.182872
$$183$$ −451817. −0.997320
$$184$$ −88655.2 −0.193046
$$185$$ 0 0
$$186$$ 609.828 0.00129248
$$187$$ 101191. 0.211611
$$188$$ −375485. −0.774815
$$189$$ 150031. 0.305512
$$190$$ 0 0
$$191$$ 356386. 0.706865 0.353433 0.935460i $$-0.385014\pi$$
0.353433 + 0.935460i $$0.385014\pi$$
$$192$$ −284036. −0.556057
$$193$$ 607666. 1.17428 0.587140 0.809486i $$-0.300254\pi$$
0.587140 + 0.809486i $$0.300254\pi$$
$$194$$ 17495.9 0.0333759
$$195$$ 0 0
$$196$$ −812500. −1.51072
$$197$$ −91261.2 −0.167541 −0.0837704 0.996485i $$-0.526696\pi$$
−0.0837704 + 0.996485i $$0.526696\pi$$
$$198$$ 4357.99 0.00789994
$$199$$ 840148. 1.50392 0.751958 0.659212i $$-0.229110\pi$$
0.751958 + 0.659212i $$0.229110\pi$$
$$200$$ 0 0
$$201$$ 36451.3 0.0636389
$$202$$ −42781.0 −0.0737687
$$203$$ −1.50389e6 −2.56139
$$204$$ −239364. −0.402701
$$205$$ 0 0
$$206$$ 24130.9 0.0396192
$$207$$ 253126. 0.410593
$$208$$ −897523. −1.43843
$$209$$ 98763.9 0.156399
$$210$$ 0 0
$$211$$ 945258. 1.46165 0.730826 0.682563i $$-0.239135\pi$$
0.730826 + 0.682563i $$0.239135\pi$$
$$212$$ −409596. −0.625916
$$213$$ 148209. 0.223834
$$214$$ −20534.1 −0.0306508
$$215$$ 0 0
$$216$$ −20681.4 −0.0301610
$$217$$ 31362.0 0.0452120
$$218$$ 72073.2 0.102715
$$219$$ −496181. −0.699084
$$220$$ 0 0
$$221$$ −746813. −1.02856
$$222$$ 34953.5 0.0476001
$$223$$ −951062. −1.28070 −0.640349 0.768084i $$-0.721210\pi$$
−0.640349 + 0.768084i $$0.721210\pi$$
$$224$$ 278808. 0.371266
$$225$$ 0 0
$$226$$ 112414. 0.146402
$$227$$ 1.13419e6 1.46090 0.730448 0.682968i $$-0.239311\pi$$
0.730448 + 0.682968i $$0.239311\pi$$
$$228$$ −233622. −0.297630
$$229$$ 1.57670e6 1.98682 0.993412 0.114595i $$-0.0365569\pi$$
0.993412 + 0.114595i $$0.0365569\pi$$
$$230$$ 0 0
$$231$$ 224121. 0.276346
$$232$$ 207307. 0.252868
$$233$$ 1.06285e6 1.28258 0.641289 0.767299i $$-0.278400\pi$$
0.641289 + 0.767299i $$0.278400\pi$$
$$234$$ −32162.9 −0.0383986
$$235$$ 0 0
$$236$$ 16461.1 0.0192389
$$237$$ 225087. 0.260303
$$238$$ 76529.4 0.0875762
$$239$$ −62094.6 −0.0703168 −0.0351584 0.999382i $$-0.511194\pi$$
−0.0351584 + 0.999382i $$0.511194\pi$$
$$240$$ 0 0
$$241$$ 90536.8 0.100411 0.0502056 0.998739i $$-0.484012\pi$$
0.0502056 + 0.998739i $$0.484012\pi$$
$$242$$ 6510.09 0.00714576
$$243$$ 59049.0 0.0641500
$$244$$ 1.59653e6 1.71674
$$245$$ 0 0
$$246$$ −38452.6 −0.0405124
$$247$$ −728899. −0.760194
$$248$$ −4323.16 −0.00446346
$$249$$ −685330. −0.700489
$$250$$ 0 0
$$251$$ −1.35882e6 −1.36138 −0.680689 0.732572i $$-0.738320\pi$$
−0.680689 + 0.732572i $$0.738320\pi$$
$$252$$ −530149. −0.525892
$$253$$ 378127. 0.371395
$$254$$ 76387.0 0.0742908
$$255$$ 0 0
$$256$$ 984389. 0.938786
$$257$$ 648294. 0.612265 0.306132 0.951989i $$-0.400965\pi$$
0.306132 + 0.951989i $$0.400965\pi$$
$$258$$ 36850.1 0.0344659
$$259$$ 1.79757e6 1.66509
$$260$$ 0 0
$$261$$ −591897. −0.537830
$$262$$ 62437.6 0.0561944
$$263$$ −1.78919e6 −1.59502 −0.797510 0.603306i $$-0.793850\pi$$
−0.797510 + 0.603306i $$0.793850\pi$$
$$264$$ −30894.4 −0.0272816
$$265$$ 0 0
$$266$$ 74693.7 0.0647261
$$267$$ 940145. 0.807080
$$268$$ −128804. −0.109545
$$269$$ 1.39327e6 1.17397 0.586984 0.809599i $$-0.300315\pi$$
0.586984 + 0.809599i $$0.300315\pi$$
$$270$$ 0 0
$$271$$ −694086. −0.574104 −0.287052 0.957915i $$-0.592675\pi$$
−0.287052 + 0.957915i $$0.592675\pi$$
$$272$$ 840522. 0.688853
$$273$$ −1.65406e6 −1.34321
$$274$$ 86452.4 0.0695666
$$275$$ 0 0
$$276$$ −894443. −0.706773
$$277$$ 1.92718e6 1.50911 0.754557 0.656234i $$-0.227852\pi$$
0.754557 + 0.656234i $$0.227852\pi$$
$$278$$ −109559. −0.0850233
$$279$$ 12343.4 0.00949343
$$280$$ 0 0
$$281$$ 473885. 0.358020 0.179010 0.983847i $$-0.442711\pi$$
0.179010 + 0.983847i $$0.442711\pi$$
$$282$$ 47249.0 0.0353810
$$283$$ 568257. 0.421773 0.210887 0.977511i $$-0.432365\pi$$
0.210887 + 0.977511i $$0.432365\pi$$
$$284$$ −523710. −0.385297
$$285$$ 0 0
$$286$$ −48045.8 −0.0347329
$$287$$ −1.97752e6 −1.41715
$$288$$ 109733. 0.0779568
$$289$$ −720474. −0.507427
$$290$$ 0 0
$$291$$ 354130. 0.245149
$$292$$ 1.75330e6 1.20337
$$293$$ −1.76731e6 −1.20266 −0.601331 0.799000i $$-0.705363\pi$$
−0.601331 + 0.799000i $$0.705363\pi$$
$$294$$ 102241. 0.0689851
$$295$$ 0 0
$$296$$ −247790. −0.164382
$$297$$ 88209.0 0.0580259
$$298$$ −146318. −0.0954457
$$299$$ −2.79065e6 −1.80521
$$300$$ 0 0
$$301$$ 1.89511e6 1.20564
$$302$$ 55934.1 0.0352906
$$303$$ −865918. −0.541839
$$304$$ 820360. 0.509120
$$305$$ 0 0
$$306$$ 30120.3 0.0183889
$$307$$ −2.67297e6 −1.61863 −0.809316 0.587374i $$-0.800162\pi$$
−0.809316 + 0.587374i $$0.800162\pi$$
$$308$$ −791951. −0.475687
$$309$$ 488428. 0.291008
$$310$$ 0 0
$$311$$ 2.23621e6 1.31102 0.655512 0.755185i $$-0.272453\pi$$
0.655512 + 0.755185i $$0.272453\pi$$
$$312$$ 228007. 0.132606
$$313$$ −1.55507e6 −0.897199 −0.448600 0.893733i $$-0.648077\pi$$
−0.448600 + 0.893733i $$0.648077\pi$$
$$314$$ 9569.29 0.00547716
$$315$$ 0 0
$$316$$ −795363. −0.448072
$$317$$ 1.64671e6 0.920383 0.460192 0.887820i $$-0.347781\pi$$
0.460192 + 0.887820i $$0.347781\pi$$
$$318$$ 51541.4 0.0285817
$$319$$ −884192. −0.486485
$$320$$ 0 0
$$321$$ −415626. −0.225133
$$322$$ 285972. 0.153703
$$323$$ 682607. 0.364053
$$324$$ −208655. −0.110425
$$325$$ 0 0
$$326$$ 170649. 0.0889322
$$327$$ 1.45881e6 0.754450
$$328$$ 272596. 0.139906
$$329$$ 2.42990e6 1.23765
$$330$$ 0 0
$$331$$ 249065. 0.124952 0.0624760 0.998046i $$-0.480100\pi$$
0.0624760 + 0.998046i $$0.480100\pi$$
$$332$$ 2.42167e6 1.20579
$$333$$ 707484. 0.349628
$$334$$ 227689. 0.111680
$$335$$ 0 0
$$336$$ 1.86161e6 0.899581
$$337$$ 712382. 0.341695 0.170847 0.985298i $$-0.445350\pi$$
0.170847 + 0.985298i $$0.445350\pi$$
$$338$$ 189494. 0.0902200
$$339$$ 2.27534e6 1.07534
$$340$$ 0 0
$$341$$ 18438.9 0.00858713
$$342$$ 29397.8 0.0135909
$$343$$ 1.79903e6 0.825664
$$344$$ −261236. −0.119024
$$345$$ 0 0
$$346$$ −115652. −0.0519353
$$347$$ −1.05963e6 −0.472421 −0.236210 0.971702i $$-0.575905\pi$$
−0.236210 + 0.971702i $$0.575905\pi$$
$$348$$ 2.09152e6 0.925793
$$349$$ −2.96255e6 −1.30197 −0.650986 0.759090i $$-0.725644\pi$$
−0.650986 + 0.759090i $$0.725644\pi$$
$$350$$ 0 0
$$351$$ −651001. −0.282042
$$352$$ 163921. 0.0705146
$$353$$ 2.45696e6 1.04945 0.524725 0.851272i $$-0.324168\pi$$
0.524725 + 0.851272i $$0.324168\pi$$
$$354$$ −2071.38 −0.000878520 0
$$355$$ 0 0
$$356$$ −3.32209e6 −1.38927
$$357$$ 1.54901e6 0.643257
$$358$$ 50020.5 0.0206272
$$359$$ 3.72854e6 1.52687 0.763436 0.645884i $$-0.223511\pi$$
0.763436 + 0.645884i $$0.223511\pi$$
$$360$$ 0 0
$$361$$ −1.80987e6 −0.730935
$$362$$ 25104.6 0.0100689
$$363$$ 131769. 0.0524864
$$364$$ 5.84477e6 2.31214
$$365$$ 0 0
$$366$$ −200899. −0.0783927
$$367$$ −141287. −0.0547566 −0.0273783 0.999625i $$-0.508716\pi$$
−0.0273783 + 0.999625i $$0.508716\pi$$
$$368$$ 3.14082e6 1.20899
$$369$$ −778309. −0.297568
$$370$$ 0 0
$$371$$ 2.65065e6 0.999809
$$372$$ −43616.4 −0.0163415
$$373$$ −826181. −0.307470 −0.153735 0.988112i $$-0.549130\pi$$
−0.153735 + 0.988112i $$0.549130\pi$$
$$374$$ 44994.5 0.0166334
$$375$$ 0 0
$$376$$ −334955. −0.122185
$$377$$ 6.52552e6 2.36462
$$378$$ 66711.1 0.0240142
$$379$$ −1.51195e6 −0.540678 −0.270339 0.962765i $$-0.587136\pi$$
−0.270339 + 0.962765i $$0.587136\pi$$
$$380$$ 0 0
$$381$$ 1.54613e6 0.545674
$$382$$ 158466. 0.0555620
$$383$$ −3.81853e6 −1.33015 −0.665074 0.746778i $$-0.731600\pi$$
−0.665074 + 0.746778i $$0.731600\pi$$
$$384$$ −516456. −0.178733
$$385$$ 0 0
$$386$$ 270197. 0.0923023
$$387$$ 745873. 0.253156
$$388$$ −1.25135e6 −0.421988
$$389$$ 3.87738e6 1.29916 0.649582 0.760292i $$-0.274944\pi$$
0.649582 + 0.760292i $$0.274944\pi$$
$$390$$ 0 0
$$391$$ 2.61342e6 0.864505
$$392$$ −724798. −0.238233
$$393$$ 1.26378e6 0.412754
$$394$$ −40579.1 −0.0131693
$$395$$ 0 0
$$396$$ −311694. −0.0998828
$$397$$ −4.70336e6 −1.49772 −0.748862 0.662725i $$-0.769400\pi$$
−0.748862 + 0.662725i $$0.769400\pi$$
$$398$$ 373570. 0.118213
$$399$$ 1.51186e6 0.475421
$$400$$ 0 0
$$401$$ −2.21418e6 −0.687625 −0.343812 0.939038i $$-0.611718\pi$$
−0.343812 + 0.939038i $$0.611718\pi$$
$$402$$ 16208.0 0.00500223
$$403$$ −136083. −0.0417388
$$404$$ 3.05980e6 0.932694
$$405$$ 0 0
$$406$$ −668701. −0.201334
$$407$$ 1.05686e6 0.316251
$$408$$ −213527. −0.0635041
$$409$$ −2.24429e6 −0.663394 −0.331697 0.943386i $$-0.607621\pi$$
−0.331697 + 0.943386i $$0.607621\pi$$
$$410$$ 0 0
$$411$$ 1.74986e6 0.510974
$$412$$ −1.72590e6 −0.500926
$$413$$ −106526. −0.0307313
$$414$$ 112552. 0.0322740
$$415$$ 0 0
$$416$$ −1.20977e6 −0.342745
$$417$$ −2.21757e6 −0.624505
$$418$$ 43915.2 0.0122935
$$419$$ −1.31098e6 −0.364806 −0.182403 0.983224i $$-0.558388\pi$$
−0.182403 + 0.983224i $$0.558388\pi$$
$$420$$ 0 0
$$421$$ 898192. 0.246981 0.123491 0.992346i $$-0.460591\pi$$
0.123491 + 0.992346i $$0.460591\pi$$
$$422$$ 420307. 0.114891
$$423$$ 956356. 0.259877
$$424$$ −365384. −0.0987040
$$425$$ 0 0
$$426$$ 65900.9 0.0175941
$$427$$ −1.03318e7 −2.74223
$$428$$ 1.46865e6 0.387533
$$429$$ −972483. −0.255117
$$430$$ 0 0
$$431$$ 6.21890e6 1.61258 0.806288 0.591523i $$-0.201473\pi$$
0.806288 + 0.591523i $$0.201473\pi$$
$$432$$ 732688. 0.188890
$$433$$ −1.00093e6 −0.256557 −0.128279 0.991738i $$-0.540945\pi$$
−0.128279 + 0.991738i $$0.540945\pi$$
$$434$$ 13945.0 0.00355382
$$435$$ 0 0
$$436$$ −5.15485e6 −1.29867
$$437$$ 2.55073e6 0.638942
$$438$$ −220626. −0.0549504
$$439$$ 4.86510e6 1.20484 0.602422 0.798178i $$-0.294203\pi$$
0.602422 + 0.798178i $$0.294203\pi$$
$$440$$ 0 0
$$441$$ 2.06943e6 0.506703
$$442$$ −332069. −0.0808485
$$443$$ 7.54601e6 1.82687 0.913435 0.406984i $$-0.133420\pi$$
0.913435 + 0.406984i $$0.133420\pi$$
$$444$$ −2.49996e6 −0.601832
$$445$$ 0 0
$$446$$ −422888. −0.100667
$$447$$ −2.96158e6 −0.701059
$$448$$ −6.49509e6 −1.52894
$$449$$ −2.70457e6 −0.633115 −0.316557 0.948573i $$-0.602527\pi$$
−0.316557 + 0.948573i $$0.602527\pi$$
$$450$$ 0 0
$$451$$ −1.16266e6 −0.269160
$$452$$ −8.04011e6 −1.85104
$$453$$ 1.13215e6 0.259214
$$454$$ 504313. 0.114831
$$455$$ 0 0
$$456$$ −208405. −0.0469349
$$457$$ −3.69700e6 −0.828055 −0.414027 0.910264i $$-0.635878\pi$$
−0.414027 + 0.910264i $$0.635878\pi$$
$$458$$ 701075. 0.156171
$$459$$ 609656. 0.135068
$$460$$ 0 0
$$461$$ 8.68496e6 1.90334 0.951669 0.307127i $$-0.0993676\pi$$
0.951669 + 0.307127i $$0.0993676\pi$$
$$462$$ 99654.9 0.0217217
$$463$$ −2.75398e6 −0.597047 −0.298523 0.954402i $$-0.596494\pi$$
−0.298523 + 0.954402i $$0.596494\pi$$
$$464$$ −7.34433e6 −1.58364
$$465$$ 0 0
$$466$$ 472596. 0.100815
$$467$$ −7.31776e6 −1.55269 −0.776347 0.630306i $$-0.782929\pi$$
−0.776347 + 0.630306i $$0.782929\pi$$
$$468$$ 2.30037e6 0.485493
$$469$$ 833537. 0.174982
$$470$$ 0 0
$$471$$ 193690. 0.0402304
$$472$$ 14684.3 0.00303388
$$473$$ 1.11421e6 0.228988
$$474$$ 100084. 0.0204607
$$475$$ 0 0
$$476$$ −5.47357e6 −1.10727
$$477$$ 1.04324e6 0.209936
$$478$$ −27610.2 −0.00552713
$$479$$ −230118. −0.0458260 −0.0229130 0.999737i $$-0.507294\pi$$
−0.0229130 + 0.999737i $$0.507294\pi$$
$$480$$ 0 0
$$481$$ −7.79985e6 −1.53718
$$482$$ 40257.0 0.00789267
$$483$$ 5.78827e6 1.12897
$$484$$ −465617. −0.0903474
$$485$$ 0 0
$$486$$ 26256.0 0.00504241
$$487$$ 8.18489e6 1.56383 0.781917 0.623383i $$-0.214242\pi$$
0.781917 + 0.623383i $$0.214242\pi$$
$$488$$ 1.42420e6 0.270721
$$489$$ 3.45405e6 0.653216
$$490$$ 0 0
$$491$$ −1.27094e6 −0.237915 −0.118957 0.992899i $$-0.537955\pi$$
−0.118957 + 0.992899i $$0.537955\pi$$
$$492$$ 2.75022e6 0.512219
$$493$$ −6.11109e6 −1.13240
$$494$$ −324103. −0.0597538
$$495$$ 0 0
$$496$$ 153158. 0.0279535
$$497$$ 3.38912e6 0.615455
$$498$$ −304730. −0.0550608
$$499$$ −9.67837e6 −1.74001 −0.870003 0.493046i $$-0.835883\pi$$
−0.870003 + 0.493046i $$0.835883\pi$$
$$500$$ 0 0
$$501$$ 4.60859e6 0.820303
$$502$$ −604198. −0.107009
$$503$$ −6.99048e6 −1.23193 −0.615966 0.787773i $$-0.711234\pi$$
−0.615966 + 0.787773i $$0.711234\pi$$
$$504$$ −472925. −0.0829308
$$505$$ 0 0
$$506$$ 168133. 0.0291929
$$507$$ 3.83549e6 0.662676
$$508$$ −5.46338e6 −0.939296
$$509$$ −6.03866e6 −1.03311 −0.516555 0.856254i $$-0.672786\pi$$
−0.516555 + 0.856254i $$0.672786\pi$$
$$510$$ 0 0
$$511$$ −1.13462e7 −1.92221
$$512$$ 2.27399e6 0.383367
$$513$$ 595032. 0.0998268
$$514$$ 288262. 0.0481261
$$515$$ 0 0
$$516$$ −2.63561e6 −0.435769
$$517$$ 1.42863e6 0.235068
$$518$$ 799287. 0.130882
$$519$$ −2.34088e6 −0.381470
$$520$$ 0 0
$$521$$ 6.75056e6 1.08955 0.544773 0.838584i $$-0.316616\pi$$
0.544773 + 0.838584i $$0.316616\pi$$
$$522$$ −263186. −0.0422752
$$523$$ −3.61437e6 −0.577801 −0.288900 0.957359i $$-0.593290\pi$$
−0.288900 + 0.957359i $$0.593290\pi$$
$$524$$ −4.46569e6 −0.710494
$$525$$ 0 0
$$526$$ −795557. −0.125374
$$527$$ 127440. 0.0199885
$$528$$ 1.09451e6 0.170858
$$529$$ 3.32937e6 0.517276
$$530$$ 0 0
$$531$$ −41926.3 −0.00645282
$$532$$ −5.34227e6 −0.818365
$$533$$ 8.58067e6 1.30829
$$534$$ 418033. 0.0634392
$$535$$ 0 0
$$536$$ −114901. −0.0172747
$$537$$ 1.01245e6 0.151509
$$538$$ 619516. 0.0922778
$$539$$ 3.09136e6 0.458330
$$540$$ 0 0
$$541$$ 7.17998e6 1.05470 0.527351 0.849647i $$-0.323185\pi$$
0.527351 + 0.849647i $$0.323185\pi$$
$$542$$ −308624. −0.0451265
$$543$$ 508136. 0.0739572
$$544$$ 1.13294e6 0.164139
$$545$$ 0 0
$$546$$ −735474. −0.105581
$$547$$ −5.53996e6 −0.791659 −0.395830 0.918324i $$-0.629543\pi$$
−0.395830 + 0.918324i $$0.629543\pi$$
$$548$$ −6.18329e6 −0.879565
$$549$$ −4.06635e6 −0.575803
$$550$$ 0 0
$$551$$ −5.96450e6 −0.836941
$$552$$ −797897. −0.111455
$$553$$ 5.14709e6 0.715730
$$554$$ 856915. 0.118621
$$555$$ 0 0
$$556$$ 7.83596e6 1.07499
$$557$$ 1.07664e7 1.47039 0.735197 0.677854i $$-0.237090\pi$$
0.735197 + 0.677854i $$0.237090\pi$$
$$558$$ 5488.45 0.000746216 0
$$559$$ −8.22307e6 −1.11302
$$560$$ 0 0
$$561$$ 910721. 0.122174
$$562$$ 210712. 0.0281416
$$563$$ 6.29087e6 0.836449 0.418224 0.908344i $$-0.362653\pi$$
0.418224 + 0.908344i $$0.362653\pi$$
$$564$$ −3.37937e6 −0.447340
$$565$$ 0 0
$$566$$ 252674. 0.0331528
$$567$$ 1.35028e6 0.176387
$$568$$ −467181. −0.0607595
$$569$$ −1.25297e7 −1.62241 −0.811205 0.584761i $$-0.801188\pi$$
−0.811205 + 0.584761i $$0.801188\pi$$
$$570$$ 0 0
$$571$$ −300370. −0.0385538 −0.0192769 0.999814i $$-0.506136\pi$$
−0.0192769 + 0.999814i $$0.506136\pi$$
$$572$$ 3.43635e6 0.439145
$$573$$ 3.20747e6 0.408109
$$574$$ −879302. −0.111393
$$575$$ 0 0
$$576$$ −2.55632e6 −0.321040
$$577$$ 4.91623e6 0.614742 0.307371 0.951590i $$-0.400551\pi$$
0.307371 + 0.951590i $$0.400551\pi$$
$$578$$ −320357. −0.0398855
$$579$$ 5.46899e6 0.677971
$$580$$ 0 0
$$581$$ −1.56715e7 −1.92607
$$582$$ 157463. 0.0192696
$$583$$ 1.55841e6 0.189894
$$584$$ 1.56405e6 0.189766
$$585$$ 0 0
$$586$$ −785830. −0.0945333
$$587$$ −1.27769e7 −1.53048 −0.765242 0.643742i $$-0.777381\pi$$
−0.765242 + 0.643742i $$0.777381\pi$$
$$588$$ −7.31250e6 −0.872213
$$589$$ 124383. 0.0147732
$$590$$ 0 0
$$591$$ −821351. −0.0967298
$$592$$ 8.77856e6 1.02948
$$593$$ 2.84036e6 0.331694 0.165847 0.986152i $$-0.446964\pi$$
0.165847 + 0.986152i $$0.446964\pi$$
$$594$$ 39221.9 0.00456103
$$595$$ 0 0
$$596$$ 1.04650e7 1.20677
$$597$$ 7.56134e6 0.868286
$$598$$ −1.24086e6 −0.141896
$$599$$ −1.61252e7 −1.83628 −0.918140 0.396257i $$-0.870309\pi$$
−0.918140 + 0.396257i $$0.870309\pi$$
$$600$$ 0 0
$$601$$ −6.84876e6 −0.773438 −0.386719 0.922198i $$-0.626392\pi$$
−0.386719 + 0.922198i $$0.626392\pi$$
$$602$$ 842657. 0.0947675
$$603$$ 328061. 0.0367419
$$604$$ −4.00055e6 −0.446197
$$605$$ 0 0
$$606$$ −385029. −0.0425904
$$607$$ −1.39670e7 −1.53862 −0.769312 0.638874i $$-0.779401\pi$$
−0.769312 + 0.638874i $$0.779401\pi$$
$$608$$ 1.10577e6 0.121312
$$609$$ −1.35350e7 −1.47882
$$610$$ 0 0
$$611$$ −1.05436e7 −1.14258
$$612$$ −2.15427e6 −0.232500
$$613$$ 7.54369e6 0.810835 0.405418 0.914132i $$-0.367126\pi$$
0.405418 + 0.914132i $$0.367126\pi$$
$$614$$ −1.18853e6 −0.127230
$$615$$ 0 0
$$616$$ −706468. −0.0750137
$$617$$ 3.93986e6 0.416647 0.208323 0.978060i $$-0.433199\pi$$
0.208323 + 0.978060i $$0.433199\pi$$
$$618$$ 217178. 0.0228742
$$619$$ −5.93266e6 −0.622333 −0.311166 0.950355i $$-0.600720\pi$$
−0.311166 + 0.950355i $$0.600720\pi$$
$$620$$ 0 0
$$621$$ 2.27813e6 0.237056
$$622$$ 994324. 0.103051
$$623$$ 2.14984e7 2.21915
$$624$$ −8.07771e6 −0.830475
$$625$$ 0 0
$$626$$ −691458. −0.0705229
$$627$$ 888875. 0.0902967
$$628$$ −684419. −0.0692505
$$629$$ 7.30448e6 0.736144
$$630$$ 0 0
$$631$$ −4.19698e6 −0.419627 −0.209813 0.977741i $$-0.567286\pi$$
−0.209813 + 0.977741i $$0.567286\pi$$
$$632$$ −709512. −0.0706589
$$633$$ 8.50732e6 0.843886
$$634$$ 732205. 0.0723452
$$635$$ 0 0
$$636$$ −3.68636e6 −0.361373
$$637$$ −2.28149e7 −2.22777
$$638$$ −393154. −0.0382394
$$639$$ 1.33388e6 0.129231
$$640$$ 0 0
$$641$$ 1.95566e7 1.87995 0.939977 0.341236i $$-0.110846\pi$$
0.939977 + 0.341236i $$0.110846\pi$$
$$642$$ −184807. −0.0176962
$$643$$ −1.89046e7 −1.80318 −0.901591 0.432590i $$-0.857600\pi$$
−0.901591 + 0.432590i $$0.857600\pi$$
$$644$$ −2.04534e7 −1.94335
$$645$$ 0 0
$$646$$ 303520. 0.0286158
$$647$$ 1.35066e7 1.26849 0.634244 0.773133i $$-0.281311\pi$$
0.634244 + 0.773133i $$0.281311\pi$$
$$648$$ −186133. −0.0174135
$$649$$ −62630.6 −0.00583680
$$650$$ 0 0
$$651$$ 282258. 0.0261032
$$652$$ −1.22052e7 −1.12441
$$653$$ −1.11108e7 −1.01968 −0.509840 0.860269i $$-0.670295\pi$$
−0.509840 + 0.860269i $$0.670295\pi$$
$$654$$ 648658. 0.0593023
$$655$$ 0 0
$$656$$ −9.65736e6 −0.876192
$$657$$ −4.46563e6 −0.403616
$$658$$ 1.08045e6 0.0972838
$$659$$ 1.35954e7 1.21949 0.609745 0.792597i $$-0.291272\pi$$
0.609745 + 0.792597i $$0.291272\pi$$
$$660$$ 0 0
$$661$$ −856154. −0.0762163 −0.0381082 0.999274i $$-0.512133\pi$$
−0.0381082 + 0.999274i $$0.512133\pi$$
$$662$$ 110746. 0.00982164
$$663$$ −6.72131e6 −0.593841
$$664$$ 2.16028e6 0.190147
$$665$$ 0 0
$$666$$ 314581. 0.0274819
$$667$$ −2.28356e7 −1.98746
$$668$$ −1.62849e7 −1.41203
$$669$$ −8.55956e6 −0.739411
$$670$$ 0 0
$$671$$ −6.07442e6 −0.520833
$$672$$ 2.50927e6 0.214350
$$673$$ 1.21181e7 1.03133 0.515665 0.856790i $$-0.327545\pi$$
0.515665 + 0.856790i $$0.327545\pi$$
$$674$$ 316759. 0.0268583
$$675$$ 0 0
$$676$$ −1.35530e7 −1.14070
$$677$$ −8.70607e6 −0.730046 −0.365023 0.930998i $$-0.618939\pi$$
−0.365023 + 0.930998i $$0.618939\pi$$
$$678$$ 1.01172e6 0.0845255
$$679$$ 8.09796e6 0.674064
$$680$$ 0 0
$$681$$ 1.02077e7 0.843449
$$682$$ 8198.80 0.000674977 0
$$683$$ 3.49312e6 0.286525 0.143262 0.989685i $$-0.454241\pi$$
0.143262 + 0.989685i $$0.454241\pi$$
$$684$$ −2.10260e6 −0.171837
$$685$$ 0 0
$$686$$ 799936. 0.0649000
$$687$$ 1.41903e7 1.14709
$$688$$ 9.25489e6 0.745419
$$689$$ −1.15014e7 −0.923003
$$690$$ 0 0
$$691$$ −2.98454e6 −0.237784 −0.118892 0.992907i $$-0.537934\pi$$
−0.118892 + 0.992907i $$0.537934\pi$$
$$692$$ 8.27170e6 0.656643
$$693$$ 2.01709e6 0.159548
$$694$$ −471160. −0.0371339
$$695$$ 0 0
$$696$$ 1.86576e6 0.145993
$$697$$ −8.03572e6 −0.626531
$$698$$ −1.31729e6 −0.102339
$$699$$ 9.56569e6 0.740497
$$700$$ 0 0
$$701$$ −3.32551e6 −0.255601 −0.127801 0.991800i $$-0.540792\pi$$
−0.127801 + 0.991800i $$0.540792\pi$$
$$702$$ −289466. −0.0221695
$$703$$ 7.12927e6 0.544072
$$704$$ −3.81870e6 −0.290392
$$705$$ 0 0
$$706$$ 1.09248e6 0.0824903
$$707$$ −1.98011e7 −1.48984
$$708$$ 148150. 0.0111076
$$709$$ 1.53451e7 1.14645 0.573225 0.819398i $$-0.305692\pi$$
0.573225 + 0.819398i $$0.305692\pi$$
$$710$$ 0 0
$$711$$ 2.02578e6 0.150286
$$712$$ −2.96350e6 −0.219081
$$713$$ 476212. 0.0350814
$$714$$ 688765. 0.0505621
$$715$$ 0 0
$$716$$ −3.57759e6 −0.260800
$$717$$ −558851. −0.0405974
$$718$$ 1.65789e6 0.120017
$$719$$ 2.09235e7 1.50943 0.754714 0.656054i $$-0.227776\pi$$
0.754714 + 0.656054i $$0.227776\pi$$
$$720$$ 0 0
$$721$$ 1.11690e7 0.800156
$$722$$ −804753. −0.0574539
$$723$$ 814831. 0.0579725
$$724$$ −1.79554e6 −0.127306
$$725$$ 0 0
$$726$$ 58590.8 0.00412561
$$727$$ −1.75630e7 −1.23243 −0.616217 0.787576i $$-0.711336\pi$$
−0.616217 + 0.787576i $$0.711336\pi$$
$$728$$ 5.21388e6 0.364614
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ 7.70083e6 0.533021
$$732$$ 1.43688e7 0.991158
$$733$$ 1.52765e7 1.05018 0.525090 0.851047i $$-0.324032\pi$$
0.525090 + 0.851047i $$0.324032\pi$$
$$734$$ −62822.8 −0.00430405
$$735$$ 0 0
$$736$$ 4.23353e6 0.288076
$$737$$ 490067. 0.0332343
$$738$$ −346074. −0.0233899
$$739$$ 2.22683e7 1.49995 0.749974 0.661467i $$-0.230066\pi$$
0.749974 + 0.661467i $$0.230066\pi$$
$$740$$ 0 0
$$741$$ −6.56009e6 −0.438898
$$742$$ 1.17860e6 0.0785884
$$743$$ −840214. −0.0558365 −0.0279182 0.999610i $$-0.508888\pi$$
−0.0279182 + 0.999610i $$0.508888\pi$$
$$744$$ −38908.4 −0.00257698
$$745$$ 0 0
$$746$$ −367360. −0.0241682
$$747$$ −6.16797e6 −0.404427
$$748$$ −3.21811e6 −0.210304
$$749$$ −9.50418e6 −0.619028
$$750$$ 0 0
$$751$$ −2.63571e7 −1.70529 −0.852644 0.522493i $$-0.825002\pi$$
−0.852644 + 0.522493i $$0.825002\pi$$
$$752$$ 1.18666e7 0.765211
$$753$$ −1.22294e7 −0.785992
$$754$$ 2.90156e6 0.185867
$$755$$ 0 0
$$756$$ −4.77134e6 −0.303624
$$757$$ −1.71589e7 −1.08830 −0.544150 0.838988i $$-0.683148\pi$$
−0.544150 + 0.838988i $$0.683148\pi$$
$$758$$ −672284. −0.0424991
$$759$$ 3.40314e6 0.214425
$$760$$ 0 0
$$761$$ −2.23433e7 −1.39857 −0.699287 0.714841i $$-0.746499\pi$$
−0.699287 + 0.714841i $$0.746499\pi$$
$$762$$ 687483. 0.0428918
$$763$$ 3.33589e7 2.07444
$$764$$ −1.13339e7 −0.702498
$$765$$ 0 0
$$766$$ −1.69790e6 −0.104554
$$767$$ 462227. 0.0283705
$$768$$ 8.85950e6 0.542008
$$769$$ −8.69614e6 −0.530287 −0.265143 0.964209i $$-0.585419\pi$$
−0.265143 + 0.964209i $$0.585419\pi$$
$$770$$ 0 0
$$771$$ 5.83464e6 0.353491
$$772$$ −1.93252e7 −1.16702
$$773$$ 5.79015e6 0.348531 0.174265 0.984699i $$-0.444245\pi$$
0.174265 + 0.984699i $$0.444245\pi$$
$$774$$ 331651. 0.0198989
$$775$$ 0 0
$$776$$ −1.11628e6 −0.0665455
$$777$$ 1.61782e7 0.961339
$$778$$ 1.72407e6 0.102119
$$779$$ −7.84296e6 −0.463059
$$780$$ 0 0
$$781$$ 1.99259e6 0.116894
$$782$$ 1.16205e6 0.0679530
$$783$$ −5.32707e6 −0.310516
$$784$$ 2.56777e7 1.49199
$$785$$ 0 0
$$786$$ 561938. 0.0324438
$$787$$ −3.16605e7 −1.82213 −0.911067 0.412258i $$-0.864740\pi$$
−0.911067 + 0.412258i $$0.864740\pi$$
$$788$$ 2.90232e6 0.166506
$$789$$ −1.61027e7 −0.920885
$$790$$ 0 0
$$791$$ 5.20305e7 2.95677
$$792$$ −278050. −0.0157511
$$793$$ 4.48305e7 2.53157
$$794$$ −2.09134e6 −0.117726
$$795$$ 0 0
$$796$$ −2.67186e7 −1.49462
$$797$$ −1.21137e7 −0.675511 −0.337755 0.941234i $$-0.609668\pi$$
−0.337755 + 0.941234i $$0.609668\pi$$
$$798$$ 672243. 0.0373697
$$799$$ 9.87397e6 0.547173
$$800$$ 0 0
$$801$$ 8.46131e6 0.465968
$$802$$ −984529. −0.0540496
$$803$$ −6.67087e6 −0.365085
$$804$$ −1.15923e6 −0.0632457
$$805$$ 0 0
$$806$$ −60508.9 −0.00328081
$$807$$ 1.25395e7 0.677790
$$808$$ 2.72952e6 0.147082
$$809$$ −1.66584e7 −0.894875 −0.447438 0.894315i $$-0.647663\pi$$
−0.447438 + 0.894315i $$0.647663\pi$$
$$810$$ 0 0
$$811$$ −2.63057e7 −1.40442 −0.702211 0.711969i $$-0.747804\pi$$
−0.702211 + 0.711969i $$0.747804\pi$$
$$812$$ 4.78271e7 2.54556
$$813$$ −6.24678e6 −0.331459
$$814$$ 469930. 0.0248584
$$815$$ 0 0
$$816$$ 7.56470e6 0.397710
$$817$$ 7.51610e6 0.393947
$$818$$ −997920. −0.0521450
$$819$$ −1.48865e7 −0.775504
$$820$$ 0 0
$$821$$ 2.92543e6 0.151472 0.0757358 0.997128i $$-0.475869\pi$$
0.0757358 + 0.997128i $$0.475869\pi$$
$$822$$ 778072. 0.0401643
$$823$$ −2.99240e7 −1.54000 −0.770000 0.638044i $$-0.779744\pi$$
−0.770000 + 0.638044i $$0.779744\pi$$
$$824$$ −1.53961e6 −0.0789937
$$825$$ 0 0
$$826$$ −47366.6 −0.00241558
$$827$$ 1.24304e6 0.0632008 0.0316004 0.999501i $$-0.489940\pi$$
0.0316004 + 0.999501i $$0.489940\pi$$
$$828$$ −8.04999e6 −0.408056
$$829$$ 7.42673e6 0.375328 0.187664 0.982233i $$-0.439908\pi$$
0.187664 + 0.982233i $$0.439908\pi$$
$$830$$ 0 0
$$831$$ 1.73446e7 0.871288
$$832$$ 2.81828e7 1.41148
$$833$$ 2.13660e7 1.06687
$$834$$ −986035. −0.0490882
$$835$$ 0 0
$$836$$ −3.14092e6 −0.155432
$$837$$ 111090. 0.00548103
$$838$$ −582926. −0.0286750
$$839$$ −9.30289e6 −0.456260 −0.228130 0.973631i $$-0.573261\pi$$
−0.228130 + 0.973631i $$0.573261\pi$$
$$840$$ 0 0
$$841$$ 3.28865e7 1.60335
$$842$$ 399379. 0.0194136
$$843$$ 4.26497e6 0.206703
$$844$$ −3.00614e7 −1.45262
$$845$$ 0 0
$$846$$ 425241. 0.0204272
$$847$$ 3.01318e6 0.144317
$$848$$ 1.29446e7 0.618157
$$849$$ 5.11431e6 0.243511
$$850$$ 0 0
$$851$$ 2.72950e7 1.29199
$$852$$ −4.71339e6 −0.222451
$$853$$ −3.06103e7 −1.44044 −0.720219 0.693746i $$-0.755959\pi$$
−0.720219 + 0.693746i $$0.755959\pi$$
$$854$$ −4.59399e6 −0.215549
$$855$$ 0 0
$$856$$ 1.31012e6 0.0611122
$$857$$ 8.04257e6 0.374061 0.187031 0.982354i $$-0.440114\pi$$
0.187031 + 0.982354i $$0.440114\pi$$
$$858$$ −432412. −0.0200530
$$859$$ 2.02293e7 0.935403 0.467702 0.883886i $$-0.345082\pi$$
0.467702 + 0.883886i $$0.345082\pi$$
$$860$$ 0 0
$$861$$ −1.77977e7 −0.818194
$$862$$ 2.76522e6 0.126754
$$863$$ 2.06034e7 0.941700 0.470850 0.882213i $$-0.343947\pi$$
0.470850 + 0.882213i $$0.343947\pi$$
$$864$$ 987593. 0.0450084
$$865$$ 0 0
$$866$$ −445062. −0.0201663
$$867$$ −6.48427e6 −0.292963
$$868$$ −997382. −0.0449327
$$869$$ 3.02616e6 0.135939
$$870$$ 0 0
$$871$$ −3.61680e6 −0.161540
$$872$$ −4.59843e6 −0.204795
$$873$$ 3.18717e6 0.141537
$$874$$ 1.13418e6 0.0502230
$$875$$ 0 0
$$876$$ 1.57797e7 0.694765
$$877$$ 2.65867e7 1.16726 0.583628 0.812021i $$-0.301633\pi$$
0.583628 + 0.812021i $$0.301633\pi$$
$$878$$ 2.16326e6 0.0947047
$$879$$ −1.59058e7 −0.694357
$$880$$ 0 0
$$881$$ 4.01120e7 1.74114 0.870572 0.492041i $$-0.163749\pi$$
0.870572 + 0.492041i $$0.163749\pi$$
$$882$$ 920166. 0.0398285
$$883$$ −3.61504e7 −1.56031 −0.780155 0.625586i $$-0.784860\pi$$
−0.780155 + 0.625586i $$0.784860\pi$$
$$884$$ 2.37504e7 1.02221
$$885$$ 0 0
$$886$$ 3.35531e6 0.143598
$$887$$ −2.25484e7 −0.962290 −0.481145 0.876641i $$-0.659779\pi$$
−0.481145 + 0.876641i $$0.659779\pi$$
$$888$$ −2.23011e6 −0.0949062
$$889$$ 3.53556e7 1.50039
$$890$$ 0 0
$$891$$ 793881. 0.0335013
$$892$$ 3.02459e7 1.27278
$$893$$ 9.63712e6 0.404407
$$894$$ −1.31686e6 −0.0551056
$$895$$ 0 0
$$896$$ −1.18099e7 −0.491446
$$897$$ −2.51159e7 −1.04224
$$898$$ −1.20258e6 −0.0497650
$$899$$ −1.11355e6 −0.0459526
$$900$$ 0 0
$$901$$ 1.07710e7 0.442021
$$902$$ −516974. −0.0211569
$$903$$ 1.70560e7 0.696078
$$904$$ −7.17226e6 −0.291900
$$905$$ 0 0
$$906$$ 503407. 0.0203751
$$907$$ 1.61295e7 0.651032 0.325516 0.945537i $$-0.394462\pi$$
0.325516 + 0.945537i $$0.394462\pi$$
$$908$$ −3.60697e7 −1.45187
$$909$$ −7.79326e6 −0.312831
$$910$$ 0 0
$$911$$ −1.72749e7 −0.689637 −0.344819 0.938669i $$-0.612060\pi$$
−0.344819 + 0.938669i $$0.612060\pi$$
$$912$$ 7.38324e6 0.293941
$$913$$ −9.21388e6 −0.365818
$$914$$ −1.64386e6 −0.0650879
$$915$$ 0 0
$$916$$ −5.01426e7 −1.97455
$$917$$ 2.88991e7 1.13491
$$918$$ 271082. 0.0106168
$$919$$ 966289. 0.0377414 0.0188707 0.999822i $$-0.493993\pi$$
0.0188707 + 0.999822i $$0.493993\pi$$
$$920$$ 0 0
$$921$$ −2.40567e7 −0.934517
$$922$$ 3.86175e6 0.149609
$$923$$ −1.47057e7 −0.568176
$$924$$ −7.12756e6 −0.274638
$$925$$ 0 0
$$926$$ −1.22455e6 −0.0469299
$$927$$ 4.39585e6 0.168013
$$928$$ −9.89946e6 −0.377348
$$929$$ −1.54446e7 −0.587133 −0.293567 0.955939i $$-0.594842\pi$$
−0.293567 + 0.955939i $$0.594842\pi$$
$$930$$ 0 0
$$931$$ 2.08534e7 0.788503
$$932$$ −3.38012e7 −1.27465
$$933$$ 2.01259e7 0.756920
$$934$$ −3.25382e6 −0.122047
$$935$$ 0 0
$$936$$ 2.05207e6 0.0765600
$$937$$ −3.01699e7 −1.12260 −0.561300 0.827612i $$-0.689699\pi$$
−0.561300 + 0.827612i $$0.689699\pi$$
$$938$$ 370630. 0.0137542
$$939$$ −1.39956e7 −0.517998
$$940$$ 0 0
$$941$$ −1.74463e7 −0.642286 −0.321143 0.947031i $$-0.604067\pi$$
−0.321143 + 0.947031i $$0.604067\pi$$
$$942$$ 86123.6 0.00316224
$$943$$ −3.00275e7 −1.09961
$$944$$ −520227. −0.0190004
$$945$$ 0 0
$$946$$ 495429. 0.0179992
$$947$$ 3.47258e7 1.25828 0.629140 0.777292i $$-0.283407\pi$$
0.629140 + 0.777292i $$0.283407\pi$$
$$948$$ −7.15827e6 −0.258694
$$949$$ 4.92324e7 1.77454
$$950$$ 0 0
$$951$$ 1.48204e7 0.531383
$$952$$ −4.88275e6 −0.174611
$$953$$ −5.30473e7 −1.89204 −0.946021 0.324104i $$-0.894937\pi$$
−0.946021 + 0.324104i $$0.894937\pi$$
$$954$$ 463872. 0.0165017
$$955$$ 0 0
$$956$$ 1.97475e6 0.0698823
$$957$$ −7.95772e6 −0.280872
$$958$$ −102322. −0.00360208
$$959$$ 4.00144e7 1.40498
$$960$$ 0 0
$$961$$ −2.86059e7 −0.999189
$$962$$ −3.46818e6 −0.120827
$$963$$ −3.74063e6 −0.129981
$$964$$ −2.87928e6 −0.0997909
$$965$$ 0 0
$$966$$ 2.57374e6 0.0887406
$$967$$ 8.74388e6 0.300703 0.150352 0.988633i $$-0.451959\pi$$
0.150352 + 0.988633i $$0.451959\pi$$
$$968$$ −415359. −0.0142474
$$969$$ 6.14346e6 0.210186
$$970$$ 0 0
$$971$$ 2.90606e7 0.989136 0.494568 0.869139i $$-0.335326\pi$$
0.494568 + 0.869139i $$0.335326\pi$$
$$972$$ −1.87789e6 −0.0637537
$$973$$ −5.07094e7 −1.71714
$$974$$ 3.63939e6 0.122923
$$975$$ 0 0
$$976$$ −5.04558e7 −1.69546
$$977$$ 4.53644e6 0.152047 0.0760236 0.997106i $$-0.475778\pi$$
0.0760236 + 0.997106i $$0.475778\pi$$
$$978$$ 1.53584e6 0.0513450
$$979$$ 1.26397e7 0.421484
$$980$$ 0 0
$$981$$ 1.31293e7 0.435582
$$982$$ −565121. −0.0187009
$$983$$ 2.56053e7 0.845174 0.422587 0.906322i $$-0.361122\pi$$
0.422587 + 0.906322i $$0.361122\pi$$
$$984$$ 2.45336e6 0.0807745
$$985$$ 0 0
$$986$$ −2.71728e6 −0.0890107
$$987$$ 2.18691e7 0.714560
$$988$$ 2.31806e7 0.755498
$$989$$ 2.87761e7 0.935494
$$990$$ 0 0
$$991$$ −7.43675e6 −0.240547 −0.120273 0.992741i $$-0.538377\pi$$
−0.120273 + 0.992741i $$0.538377\pi$$
$$992$$ 206442. 0.00666070
$$993$$ 2.24159e6 0.0721410
$$994$$ 1.50697e6 0.0483769
$$995$$ 0 0
$$996$$ 2.17951e7 0.696161
$$997$$ −2.77690e7 −0.884752 −0.442376 0.896830i $$-0.645864\pi$$
−0.442376 + 0.896830i $$0.645864\pi$$
$$998$$ −4.30347e6 −0.136770
$$999$$ 6.36736e6 0.201858
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 825.6.a.t.1.6 10
5.4 even 2 825.6.a.u.1.5 yes 10

By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.6 10 1.1 even 1 trivial
825.6.a.u.1.5 yes 10 5.4 even 2