Properties

Label 825.6.a.q.1.1
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-10.0327\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.03274 q^{2} -9.00000 q^{3} +49.5905 q^{4} +81.2947 q^{6} -36.7509 q^{7} -158.890 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-9.03274 q^{2} -9.00000 q^{3} +49.5905 q^{4} +81.2947 q^{6} -36.7509 q^{7} -158.890 q^{8} +81.0000 q^{9} -121.000 q^{11} -446.314 q^{12} +982.172 q^{13} +331.961 q^{14} -151.680 q^{16} +870.751 q^{17} -731.652 q^{18} -2036.24 q^{19} +330.758 q^{21} +1092.96 q^{22} -132.236 q^{23} +1430.01 q^{24} -8871.71 q^{26} -729.000 q^{27} -1822.49 q^{28} +2731.80 q^{29} -1403.06 q^{31} +6454.57 q^{32} +1089.00 q^{33} -7865.28 q^{34} +4016.83 q^{36} -693.003 q^{37} +18392.8 q^{38} -8839.55 q^{39} -8864.32 q^{41} -2987.65 q^{42} +22753.8 q^{43} -6000.45 q^{44} +1194.45 q^{46} -22593.6 q^{47} +1365.12 q^{48} -15456.4 q^{49} -7836.76 q^{51} +48706.4 q^{52} -6724.75 q^{53} +6584.87 q^{54} +5839.36 q^{56} +18326.1 q^{57} -24675.7 q^{58} +4319.48 q^{59} -45516.1 q^{61} +12673.5 q^{62} -2976.82 q^{63} -53448.8 q^{64} -9836.66 q^{66} -50714.9 q^{67} +43181.0 q^{68} +1190.12 q^{69} +50081.0 q^{71} -12870.1 q^{72} -30069.6 q^{73} +6259.72 q^{74} -100978. q^{76} +4446.86 q^{77} +79845.4 q^{78} +2221.45 q^{79} +6561.00 q^{81} +80069.2 q^{82} +26617.8 q^{83} +16402.4 q^{84} -205529. q^{86} -24586.2 q^{87} +19225.7 q^{88} +147093. q^{89} -36095.7 q^{91} -6557.65 q^{92} +12627.5 q^{93} +204082. q^{94} -58091.2 q^{96} +62580.9 q^{97} +139613. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22} + 5972 q^{23} - 1863 q^{24} - 14579 q^{26} - 5832 q^{27} + 5942 q^{28} + 1032 q^{29} - 4682 q^{31} + 3863 q^{32} + 8712 q^{33} + 2206 q^{34} + 8667 q^{36} - 17200 q^{37} + 11011 q^{38} - 3438 q^{39} - 13220 q^{41} - 9432 q^{42} + 22872 q^{43} - 12947 q^{44} + 9101 q^{46} + 6700 q^{47} + 2925 q^{48} - 43466 q^{49} - 2592 q^{51} + 5009 q^{52} + 6224 q^{53} - 6561 q^{54} + 20992 q^{56} + 8892 q^{57} + 33015 q^{58} - 77556 q^{59} + 11554 q^{61} + 12135 q^{62} - 5346 q^{63} - 149917 q^{64} + 9801 q^{66} + 20894 q^{67} + 91776 q^{68} - 53748 q^{69} - 21648 q^{71} + 16767 q^{72} + 64660 q^{73} - 179522 q^{74} + 24401 q^{76} + 7986 q^{77} + 131211 q^{78} - 22660 q^{79} + 52488 q^{81} - 56080 q^{82} + 100390 q^{83} - 53478 q^{84} + 47271 q^{86} - 9288 q^{87} - 25047 q^{88} - 25578 q^{89} + 73250 q^{91} + 95311 q^{92} + 42138 q^{93} - 170120 q^{94} - 34767 q^{96} + 142828 q^{97} + 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.03274 −1.59678 −0.798389 0.602142i \(-0.794314\pi\)
−0.798389 + 0.602142i \(0.794314\pi\)
\(3\) −9.00000 −0.577350
\(4\) 49.5905 1.54970
\(5\) 0 0
\(6\) 81.2947 0.921901
\(7\) −36.7509 −0.283480 −0.141740 0.989904i \(-0.545270\pi\)
−0.141740 + 0.989904i \(0.545270\pi\)
\(8\) −158.890 −0.877753
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −446.314 −0.894721
\(13\) 982.172 1.61187 0.805934 0.592006i \(-0.201664\pi\)
0.805934 + 0.592006i \(0.201664\pi\)
\(14\) 331.961 0.452655
\(15\) 0 0
\(16\) −151.680 −0.148125
\(17\) 870.751 0.730755 0.365378 0.930859i \(-0.380940\pi\)
0.365378 + 0.930859i \(0.380940\pi\)
\(18\) −731.652 −0.532260
\(19\) −2036.24 −1.29403 −0.647015 0.762477i \(-0.723983\pi\)
−0.647015 + 0.762477i \(0.723983\pi\)
\(20\) 0 0
\(21\) 330.758 0.163667
\(22\) 1092.96 0.481447
\(23\) −132.236 −0.0521231 −0.0260615 0.999660i \(-0.508297\pi\)
−0.0260615 + 0.999660i \(0.508297\pi\)
\(24\) 1430.01 0.506771
\(25\) 0 0
\(26\) −8871.71 −2.57380
\(27\) −729.000 −0.192450
\(28\) −1822.49 −0.439310
\(29\) 2731.80 0.603190 0.301595 0.953436i \(-0.402481\pi\)
0.301595 + 0.953436i \(0.402481\pi\)
\(30\) 0 0
\(31\) −1403.06 −0.262224 −0.131112 0.991368i \(-0.541855\pi\)
−0.131112 + 0.991368i \(0.541855\pi\)
\(32\) 6454.57 1.11428
\(33\) 1089.00 0.174078
\(34\) −7865.28 −1.16685
\(35\) 0 0
\(36\) 4016.83 0.516567
\(37\) −693.003 −0.0832206 −0.0416103 0.999134i \(-0.513249\pi\)
−0.0416103 + 0.999134i \(0.513249\pi\)
\(38\) 18392.8 2.06628
\(39\) −8839.55 −0.930612
\(40\) 0 0
\(41\) −8864.32 −0.823542 −0.411771 0.911287i \(-0.635090\pi\)
−0.411771 + 0.911287i \(0.635090\pi\)
\(42\) −2987.65 −0.261341
\(43\) 22753.8 1.87665 0.938325 0.345755i \(-0.112377\pi\)
0.938325 + 0.345755i \(0.112377\pi\)
\(44\) −6000.45 −0.467253
\(45\) 0 0
\(46\) 1194.45 0.0832290
\(47\) −22593.6 −1.49190 −0.745951 0.666001i \(-0.768005\pi\)
−0.745951 + 0.666001i \(0.768005\pi\)
\(48\) 1365.12 0.0855199
\(49\) −15456.4 −0.919639
\(50\) 0 0
\(51\) −7836.76 −0.421902
\(52\) 48706.4 2.49791
\(53\) −6724.75 −0.328842 −0.164421 0.986390i \(-0.552575\pi\)
−0.164421 + 0.986390i \(0.552575\pi\)
\(54\) 6584.87 0.307300
\(55\) 0 0
\(56\) 5839.36 0.248826
\(57\) 18326.1 0.747109
\(58\) −24675.7 −0.963161
\(59\) 4319.48 0.161548 0.0807741 0.996732i \(-0.474261\pi\)
0.0807741 + 0.996732i \(0.474261\pi\)
\(60\) 0 0
\(61\) −45516.1 −1.56618 −0.783088 0.621911i \(-0.786357\pi\)
−0.783088 + 0.621911i \(0.786357\pi\)
\(62\) 12673.5 0.418713
\(63\) −2976.82 −0.0944934
\(64\) −53448.8 −1.63113
\(65\) 0 0
\(66\) −9836.66 −0.277964
\(67\) −50714.9 −1.38022 −0.690111 0.723704i \(-0.742438\pi\)
−0.690111 + 0.723704i \(0.742438\pi\)
\(68\) 43181.0 1.13245
\(69\) 1190.12 0.0300933
\(70\) 0 0
\(71\) 50081.0 1.17904 0.589519 0.807755i \(-0.299318\pi\)
0.589519 + 0.807755i \(0.299318\pi\)
\(72\) −12870.1 −0.292584
\(73\) −30069.6 −0.660420 −0.330210 0.943907i \(-0.607120\pi\)
−0.330210 + 0.943907i \(0.607120\pi\)
\(74\) 6259.72 0.132885
\(75\) 0 0
\(76\) −100978. −2.00536
\(77\) 4446.86 0.0854725
\(78\) 79845.4 1.48598
\(79\) 2221.45 0.0400469 0.0200234 0.999800i \(-0.493626\pi\)
0.0200234 + 0.999800i \(0.493626\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 80069.2 1.31502
\(83\) 26617.8 0.424108 0.212054 0.977258i \(-0.431985\pi\)
0.212054 + 0.977258i \(0.431985\pi\)
\(84\) 16402.4 0.253636
\(85\) 0 0
\(86\) −205529. −2.99659
\(87\) −24586.2 −0.348252
\(88\) 19225.7 0.264653
\(89\) 147093. 1.96841 0.984205 0.177031i \(-0.0566493\pi\)
0.984205 + 0.177031i \(0.0566493\pi\)
\(90\) 0 0
\(91\) −36095.7 −0.456932
\(92\) −6557.65 −0.0807753
\(93\) 12627.5 0.151395
\(94\) 204082. 2.38224
\(95\) 0 0
\(96\) −58091.2 −0.643327
\(97\) 62580.9 0.675325 0.337662 0.941267i \(-0.390364\pi\)
0.337662 + 0.941267i \(0.390364\pi\)
\(98\) 139613. 1.46846
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 29422.4 0.286996 0.143498 0.989651i \(-0.454165\pi\)
0.143498 + 0.989651i \(0.454165\pi\)
\(102\) 70787.5 0.673684
\(103\) −39833.8 −0.369963 −0.184982 0.982742i \(-0.559223\pi\)
−0.184982 + 0.982742i \(0.559223\pi\)
\(104\) −156058. −1.41482
\(105\) 0 0
\(106\) 60743.0 0.525087
\(107\) 86367.2 0.729272 0.364636 0.931150i \(-0.381193\pi\)
0.364636 + 0.931150i \(0.381193\pi\)
\(108\) −36151.5 −0.298240
\(109\) 65554.0 0.528485 0.264243 0.964456i \(-0.414878\pi\)
0.264243 + 0.964456i \(0.414878\pi\)
\(110\) 0 0
\(111\) 6237.03 0.0480474
\(112\) 5574.36 0.0419904
\(113\) −42275.3 −0.311451 −0.155726 0.987800i \(-0.549772\pi\)
−0.155726 + 0.987800i \(0.549772\pi\)
\(114\) −165535. −1.19297
\(115\) 0 0
\(116\) 135471. 0.934765
\(117\) 79556.0 0.537289
\(118\) −39016.8 −0.257957
\(119\) −32000.9 −0.207155
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 411136. 2.50084
\(123\) 79778.9 0.475472
\(124\) −69578.4 −0.406369
\(125\) 0 0
\(126\) 26888.9 0.150885
\(127\) 229543. 1.26286 0.631428 0.775434i \(-0.282469\pi\)
0.631428 + 0.775434i \(0.282469\pi\)
\(128\) 276243. 1.49027
\(129\) −204784. −1.08348
\(130\) 0 0
\(131\) 128823. 0.655868 0.327934 0.944701i \(-0.393648\pi\)
0.327934 + 0.944701i \(0.393648\pi\)
\(132\) 54004.0 0.269769
\(133\) 74833.5 0.366832
\(134\) 458095. 2.20391
\(135\) 0 0
\(136\) −138354. −0.641423
\(137\) 393234. 1.78999 0.894993 0.446080i \(-0.147180\pi\)
0.894993 + 0.446080i \(0.147180\pi\)
\(138\) −10750.1 −0.0480523
\(139\) 52591.4 0.230875 0.115438 0.993315i \(-0.463173\pi\)
0.115438 + 0.993315i \(0.463173\pi\)
\(140\) 0 0
\(141\) 203342. 0.861350
\(142\) −452369. −1.88266
\(143\) −118843. −0.485996
\(144\) −12286.1 −0.0493749
\(145\) 0 0
\(146\) 271611. 1.05454
\(147\) 139107. 0.530954
\(148\) −34366.4 −0.128967
\(149\) −71304.8 −0.263120 −0.131560 0.991308i \(-0.541999\pi\)
−0.131560 + 0.991308i \(0.541999\pi\)
\(150\) 0 0
\(151\) −438898. −1.56647 −0.783234 0.621727i \(-0.786431\pi\)
−0.783234 + 0.621727i \(0.786431\pi\)
\(152\) 323539. 1.13584
\(153\) 70530.9 0.243585
\(154\) −40167.3 −0.136481
\(155\) 0 0
\(156\) −438358. −1.44217
\(157\) 219116. 0.709454 0.354727 0.934970i \(-0.384574\pi\)
0.354727 + 0.934970i \(0.384574\pi\)
\(158\) −20065.8 −0.0639460
\(159\) 60522.8 0.189857
\(160\) 0 0
\(161\) 4859.79 0.0147759
\(162\) −59263.8 −0.177420
\(163\) −359538. −1.05993 −0.529963 0.848021i \(-0.677794\pi\)
−0.529963 + 0.848021i \(0.677794\pi\)
\(164\) −439586. −1.27625
\(165\) 0 0
\(166\) −240431. −0.677206
\(167\) 119110. 0.330489 0.165245 0.986253i \(-0.447159\pi\)
0.165245 + 0.986253i \(0.447159\pi\)
\(168\) −52554.2 −0.143660
\(169\) 593370. 1.59812
\(170\) 0 0
\(171\) −164935. −0.431344
\(172\) 1.12837e6 2.90825
\(173\) −394947. −1.00328 −0.501642 0.865076i \(-0.667270\pi\)
−0.501642 + 0.865076i \(0.667270\pi\)
\(174\) 222081. 0.556081
\(175\) 0 0
\(176\) 18353.3 0.0446613
\(177\) −38875.4 −0.0932699
\(178\) −1.32865e6 −3.14312
\(179\) −410121. −0.956707 −0.478354 0.878167i \(-0.658766\pi\)
−0.478354 + 0.878167i \(0.658766\pi\)
\(180\) 0 0
\(181\) 450092. 1.02119 0.510593 0.859822i \(-0.329426\pi\)
0.510593 + 0.859822i \(0.329426\pi\)
\(182\) 326043. 0.729620
\(183\) 409645. 0.904232
\(184\) 21011.0 0.0457512
\(185\) 0 0
\(186\) −114061. −0.241744
\(187\) −105361. −0.220331
\(188\) −1.12043e6 −2.31200
\(189\) 26791.4 0.0545558
\(190\) 0 0
\(191\) −936179. −1.85684 −0.928422 0.371526i \(-0.878835\pi\)
−0.928422 + 0.371526i \(0.878835\pi\)
\(192\) 481039. 0.941732
\(193\) 448417. 0.866540 0.433270 0.901264i \(-0.357360\pi\)
0.433270 + 0.901264i \(0.357360\pi\)
\(194\) −565278. −1.07834
\(195\) 0 0
\(196\) −766489. −1.42517
\(197\) 554289. 1.01759 0.508793 0.860889i \(-0.330092\pi\)
0.508793 + 0.860889i \(0.330092\pi\)
\(198\) 88529.9 0.160482
\(199\) 228423. 0.408891 0.204445 0.978878i \(-0.434461\pi\)
0.204445 + 0.978878i \(0.434461\pi\)
\(200\) 0 0
\(201\) 456435. 0.796872
\(202\) −265765. −0.458269
\(203\) −100396. −0.170992
\(204\) −388629. −0.653822
\(205\) 0 0
\(206\) 359808. 0.590749
\(207\) −10711.1 −0.0173744
\(208\) −148976. −0.238757
\(209\) 246385. 0.390165
\(210\) 0 0
\(211\) −735489. −1.13729 −0.568643 0.822584i \(-0.692531\pi\)
−0.568643 + 0.822584i \(0.692531\pi\)
\(212\) −333484. −0.509607
\(213\) −450729. −0.680717
\(214\) −780133. −1.16449
\(215\) 0 0
\(216\) 115831. 0.168924
\(217\) 51563.7 0.0743352
\(218\) −592132. −0.843874
\(219\) 270626. 0.381294
\(220\) 0 0
\(221\) 855228. 1.17788
\(222\) −56337.5 −0.0767211
\(223\) −978404. −1.31752 −0.658758 0.752355i \(-0.728918\pi\)
−0.658758 + 0.752355i \(0.728918\pi\)
\(224\) −237211. −0.315875
\(225\) 0 0
\(226\) 381862. 0.497319
\(227\) 79238.8 0.102064 0.0510321 0.998697i \(-0.483749\pi\)
0.0510321 + 0.998697i \(0.483749\pi\)
\(228\) 908802. 1.15780
\(229\) 530652. 0.668684 0.334342 0.942452i \(-0.391486\pi\)
0.334342 + 0.942452i \(0.391486\pi\)
\(230\) 0 0
\(231\) −40021.7 −0.0493476
\(232\) −434057. −0.529452
\(233\) 795568. 0.960035 0.480018 0.877259i \(-0.340630\pi\)
0.480018 + 0.877259i \(0.340630\pi\)
\(234\) −718609. −0.857932
\(235\) 0 0
\(236\) 214205. 0.250352
\(237\) −19993.0 −0.0231211
\(238\) 289056. 0.330780
\(239\) 838899. 0.949981 0.474991 0.879991i \(-0.342451\pi\)
0.474991 + 0.879991i \(0.342451\pi\)
\(240\) 0 0
\(241\) −1.57587e6 −1.74775 −0.873873 0.486154i \(-0.838399\pi\)
−0.873873 + 0.486154i \(0.838399\pi\)
\(242\) −132248. −0.145162
\(243\) −59049.0 −0.0641500
\(244\) −2.25717e6 −2.42711
\(245\) 0 0
\(246\) −720623. −0.759224
\(247\) −1.99994e6 −2.08581
\(248\) 222933. 0.230168
\(249\) −239560. −0.244859
\(250\) 0 0
\(251\) 1.47001e6 1.47278 0.736388 0.676560i \(-0.236530\pi\)
0.736388 + 0.676560i \(0.236530\pi\)
\(252\) −147622. −0.146437
\(253\) 16000.6 0.0157157
\(254\) −2.07340e6 −2.01650
\(255\) 0 0
\(256\) −784869. −0.748510
\(257\) 935502. 0.883511 0.441755 0.897136i \(-0.354356\pi\)
0.441755 + 0.897136i \(0.354356\pi\)
\(258\) 1.84977e6 1.73008
\(259\) 25468.5 0.0235914
\(260\) 0 0
\(261\) 221276. 0.201063
\(262\) −1.16363e6 −1.04728
\(263\) −958944. −0.854877 −0.427439 0.904044i \(-0.640584\pi\)
−0.427439 + 0.904044i \(0.640584\pi\)
\(264\) −173032. −0.152797
\(265\) 0 0
\(266\) −675952. −0.585750
\(267\) −1.32383e6 −1.13646
\(268\) −2.51498e6 −2.13893
\(269\) −1.60572e6 −1.35297 −0.676486 0.736456i \(-0.736498\pi\)
−0.676486 + 0.736456i \(0.736498\pi\)
\(270\) 0 0
\(271\) −1.41235e6 −1.16820 −0.584101 0.811681i \(-0.698553\pi\)
−0.584101 + 0.811681i \(0.698553\pi\)
\(272\) −132075. −0.108243
\(273\) 324861. 0.263810
\(274\) −3.55198e6 −2.85821
\(275\) 0 0
\(276\) 59018.8 0.0466356
\(277\) −1.14183e6 −0.894136 −0.447068 0.894500i \(-0.647532\pi\)
−0.447068 + 0.894500i \(0.647532\pi\)
\(278\) −475044. −0.368657
\(279\) −113648. −0.0874079
\(280\) 0 0
\(281\) −403724. −0.305013 −0.152507 0.988302i \(-0.548735\pi\)
−0.152507 + 0.988302i \(0.548735\pi\)
\(282\) −1.83674e6 −1.37539
\(283\) −890415. −0.660886 −0.330443 0.943826i \(-0.607198\pi\)
−0.330443 + 0.943826i \(0.607198\pi\)
\(284\) 2.48354e6 1.82716
\(285\) 0 0
\(286\) 1.07348e6 0.776029
\(287\) 325772. 0.233458
\(288\) 522821. 0.371425
\(289\) −661649. −0.465997
\(290\) 0 0
\(291\) −563229. −0.389899
\(292\) −1.49117e6 −1.02345
\(293\) 262945. 0.178935 0.0894676 0.995990i \(-0.471483\pi\)
0.0894676 + 0.995990i \(0.471483\pi\)
\(294\) −1.25652e6 −0.847816
\(295\) 0 0
\(296\) 110111. 0.0730472
\(297\) 88209.0 0.0580259
\(298\) 644078. 0.420144
\(299\) −129879. −0.0840155
\(300\) 0 0
\(301\) −836223. −0.531993
\(302\) 3.96446e6 2.50130
\(303\) −264802. −0.165697
\(304\) 308856. 0.191678
\(305\) 0 0
\(306\) −637087. −0.388951
\(307\) −237821. −0.144014 −0.0720070 0.997404i \(-0.522940\pi\)
−0.0720070 + 0.997404i \(0.522940\pi\)
\(308\) 220522. 0.132457
\(309\) 358504. 0.213598
\(310\) 0 0
\(311\) 2.10387e6 1.23344 0.616721 0.787182i \(-0.288461\pi\)
0.616721 + 0.787182i \(0.288461\pi\)
\(312\) 1.40452e6 0.816848
\(313\) 878264. 0.506716 0.253358 0.967373i \(-0.418465\pi\)
0.253358 + 0.967373i \(0.418465\pi\)
\(314\) −1.97922e6 −1.13284
\(315\) 0 0
\(316\) 110163. 0.0620607
\(317\) −1.08602e6 −0.607000 −0.303500 0.952831i \(-0.598155\pi\)
−0.303500 + 0.952831i \(0.598155\pi\)
\(318\) −546687. −0.303159
\(319\) −330548. −0.181869
\(320\) 0 0
\(321\) −777305. −0.421045
\(322\) −43897.2 −0.0235938
\(323\) −1.77306e6 −0.945620
\(324\) 325363. 0.172189
\(325\) 0 0
\(326\) 3.24761e6 1.69247
\(327\) −589986. −0.305121
\(328\) 1.40846e6 0.722867
\(329\) 830334. 0.422925
\(330\) 0 0
\(331\) −2.62891e6 −1.31888 −0.659441 0.751757i \(-0.729207\pi\)
−0.659441 + 0.751757i \(0.729207\pi\)
\(332\) 1.31999e6 0.657241
\(333\) −56133.3 −0.0277402
\(334\) −1.07589e6 −0.527718
\(335\) 0 0
\(336\) −50169.3 −0.0242432
\(337\) −1.62644e6 −0.780125 −0.390062 0.920788i \(-0.627547\pi\)
−0.390062 + 0.920788i \(0.627547\pi\)
\(338\) −5.35976e6 −2.55184
\(339\) 380477. 0.179816
\(340\) 0 0
\(341\) 169770. 0.0790634
\(342\) 1.48982e6 0.688760
\(343\) 1.18571e6 0.544180
\(344\) −3.61536e6 −1.64724
\(345\) 0 0
\(346\) 3.56746e6 1.60202
\(347\) −1.10198e6 −0.491304 −0.245652 0.969358i \(-0.579002\pi\)
−0.245652 + 0.969358i \(0.579002\pi\)
\(348\) −1.21924e6 −0.539687
\(349\) −437848. −0.192424 −0.0962120 0.995361i \(-0.530673\pi\)
−0.0962120 + 0.995361i \(0.530673\pi\)
\(350\) 0 0
\(351\) −716004. −0.310204
\(352\) −781003. −0.335967
\(353\) 498788. 0.213049 0.106524 0.994310i \(-0.466028\pi\)
0.106524 + 0.994310i \(0.466028\pi\)
\(354\) 351151. 0.148931
\(355\) 0 0
\(356\) 7.29439e6 3.05045
\(357\) 288008. 0.119601
\(358\) 3.70451e6 1.52765
\(359\) −509877. −0.208799 −0.104400 0.994535i \(-0.533292\pi\)
−0.104400 + 0.994535i \(0.533292\pi\)
\(360\) 0 0
\(361\) 1.67017e6 0.674516
\(362\) −4.06557e6 −1.63061
\(363\) −131769. −0.0524864
\(364\) −1.79000e6 −0.708109
\(365\) 0 0
\(366\) −3.70022e6 −1.44386
\(367\) −3.66059e6 −1.41869 −0.709343 0.704864i \(-0.751008\pi\)
−0.709343 + 0.704864i \(0.751008\pi\)
\(368\) 20057.5 0.00772072
\(369\) −718010. −0.274514
\(370\) 0 0
\(371\) 247141. 0.0932201
\(372\) 626206. 0.234617
\(373\) −3.42147e6 −1.27333 −0.636665 0.771140i \(-0.719687\pi\)
−0.636665 + 0.771140i \(0.719687\pi\)
\(374\) 951698. 0.351820
\(375\) 0 0
\(376\) 3.58990e6 1.30952
\(377\) 2.68310e6 0.972263
\(378\) −242000. −0.0871135
\(379\) −4.82172e6 −1.72427 −0.862133 0.506682i \(-0.830872\pi\)
−0.862133 + 0.506682i \(0.830872\pi\)
\(380\) 0 0
\(381\) −2.06588e6 −0.729110
\(382\) 8.45627e6 2.96497
\(383\) −4.40152e6 −1.53323 −0.766613 0.642110i \(-0.778059\pi\)
−0.766613 + 0.642110i \(0.778059\pi\)
\(384\) −2.48618e6 −0.860410
\(385\) 0 0
\(386\) −4.05043e6 −1.38367
\(387\) 1.84306e6 0.625550
\(388\) 3.10342e6 1.04655
\(389\) 1.88689e6 0.632228 0.316114 0.948721i \(-0.397622\pi\)
0.316114 + 0.948721i \(0.397622\pi\)
\(390\) 0 0
\(391\) −115145. −0.0380892
\(392\) 2.45587e6 0.807216
\(393\) −1.15941e6 −0.378666
\(394\) −5.00675e6 −1.62486
\(395\) 0 0
\(396\) −486036. −0.155751
\(397\) −861255. −0.274255 −0.137128 0.990553i \(-0.543787\pi\)
−0.137128 + 0.990553i \(0.543787\pi\)
\(398\) −2.06329e6 −0.652908
\(399\) −673502. −0.211791
\(400\) 0 0
\(401\) −763714. −0.237176 −0.118588 0.992944i \(-0.537837\pi\)
−0.118588 + 0.992944i \(0.537837\pi\)
\(402\) −4.12286e6 −1.27243
\(403\) −1.37805e6 −0.422670
\(404\) 1.45907e6 0.444758
\(405\) 0 0
\(406\) 906853. 0.273037
\(407\) 83853.4 0.0250920
\(408\) 1.24519e6 0.370326
\(409\) −10162.5 −0.00300395 −0.00150198 0.999999i \(-0.500478\pi\)
−0.00150198 + 0.999999i \(0.500478\pi\)
\(410\) 0 0
\(411\) −3.53911e6 −1.03345
\(412\) −1.97538e6 −0.573333
\(413\) −158745. −0.0457957
\(414\) 96750.8 0.0277430
\(415\) 0 0
\(416\) 6.33950e6 1.79606
\(417\) −473322. −0.133296
\(418\) −2.22553e6 −0.623007
\(419\) 6.49507e6 1.80738 0.903688 0.428191i \(-0.140849\pi\)
0.903688 + 0.428191i \(0.140849\pi\)
\(420\) 0 0
\(421\) 3.07343e6 0.845120 0.422560 0.906335i \(-0.361132\pi\)
0.422560 + 0.906335i \(0.361132\pi\)
\(422\) 6.64348e6 1.81599
\(423\) −1.83008e6 −0.497301
\(424\) 1.06850e6 0.288642
\(425\) 0 0
\(426\) 4.07132e6 1.08696
\(427\) 1.67276e6 0.443980
\(428\) 4.28299e6 1.13015
\(429\) 1.06959e6 0.280590
\(430\) 0 0
\(431\) −2.01477e6 −0.522436 −0.261218 0.965280i \(-0.584124\pi\)
−0.261218 + 0.965280i \(0.584124\pi\)
\(432\) 110575. 0.0285066
\(433\) 45058.6 0.0115494 0.00577469 0.999983i \(-0.498162\pi\)
0.00577469 + 0.999983i \(0.498162\pi\)
\(434\) −465762. −0.118697
\(435\) 0 0
\(436\) 3.25085e6 0.818995
\(437\) 269264. 0.0674489
\(438\) −2.44450e6 −0.608842
\(439\) 5.65439e6 1.40031 0.700155 0.713991i \(-0.253114\pi\)
0.700155 + 0.713991i \(0.253114\pi\)
\(440\) 0 0
\(441\) −1.25197e6 −0.306546
\(442\) −7.72506e6 −1.88081
\(443\) −2.74516e6 −0.664597 −0.332299 0.943174i \(-0.607824\pi\)
−0.332299 + 0.943174i \(0.607824\pi\)
\(444\) 309297. 0.0744592
\(445\) 0 0
\(446\) 8.83767e6 2.10378
\(447\) 641743. 0.151912
\(448\) 1.96429e6 0.462392
\(449\) 4.63732e6 1.08555 0.542777 0.839877i \(-0.317373\pi\)
0.542777 + 0.839877i \(0.317373\pi\)
\(450\) 0 0
\(451\) 1.07258e6 0.248307
\(452\) −2.09645e6 −0.482657
\(453\) 3.95009e6 0.904401
\(454\) −715744. −0.162974
\(455\) 0 0
\(456\) −2.91185e6 −0.655777
\(457\) 2.10576e6 0.471649 0.235824 0.971796i \(-0.424221\pi\)
0.235824 + 0.971796i \(0.424221\pi\)
\(458\) −4.79325e6 −1.06774
\(459\) −634778. −0.140634
\(460\) 0 0
\(461\) −4.02518e6 −0.882131 −0.441066 0.897475i \(-0.645399\pi\)
−0.441066 + 0.897475i \(0.645399\pi\)
\(462\) 361506. 0.0787971
\(463\) 5.62367e6 1.21918 0.609590 0.792717i \(-0.291334\pi\)
0.609590 + 0.792717i \(0.291334\pi\)
\(464\) −414359. −0.0893474
\(465\) 0 0
\(466\) −7.18616e6 −1.53296
\(467\) 3.94737e6 0.837559 0.418779 0.908088i \(-0.362458\pi\)
0.418779 + 0.908088i \(0.362458\pi\)
\(468\) 3.94522e6 0.832638
\(469\) 1.86382e6 0.391266
\(470\) 0 0
\(471\) −1.97204e6 −0.409603
\(472\) −686324. −0.141799
\(473\) −2.75321e6 −0.565831
\(474\) 180592. 0.0369192
\(475\) 0 0
\(476\) −1.58694e6 −0.321028
\(477\) −544705. −0.109614
\(478\) −7.57756e6 −1.51691
\(479\) 1.96093e6 0.390503 0.195251 0.980753i \(-0.437448\pi\)
0.195251 + 0.980753i \(0.437448\pi\)
\(480\) 0 0
\(481\) −680649. −0.134141
\(482\) 1.42344e7 2.79076
\(483\) −43738.1 −0.00853085
\(484\) 726054. 0.140882
\(485\) 0 0
\(486\) 533375. 0.102433
\(487\) 4.10569e6 0.784447 0.392223 0.919870i \(-0.371706\pi\)
0.392223 + 0.919870i \(0.371706\pi\)
\(488\) 7.23207e6 1.37472
\(489\) 3.23584e6 0.611948
\(490\) 0 0
\(491\) −4.39637e6 −0.822983 −0.411492 0.911414i \(-0.634992\pi\)
−0.411492 + 0.911414i \(0.634992\pi\)
\(492\) 3.95627e6 0.736841
\(493\) 2.37872e6 0.440784
\(494\) 1.80649e7 3.33057
\(495\) 0 0
\(496\) 212816. 0.0388418
\(497\) −1.84052e6 −0.334234
\(498\) 2.16388e6 0.390985
\(499\) −7.10912e6 −1.27810 −0.639049 0.769166i \(-0.720672\pi\)
−0.639049 + 0.769166i \(0.720672\pi\)
\(500\) 0 0
\(501\) −1.07199e6 −0.190808
\(502\) −1.32782e7 −2.35170
\(503\) −1.86781e6 −0.329165 −0.164582 0.986363i \(-0.552628\pi\)
−0.164582 + 0.986363i \(0.552628\pi\)
\(504\) 472988. 0.0829419
\(505\) 0 0
\(506\) −144529. −0.0250945
\(507\) −5.34033e6 −0.922673
\(508\) 1.13831e7 1.95705
\(509\) −3.38994e6 −0.579959 −0.289979 0.957033i \(-0.593648\pi\)
−0.289979 + 0.957033i \(0.593648\pi\)
\(510\) 0 0
\(511\) 1.10508e6 0.187216
\(512\) −1.75024e6 −0.295069
\(513\) 1.48442e6 0.249036
\(514\) −8.45015e6 −1.41077
\(515\) 0 0
\(516\) −1.01554e7 −1.67908
\(517\) 2.73382e6 0.449825
\(518\) −230050. −0.0376702
\(519\) 3.55452e6 0.579246
\(520\) 0 0
\(521\) −1.99566e6 −0.322102 −0.161051 0.986946i \(-0.551488\pi\)
−0.161051 + 0.986946i \(0.551488\pi\)
\(522\) −1.99873e6 −0.321054
\(523\) 6.85423e6 1.09573 0.547866 0.836566i \(-0.315440\pi\)
0.547866 + 0.836566i \(0.315440\pi\)
\(524\) 6.38842e6 1.01640
\(525\) 0 0
\(526\) 8.66190e6 1.36505
\(527\) −1.22172e6 −0.191621
\(528\) −165179. −0.0257852
\(529\) −6.41886e6 −0.997283
\(530\) 0 0
\(531\) 349878. 0.0538494
\(532\) 3.71103e6 0.568480
\(533\) −8.70629e6 −1.32744
\(534\) 1.19578e7 1.81468
\(535\) 0 0
\(536\) 8.05811e6 1.21149
\(537\) 3.69109e6 0.552355
\(538\) 1.45040e7 2.16040
\(539\) 1.87022e6 0.277282
\(540\) 0 0
\(541\) −1.24540e7 −1.82942 −0.914712 0.404107i \(-0.867582\pi\)
−0.914712 + 0.404107i \(0.867582\pi\)
\(542\) 1.27574e7 1.86536
\(543\) −4.05083e6 −0.589582
\(544\) 5.62033e6 0.814263
\(545\) 0 0
\(546\) −2.93439e6 −0.421246
\(547\) 253083. 0.0361655 0.0180828 0.999836i \(-0.494244\pi\)
0.0180828 + 0.999836i \(0.494244\pi\)
\(548\) 1.95007e7 2.77395
\(549\) −3.68681e6 −0.522059
\(550\) 0 0
\(551\) −5.56260e6 −0.780547
\(552\) −189099. −0.0264145
\(553\) −81640.2 −0.0113525
\(554\) 1.03139e7 1.42774
\(555\) 0 0
\(556\) 2.60803e6 0.357788
\(557\) −1.39450e7 −1.90450 −0.952251 0.305317i \(-0.901237\pi\)
−0.952251 + 0.305317i \(0.901237\pi\)
\(558\) 1.02655e6 0.139571
\(559\) 2.23482e7 3.02491
\(560\) 0 0
\(561\) 948248. 0.127208
\(562\) 3.64673e6 0.487038
\(563\) 2.56143e6 0.340574 0.170287 0.985395i \(-0.445531\pi\)
0.170287 + 0.985395i \(0.445531\pi\)
\(564\) 1.00838e7 1.33484
\(565\) 0 0
\(566\) 8.04289e6 1.05529
\(567\) −241123. −0.0314978
\(568\) −7.95739e6 −1.03490
\(569\) 6.29287e6 0.814832 0.407416 0.913243i \(-0.366430\pi\)
0.407416 + 0.913243i \(0.366430\pi\)
\(570\) 0 0
\(571\) −5.95809e6 −0.764745 −0.382372 0.924008i \(-0.624893\pi\)
−0.382372 + 0.924008i \(0.624893\pi\)
\(572\) −5.89347e6 −0.753150
\(573\) 8.42561e6 1.07205
\(574\) −2.94261e6 −0.372781
\(575\) 0 0
\(576\) −4.32935e6 −0.543709
\(577\) 3.35274e6 0.419238 0.209619 0.977783i \(-0.432778\pi\)
0.209619 + 0.977783i \(0.432778\pi\)
\(578\) 5.97651e6 0.744094
\(579\) −4.03575e6 −0.500297
\(580\) 0 0
\(581\) −978226. −0.120226
\(582\) 5.08750e6 0.622583
\(583\) 813695. 0.0991495
\(584\) 4.77777e6 0.579686
\(585\) 0 0
\(586\) −2.37512e6 −0.285720
\(587\) −9.25847e6 −1.10903 −0.554516 0.832173i \(-0.687096\pi\)
−0.554516 + 0.832173i \(0.687096\pi\)
\(588\) 6.89840e6 0.822820
\(589\) 2.85696e6 0.339326
\(590\) 0 0
\(591\) −4.98860e6 −0.587503
\(592\) 105115. 0.0123270
\(593\) −2.34931e6 −0.274349 −0.137175 0.990547i \(-0.543802\pi\)
−0.137175 + 0.990547i \(0.543802\pi\)
\(594\) −796769. −0.0926545
\(595\) 0 0
\(596\) −3.53604e6 −0.407757
\(597\) −2.05581e6 −0.236073
\(598\) 1.17316e6 0.134154
\(599\) −3.60123e6 −0.410094 −0.205047 0.978752i \(-0.565735\pi\)
−0.205047 + 0.978752i \(0.565735\pi\)
\(600\) 0 0
\(601\) −623839. −0.0704509 −0.0352255 0.999379i \(-0.511215\pi\)
−0.0352255 + 0.999379i \(0.511215\pi\)
\(602\) 7.55339e6 0.849475
\(603\) −4.10791e6 −0.460074
\(604\) −2.17652e7 −2.42756
\(605\) 0 0
\(606\) 2.39189e6 0.264582
\(607\) −8.09841e6 −0.892130 −0.446065 0.895000i \(-0.647175\pi\)
−0.446065 + 0.895000i \(0.647175\pi\)
\(608\) −1.31431e7 −1.44191
\(609\) 903565. 0.0987225
\(610\) 0 0
\(611\) −2.21908e7 −2.40475
\(612\) 3.49766e6 0.377484
\(613\) −1.20057e7 −1.29044 −0.645220 0.763997i \(-0.723234\pi\)
−0.645220 + 0.763997i \(0.723234\pi\)
\(614\) 2.14818e6 0.229959
\(615\) 0 0
\(616\) −706562. −0.0750237
\(617\) 5.30122e6 0.560613 0.280307 0.959911i \(-0.409564\pi\)
0.280307 + 0.959911i \(0.409564\pi\)
\(618\) −3.23828e6 −0.341069
\(619\) 5.80612e6 0.609059 0.304529 0.952503i \(-0.401501\pi\)
0.304529 + 0.952503i \(0.401501\pi\)
\(620\) 0 0
\(621\) 96400.0 0.0100311
\(622\) −1.90038e7 −1.96953
\(623\) −5.40578e6 −0.558005
\(624\) 1.34078e6 0.137847
\(625\) 0 0
\(626\) −7.93314e6 −0.809113
\(627\) −2.21746e6 −0.225262
\(628\) 1.08660e7 1.09944
\(629\) −603433. −0.0608139
\(630\) 0 0
\(631\) −3.53347e6 −0.353288 −0.176644 0.984275i \(-0.556524\pi\)
−0.176644 + 0.984275i \(0.556524\pi\)
\(632\) −352967. −0.0351513
\(633\) 6.61940e6 0.656613
\(634\) 9.80972e6 0.969245
\(635\) 0 0
\(636\) 3.00135e6 0.294222
\(637\) −1.51808e7 −1.48234
\(638\) 2.98576e6 0.290404
\(639\) 4.05656e6 0.393012
\(640\) 0 0
\(641\) −1.27847e7 −1.22898 −0.614489 0.788925i \(-0.710638\pi\)
−0.614489 + 0.788925i \(0.710638\pi\)
\(642\) 7.02120e6 0.672316
\(643\) −2.85298e6 −0.272127 −0.136063 0.990700i \(-0.543445\pi\)
−0.136063 + 0.990700i \(0.543445\pi\)
\(644\) 240999. 0.0228982
\(645\) 0 0
\(646\) 1.60156e7 1.50995
\(647\) 1.79371e7 1.68458 0.842291 0.539022i \(-0.181206\pi\)
0.842291 + 0.539022i \(0.181206\pi\)
\(648\) −1.04248e6 −0.0975281
\(649\) −522658. −0.0487086
\(650\) 0 0
\(651\) −464073. −0.0429175
\(652\) −1.78296e7 −1.64257
\(653\) 1.68822e7 1.54934 0.774669 0.632367i \(-0.217917\pi\)
0.774669 + 0.632367i \(0.217917\pi\)
\(654\) 5.32919e6 0.487211
\(655\) 0 0
\(656\) 1.34454e6 0.121987
\(657\) −2.43564e6 −0.220140
\(658\) −7.50019e6 −0.675317
\(659\) −1.75716e7 −1.57615 −0.788074 0.615580i \(-0.788922\pi\)
−0.788074 + 0.615580i \(0.788922\pi\)
\(660\) 0 0
\(661\) −1.16563e7 −1.03766 −0.518830 0.854877i \(-0.673632\pi\)
−0.518830 + 0.854877i \(0.673632\pi\)
\(662\) 2.37463e7 2.10596
\(663\) −7.69705e6 −0.680050
\(664\) −4.22930e6 −0.372262
\(665\) 0 0
\(666\) 507037. 0.0442950
\(667\) −361243. −0.0314401
\(668\) 5.90672e6 0.512160
\(669\) 8.80564e6 0.760668
\(670\) 0 0
\(671\) 5.50745e6 0.472220
\(672\) 2.13490e6 0.182371
\(673\) 9.13712e6 0.777627 0.388814 0.921316i \(-0.372885\pi\)
0.388814 + 0.921316i \(0.372885\pi\)
\(674\) 1.46912e7 1.24569
\(675\) 0 0
\(676\) 2.94255e7 2.47661
\(677\) 7.68533e6 0.644453 0.322226 0.946663i \(-0.395569\pi\)
0.322226 + 0.946663i \(0.395569\pi\)
\(678\) −3.43675e6 −0.287127
\(679\) −2.29990e6 −0.191441
\(680\) 0 0
\(681\) −713149. −0.0589268
\(682\) −1.53349e6 −0.126247
\(683\) −1.21807e7 −0.999130 −0.499565 0.866277i \(-0.666507\pi\)
−0.499565 + 0.866277i \(0.666507\pi\)
\(684\) −8.17922e6 −0.668454
\(685\) 0 0
\(686\) −1.07102e7 −0.868934
\(687\) −4.77587e6 −0.386065
\(688\) −3.45129e6 −0.277978
\(689\) −6.60487e6 −0.530049
\(690\) 0 0
\(691\) 1.96826e7 1.56815 0.784073 0.620668i \(-0.213139\pi\)
0.784073 + 0.620668i \(0.213139\pi\)
\(692\) −1.95856e7 −1.55479
\(693\) 360195. 0.0284908
\(694\) 9.95392e6 0.784505
\(695\) 0 0
\(696\) 3.90651e6 0.305679
\(697\) −7.71862e6 −0.601808
\(698\) 3.95497e6 0.307259
\(699\) −7.16011e6 −0.554277
\(700\) 0 0
\(701\) 1.91355e7 1.47077 0.735385 0.677649i \(-0.237001\pi\)
0.735385 + 0.677649i \(0.237001\pi\)
\(702\) 6.46748e6 0.495327
\(703\) 1.41112e6 0.107690
\(704\) 6.46730e6 0.491803
\(705\) 0 0
\(706\) −4.50542e6 −0.340192
\(707\) −1.08130e6 −0.0813576
\(708\) −1.92785e6 −0.144541
\(709\) 1.39074e7 1.03904 0.519519 0.854459i \(-0.326111\pi\)
0.519519 + 0.854459i \(0.326111\pi\)
\(710\) 0 0
\(711\) 179937. 0.0133490
\(712\) −2.33716e7 −1.72778
\(713\) 185535. 0.0136679
\(714\) −2.60150e6 −0.190976
\(715\) 0 0
\(716\) −2.03381e7 −1.48261
\(717\) −7.55009e6 −0.548472
\(718\) 4.60559e6 0.333406
\(719\) −2.17349e7 −1.56796 −0.783982 0.620783i \(-0.786815\pi\)
−0.783982 + 0.620783i \(0.786815\pi\)
\(720\) 0 0
\(721\) 1.46393e6 0.104877
\(722\) −1.50862e7 −1.07705
\(723\) 1.41828e7 1.00906
\(724\) 2.23203e7 1.58254
\(725\) 0 0
\(726\) 1.19024e6 0.0838092
\(727\) −7.28585e6 −0.511263 −0.255632 0.966774i \(-0.582283\pi\)
−0.255632 + 0.966774i \(0.582283\pi\)
\(728\) 5.73526e6 0.401074
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 1.98129e7 1.37137
\(732\) 2.03145e7 1.40129
\(733\) −2.16351e7 −1.48730 −0.743651 0.668568i \(-0.766908\pi\)
−0.743651 + 0.668568i \(0.766908\pi\)
\(734\) 3.30652e7 2.26533
\(735\) 0 0
\(736\) −853527. −0.0580795
\(737\) 6.13651e6 0.416153
\(738\) 6.48560e6 0.438338
\(739\) 9.01968e6 0.607547 0.303774 0.952744i \(-0.401753\pi\)
0.303774 + 0.952744i \(0.401753\pi\)
\(740\) 0 0
\(741\) 1.79994e7 1.20424
\(742\) −2.23236e6 −0.148852
\(743\) 8.05463e6 0.535271 0.267635 0.963520i \(-0.413758\pi\)
0.267635 + 0.963520i \(0.413758\pi\)
\(744\) −2.00639e6 −0.132887
\(745\) 0 0
\(746\) 3.09053e7 2.03323
\(747\) 2.15604e6 0.141369
\(748\) −5.22490e6 −0.341447
\(749\) −3.17407e6 −0.206734
\(750\) 0 0
\(751\) 1.87665e7 1.21418 0.607089 0.794634i \(-0.292337\pi\)
0.607089 + 0.794634i \(0.292337\pi\)
\(752\) 3.42699e6 0.220988
\(753\) −1.32301e7 −0.850307
\(754\) −2.42358e7 −1.55249
\(755\) 0 0
\(756\) 1.32860e6 0.0845452
\(757\) 1.31191e7 0.832082 0.416041 0.909346i \(-0.363417\pi\)
0.416041 + 0.909346i \(0.363417\pi\)
\(758\) 4.35534e7 2.75327
\(759\) −144005. −0.00907346
\(760\) 0 0
\(761\) 4.13520e6 0.258842 0.129421 0.991590i \(-0.458688\pi\)
0.129421 + 0.991590i \(0.458688\pi\)
\(762\) 1.86606e7 1.16423
\(763\) −2.40917e6 −0.149815
\(764\) −4.64256e7 −2.87756
\(765\) 0 0
\(766\) 3.97578e7 2.44822
\(767\) 4.24248e6 0.260394
\(768\) 7.06382e6 0.432152
\(769\) −1.72322e7 −1.05081 −0.525405 0.850852i \(-0.676086\pi\)
−0.525405 + 0.850852i \(0.676086\pi\)
\(770\) 0 0
\(771\) −8.41951e6 −0.510095
\(772\) 2.22372e7 1.34288
\(773\) 5.16026e6 0.310615 0.155308 0.987866i \(-0.450363\pi\)
0.155308 + 0.987866i \(0.450363\pi\)
\(774\) −1.66479e7 −0.998865
\(775\) 0 0
\(776\) −9.94351e6 −0.592769
\(777\) −229216. −0.0136205
\(778\) −1.70438e7 −1.00953
\(779\) 1.80499e7 1.06569
\(780\) 0 0
\(781\) −6.05981e6 −0.355493
\(782\) 1.04007e6 0.0608200
\(783\) −1.99148e6 −0.116084
\(784\) 2.34442e6 0.136221
\(785\) 0 0
\(786\) 1.04727e7 0.604645
\(787\) −2.48500e7 −1.43018 −0.715088 0.699035i \(-0.753613\pi\)
−0.715088 + 0.699035i \(0.753613\pi\)
\(788\) 2.74875e7 1.57695
\(789\) 8.63050e6 0.493564
\(790\) 0 0
\(791\) 1.55365e6 0.0882902
\(792\) 1.55728e6 0.0882175
\(793\) −4.47047e7 −2.52447
\(794\) 7.77949e6 0.437925
\(795\) 0 0
\(796\) 1.13276e7 0.633659
\(797\) −2.02342e7 −1.12834 −0.564170 0.825659i \(-0.690804\pi\)
−0.564170 + 0.825659i \(0.690804\pi\)
\(798\) 6.08357e6 0.338183
\(799\) −1.96734e7 −1.09022
\(800\) 0 0
\(801\) 1.19145e7 0.656137
\(802\) 6.89844e6 0.378717
\(803\) 3.63842e6 0.199124
\(804\) 2.26348e7 1.23491
\(805\) 0 0
\(806\) 1.24475e7 0.674910
\(807\) 1.44515e7 0.781138
\(808\) −4.67494e6 −0.251911
\(809\) −1.24931e7 −0.671116 −0.335558 0.942020i \(-0.608925\pi\)
−0.335558 + 0.942020i \(0.608925\pi\)
\(810\) 0 0
\(811\) 2.70382e7 1.44353 0.721765 0.692138i \(-0.243331\pi\)
0.721765 + 0.692138i \(0.243331\pi\)
\(812\) −4.97869e6 −0.264987
\(813\) 1.27111e7 0.674461
\(814\) −757426. −0.0400663
\(815\) 0 0
\(816\) 1.18868e6 0.0624941
\(817\) −4.63322e7 −2.42844
\(818\) 91795.4 0.00479665
\(819\) −2.92375e6 −0.152311
\(820\) 0 0
\(821\) 6.49168e6 0.336124 0.168062 0.985776i \(-0.446249\pi\)
0.168062 + 0.985776i \(0.446249\pi\)
\(822\) 3.19678e7 1.65019
\(823\) 1.67550e7 0.862274 0.431137 0.902286i \(-0.358113\pi\)
0.431137 + 0.902286i \(0.358113\pi\)
\(824\) 6.32920e6 0.324736
\(825\) 0 0
\(826\) 1.43390e6 0.0731256
\(827\) −3.57022e7 −1.81523 −0.907614 0.419807i \(-0.862098\pi\)
−0.907614 + 0.419807i \(0.862098\pi\)
\(828\) −531169. −0.0269251
\(829\) 3.44144e7 1.73922 0.869608 0.493743i \(-0.164372\pi\)
0.869608 + 0.493743i \(0.164372\pi\)
\(830\) 0 0
\(831\) 1.02765e7 0.516230
\(832\) −5.24959e7 −2.62916
\(833\) −1.34587e7 −0.672031
\(834\) 4.27540e6 0.212844
\(835\) 0 0
\(836\) 1.22183e7 0.604640
\(837\) 1.02283e6 0.0504650
\(838\) −5.86683e7 −2.88598
\(839\) −1.15982e7 −0.568834 −0.284417 0.958701i \(-0.591800\pi\)
−0.284417 + 0.958701i \(0.591800\pi\)
\(840\) 0 0
\(841\) −1.30484e7 −0.636162
\(842\) −2.77615e7 −1.34947
\(843\) 3.63351e6 0.176099
\(844\) −3.64732e7 −1.76246
\(845\) 0 0
\(846\) 1.65306e7 0.794079
\(847\) −538070. −0.0257709
\(848\) 1.02001e6 0.0487096
\(849\) 8.01374e6 0.381563
\(850\) 0 0
\(851\) 91640.0 0.00433771
\(852\) −2.23519e7 −1.05491
\(853\) 1.81651e6 0.0854801 0.0427401 0.999086i \(-0.486391\pi\)
0.0427401 + 0.999086i \(0.486391\pi\)
\(854\) −1.51096e7 −0.708938
\(855\) 0 0
\(856\) −1.37229e7 −0.640121
\(857\) −3.25920e7 −1.51586 −0.757930 0.652336i \(-0.773789\pi\)
−0.757930 + 0.652336i \(0.773789\pi\)
\(858\) −9.66129e6 −0.448040
\(859\) −3.15261e6 −0.145776 −0.0728882 0.997340i \(-0.523222\pi\)
−0.0728882 + 0.997340i \(0.523222\pi\)
\(860\) 0 0
\(861\) −2.93195e6 −0.134787
\(862\) 1.81989e7 0.834214
\(863\) −1.61013e7 −0.735924 −0.367962 0.929841i \(-0.619944\pi\)
−0.367962 + 0.929841i \(0.619944\pi\)
\(864\) −4.70538e6 −0.214442
\(865\) 0 0
\(866\) −407003. −0.0184418
\(867\) 5.95484e6 0.269043
\(868\) 2.55707e6 0.115197
\(869\) −268795. −0.0120746
\(870\) 0 0
\(871\) −4.98108e7 −2.22474
\(872\) −1.04159e7 −0.463880
\(873\) 5.06906e6 0.225108
\(874\) −2.43219e6 −0.107701
\(875\) 0 0
\(876\) 1.34205e7 0.590892
\(877\) 3.99907e7 1.75574 0.877869 0.478901i \(-0.158965\pi\)
0.877869 + 0.478901i \(0.158965\pi\)
\(878\) −5.10746e7 −2.23599
\(879\) −2.36651e6 −0.103308
\(880\) 0 0
\(881\) −3.04686e7 −1.32255 −0.661275 0.750144i \(-0.729984\pi\)
−0.661275 + 0.750144i \(0.729984\pi\)
\(882\) 1.13087e7 0.489487
\(883\) −6.10260e6 −0.263398 −0.131699 0.991290i \(-0.542043\pi\)
−0.131699 + 0.991290i \(0.542043\pi\)
\(884\) 4.24112e7 1.82536
\(885\) 0 0
\(886\) 2.47963e7 1.06121
\(887\) 2.31117e7 0.986329 0.493165 0.869936i \(-0.335840\pi\)
0.493165 + 0.869936i \(0.335840\pi\)
\(888\) −991003. −0.0421738
\(889\) −8.43589e6 −0.357995
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −4.85195e7 −2.04176
\(893\) 4.60059e7 1.93057
\(894\) −5.79670e6 −0.242570
\(895\) 0 0
\(896\) −1.01522e7 −0.422463
\(897\) 1.16891e6 0.0485064
\(898\) −4.18877e7 −1.73339
\(899\) −3.83288e6 −0.158171
\(900\) 0 0
\(901\) −5.85559e6 −0.240303
\(902\) −9.68837e6 −0.396492
\(903\) 7.52601e6 0.307146
\(904\) 6.71713e6 0.273377
\(905\) 0 0
\(906\) −3.56801e7 −1.44413
\(907\) −3.90825e7 −1.57748 −0.788740 0.614727i \(-0.789266\pi\)
−0.788740 + 0.614727i \(0.789266\pi\)
\(908\) 3.92949e6 0.158169
\(909\) 2.38322e6 0.0956652
\(910\) 0 0
\(911\) −4.59143e7 −1.83296 −0.916478 0.400084i \(-0.868981\pi\)
−0.916478 + 0.400084i \(0.868981\pi\)
\(912\) −2.77971e6 −0.110665
\(913\) −3.22075e6 −0.127873
\(914\) −1.90208e7 −0.753119
\(915\) 0 0
\(916\) 2.63153e7 1.03626
\(917\) −4.73437e6 −0.185926
\(918\) 5.73379e6 0.224561
\(919\) 3.21024e7 1.25386 0.626929 0.779077i \(-0.284312\pi\)
0.626929 + 0.779077i \(0.284312\pi\)
\(920\) 0 0
\(921\) 2.14039e6 0.0831465
\(922\) 3.63584e7 1.40857
\(923\) 4.91882e7 1.90045
\(924\) −1.98470e6 −0.0764740
\(925\) 0 0
\(926\) −5.07972e7 −1.94676
\(927\) −3.22654e6 −0.123321
\(928\) 1.76326e7 0.672120
\(929\) −2.18468e7 −0.830517 −0.415258 0.909703i \(-0.636309\pi\)
−0.415258 + 0.909703i \(0.636309\pi\)
\(930\) 0 0
\(931\) 3.14729e7 1.19004
\(932\) 3.94526e7 1.48777
\(933\) −1.89349e7 −0.712128
\(934\) −3.56556e7 −1.33740
\(935\) 0 0
\(936\) −1.26407e7 −0.471607
\(937\) 3.42340e7 1.27382 0.636910 0.770938i \(-0.280212\pi\)
0.636910 + 0.770938i \(0.280212\pi\)
\(938\) −1.68354e7 −0.624764
\(939\) −7.90438e6 −0.292552
\(940\) 0 0
\(941\) −4.75690e7 −1.75126 −0.875628 0.482986i \(-0.839552\pi\)
−0.875628 + 0.482986i \(0.839552\pi\)
\(942\) 1.78129e7 0.654046
\(943\) 1.17218e6 0.0429256
\(944\) −655178. −0.0239293
\(945\) 0 0
\(946\) 2.48691e7 0.903507
\(947\) 1.58351e6 0.0573781 0.0286890 0.999588i \(-0.490867\pi\)
0.0286890 + 0.999588i \(0.490867\pi\)
\(948\) −991464. −0.0358308
\(949\) −2.95335e7 −1.06451
\(950\) 0 0
\(951\) 9.77416e6 0.350452
\(952\) 5.08463e6 0.181831
\(953\) −2.03382e7 −0.725404 −0.362702 0.931905i \(-0.618146\pi\)
−0.362702 + 0.931905i \(0.618146\pi\)
\(954\) 4.92018e6 0.175029
\(955\) 0 0
\(956\) 4.16014e7 1.47219
\(957\) 2.97493e6 0.105002
\(958\) −1.77126e7 −0.623547
\(959\) −1.44517e7 −0.507426
\(960\) 0 0
\(961\) −2.66606e7 −0.931239
\(962\) 6.14812e6 0.214193
\(963\) 6.99574e6 0.243091
\(964\) −7.81482e7 −2.70849
\(965\) 0 0
\(966\) 395075. 0.0136219
\(967\) −5.33702e7 −1.83541 −0.917704 0.397264i \(-0.869960\pi\)
−0.917704 + 0.397264i \(0.869960\pi\)
\(968\) −2.32631e6 −0.0797957
\(969\) 1.59575e7 0.545954
\(970\) 0 0
\(971\) −3.36587e7 −1.14564 −0.572822 0.819680i \(-0.694151\pi\)
−0.572822 + 0.819680i \(0.694151\pi\)
\(972\) −2.92827e6 −0.0994135
\(973\) −1.93278e6 −0.0654486
\(974\) −3.70856e7 −1.25259
\(975\) 0 0
\(976\) 6.90387e6 0.231990
\(977\) 3.96940e7 1.33042 0.665210 0.746657i \(-0.268342\pi\)
0.665210 + 0.746657i \(0.268342\pi\)
\(978\) −2.92285e7 −0.977146
\(979\) −1.77982e7 −0.593498
\(980\) 0 0
\(981\) 5.30987e6 0.176162
\(982\) 3.97113e7 1.31412
\(983\) −2.04490e7 −0.674975 −0.337487 0.941330i \(-0.609577\pi\)
−0.337487 + 0.941330i \(0.609577\pi\)
\(984\) −1.26761e7 −0.417347
\(985\) 0 0
\(986\) −2.14864e7 −0.703835
\(987\) −7.47300e6 −0.244176
\(988\) −9.91778e7 −3.23238
\(989\) −3.00887e6 −0.0978168
\(990\) 0 0
\(991\) −3.49505e7 −1.13050 −0.565248 0.824921i \(-0.691220\pi\)
−0.565248 + 0.824921i \(0.691220\pi\)
\(992\) −9.05616e6 −0.292190
\(993\) 2.36602e7 0.761456
\(994\) 1.66250e7 0.533697
\(995\) 0 0
\(996\) −1.18799e7 −0.379458
\(997\) −1.27768e7 −0.407085 −0.203542 0.979066i \(-0.565245\pi\)
−0.203542 + 0.979066i \(0.565245\pi\)
\(998\) 6.42148e7 2.04084
\(999\) 505199. 0.0160158
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 825.6.a.q.1.1 yes 8
5.4 even 2 825.6.a.p.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.p.1.8 8 5.4 even 2
825.6.a.q.1.1 yes 8 1.1 even 1 trivial