Properties

Label 1075.6.a.a.1.4
Level $1075$
Weight $6$
Character 1075.1
Self dual yes
Analytic conductor $172.413$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.412606299\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.58275\) of defining polynomial
Character \(\chi\) \(=\) 1075.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.582753 q^{2} -3.05838 q^{3} -31.6604 q^{4} +1.78228 q^{6} +103.690 q^{7} +37.0983 q^{8} -233.646 q^{9} +O(q^{10})\) \(q-0.582753 q^{2} -3.05838 q^{3} -31.6604 q^{4} +1.78228 q^{6} +103.690 q^{7} +37.0983 q^{8} -233.646 q^{9} -158.323 q^{11} +96.8296 q^{12} +578.882 q^{13} -60.4256 q^{14} +991.514 q^{16} -253.871 q^{17} +136.158 q^{18} -3092.51 q^{19} -317.124 q^{21} +92.2633 q^{22} -4163.45 q^{23} -113.461 q^{24} -337.345 q^{26} +1457.77 q^{27} -3282.87 q^{28} +6771.63 q^{29} +6264.06 q^{31} -1764.95 q^{32} +484.213 q^{33} +147.944 q^{34} +7397.34 q^{36} +3294.82 q^{37} +1802.17 q^{38} -1770.44 q^{39} -6150.84 q^{41} +184.805 q^{42} +1849.00 q^{43} +5012.58 q^{44} +2426.26 q^{46} -8157.17 q^{47} -3032.43 q^{48} -6055.38 q^{49} +776.436 q^{51} -18327.6 q^{52} +30457.8 q^{53} -849.517 q^{54} +3846.72 q^{56} +9458.07 q^{57} -3946.19 q^{58} -45236.1 q^{59} -7251.18 q^{61} -3650.40 q^{62} -24226.8 q^{63} -30699.9 q^{64} -282.176 q^{66} -19685.7 q^{67} +8037.67 q^{68} +12733.4 q^{69} -48132.2 q^{71} -8667.87 q^{72} -42502.1 q^{73} -1920.07 q^{74} +97910.0 q^{76} -16416.5 q^{77} +1031.73 q^{78} -65151.9 q^{79} +52317.6 q^{81} +3584.42 q^{82} -70403.6 q^{83} +10040.3 q^{84} -1077.51 q^{86} -20710.2 q^{87} -5873.52 q^{88} +29645.1 q^{89} +60024.2 q^{91} +131816. q^{92} -19157.9 q^{93} +4753.61 q^{94} +5397.90 q^{96} +89440.4 q^{97} +3528.79 q^{98} +36991.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.582753 −0.103017 −0.0515085 0.998673i \(-0.516403\pi\)
−0.0515085 + 0.998673i \(0.516403\pi\)
\(3\) −3.05838 −0.196195 −0.0980976 0.995177i \(-0.531276\pi\)
−0.0980976 + 0.995177i \(0.531276\pi\)
\(4\) −31.6604 −0.989387
\(5\) 0 0
\(6\) 1.78228 0.0202115
\(7\) 103.690 0.799819 0.399910 0.916555i \(-0.369042\pi\)
0.399910 + 0.916555i \(0.369042\pi\)
\(8\) 37.0983 0.204941
\(9\) −233.646 −0.961507
\(10\) 0 0
\(11\) −158.323 −0.394515 −0.197257 0.980352i \(-0.563203\pi\)
−0.197257 + 0.980352i \(0.563203\pi\)
\(12\) 96.8296 0.194113
\(13\) 578.882 0.950017 0.475009 0.879981i \(-0.342445\pi\)
0.475009 + 0.879981i \(0.342445\pi\)
\(14\) −60.4256 −0.0823950
\(15\) 0 0
\(16\) 991.514 0.968275
\(17\) −253.871 −0.213055 −0.106527 0.994310i \(-0.533973\pi\)
−0.106527 + 0.994310i \(0.533973\pi\)
\(18\) 136.158 0.0990517
\(19\) −3092.51 −1.96529 −0.982645 0.185494i \(-0.940612\pi\)
−0.982645 + 0.185494i \(0.940612\pi\)
\(20\) 0 0
\(21\) −317.124 −0.156921
\(22\) 92.2633 0.0406417
\(23\) −4163.45 −1.64109 −0.820547 0.571579i \(-0.806331\pi\)
−0.820547 + 0.571579i \(0.806331\pi\)
\(24\) −113.461 −0.0402084
\(25\) 0 0
\(26\) −337.345 −0.0978680
\(27\) 1457.77 0.384839
\(28\) −3282.87 −0.791331
\(29\) 6771.63 1.49520 0.747598 0.664151i \(-0.231207\pi\)
0.747598 + 0.664151i \(0.231207\pi\)
\(30\) 0 0
\(31\) 6264.06 1.17072 0.585358 0.810775i \(-0.300954\pi\)
0.585358 + 0.810775i \(0.300954\pi\)
\(32\) −1764.95 −0.304690
\(33\) 484.213 0.0774019
\(34\) 147.944 0.0219483
\(35\) 0 0
\(36\) 7397.34 0.951303
\(37\) 3294.82 0.395665 0.197833 0.980236i \(-0.436610\pi\)
0.197833 + 0.980236i \(0.436610\pi\)
\(38\) 1802.17 0.202459
\(39\) −1770.44 −0.186389
\(40\) 0 0
\(41\) −6150.84 −0.571445 −0.285723 0.958312i \(-0.592234\pi\)
−0.285723 + 0.958312i \(0.592234\pi\)
\(42\) 184.805 0.0161655
\(43\) 1849.00 0.152499
\(44\) 5012.58 0.390328
\(45\) 0 0
\(46\) 2426.26 0.169061
\(47\) −8157.17 −0.538635 −0.269318 0.963051i \(-0.586798\pi\)
−0.269318 + 0.963051i \(0.586798\pi\)
\(48\) −3032.43 −0.189971
\(49\) −6055.38 −0.360289
\(50\) 0 0
\(51\) 776.436 0.0418004
\(52\) −18327.6 −0.939935
\(53\) 30457.8 1.48939 0.744697 0.667403i \(-0.232594\pi\)
0.744697 + 0.667403i \(0.232594\pi\)
\(54\) −849.517 −0.0396449
\(55\) 0 0
\(56\) 3846.72 0.163916
\(57\) 9458.07 0.385581
\(58\) −3946.19 −0.154031
\(59\) −45236.1 −1.69182 −0.845912 0.533322i \(-0.820943\pi\)
−0.845912 + 0.533322i \(0.820943\pi\)
\(60\) 0 0
\(61\) −7251.18 −0.249508 −0.124754 0.992188i \(-0.539814\pi\)
−0.124754 + 0.992188i \(0.539814\pi\)
\(62\) −3650.40 −0.120604
\(63\) −24226.8 −0.769032
\(64\) −30699.9 −0.936887
\(65\) 0 0
\(66\) −282.176 −0.00797372
\(67\) −19685.7 −0.535751 −0.267876 0.963454i \(-0.586322\pi\)
−0.267876 + 0.963454i \(0.586322\pi\)
\(68\) 8037.67 0.210794
\(69\) 12733.4 0.321975
\(70\) 0 0
\(71\) −48132.2 −1.13316 −0.566579 0.824008i \(-0.691733\pi\)
−0.566579 + 0.824008i \(0.691733\pi\)
\(72\) −8667.87 −0.197052
\(73\) −42502.1 −0.933475 −0.466738 0.884396i \(-0.654571\pi\)
−0.466738 + 0.884396i \(0.654571\pi\)
\(74\) −1920.07 −0.0407603
\(75\) 0 0
\(76\) 97910.0 1.94443
\(77\) −16416.5 −0.315540
\(78\) 1031.73 0.0192012
\(79\) −65151.9 −1.17452 −0.587258 0.809400i \(-0.699793\pi\)
−0.587258 + 0.809400i \(0.699793\pi\)
\(80\) 0 0
\(81\) 52317.6 0.886004
\(82\) 3584.42 0.0588686
\(83\) −70403.6 −1.12176 −0.560880 0.827897i \(-0.689537\pi\)
−0.560880 + 0.827897i \(0.689537\pi\)
\(84\) 10040.3 0.155255
\(85\) 0 0
\(86\) −1077.51 −0.0157100
\(87\) −20710.2 −0.293350
\(88\) −5873.52 −0.0808522
\(89\) 29645.1 0.396714 0.198357 0.980130i \(-0.436439\pi\)
0.198357 + 0.980130i \(0.436439\pi\)
\(90\) 0 0
\(91\) 60024.2 0.759842
\(92\) 131816. 1.62368
\(93\) −19157.9 −0.229689
\(94\) 4753.61 0.0554887
\(95\) 0 0
\(96\) 5397.90 0.0597787
\(97\) 89440.4 0.965171 0.482585 0.875849i \(-0.339698\pi\)
0.482585 + 0.875849i \(0.339698\pi\)
\(98\) 3528.79 0.0371160
\(99\) 36991.6 0.379329
\(100\) 0 0
\(101\) −28633.0 −0.279295 −0.139647 0.990201i \(-0.544597\pi\)
−0.139647 + 0.990201i \(0.544597\pi\)
\(102\) −452.470 −0.00430615
\(103\) −30124.9 −0.279790 −0.139895 0.990166i \(-0.544676\pi\)
−0.139895 + 0.990166i \(0.544676\pi\)
\(104\) 21475.5 0.194697
\(105\) 0 0
\(106\) −17749.4 −0.153433
\(107\) −83073.3 −0.701459 −0.350729 0.936477i \(-0.614066\pi\)
−0.350729 + 0.936477i \(0.614066\pi\)
\(108\) −46153.5 −0.380754
\(109\) 58783.5 0.473903 0.236951 0.971522i \(-0.423852\pi\)
0.236951 + 0.971522i \(0.423852\pi\)
\(110\) 0 0
\(111\) −10076.8 −0.0776276
\(112\) 102810. 0.774445
\(113\) 232647. 1.71396 0.856981 0.515347i \(-0.172337\pi\)
0.856981 + 0.515347i \(0.172337\pi\)
\(114\) −5511.72 −0.0397214
\(115\) 0 0
\(116\) −214392. −1.47933
\(117\) −135254. −0.913449
\(118\) 26361.5 0.174287
\(119\) −26323.9 −0.170405
\(120\) 0 0
\(121\) −135985. −0.844358
\(122\) 4225.64 0.0257036
\(123\) 18811.6 0.112115
\(124\) −198323. −1.15829
\(125\) 0 0
\(126\) 14118.2 0.0792234
\(127\) −109116. −0.600313 −0.300156 0.953890i \(-0.597039\pi\)
−0.300156 + 0.953890i \(0.597039\pi\)
\(128\) 74368.9 0.401205
\(129\) −5654.95 −0.0299195
\(130\) 0 0
\(131\) 139424. 0.709839 0.354920 0.934897i \(-0.384508\pi\)
0.354920 + 0.934897i \(0.384508\pi\)
\(132\) −15330.4 −0.0765805
\(133\) −320662. −1.57188
\(134\) 11471.9 0.0551915
\(135\) 0 0
\(136\) −9418.19 −0.0436637
\(137\) 64194.6 0.292211 0.146106 0.989269i \(-0.453326\pi\)
0.146106 + 0.989269i \(0.453326\pi\)
\(138\) −7420.43 −0.0331689
\(139\) −281632. −1.23636 −0.618181 0.786036i \(-0.712130\pi\)
−0.618181 + 0.786036i \(0.712130\pi\)
\(140\) 0 0
\(141\) 24947.7 0.105678
\(142\) 28049.2 0.116735
\(143\) −91650.4 −0.374796
\(144\) −231664. −0.931004
\(145\) 0 0
\(146\) 24768.2 0.0961639
\(147\) 18519.7 0.0706871
\(148\) −104315. −0.391466
\(149\) 225391. 0.831710 0.415855 0.909431i \(-0.363482\pi\)
0.415855 + 0.909431i \(0.363482\pi\)
\(150\) 0 0
\(151\) 388390. 1.38620 0.693099 0.720842i \(-0.256245\pi\)
0.693099 + 0.720842i \(0.256245\pi\)
\(152\) −114727. −0.402768
\(153\) 59316.1 0.204854
\(154\) 9566.78 0.0325060
\(155\) 0 0
\(156\) 56052.9 0.184411
\(157\) 369612. 1.19673 0.598366 0.801223i \(-0.295817\pi\)
0.598366 + 0.801223i \(0.295817\pi\)
\(158\) 37967.4 0.120995
\(159\) −93151.7 −0.292212
\(160\) 0 0
\(161\) −431708. −1.31258
\(162\) −30488.2 −0.0912735
\(163\) −522204. −1.53947 −0.769735 0.638363i \(-0.779612\pi\)
−0.769735 + 0.638363i \(0.779612\pi\)
\(164\) 194738. 0.565381
\(165\) 0 0
\(166\) 41027.9 0.115560
\(167\) 17232.5 0.0478141 0.0239071 0.999714i \(-0.492389\pi\)
0.0239071 + 0.999714i \(0.492389\pi\)
\(168\) −11764.7 −0.0321595
\(169\) −36189.0 −0.0974674
\(170\) 0 0
\(171\) 722553. 1.88964
\(172\) −58540.1 −0.150880
\(173\) 571643. 1.45214 0.726072 0.687618i \(-0.241344\pi\)
0.726072 + 0.687618i \(0.241344\pi\)
\(174\) 12068.9 0.0302201
\(175\) 0 0
\(176\) −156980. −0.381999
\(177\) 138349. 0.331928
\(178\) −17275.7 −0.0408683
\(179\) −191963. −0.447802 −0.223901 0.974612i \(-0.571879\pi\)
−0.223901 + 0.974612i \(0.571879\pi\)
\(180\) 0 0
\(181\) 110804. 0.251396 0.125698 0.992069i \(-0.459883\pi\)
0.125698 + 0.992069i \(0.459883\pi\)
\(182\) −34979.3 −0.0782767
\(183\) 22176.9 0.0489522
\(184\) −154457. −0.336327
\(185\) 0 0
\(186\) 11164.3 0.0236619
\(187\) 40193.8 0.0840533
\(188\) 258259. 0.532919
\(189\) 151156. 0.307801
\(190\) 0 0
\(191\) 295668. 0.586437 0.293218 0.956046i \(-0.405274\pi\)
0.293218 + 0.956046i \(0.405274\pi\)
\(192\) 93892.0 0.183813
\(193\) 375944. 0.726491 0.363246 0.931693i \(-0.381669\pi\)
0.363246 + 0.931693i \(0.381669\pi\)
\(194\) −52121.6 −0.0994291
\(195\) 0 0
\(196\) 191716. 0.356466
\(197\) 732825. 1.34535 0.672674 0.739939i \(-0.265146\pi\)
0.672674 + 0.739939i \(0.265146\pi\)
\(198\) −21557.0 −0.0390773
\(199\) 589939. 1.05603 0.528013 0.849236i \(-0.322937\pi\)
0.528013 + 0.849236i \(0.322937\pi\)
\(200\) 0 0
\(201\) 60206.3 0.105112
\(202\) 16685.9 0.0287722
\(203\) 702150. 1.19589
\(204\) −24582.3 −0.0413568
\(205\) 0 0
\(206\) 17555.3 0.0288231
\(207\) 972774. 1.57792
\(208\) 573969. 0.919878
\(209\) 489616. 0.775336
\(210\) 0 0
\(211\) −371450. −0.574374 −0.287187 0.957875i \(-0.592720\pi\)
−0.287187 + 0.957875i \(0.592720\pi\)
\(212\) −964307. −1.47359
\(213\) 147207. 0.222320
\(214\) 48411.2 0.0722622
\(215\) 0 0
\(216\) 54080.6 0.0788692
\(217\) 649520. 0.936361
\(218\) −34256.2 −0.0488201
\(219\) 129988. 0.183143
\(220\) 0 0
\(221\) −146962. −0.202406
\(222\) 5872.30 0.00799697
\(223\) 835552. 1.12515 0.562576 0.826746i \(-0.309810\pi\)
0.562576 + 0.826746i \(0.309810\pi\)
\(224\) −183008. −0.243697
\(225\) 0 0
\(226\) −135576. −0.176567
\(227\) 363342. 0.468006 0.234003 0.972236i \(-0.424818\pi\)
0.234003 + 0.972236i \(0.424818\pi\)
\(228\) −299446. −0.381489
\(229\) −106091. −0.133688 −0.0668438 0.997763i \(-0.521293\pi\)
−0.0668438 + 0.997763i \(0.521293\pi\)
\(230\) 0 0
\(231\) 50208.0 0.0619075
\(232\) 251216. 0.306427
\(233\) −713760. −0.861315 −0.430658 0.902515i \(-0.641718\pi\)
−0.430658 + 0.902515i \(0.641718\pi\)
\(234\) 78819.4 0.0941008
\(235\) 0 0
\(236\) 1.43219e6 1.67387
\(237\) 199259. 0.230435
\(238\) 15340.3 0.0175547
\(239\) −70400.0 −0.0797220 −0.0398610 0.999205i \(-0.512692\pi\)
−0.0398610 + 0.999205i \(0.512692\pi\)
\(240\) 0 0
\(241\) 44600.6 0.0494650 0.0247325 0.999694i \(-0.492127\pi\)
0.0247325 + 0.999694i \(0.492127\pi\)
\(242\) 79245.5 0.0869833
\(243\) −514245. −0.558668
\(244\) 229575. 0.246860
\(245\) 0 0
\(246\) −10962.5 −0.0115497
\(247\) −1.79020e6 −1.86706
\(248\) 232386. 0.239928
\(249\) 215321. 0.220084
\(250\) 0 0
\(251\) 1.05989e6 1.06188 0.530942 0.847408i \(-0.321838\pi\)
0.530942 + 0.847408i \(0.321838\pi\)
\(252\) 767030. 0.760871
\(253\) 659170. 0.647435
\(254\) 63587.4 0.0618425
\(255\) 0 0
\(256\) 939058. 0.895556
\(257\) 966091. 0.912400 0.456200 0.889877i \(-0.349210\pi\)
0.456200 + 0.889877i \(0.349210\pi\)
\(258\) 3295.44 0.00308222
\(259\) 341640. 0.316460
\(260\) 0 0
\(261\) −1.58217e6 −1.43764
\(262\) −81249.8 −0.0731255
\(263\) −855270. −0.762454 −0.381227 0.924481i \(-0.624498\pi\)
−0.381227 + 0.924481i \(0.624498\pi\)
\(264\) 17963.5 0.0158628
\(265\) 0 0
\(266\) 186867. 0.161930
\(267\) −90665.9 −0.0778334
\(268\) 623256. 0.530065
\(269\) 1.55224e6 1.30791 0.653955 0.756533i \(-0.273108\pi\)
0.653955 + 0.756533i \(0.273108\pi\)
\(270\) 0 0
\(271\) −4177.31 −0.00345520 −0.00172760 0.999999i \(-0.500550\pi\)
−0.00172760 + 0.999999i \(0.500550\pi\)
\(272\) −251717. −0.206296
\(273\) −183577. −0.149077
\(274\) −37409.5 −0.0301027
\(275\) 0 0
\(276\) −403145. −0.318558
\(277\) 1.59749e6 1.25095 0.625475 0.780244i \(-0.284905\pi\)
0.625475 + 0.780244i \(0.284905\pi\)
\(278\) 164122. 0.127366
\(279\) −1.46357e6 −1.12565
\(280\) 0 0
\(281\) 1.60304e6 1.21110 0.605549 0.795808i \(-0.292954\pi\)
0.605549 + 0.795808i \(0.292954\pi\)
\(282\) −14538.4 −0.0108866
\(283\) −2.04664e6 −1.51906 −0.759530 0.650472i \(-0.774571\pi\)
−0.759530 + 0.650472i \(0.774571\pi\)
\(284\) 1.52389e6 1.12113
\(285\) 0 0
\(286\) 53409.5 0.0386103
\(287\) −637780. −0.457053
\(288\) 412374. 0.292961
\(289\) −1.35541e6 −0.954608
\(290\) 0 0
\(291\) −273543. −0.189362
\(292\) 1.34563e6 0.923569
\(293\) −1.35028e6 −0.918869 −0.459434 0.888212i \(-0.651948\pi\)
−0.459434 + 0.888212i \(0.651948\pi\)
\(294\) −10792.4 −0.00728198
\(295\) 0 0
\(296\) 122232. 0.0810880
\(297\) −230798. −0.151824
\(298\) −131347. −0.0856803
\(299\) −2.41014e6 −1.55907
\(300\) 0 0
\(301\) 191723. 0.121971
\(302\) −226335. −0.142802
\(303\) 87570.6 0.0547964
\(304\) −3.06626e6 −1.90294
\(305\) 0 0
\(306\) −34566.6 −0.0211035
\(307\) 361103. 0.218668 0.109334 0.994005i \(-0.465128\pi\)
0.109334 + 0.994005i \(0.465128\pi\)
\(308\) 519754. 0.312192
\(309\) 92133.4 0.0548935
\(310\) 0 0
\(311\) 2.12425e6 1.24539 0.622694 0.782466i \(-0.286038\pi\)
0.622694 + 0.782466i \(0.286038\pi\)
\(312\) −65680.3 −0.0381987
\(313\) 541717. 0.312545 0.156272 0.987714i \(-0.450052\pi\)
0.156272 + 0.987714i \(0.450052\pi\)
\(314\) −215392. −0.123284
\(315\) 0 0
\(316\) 2.06273e6 1.16205
\(317\) 3.06921e6 1.71545 0.857725 0.514110i \(-0.171878\pi\)
0.857725 + 0.514110i \(0.171878\pi\)
\(318\) 54284.4 0.0301028
\(319\) −1.07211e6 −0.589877
\(320\) 0 0
\(321\) 254070. 0.137623
\(322\) 251579. 0.135218
\(323\) 785100. 0.418715
\(324\) −1.65640e6 −0.876601
\(325\) 0 0
\(326\) 304316. 0.158592
\(327\) −179782. −0.0929775
\(328\) −228185. −0.117112
\(329\) −845817. −0.430811
\(330\) 0 0
\(331\) −282184. −0.141567 −0.0707836 0.997492i \(-0.522550\pi\)
−0.0707836 + 0.997492i \(0.522550\pi\)
\(332\) 2.22901e6 1.10986
\(333\) −769823. −0.380435
\(334\) −10042.3 −0.00492567
\(335\) 0 0
\(336\) −314432. −0.151942
\(337\) −1.04584e6 −0.501640 −0.250820 0.968034i \(-0.580700\pi\)
−0.250820 + 0.968034i \(0.580700\pi\)
\(338\) 21089.2 0.0100408
\(339\) −711523. −0.336271
\(340\) 0 0
\(341\) −991746. −0.461864
\(342\) −421070. −0.194665
\(343\) −2.37060e6 −1.08799
\(344\) 68594.7 0.0312532
\(345\) 0 0
\(346\) −333127. −0.149596
\(347\) −1.90831e6 −0.850796 −0.425398 0.905006i \(-0.639866\pi\)
−0.425398 + 0.905006i \(0.639866\pi\)
\(348\) 655694. 0.290237
\(349\) 1.46350e6 0.643175 0.321588 0.946880i \(-0.395784\pi\)
0.321588 + 0.946880i \(0.395784\pi\)
\(350\) 0 0
\(351\) 843874. 0.365603
\(352\) 279433. 0.120205
\(353\) −867374. −0.370484 −0.185242 0.982693i \(-0.559307\pi\)
−0.185242 + 0.982693i \(0.559307\pi\)
\(354\) −80623.4 −0.0341942
\(355\) 0 0
\(356\) −938575. −0.392504
\(357\) 80508.6 0.0334327
\(358\) 111867. 0.0461313
\(359\) 3.45466e6 1.41472 0.707359 0.706855i \(-0.249887\pi\)
0.707359 + 0.706855i \(0.249887\pi\)
\(360\) 0 0
\(361\) 7.08751e6 2.86237
\(362\) −64571.2 −0.0258981
\(363\) 415893. 0.165659
\(364\) −1.90039e6 −0.751778
\(365\) 0 0
\(366\) −12923.6 −0.00504292
\(367\) 2.78797e6 1.08050 0.540248 0.841506i \(-0.318330\pi\)
0.540248 + 0.841506i \(0.318330\pi\)
\(368\) −4.12811e6 −1.58903
\(369\) 1.43712e6 0.549449
\(370\) 0 0
\(371\) 3.15817e6 1.19125
\(372\) 606546. 0.227251
\(373\) 1.65578e6 0.616212 0.308106 0.951352i \(-0.400305\pi\)
0.308106 + 0.951352i \(0.400305\pi\)
\(374\) −23423.0 −0.00865892
\(375\) 0 0
\(376\) −302617. −0.110388
\(377\) 3.91997e6 1.42046
\(378\) −88086.4 −0.0317088
\(379\) −2.35538e6 −0.842291 −0.421146 0.906993i \(-0.638372\pi\)
−0.421146 + 0.906993i \(0.638372\pi\)
\(380\) 0 0
\(381\) 333717. 0.117779
\(382\) −172301. −0.0604130
\(383\) 4.13864e6 1.44165 0.720827 0.693115i \(-0.243762\pi\)
0.720827 + 0.693115i \(0.243762\pi\)
\(384\) −227449. −0.0787146
\(385\) 0 0
\(386\) −219083. −0.0748410
\(387\) −432012. −0.146629
\(388\) −2.83172e6 −0.954928
\(389\) −5.27781e6 −1.76840 −0.884198 0.467112i \(-0.845294\pi\)
−0.884198 + 0.467112i \(0.845294\pi\)
\(390\) 0 0
\(391\) 1.05698e6 0.349643
\(392\) −224644. −0.0738380
\(393\) −426412. −0.139267
\(394\) −427056. −0.138594
\(395\) 0 0
\(396\) −1.17117e6 −0.375303
\(397\) −3.71439e6 −1.18280 −0.591401 0.806378i \(-0.701425\pi\)
−0.591401 + 0.806378i \(0.701425\pi\)
\(398\) −343789. −0.108789
\(399\) 980707. 0.308395
\(400\) 0 0
\(401\) 3.60034e6 1.11810 0.559052 0.829132i \(-0.311165\pi\)
0.559052 + 0.829132i \(0.311165\pi\)
\(402\) −35085.4 −0.0108283
\(403\) 3.62615e6 1.11220
\(404\) 906532. 0.276331
\(405\) 0 0
\(406\) −409180. −0.123197
\(407\) −521647. −0.156096
\(408\) 28804.4 0.00856661
\(409\) 1.65305e6 0.488627 0.244314 0.969696i \(-0.421437\pi\)
0.244314 + 0.969696i \(0.421437\pi\)
\(410\) 0 0
\(411\) −196331. −0.0573304
\(412\) 953765. 0.276821
\(413\) −4.69053e6 −1.35315
\(414\) −566887. −0.162553
\(415\) 0 0
\(416\) −1.02170e6 −0.289461
\(417\) 861339. 0.242568
\(418\) −285325. −0.0798728
\(419\) 1.25123e6 0.348180 0.174090 0.984730i \(-0.444302\pi\)
0.174090 + 0.984730i \(0.444302\pi\)
\(420\) 0 0
\(421\) 3.12108e6 0.858221 0.429111 0.903252i \(-0.358827\pi\)
0.429111 + 0.903252i \(0.358827\pi\)
\(422\) 216464. 0.0591703
\(423\) 1.90589e6 0.517902
\(424\) 1.12993e6 0.305238
\(425\) 0 0
\(426\) −85785.1 −0.0229028
\(427\) −751875. −0.199561
\(428\) 2.63013e6 0.694015
\(429\) 280302. 0.0735331
\(430\) 0 0
\(431\) 246469. 0.0639102 0.0319551 0.999489i \(-0.489827\pi\)
0.0319551 + 0.999489i \(0.489827\pi\)
\(432\) 1.44540e6 0.372630
\(433\) 7.56031e6 1.93785 0.968924 0.247357i \(-0.0795621\pi\)
0.968924 + 0.247357i \(0.0795621\pi\)
\(434\) −378510. −0.0964612
\(435\) 0 0
\(436\) −1.86111e6 −0.468873
\(437\) 1.28755e7 3.22523
\(438\) −75750.6 −0.0188669
\(439\) −3.76145e6 −0.931525 −0.465763 0.884910i \(-0.654220\pi\)
−0.465763 + 0.884910i \(0.654220\pi\)
\(440\) 0 0
\(441\) 1.41482e6 0.346421
\(442\) 85642.2 0.0208513
\(443\) 1.02986e6 0.249327 0.124664 0.992199i \(-0.460215\pi\)
0.124664 + 0.992199i \(0.460215\pi\)
\(444\) 319036. 0.0768038
\(445\) 0 0
\(446\) −486920. −0.115910
\(447\) −689333. −0.163177
\(448\) −3.18327e6 −0.749340
\(449\) −4.29961e6 −1.00650 −0.503249 0.864141i \(-0.667862\pi\)
−0.503249 + 0.864141i \(0.667862\pi\)
\(450\) 0 0
\(451\) 973820. 0.225443
\(452\) −7.36570e6 −1.69577
\(453\) −1.18784e6 −0.271966
\(454\) −211739. −0.0482126
\(455\) 0 0
\(456\) 350878. 0.0790213
\(457\) 1.42831e6 0.319913 0.159957 0.987124i \(-0.448865\pi\)
0.159957 + 0.987124i \(0.448865\pi\)
\(458\) 61825.0 0.0137721
\(459\) −370085. −0.0819917
\(460\) 0 0
\(461\) −3.06394e6 −0.671472 −0.335736 0.941956i \(-0.608985\pi\)
−0.335736 + 0.941956i \(0.608985\pi\)
\(462\) −29258.9 −0.00637753
\(463\) 3.51890e6 0.762877 0.381438 0.924394i \(-0.375429\pi\)
0.381438 + 0.924394i \(0.375429\pi\)
\(464\) 6.71416e6 1.44776
\(465\) 0 0
\(466\) 415945. 0.0887302
\(467\) 6.90153e6 1.46438 0.732188 0.681102i \(-0.238499\pi\)
0.732188 + 0.681102i \(0.238499\pi\)
\(468\) 4.28218e6 0.903755
\(469\) −2.04121e6 −0.428504
\(470\) 0 0
\(471\) −1.13041e6 −0.234793
\(472\) −1.67818e6 −0.346724
\(473\) −292740. −0.0601629
\(474\) −116119. −0.0237387
\(475\) 0 0
\(476\) 833426. 0.168597
\(477\) −7.11636e6 −1.43206
\(478\) 41025.8 0.00821272
\(479\) −4.37684e6 −0.871609 −0.435805 0.900041i \(-0.643536\pi\)
−0.435805 + 0.900041i \(0.643536\pi\)
\(480\) 0 0
\(481\) 1.90731e6 0.375889
\(482\) −25991.1 −0.00509574
\(483\) 1.32033e6 0.257522
\(484\) 4.30533e6 0.835398
\(485\) 0 0
\(486\) 299677. 0.0575524
\(487\) −4.08443e6 −0.780385 −0.390192 0.920733i \(-0.627591\pi\)
−0.390192 + 0.920733i \(0.627591\pi\)
\(488\) −269006. −0.0511343
\(489\) 1.59710e6 0.302037
\(490\) 0 0
\(491\) −4.67807e6 −0.875716 −0.437858 0.899044i \(-0.644263\pi\)
−0.437858 + 0.899044i \(0.644263\pi\)
\(492\) −595583. −0.110925
\(493\) −1.71912e6 −0.318559
\(494\) 1.04324e6 0.192339
\(495\) 0 0
\(496\) 6.21090e6 1.13357
\(497\) −4.99083e6 −0.906321
\(498\) −125479. −0.0226724
\(499\) 9.66466e6 1.73754 0.868771 0.495214i \(-0.164910\pi\)
0.868771 + 0.495214i \(0.164910\pi\)
\(500\) 0 0
\(501\) −52703.5 −0.00938091
\(502\) −617655. −0.109392
\(503\) 1.03319e7 1.82079 0.910396 0.413738i \(-0.135777\pi\)
0.910396 + 0.413738i \(0.135777\pi\)
\(504\) −898772. −0.157606
\(505\) 0 0
\(506\) −384133. −0.0666969
\(507\) 110680. 0.0191227
\(508\) 3.45464e6 0.593942
\(509\) −4.20006e6 −0.718557 −0.359278 0.933230i \(-0.616977\pi\)
−0.359278 + 0.933230i \(0.616977\pi\)
\(510\) 0 0
\(511\) −4.40704e6 −0.746611
\(512\) −2.92704e6 −0.493463
\(513\) −4.50815e6 −0.756320
\(514\) −562992. −0.0939928
\(515\) 0 0
\(516\) 179038. 0.0296020
\(517\) 1.29147e6 0.212500
\(518\) −199092. −0.0326008
\(519\) −1.74830e6 −0.284904
\(520\) 0 0
\(521\) 6.46853e6 1.04403 0.522013 0.852938i \(-0.325181\pi\)
0.522013 + 0.852938i \(0.325181\pi\)
\(522\) 922012. 0.148102
\(523\) 3.00248e6 0.479983 0.239991 0.970775i \(-0.422855\pi\)
0.239991 + 0.970775i \(0.422855\pi\)
\(524\) −4.41423e6 −0.702306
\(525\) 0 0
\(526\) 498411. 0.0785458
\(527\) −1.59027e6 −0.249427
\(528\) 480104. 0.0749463
\(529\) 1.08979e7 1.69319
\(530\) 0 0
\(531\) 1.05692e7 1.62670
\(532\) 1.01523e7 1.55520
\(533\) −3.56061e6 −0.542883
\(534\) 52835.8 0.00801817
\(535\) 0 0
\(536\) −730304. −0.109797
\(537\) 587098. 0.0878566
\(538\) −904572. −0.134737
\(539\) 958708. 0.142139
\(540\) 0 0
\(541\) −5.21767e6 −0.766449 −0.383224 0.923655i \(-0.625186\pi\)
−0.383224 + 0.923655i \(0.625186\pi\)
\(542\) 2434.34 0.000355945 0
\(543\) −338880. −0.0493227
\(544\) 448071. 0.0649157
\(545\) 0 0
\(546\) 106980. 0.0153575
\(547\) 7.86187e6 1.12346 0.561730 0.827321i \(-0.310136\pi\)
0.561730 + 0.827321i \(0.310136\pi\)
\(548\) −2.03243e6 −0.289110
\(549\) 1.69421e6 0.239903
\(550\) 0 0
\(551\) −2.09413e7 −2.93850
\(552\) 472387. 0.0659858
\(553\) −6.75560e6 −0.939401
\(554\) −930944. −0.128869
\(555\) 0 0
\(556\) 8.91659e6 1.22324
\(557\) −1.21531e7 −1.65978 −0.829889 0.557929i \(-0.811596\pi\)
−0.829889 + 0.557929i \(0.811596\pi\)
\(558\) 852902. 0.115961
\(559\) 1.07035e6 0.144876
\(560\) 0 0
\(561\) −122928. −0.0164909
\(562\) −934177. −0.124764
\(563\) −5.15518e6 −0.685446 −0.342723 0.939437i \(-0.611349\pi\)
−0.342723 + 0.939437i \(0.611349\pi\)
\(564\) −789856. −0.104556
\(565\) 0 0
\(566\) 1.19268e6 0.156489
\(567\) 5.42482e6 0.708643
\(568\) −1.78562e6 −0.232230
\(569\) −1.42079e6 −0.183972 −0.0919858 0.995760i \(-0.529321\pi\)
−0.0919858 + 0.995760i \(0.529321\pi\)
\(570\) 0 0
\(571\) −1.14786e7 −1.47332 −0.736661 0.676262i \(-0.763599\pi\)
−0.736661 + 0.676262i \(0.763599\pi\)
\(572\) 2.90169e6 0.370818
\(573\) −904266. −0.115056
\(574\) 371668. 0.0470842
\(575\) 0 0
\(576\) 7.17292e6 0.900824
\(577\) −1.00683e7 −1.25897 −0.629484 0.777014i \(-0.716734\pi\)
−0.629484 + 0.777014i \(0.716734\pi\)
\(578\) 789867. 0.0983409
\(579\) −1.14978e6 −0.142534
\(580\) 0 0
\(581\) −7.30015e6 −0.897205
\(582\) 159408. 0.0195075
\(583\) −4.82218e6 −0.587587
\(584\) −1.57675e6 −0.191307
\(585\) 0 0
\(586\) 786877. 0.0946592
\(587\) 1.33560e7 1.59985 0.799927 0.600097i \(-0.204871\pi\)
0.799927 + 0.600097i \(0.204871\pi\)
\(588\) −586340. −0.0699369
\(589\) −1.93716e7 −2.30080
\(590\) 0 0
\(591\) −2.24126e6 −0.263951
\(592\) 3.26686e6 0.383113
\(593\) −1.20725e7 −1.40981 −0.704906 0.709301i \(-0.749011\pi\)
−0.704906 + 0.709301i \(0.749011\pi\)
\(594\) 134498. 0.0156405
\(595\) 0 0
\(596\) −7.13598e6 −0.822883
\(597\) −1.80426e6 −0.207187
\(598\) 1.40452e6 0.160611
\(599\) 1.59735e7 1.81900 0.909502 0.415700i \(-0.136463\pi\)
0.909502 + 0.415700i \(0.136463\pi\)
\(600\) 0 0
\(601\) 7.82358e6 0.883526 0.441763 0.897132i \(-0.354353\pi\)
0.441763 + 0.897132i \(0.354353\pi\)
\(602\) −111727. −0.0125651
\(603\) 4.59948e6 0.515129
\(604\) −1.22966e7 −1.37149
\(605\) 0 0
\(606\) −51032.0 −0.00564496
\(607\) −3.67967e6 −0.405356 −0.202678 0.979245i \(-0.564964\pi\)
−0.202678 + 0.979245i \(0.564964\pi\)
\(608\) 5.45813e6 0.598804
\(609\) −2.14744e6 −0.234627
\(610\) 0 0
\(611\) −4.72204e6 −0.511713
\(612\) −1.87797e6 −0.202680
\(613\) −4.33193e6 −0.465619 −0.232810 0.972522i \(-0.574792\pi\)
−0.232810 + 0.972522i \(0.574792\pi\)
\(614\) −210434. −0.0225265
\(615\) 0 0
\(616\) −609025. −0.0646671
\(617\) 6.45241e6 0.682353 0.341176 0.939999i \(-0.389175\pi\)
0.341176 + 0.939999i \(0.389175\pi\)
\(618\) −53691.0 −0.00565497
\(619\) 7.05387e6 0.739947 0.369974 0.929042i \(-0.379367\pi\)
0.369974 + 0.929042i \(0.379367\pi\)
\(620\) 0 0
\(621\) −6.06933e6 −0.631556
\(622\) −1.23791e6 −0.128296
\(623\) 3.07390e6 0.317299
\(624\) −1.75542e6 −0.180476
\(625\) 0 0
\(626\) −315687. −0.0321974
\(627\) −1.49743e6 −0.152117
\(628\) −1.17021e7 −1.18403
\(629\) −836461. −0.0842984
\(630\) 0 0
\(631\) 1.46818e7 1.46793 0.733966 0.679186i \(-0.237667\pi\)
0.733966 + 0.679186i \(0.237667\pi\)
\(632\) −2.41702e6 −0.240706
\(633\) 1.13604e6 0.112689
\(634\) −1.78859e6 −0.176721
\(635\) 0 0
\(636\) 2.94922e6 0.289111
\(637\) −3.50535e6 −0.342281
\(638\) 624773. 0.0607674
\(639\) 1.12459e7 1.08954
\(640\) 0 0
\(641\) 3.24675e6 0.312107 0.156054 0.987749i \(-0.450123\pi\)
0.156054 + 0.987749i \(0.450123\pi\)
\(642\) −148060. −0.0141775
\(643\) 7.57964e6 0.722971 0.361486 0.932378i \(-0.382270\pi\)
0.361486 + 0.932378i \(0.382270\pi\)
\(644\) 1.36680e7 1.29865
\(645\) 0 0
\(646\) −457519. −0.0431348
\(647\) −6.13290e6 −0.575977 −0.287988 0.957634i \(-0.592986\pi\)
−0.287988 + 0.957634i \(0.592986\pi\)
\(648\) 1.94089e6 0.181578
\(649\) 7.16193e6 0.667449
\(650\) 0 0
\(651\) −1.98648e6 −0.183710
\(652\) 1.65332e7 1.52313
\(653\) 1.13174e6 0.103864 0.0519318 0.998651i \(-0.483462\pi\)
0.0519318 + 0.998651i \(0.483462\pi\)
\(654\) 104769. 0.00957827
\(655\) 0 0
\(656\) −6.09864e6 −0.553316
\(657\) 9.93045e6 0.897544
\(658\) 492902. 0.0443809
\(659\) 1.03411e7 0.927586 0.463793 0.885944i \(-0.346488\pi\)
0.463793 + 0.885944i \(0.346488\pi\)
\(660\) 0 0
\(661\) −8.30708e6 −0.739511 −0.369756 0.929129i \(-0.620559\pi\)
−0.369756 + 0.929129i \(0.620559\pi\)
\(662\) 164444. 0.0145838
\(663\) 449465. 0.0397111
\(664\) −2.61185e6 −0.229894
\(665\) 0 0
\(666\) 448616. 0.0391913
\(667\) −2.81933e7 −2.45376
\(668\) −545587. −0.0473067
\(669\) −2.55544e6 −0.220750
\(670\) 0 0
\(671\) 1.14803e6 0.0984344
\(672\) 559708. 0.0478121
\(673\) 1.13225e7 0.963617 0.481809 0.876276i \(-0.339980\pi\)
0.481809 + 0.876276i \(0.339980\pi\)
\(674\) 609468. 0.0516775
\(675\) 0 0
\(676\) 1.14576e6 0.0964331
\(677\) −2.12894e7 −1.78522 −0.892609 0.450831i \(-0.851128\pi\)
−0.892609 + 0.450831i \(0.851128\pi\)
\(678\) 414642. 0.0346417
\(679\) 9.27407e6 0.771962
\(680\) 0 0
\(681\) −1.11124e6 −0.0918205
\(682\) 577943. 0.0475799
\(683\) 1.18724e7 0.973841 0.486920 0.873446i \(-0.338120\pi\)
0.486920 + 0.873446i \(0.338120\pi\)
\(684\) −2.28763e7 −1.86959
\(685\) 0 0
\(686\) 1.38147e6 0.112081
\(687\) 324468. 0.0262289
\(688\) 1.83331e6 0.147661
\(689\) 1.76315e7 1.41495
\(690\) 0 0
\(691\) −1.96927e7 −1.56896 −0.784478 0.620157i \(-0.787069\pi\)
−0.784478 + 0.620157i \(0.787069\pi\)
\(692\) −1.80985e7 −1.43673
\(693\) 3.83566e6 0.303394
\(694\) 1.11207e6 0.0876465
\(695\) 0 0
\(696\) −768314. −0.0601195
\(697\) 1.56152e6 0.121749
\(698\) −852859. −0.0662581
\(699\) 2.18295e6 0.168986
\(700\) 0 0
\(701\) −1.36097e7 −1.04605 −0.523027 0.852316i \(-0.675197\pi\)
−0.523027 + 0.852316i \(0.675197\pi\)
\(702\) −491770. −0.0376634
\(703\) −1.01893e7 −0.777597
\(704\) 4.86051e6 0.369615
\(705\) 0 0
\(706\) 505464. 0.0381662
\(707\) −2.96895e6 −0.223385
\(708\) −4.38019e6 −0.328405
\(709\) 1.30274e7 0.973290 0.486645 0.873600i \(-0.338220\pi\)
0.486645 + 0.873600i \(0.338220\pi\)
\(710\) 0 0
\(711\) 1.52225e7 1.12931
\(712\) 1.09978e6 0.0813029
\(713\) −2.60801e7 −1.92125
\(714\) −46916.6 −0.00344414
\(715\) 0 0
\(716\) 6.07764e6 0.443050
\(717\) 215310. 0.0156411
\(718\) −2.01321e6 −0.145740
\(719\) −9.11723e6 −0.657719 −0.328860 0.944379i \(-0.606664\pi\)
−0.328860 + 0.944379i \(0.606664\pi\)
\(720\) 0 0
\(721\) −3.12365e6 −0.223781
\(722\) −4.13026e6 −0.294873
\(723\) −136406. −0.00970480
\(724\) −3.50809e6 −0.248728
\(725\) 0 0
\(726\) −242363. −0.0170657
\(727\) 1.52424e7 1.06959 0.534797 0.844981i \(-0.320388\pi\)
0.534797 + 0.844981i \(0.320388\pi\)
\(728\) 2.22680e6 0.155723
\(729\) −1.11404e7 −0.776396
\(730\) 0 0
\(731\) −469408. −0.0324906
\(732\) −702128. −0.0484327
\(733\) 1.61394e7 1.10950 0.554751 0.832016i \(-0.312813\pi\)
0.554751 + 0.832016i \(0.312813\pi\)
\(734\) −1.62470e6 −0.111310
\(735\) 0 0
\(736\) 7.34828e6 0.500025
\(737\) 3.11670e6 0.211362
\(738\) −837486. −0.0566026
\(739\) −9.24471e6 −0.622704 −0.311352 0.950295i \(-0.600782\pi\)
−0.311352 + 0.950295i \(0.600782\pi\)
\(740\) 0 0
\(741\) 5.47510e6 0.366308
\(742\) −1.84043e6 −0.122719
\(743\) 3.89612e6 0.258917 0.129459 0.991585i \(-0.458676\pi\)
0.129459 + 0.991585i \(0.458676\pi\)
\(744\) −710724. −0.0470727
\(745\) 0 0
\(746\) −964909. −0.0634804
\(747\) 1.64495e7 1.07858
\(748\) −1.27255e6 −0.0831613
\(749\) −8.61387e6 −0.561040
\(750\) 0 0
\(751\) −2.36303e7 −1.52887 −0.764433 0.644704i \(-0.776981\pi\)
−0.764433 + 0.644704i \(0.776981\pi\)
\(752\) −8.08795e6 −0.521547
\(753\) −3.24155e6 −0.208337
\(754\) −2.28437e6 −0.146332
\(755\) 0 0
\(756\) −4.78565e6 −0.304535
\(757\) −2.86009e6 −0.181401 −0.0907005 0.995878i \(-0.528911\pi\)
−0.0907005 + 0.995878i \(0.528911\pi\)
\(758\) 1.37260e6 0.0867704
\(759\) −2.01599e6 −0.127024
\(760\) 0 0
\(761\) −2.90585e7 −1.81891 −0.909456 0.415800i \(-0.863502\pi\)
−0.909456 + 0.415800i \(0.863502\pi\)
\(762\) −194475. −0.0121332
\(763\) 6.09526e6 0.379036
\(764\) −9.36097e6 −0.580213
\(765\) 0 0
\(766\) −2.41180e6 −0.148515
\(767\) −2.61864e7 −1.60726
\(768\) −2.87200e6 −0.175704
\(769\) −1.85731e7 −1.13258 −0.566290 0.824206i \(-0.691622\pi\)
−0.566290 + 0.824206i \(0.691622\pi\)
\(770\) 0 0
\(771\) −2.95468e6 −0.179009
\(772\) −1.19026e7 −0.718781
\(773\) −1.38698e7 −0.834872 −0.417436 0.908706i \(-0.637071\pi\)
−0.417436 + 0.908706i \(0.637071\pi\)
\(774\) 251756. 0.0151052
\(775\) 0 0
\(776\) 3.31808e6 0.197803
\(777\) −1.04487e6 −0.0620881
\(778\) 3.07566e6 0.182175
\(779\) 1.90215e7 1.12306
\(780\) 0 0
\(781\) 7.62045e6 0.447047
\(782\) −615958. −0.0360192
\(783\) 9.87145e6 0.575409
\(784\) −6.00400e6 −0.348859
\(785\) 0 0
\(786\) 248493. 0.0143469
\(787\) 1.95803e7 1.12689 0.563445 0.826154i \(-0.309476\pi\)
0.563445 + 0.826154i \(0.309476\pi\)
\(788\) −2.32015e7 −1.33107
\(789\) 2.61574e6 0.149590
\(790\) 0 0
\(791\) 2.41232e7 1.37086
\(792\) 1.37233e6 0.0777400
\(793\) −4.19757e6 −0.237037
\(794\) 2.16457e6 0.121849
\(795\) 0 0
\(796\) −1.86777e7 −1.04482
\(797\) 2.66479e7 1.48600 0.742998 0.669293i \(-0.233403\pi\)
0.742998 + 0.669293i \(0.233403\pi\)
\(798\) −571510. −0.0317699
\(799\) 2.07087e6 0.114759
\(800\) 0 0
\(801\) −6.92646e6 −0.381443
\(802\) −2.09811e6 −0.115184
\(803\) 6.72907e6 0.368270
\(804\) −1.90615e6 −0.103996
\(805\) 0 0
\(806\) −2.11315e6 −0.114576
\(807\) −4.74734e6 −0.256606
\(808\) −1.06223e6 −0.0572390
\(809\) 4.31569e6 0.231835 0.115917 0.993259i \(-0.463019\pi\)
0.115917 + 0.993259i \(0.463019\pi\)
\(810\) 0 0
\(811\) 1.07890e7 0.576007 0.288003 0.957629i \(-0.407009\pi\)
0.288003 + 0.957629i \(0.407009\pi\)
\(812\) −2.22304e7 −1.18320
\(813\) 12775.8 0.000677894 0
\(814\) 303991. 0.0160805
\(815\) 0 0
\(816\) 769847. 0.0404743
\(817\) −5.71805e6 −0.299704
\(818\) −963319. −0.0503369
\(819\) −1.40244e7 −0.730594
\(820\) 0 0
\(821\) −2.12759e7 −1.10162 −0.550808 0.834632i \(-0.685680\pi\)
−0.550808 + 0.834632i \(0.685680\pi\)
\(822\) 114413. 0.00590602
\(823\) −7.04803e6 −0.362717 −0.181359 0.983417i \(-0.558050\pi\)
−0.181359 + 0.983417i \(0.558050\pi\)
\(824\) −1.11758e6 −0.0573404
\(825\) 0 0
\(826\) 2.73342e6 0.139398
\(827\) −2.00838e7 −1.02113 −0.510566 0.859839i \(-0.670564\pi\)
−0.510566 + 0.859839i \(0.670564\pi\)
\(828\) −3.07984e7 −1.56118
\(829\) 2.04193e7 1.03194 0.515970 0.856606i \(-0.327431\pi\)
0.515970 + 0.856606i \(0.327431\pi\)
\(830\) 0 0
\(831\) −4.88575e6 −0.245431
\(832\) −1.77716e7 −0.890059
\(833\) 1.53729e6 0.0767614
\(834\) −501948. −0.0249887
\(835\) 0 0
\(836\) −1.55014e7 −0.767108
\(837\) 9.13153e6 0.450537
\(838\) −729160. −0.0358685
\(839\) −4.92445e6 −0.241520 −0.120760 0.992682i \(-0.538533\pi\)
−0.120760 + 0.992682i \(0.538533\pi\)
\(840\) 0 0
\(841\) 2.53438e7 1.23561
\(842\) −1.81882e6 −0.0884114
\(843\) −4.90271e6 −0.237612
\(844\) 1.17603e7 0.568278
\(845\) 0 0
\(846\) −1.11066e6 −0.0533528
\(847\) −1.41003e7 −0.675334
\(848\) 3.01994e7 1.44214
\(849\) 6.25940e6 0.298032
\(850\) 0 0
\(851\) −1.37178e7 −0.649324
\(852\) −4.66062e6 −0.219961
\(853\) 1.25088e7 0.588632 0.294316 0.955708i \(-0.404908\pi\)
0.294316 + 0.955708i \(0.404908\pi\)
\(854\) 438157. 0.0205582
\(855\) 0 0
\(856\) −3.08188e6 −0.143758
\(857\) −2.21879e7 −1.03196 −0.515982 0.856600i \(-0.672573\pi\)
−0.515982 + 0.856600i \(0.672573\pi\)
\(858\) −163347. −0.00757517
\(859\) −1.40835e7 −0.651218 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(860\) 0 0
\(861\) 1.95058e6 0.0896716
\(862\) −143631. −0.00658384
\(863\) 7.94491e6 0.363130 0.181565 0.983379i \(-0.441884\pi\)
0.181565 + 0.983379i \(0.441884\pi\)
\(864\) −2.57289e6 −0.117256
\(865\) 0 0
\(866\) −4.40579e6 −0.199632
\(867\) 4.14535e6 0.187290
\(868\) −2.05641e7 −0.926424
\(869\) 1.03151e7 0.463364
\(870\) 0 0
\(871\) −1.13957e7 −0.508973
\(872\) 2.18077e6 0.0971221
\(873\) −2.08974e7 −0.928019
\(874\) −7.50323e6 −0.332253
\(875\) 0 0
\(876\) −4.11546e6 −0.181200
\(877\) 4.03970e6 0.177358 0.0886788 0.996060i \(-0.471736\pi\)
0.0886788 + 0.996060i \(0.471736\pi\)
\(878\) 2.19200e6 0.0959630
\(879\) 4.12966e6 0.180278
\(880\) 0 0
\(881\) −1.99840e7 −0.867447 −0.433724 0.901046i \(-0.642801\pi\)
−0.433724 + 0.901046i \(0.642801\pi\)
\(882\) −824489. −0.0356873
\(883\) −1.66868e7 −0.720230 −0.360115 0.932908i \(-0.617263\pi\)
−0.360115 + 0.932908i \(0.617263\pi\)
\(884\) 4.65286e6 0.200258
\(885\) 0 0
\(886\) −600155. −0.0256850
\(887\) 1.79964e6 0.0768027 0.0384014 0.999262i \(-0.487773\pi\)
0.0384014 + 0.999262i \(0.487773\pi\)
\(888\) −373833. −0.0159091
\(889\) −1.13142e7 −0.480141
\(890\) 0 0
\(891\) −8.28310e6 −0.349541
\(892\) −2.64539e7 −1.11321
\(893\) 2.52261e7 1.05858
\(894\) 401711. 0.0168101
\(895\) 0 0
\(896\) 7.71131e6 0.320892
\(897\) 7.37114e6 0.305882
\(898\) 2.50561e6 0.103687
\(899\) 4.24179e7 1.75045
\(900\) 0 0
\(901\) −7.73238e6 −0.317323
\(902\) −567496. −0.0232245
\(903\) −586362. −0.0239302
\(904\) 8.63080e6 0.351261
\(905\) 0 0
\(906\) 692219. 0.0280171
\(907\) 3.64748e6 0.147223 0.0736113 0.997287i \(-0.476548\pi\)
0.0736113 + 0.997287i \(0.476548\pi\)
\(908\) −1.15036e7 −0.463039
\(909\) 6.68999e6 0.268544
\(910\) 0 0
\(911\) 4.92223e6 0.196502 0.0982508 0.995162i \(-0.468675\pi\)
0.0982508 + 0.995162i \(0.468675\pi\)
\(912\) 9.37781e6 0.373348
\(913\) 1.11465e7 0.442551
\(914\) −832352. −0.0329565
\(915\) 0 0
\(916\) 3.35890e6 0.132269
\(917\) 1.44569e7 0.567743
\(918\) 215668. 0.00844655
\(919\) 1.52960e7 0.597432 0.298716 0.954342i \(-0.403442\pi\)
0.298716 + 0.954342i \(0.403442\pi\)
\(920\) 0 0
\(921\) −1.10439e6 −0.0429016
\(922\) 1.78552e6 0.0691731
\(923\) −2.78629e7 −1.07652
\(924\) −1.58961e6 −0.0612505
\(925\) 0 0
\(926\) −2.05065e6 −0.0785893
\(927\) 7.03856e6 0.269020
\(928\) −1.19516e7 −0.455571
\(929\) 3.10755e7 1.18135 0.590674 0.806910i \(-0.298862\pi\)
0.590674 + 0.806910i \(0.298862\pi\)
\(930\) 0 0
\(931\) 1.87263e7 0.708073
\(932\) 2.25979e7 0.852175
\(933\) −6.49676e6 −0.244339
\(934\) −4.02188e6 −0.150856
\(935\) 0 0
\(936\) −5.01767e6 −0.187203
\(937\) 3.19233e7 1.18784 0.593922 0.804523i \(-0.297579\pi\)
0.593922 + 0.804523i \(0.297579\pi\)
\(938\) 1.18952e6 0.0441432
\(939\) −1.65678e6 −0.0613198
\(940\) 0 0
\(941\) −4.89415e7 −1.80179 −0.900894 0.434040i \(-0.857088\pi\)
−0.900894 + 0.434040i \(0.857088\pi\)
\(942\) 658752. 0.0241877
\(943\) 2.56087e7 0.937795
\(944\) −4.48522e7 −1.63815
\(945\) 0 0
\(946\) 170595. 0.00619781
\(947\) 1.45580e7 0.527505 0.263753 0.964590i \(-0.415040\pi\)
0.263753 + 0.964590i \(0.415040\pi\)
\(948\) −6.30863e6 −0.227989
\(949\) −2.46037e7 −0.886818
\(950\) 0 0
\(951\) −9.38680e6 −0.336563
\(952\) −976572. −0.0349230
\(953\) 2.71191e7 0.967261 0.483631 0.875272i \(-0.339318\pi\)
0.483631 + 0.875272i \(0.339318\pi\)
\(954\) 4.14708e6 0.147527
\(955\) 0 0
\(956\) 2.22889e6 0.0788759
\(957\) 3.27891e6 0.115731
\(958\) 2.55062e6 0.0897907
\(959\) 6.65633e6 0.233716
\(960\) 0 0
\(961\) 1.06093e7 0.370576
\(962\) −1.11149e6 −0.0387229
\(963\) 1.94098e7 0.674458
\(964\) −1.41207e6 −0.0489401
\(965\) 0 0
\(966\) −769424. −0.0265291
\(967\) −1.16883e7 −0.401963 −0.200982 0.979595i \(-0.564413\pi\)
−0.200982 + 0.979595i \(0.564413\pi\)
\(968\) −5.04480e6 −0.173044
\(969\) −2.40113e6 −0.0821499
\(970\) 0 0
\(971\) 3.07622e6 0.104706 0.0523528 0.998629i \(-0.483328\pi\)
0.0523528 + 0.998629i \(0.483328\pi\)
\(972\) 1.62812e7 0.552739
\(973\) −2.92024e7 −0.988865
\(974\) 2.38021e6 0.0803930
\(975\) 0 0
\(976\) −7.18964e6 −0.241592
\(977\) −3.81186e7 −1.27762 −0.638809 0.769365i \(-0.720573\pi\)
−0.638809 + 0.769365i \(0.720573\pi\)
\(978\) −930714. −0.0311150
\(979\) −4.69350e6 −0.156509
\(980\) 0 0
\(981\) −1.37345e7 −0.455661
\(982\) 2.72616e6 0.0902137
\(983\) 3.14497e7 1.03809 0.519043 0.854748i \(-0.326289\pi\)
0.519043 + 0.854748i \(0.326289\pi\)
\(984\) 697878. 0.0229769
\(985\) 0 0
\(986\) 1.00182e6 0.0328170
\(987\) 2.58683e6 0.0845231
\(988\) 5.66783e7 1.84725
\(989\) −7.69821e6 −0.250264
\(990\) 0 0
\(991\) 1.08378e7 0.350555 0.175277 0.984519i \(-0.443918\pi\)
0.175277 + 0.984519i \(0.443918\pi\)
\(992\) −1.10558e7 −0.356705
\(993\) 863027. 0.0277748
\(994\) 2.90842e6 0.0933665
\(995\) 0 0
\(996\) −6.81715e6 −0.217748
\(997\) 1.37214e7 0.437179 0.218590 0.975817i \(-0.429854\pi\)
0.218590 + 0.975817i \(0.429854\pi\)
\(998\) −5.63211e6 −0.178997
\(999\) 4.80308e6 0.152267
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 1075.6.a.a.1.4 8
5.4 even 2 43.6.a.a.1.5 8
15.14 odd 2 387.6.a.c.1.4 8
20.19 odd 2 688.6.a.e.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.5 8 5.4 even 2
387.6.a.c.1.4 8 15.14 odd 2
688.6.a.e.1.4 8 20.19 odd 2
1075.6.a.a.1.4 8 1.1 even 1 trivial