Properties

Label 825.6.a.m.1.4
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.60892\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.60892 q^{2} +9.00000 q^{3} -25.1935 q^{4} -23.4803 q^{6} +174.969 q^{7} +149.214 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-2.60892 q^{2} +9.00000 q^{3} -25.1935 q^{4} -23.4803 q^{6} +174.969 q^{7} +149.214 q^{8} +81.0000 q^{9} +121.000 q^{11} -226.742 q^{12} -815.527 q^{13} -456.481 q^{14} +416.906 q^{16} +462.991 q^{17} -211.323 q^{18} -1996.09 q^{19} +1574.72 q^{21} -315.680 q^{22} +1342.70 q^{23} +1342.92 q^{24} +2127.65 q^{26} +729.000 q^{27} -4408.08 q^{28} +6586.33 q^{29} -4492.74 q^{31} -5862.51 q^{32} +1089.00 q^{33} -1207.91 q^{34} -2040.67 q^{36} -9089.95 q^{37} +5207.64 q^{38} -7339.74 q^{39} -1108.16 q^{41} -4108.32 q^{42} -20231.4 q^{43} -3048.42 q^{44} -3503.01 q^{46} -9098.00 q^{47} +3752.15 q^{48} +13807.1 q^{49} +4166.92 q^{51} +20546.0 q^{52} -38340.2 q^{53} -1901.91 q^{54} +26107.7 q^{56} -17964.8 q^{57} -17183.2 q^{58} +29309.0 q^{59} +32390.1 q^{61} +11721.2 q^{62} +14172.5 q^{63} +1953.85 q^{64} -2841.12 q^{66} +64412.9 q^{67} -11664.4 q^{68} +12084.3 q^{69} -63314.4 q^{71} +12086.3 q^{72} -4121.83 q^{73} +23715.0 q^{74} +50288.4 q^{76} +21171.2 q^{77} +19148.8 q^{78} -53233.3 q^{79} +6561.00 q^{81} +2891.10 q^{82} +119847. q^{83} -39672.7 q^{84} +52782.1 q^{86} +59277.0 q^{87} +18054.8 q^{88} +66306.8 q^{89} -142692. q^{91} -33827.4 q^{92} -40434.6 q^{93} +23736.0 q^{94} -52762.6 q^{96} +180567. q^{97} -36021.7 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.60892 −0.461197 −0.230598 0.973049i \(-0.574068\pi\)
−0.230598 + 0.973049i \(0.574068\pi\)
\(3\) 9.00000 0.577350
\(4\) −25.1935 −0.787297
\(5\) 0 0
\(6\) −23.4803 −0.266272
\(7\) 174.969 1.34963 0.674817 0.737985i \(-0.264223\pi\)
0.674817 + 0.737985i \(0.264223\pi\)
\(8\) 149.214 0.824296
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −226.742 −0.454546
\(13\) −815.527 −1.33838 −0.669191 0.743091i \(-0.733359\pi\)
−0.669191 + 0.743091i \(0.733359\pi\)
\(14\) −456.481 −0.622447
\(15\) 0 0
\(16\) 416.906 0.407135
\(17\) 462.991 0.388553 0.194276 0.980947i \(-0.437764\pi\)
0.194276 + 0.980947i \(0.437764\pi\)
\(18\) −211.323 −0.153732
\(19\) −1996.09 −1.26851 −0.634257 0.773122i \(-0.718694\pi\)
−0.634257 + 0.773122i \(0.718694\pi\)
\(20\) 0 0
\(21\) 1574.72 0.779211
\(22\) −315.680 −0.139056
\(23\) 1342.70 0.529250 0.264625 0.964351i \(-0.414752\pi\)
0.264625 + 0.964351i \(0.414752\pi\)
\(24\) 1342.92 0.475908
\(25\) 0 0
\(26\) 2127.65 0.617257
\(27\) 729.000 0.192450
\(28\) −4408.08 −1.06256
\(29\) 6586.33 1.45428 0.727141 0.686488i \(-0.240849\pi\)
0.727141 + 0.686488i \(0.240849\pi\)
\(30\) 0 0
\(31\) −4492.74 −0.839667 −0.419833 0.907601i \(-0.637911\pi\)
−0.419833 + 0.907601i \(0.637911\pi\)
\(32\) −5862.51 −1.01207
\(33\) 1089.00 0.174078
\(34\) −1207.91 −0.179199
\(35\) 0 0
\(36\) −2040.67 −0.262432
\(37\) −9089.95 −1.09158 −0.545792 0.837921i \(-0.683771\pi\)
−0.545792 + 0.837921i \(0.683771\pi\)
\(38\) 5207.64 0.585035
\(39\) −7339.74 −0.772715
\(40\) 0 0
\(41\) −1108.16 −0.102954 −0.0514769 0.998674i \(-0.516393\pi\)
−0.0514769 + 0.998674i \(0.516393\pi\)
\(42\) −4108.32 −0.359370
\(43\) −20231.4 −1.66861 −0.834304 0.551304i \(-0.814130\pi\)
−0.834304 + 0.551304i \(0.814130\pi\)
\(44\) −3048.42 −0.237379
\(45\) 0 0
\(46\) −3503.01 −0.244088
\(47\) −9098.00 −0.600761 −0.300380 0.953820i \(-0.597114\pi\)
−0.300380 + 0.953820i \(0.597114\pi\)
\(48\) 3752.15 0.235059
\(49\) 13807.1 0.821510
\(50\) 0 0
\(51\) 4166.92 0.224331
\(52\) 20546.0 1.05370
\(53\) −38340.2 −1.87484 −0.937422 0.348196i \(-0.886794\pi\)
−0.937422 + 0.348196i \(0.886794\pi\)
\(54\) −1901.91 −0.0887574
\(55\) 0 0
\(56\) 26107.7 1.11250
\(57\) −17964.8 −0.732377
\(58\) −17183.2 −0.670711
\(59\) 29309.0 1.09615 0.548076 0.836429i \(-0.315361\pi\)
0.548076 + 0.836429i \(0.315361\pi\)
\(60\) 0 0
\(61\) 32390.1 1.11452 0.557260 0.830338i \(-0.311853\pi\)
0.557260 + 0.830338i \(0.311853\pi\)
\(62\) 11721.2 0.387252
\(63\) 14172.5 0.449878
\(64\) 1953.85 0.0596267
\(65\) 0 0
\(66\) −2841.12 −0.0802841
\(67\) 64412.9 1.75302 0.876508 0.481387i \(-0.159867\pi\)
0.876508 + 0.481387i \(0.159867\pi\)
\(68\) −11664.4 −0.305907
\(69\) 12084.3 0.305562
\(70\) 0 0
\(71\) −63314.4 −1.49058 −0.745292 0.666738i \(-0.767690\pi\)
−0.745292 + 0.666738i \(0.767690\pi\)
\(72\) 12086.3 0.274765
\(73\) −4121.83 −0.0905279 −0.0452640 0.998975i \(-0.514413\pi\)
−0.0452640 + 0.998975i \(0.514413\pi\)
\(74\) 23715.0 0.503435
\(75\) 0 0
\(76\) 50288.4 0.998698
\(77\) 21171.2 0.406930
\(78\) 19148.8 0.356374
\(79\) −53233.3 −0.959656 −0.479828 0.877363i \(-0.659301\pi\)
−0.479828 + 0.877363i \(0.659301\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 2891.10 0.0474820
\(83\) 119847. 1.90956 0.954780 0.297315i \(-0.0960910\pi\)
0.954780 + 0.297315i \(0.0960910\pi\)
\(84\) −39672.7 −0.613471
\(85\) 0 0
\(86\) 52782.1 0.769557
\(87\) 59277.0 0.839630
\(88\) 18054.8 0.248535
\(89\) 66306.8 0.887326 0.443663 0.896194i \(-0.353679\pi\)
0.443663 + 0.896194i \(0.353679\pi\)
\(90\) 0 0
\(91\) −142692. −1.80632
\(92\) −33827.4 −0.416677
\(93\) −40434.6 −0.484782
\(94\) 23736.0 0.277069
\(95\) 0 0
\(96\) −52762.6 −0.584316
\(97\) 180567. 1.94854 0.974268 0.225394i \(-0.0723670\pi\)
0.974268 + 0.225394i \(0.0723670\pi\)
\(98\) −36021.7 −0.378878
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −113813. −1.11017 −0.555084 0.831794i \(-0.687314\pi\)
−0.555084 + 0.831794i \(0.687314\pi\)
\(102\) −10871.2 −0.103461
\(103\) −192758. −1.79027 −0.895135 0.445795i \(-0.852921\pi\)
−0.895135 + 0.445795i \(0.852921\pi\)
\(104\) −121688. −1.10322
\(105\) 0 0
\(106\) 100027. 0.864672
\(107\) 139114. 1.17465 0.587327 0.809350i \(-0.300180\pi\)
0.587327 + 0.809350i \(0.300180\pi\)
\(108\) −18366.1 −0.151515
\(109\) 97360.0 0.784900 0.392450 0.919773i \(-0.371628\pi\)
0.392450 + 0.919773i \(0.371628\pi\)
\(110\) 0 0
\(111\) −81809.6 −0.630227
\(112\) 72945.6 0.549483
\(113\) 20952.6 0.154362 0.0771811 0.997017i \(-0.475408\pi\)
0.0771811 + 0.997017i \(0.475408\pi\)
\(114\) 46868.7 0.337770
\(115\) 0 0
\(116\) −165933. −1.14495
\(117\) −66057.7 −0.446127
\(118\) −76464.9 −0.505542
\(119\) 81009.0 0.524404
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −84503.4 −0.514013
\(123\) −9973.43 −0.0594404
\(124\) 113188. 0.661067
\(125\) 0 0
\(126\) −36974.9 −0.207482
\(127\) −347758. −1.91323 −0.956616 0.291351i \(-0.905895\pi\)
−0.956616 + 0.291351i \(0.905895\pi\)
\(128\) 182503. 0.984566
\(129\) −182083. −0.963372
\(130\) 0 0
\(131\) −160486. −0.817070 −0.408535 0.912743i \(-0.633960\pi\)
−0.408535 + 0.912743i \(0.633960\pi\)
\(132\) −27435.7 −0.137051
\(133\) −349253. −1.71203
\(134\) −168048. −0.808486
\(135\) 0 0
\(136\) 69084.5 0.320282
\(137\) 142596. 0.649090 0.324545 0.945870i \(-0.394789\pi\)
0.324545 + 0.945870i \(0.394789\pi\)
\(138\) −31527.1 −0.140924
\(139\) −124192. −0.545201 −0.272600 0.962127i \(-0.587884\pi\)
−0.272600 + 0.962127i \(0.587884\pi\)
\(140\) 0 0
\(141\) −81882.0 −0.346849
\(142\) 165182. 0.687453
\(143\) −98678.7 −0.403537
\(144\) 33769.4 0.135712
\(145\) 0 0
\(146\) 10753.5 0.0417512
\(147\) 124264. 0.474299
\(148\) 229008. 0.859402
\(149\) −332027. −1.22520 −0.612601 0.790393i \(-0.709877\pi\)
−0.612601 + 0.790393i \(0.709877\pi\)
\(150\) 0 0
\(151\) −221114. −0.789177 −0.394589 0.918858i \(-0.629113\pi\)
−0.394589 + 0.918858i \(0.629113\pi\)
\(152\) −297843. −1.04563
\(153\) 37502.2 0.129518
\(154\) −55234.1 −0.187675
\(155\) 0 0
\(156\) 184914. 0.608356
\(157\) −178933. −0.579350 −0.289675 0.957125i \(-0.593547\pi\)
−0.289675 + 0.957125i \(0.593547\pi\)
\(158\) 138882. 0.442590
\(159\) −345062. −1.08244
\(160\) 0 0
\(161\) 234931. 0.714293
\(162\) −17117.1 −0.0512441
\(163\) −222185. −0.655007 −0.327504 0.944850i \(-0.606207\pi\)
−0.327504 + 0.944850i \(0.606207\pi\)
\(164\) 27918.4 0.0810553
\(165\) 0 0
\(166\) −312673. −0.880683
\(167\) −286150. −0.793968 −0.396984 0.917826i \(-0.629943\pi\)
−0.396984 + 0.917826i \(0.629943\pi\)
\(168\) 234970. 0.642301
\(169\) 293791. 0.791265
\(170\) 0 0
\(171\) −161683. −0.422838
\(172\) 509700. 1.31369
\(173\) −585626. −1.48767 −0.743833 0.668366i \(-0.766994\pi\)
−0.743833 + 0.668366i \(0.766994\pi\)
\(174\) −154649. −0.387235
\(175\) 0 0
\(176\) 50445.6 0.122756
\(177\) 263781. 0.632863
\(178\) −172989. −0.409232
\(179\) −511386. −1.19293 −0.596467 0.802638i \(-0.703429\pi\)
−0.596467 + 0.802638i \(0.703429\pi\)
\(180\) 0 0
\(181\) −16277.3 −0.0369306 −0.0184653 0.999830i \(-0.505878\pi\)
−0.0184653 + 0.999830i \(0.505878\pi\)
\(182\) 372272. 0.833071
\(183\) 291511. 0.643469
\(184\) 200350. 0.436258
\(185\) 0 0
\(186\) 105491. 0.223580
\(187\) 56021.9 0.117153
\(188\) 229211. 0.472977
\(189\) 127552. 0.259737
\(190\) 0 0
\(191\) −219087. −0.434543 −0.217271 0.976111i \(-0.569716\pi\)
−0.217271 + 0.976111i \(0.569716\pi\)
\(192\) 17584.6 0.0344255
\(193\) 704505. 1.36142 0.680708 0.732555i \(-0.261672\pi\)
0.680708 + 0.732555i \(0.261672\pi\)
\(194\) −471085. −0.898658
\(195\) 0 0
\(196\) −347850. −0.646773
\(197\) 670027. 1.23006 0.615031 0.788503i \(-0.289144\pi\)
0.615031 + 0.788503i \(0.289144\pi\)
\(198\) −25570.1 −0.0463520
\(199\) −591128. −1.05815 −0.529077 0.848574i \(-0.677462\pi\)
−0.529077 + 0.848574i \(0.677462\pi\)
\(200\) 0 0
\(201\) 579716. 1.01210
\(202\) 296930. 0.512006
\(203\) 1.15240e6 1.96275
\(204\) −104979. −0.176615
\(205\) 0 0
\(206\) 502890. 0.825667
\(207\) 108759. 0.176417
\(208\) −339998. −0.544901
\(209\) −241526. −0.382471
\(210\) 0 0
\(211\) −1070.59 −0.00165545 −0.000827723 1.00000i \(-0.500263\pi\)
−0.000827723 1.00000i \(0.500263\pi\)
\(212\) 965925. 1.47606
\(213\) −569829. −0.860589
\(214\) −362937. −0.541747
\(215\) 0 0
\(216\) 108777. 0.158636
\(217\) −786090. −1.13324
\(218\) −254005. −0.361993
\(219\) −37096.4 −0.0522663
\(220\) 0 0
\(221\) −377581. −0.520032
\(222\) 213435. 0.290659
\(223\) −515555. −0.694246 −0.347123 0.937820i \(-0.612841\pi\)
−0.347123 + 0.937820i \(0.612841\pi\)
\(224\) −1.02576e6 −1.36592
\(225\) 0 0
\(226\) −54663.6 −0.0711914
\(227\) −69874.2 −0.0900020 −0.0450010 0.998987i \(-0.514329\pi\)
−0.0450010 + 0.998987i \(0.514329\pi\)
\(228\) 452596. 0.576599
\(229\) 502115. 0.632725 0.316362 0.948638i \(-0.397538\pi\)
0.316362 + 0.948638i \(0.397538\pi\)
\(230\) 0 0
\(231\) 190541. 0.234941
\(232\) 982770. 1.19876
\(233\) −1.52257e6 −1.83733 −0.918664 0.395041i \(-0.870730\pi\)
−0.918664 + 0.395041i \(0.870730\pi\)
\(234\) 172339. 0.205752
\(235\) 0 0
\(236\) −738396. −0.862997
\(237\) −479099. −0.554058
\(238\) −211346. −0.241853
\(239\) −601975. −0.681685 −0.340843 0.940120i \(-0.610712\pi\)
−0.340843 + 0.940120i \(0.610712\pi\)
\(240\) 0 0
\(241\) −960570. −1.06534 −0.532668 0.846324i \(-0.678811\pi\)
−0.532668 + 0.846324i \(0.678811\pi\)
\(242\) −38197.3 −0.0419270
\(243\) 59049.0 0.0641500
\(244\) −816021. −0.877459
\(245\) 0 0
\(246\) 26019.9 0.0274137
\(247\) 1.62786e6 1.69776
\(248\) −670377. −0.692134
\(249\) 1.07863e6 1.10248
\(250\) 0 0
\(251\) −999437. −1.00132 −0.500658 0.865645i \(-0.666909\pi\)
−0.500658 + 0.865645i \(0.666909\pi\)
\(252\) −357055. −0.354188
\(253\) 162467. 0.159575
\(254\) 907274. 0.882377
\(255\) 0 0
\(256\) −538659. −0.513705
\(257\) −222038. −0.209698 −0.104849 0.994488i \(-0.533436\pi\)
−0.104849 + 0.994488i \(0.533436\pi\)
\(258\) 475039. 0.444304
\(259\) −1.59046e6 −1.47324
\(260\) 0 0
\(261\) 533493. 0.484761
\(262\) 418696. 0.376830
\(263\) 501628. 0.447190 0.223595 0.974682i \(-0.428221\pi\)
0.223595 + 0.974682i \(0.428221\pi\)
\(264\) 162494. 0.143492
\(265\) 0 0
\(266\) 911175. 0.789583
\(267\) 596761. 0.512298
\(268\) −1.62279e6 −1.38015
\(269\) −1.50544e6 −1.26848 −0.634238 0.773138i \(-0.718686\pi\)
−0.634238 + 0.773138i \(0.718686\pi\)
\(270\) 0 0
\(271\) 935698. 0.773949 0.386975 0.922090i \(-0.373520\pi\)
0.386975 + 0.922090i \(0.373520\pi\)
\(272\) 193024. 0.158193
\(273\) −1.28423e6 −1.04288
\(274\) −372021. −0.299358
\(275\) 0 0
\(276\) −304447. −0.240569
\(277\) −1.06811e6 −0.836404 −0.418202 0.908354i \(-0.637340\pi\)
−0.418202 + 0.908354i \(0.637340\pi\)
\(278\) 324007. 0.251445
\(279\) −363912. −0.279889
\(280\) 0 0
\(281\) 683491. 0.516377 0.258189 0.966095i \(-0.416874\pi\)
0.258189 + 0.966095i \(0.416874\pi\)
\(282\) 213624. 0.159966
\(283\) −272388. −0.202172 −0.101086 0.994878i \(-0.532232\pi\)
−0.101086 + 0.994878i \(0.532232\pi\)
\(284\) 1.59511e6 1.17353
\(285\) 0 0
\(286\) 257445. 0.186110
\(287\) −193893. −0.138950
\(288\) −474863. −0.337355
\(289\) −1.20550e6 −0.849027
\(290\) 0 0
\(291\) 1.62510e6 1.12499
\(292\) 103843. 0.0712724
\(293\) −1.53914e6 −1.04739 −0.523697 0.851905i \(-0.675447\pi\)
−0.523697 + 0.851905i \(0.675447\pi\)
\(294\) −324196. −0.218745
\(295\) 0 0
\(296\) −1.35634e6 −0.899789
\(297\) 88209.0 0.0580259
\(298\) 866233. 0.565059
\(299\) −1.09501e6 −0.708338
\(300\) 0 0
\(301\) −3.53986e6 −2.25201
\(302\) 576870. 0.363966
\(303\) −1.02432e6 −0.640956
\(304\) −832180. −0.516456
\(305\) 0 0
\(306\) −97840.5 −0.0597331
\(307\) 800678. 0.484855 0.242428 0.970170i \(-0.422056\pi\)
0.242428 + 0.970170i \(0.422056\pi\)
\(308\) −533378. −0.320375
\(309\) −1.73482e6 −1.03361
\(310\) 0 0
\(311\) 3.17233e6 1.85985 0.929925 0.367750i \(-0.119872\pi\)
0.929925 + 0.367750i \(0.119872\pi\)
\(312\) −1.09519e6 −0.636946
\(313\) −996401. −0.574875 −0.287437 0.957799i \(-0.592803\pi\)
−0.287437 + 0.957799i \(0.592803\pi\)
\(314\) 466822. 0.267194
\(315\) 0 0
\(316\) 1.34113e6 0.755535
\(317\) 1.73853e6 0.971702 0.485851 0.874042i \(-0.338510\pi\)
0.485851 + 0.874042i \(0.338510\pi\)
\(318\) 900241. 0.499219
\(319\) 796946. 0.438483
\(320\) 0 0
\(321\) 1.25202e6 0.678187
\(322\) −612918. −0.329430
\(323\) −924170. −0.492885
\(324\) −165295. −0.0874775
\(325\) 0 0
\(326\) 579664. 0.302087
\(327\) 876240. 0.453162
\(328\) −165352. −0.0848644
\(329\) −1.59187e6 −0.810807
\(330\) 0 0
\(331\) −530501. −0.266144 −0.133072 0.991106i \(-0.542484\pi\)
−0.133072 + 0.991106i \(0.542484\pi\)
\(332\) −3.01938e6 −1.50339
\(333\) −736286. −0.363861
\(334\) 746544. 0.366176
\(335\) 0 0
\(336\) 656510. 0.317244
\(337\) 1110.26 0.000532538 0 0.000266269 1.00000i \(-0.499915\pi\)
0.000266269 1.00000i \(0.499915\pi\)
\(338\) −766478. −0.364929
\(339\) 188573. 0.0891211
\(340\) 0 0
\(341\) −543621. −0.253169
\(342\) 421819. 0.195012
\(343\) −524885. −0.240896
\(344\) −3.01880e6 −1.37543
\(345\) 0 0
\(346\) 1.52785e6 0.686107
\(347\) −1.08849e6 −0.485289 −0.242644 0.970115i \(-0.578015\pi\)
−0.242644 + 0.970115i \(0.578015\pi\)
\(348\) −1.49340e6 −0.661039
\(349\) −591038. −0.259748 −0.129874 0.991531i \(-0.541457\pi\)
−0.129874 + 0.991531i \(0.541457\pi\)
\(350\) 0 0
\(351\) −594519. −0.257572
\(352\) −709363. −0.305149
\(353\) −980027. −0.418602 −0.209301 0.977851i \(-0.567119\pi\)
−0.209301 + 0.977851i \(0.567119\pi\)
\(354\) −688184. −0.291875
\(355\) 0 0
\(356\) −1.67050e6 −0.698589
\(357\) 729081. 0.302765
\(358\) 1.33417e6 0.550177
\(359\) −2.87712e6 −1.17821 −0.589104 0.808057i \(-0.700519\pi\)
−0.589104 + 0.808057i \(0.700519\pi\)
\(360\) 0 0
\(361\) 1.50826e6 0.609129
\(362\) 42466.3 0.0170323
\(363\) 131769. 0.0524864
\(364\) 3.59491e6 1.42211
\(365\) 0 0
\(366\) −760530. −0.296766
\(367\) −3.28496e6 −1.27311 −0.636554 0.771232i \(-0.719641\pi\)
−0.636554 + 0.771232i \(0.719641\pi\)
\(368\) 559781. 0.215476
\(369\) −89760.9 −0.0343179
\(370\) 0 0
\(371\) −6.70835e6 −2.53035
\(372\) 1.01869e6 0.381667
\(373\) −1.03382e6 −0.384744 −0.192372 0.981322i \(-0.561618\pi\)
−0.192372 + 0.981322i \(0.561618\pi\)
\(374\) −146157. −0.0540306
\(375\) 0 0
\(376\) −1.35755e6 −0.495205
\(377\) −5.37133e6 −1.94638
\(378\) −332774. −0.119790
\(379\) 3.41421e6 1.22094 0.610468 0.792041i \(-0.290982\pi\)
0.610468 + 0.792041i \(0.290982\pi\)
\(380\) 0 0
\(381\) −3.12982e6 −1.10461
\(382\) 571580. 0.200410
\(383\) 1.82511e6 0.635758 0.317879 0.948131i \(-0.397029\pi\)
0.317879 + 0.948131i \(0.397029\pi\)
\(384\) 1.64253e6 0.568439
\(385\) 0 0
\(386\) −1.83800e6 −0.627881
\(387\) −1.63874e6 −0.556203
\(388\) −4.54911e6 −1.53408
\(389\) −1.31381e6 −0.440210 −0.220105 0.975476i \(-0.570640\pi\)
−0.220105 + 0.975476i \(0.570640\pi\)
\(390\) 0 0
\(391\) 621659. 0.205641
\(392\) 2.06021e6 0.677168
\(393\) −1.44438e6 −0.471736
\(394\) −1.74805e6 −0.567301
\(395\) 0 0
\(396\) −246922. −0.0791264
\(397\) 4.07499e6 1.29763 0.648814 0.760947i \(-0.275265\pi\)
0.648814 + 0.760947i \(0.275265\pi\)
\(398\) 1.54221e6 0.488017
\(399\) −3.14328e6 −0.988441
\(400\) 0 0
\(401\) 1.88854e6 0.586495 0.293247 0.956037i \(-0.405264\pi\)
0.293247 + 0.956037i \(0.405264\pi\)
\(402\) −1.51244e6 −0.466779
\(403\) 3.66395e6 1.12379
\(404\) 2.86735e6 0.874033
\(405\) 0 0
\(406\) −3.00653e6 −0.905213
\(407\) −1.09988e6 −0.329125
\(408\) 621760. 0.184915
\(409\) −2.77474e6 −0.820188 −0.410094 0.912043i \(-0.634504\pi\)
−0.410094 + 0.912043i \(0.634504\pi\)
\(410\) 0 0
\(411\) 1.28336e6 0.374752
\(412\) 4.85624e6 1.40948
\(413\) 5.12816e6 1.47940
\(414\) −283744. −0.0813628
\(415\) 0 0
\(416\) 4.78103e6 1.35453
\(417\) −1.11773e6 −0.314772
\(418\) 630124. 0.176395
\(419\) 1.74694e6 0.486119 0.243060 0.970011i \(-0.421849\pi\)
0.243060 + 0.970011i \(0.421849\pi\)
\(420\) 0 0
\(421\) 321171. 0.0883143 0.0441572 0.999025i \(-0.485940\pi\)
0.0441572 + 0.999025i \(0.485940\pi\)
\(422\) 2793.08 0.000763487 0
\(423\) −736938. −0.200254
\(424\) −5.72088e6 −1.54543
\(425\) 0 0
\(426\) 1.48664e6 0.396901
\(427\) 5.66726e6 1.50419
\(428\) −3.50476e6 −0.924802
\(429\) −888109. −0.232982
\(430\) 0 0
\(431\) 1.16098e6 0.301046 0.150523 0.988607i \(-0.451904\pi\)
0.150523 + 0.988607i \(0.451904\pi\)
\(432\) 303924. 0.0783531
\(433\) 1.65899e6 0.425229 0.212615 0.977136i \(-0.431802\pi\)
0.212615 + 0.977136i \(0.431802\pi\)
\(434\) 2.05085e6 0.522648
\(435\) 0 0
\(436\) −2.45284e6 −0.617950
\(437\) −2.68015e6 −0.671361
\(438\) 96781.8 0.0241051
\(439\) 7.30846e6 1.80994 0.904971 0.425473i \(-0.139892\pi\)
0.904971 + 0.425473i \(0.139892\pi\)
\(440\) 0 0
\(441\) 1.11838e6 0.273837
\(442\) 985081. 0.239837
\(443\) 3.34639e6 0.810153 0.405077 0.914283i \(-0.367245\pi\)
0.405077 + 0.914283i \(0.367245\pi\)
\(444\) 2.06107e6 0.496176
\(445\) 0 0
\(446\) 1.34504e6 0.320184
\(447\) −2.98824e6 −0.707370
\(448\) 341863. 0.0804742
\(449\) −2.28621e6 −0.535180 −0.267590 0.963533i \(-0.586227\pi\)
−0.267590 + 0.963533i \(0.586227\pi\)
\(450\) 0 0
\(451\) −134087. −0.0310417
\(452\) −527869. −0.121529
\(453\) −1.99003e6 −0.455632
\(454\) 182297. 0.0415087
\(455\) 0 0
\(456\) −2.68059e6 −0.603696
\(457\) 3.57185e6 0.800024 0.400012 0.916510i \(-0.369006\pi\)
0.400012 + 0.916510i \(0.369006\pi\)
\(458\) −1.30998e6 −0.291811
\(459\) 337520. 0.0747770
\(460\) 0 0
\(461\) −7.95525e6 −1.74342 −0.871709 0.490024i \(-0.836988\pi\)
−0.871709 + 0.490024i \(0.836988\pi\)
\(462\) −497107. −0.108354
\(463\) 5.70155e6 1.23606 0.618032 0.786153i \(-0.287930\pi\)
0.618032 + 0.786153i \(0.287930\pi\)
\(464\) 2.74588e6 0.592089
\(465\) 0 0
\(466\) 3.97226e6 0.847370
\(467\) 513186. 0.108889 0.0544444 0.998517i \(-0.482661\pi\)
0.0544444 + 0.998517i \(0.482661\pi\)
\(468\) 1.66423e6 0.351235
\(469\) 1.12703e7 2.36593
\(470\) 0 0
\(471\) −1.61040e6 −0.334488
\(472\) 4.37329e6 0.903553
\(473\) −2.44800e6 −0.503105
\(474\) 1.24993e6 0.255530
\(475\) 0 0
\(476\) −2.04090e6 −0.412862
\(477\) −3.10556e6 −0.624948
\(478\) 1.57051e6 0.314391
\(479\) 5.59628e6 1.11445 0.557225 0.830361i \(-0.311866\pi\)
0.557225 + 0.830361i \(0.311866\pi\)
\(480\) 0 0
\(481\) 7.41310e6 1.46096
\(482\) 2.50605e6 0.491330
\(483\) 2.11438e6 0.412397
\(484\) −368858. −0.0715725
\(485\) 0 0
\(486\) −154054. −0.0295858
\(487\) 425345. 0.0812678 0.0406339 0.999174i \(-0.487062\pi\)
0.0406339 + 0.999174i \(0.487062\pi\)
\(488\) 4.83304e6 0.918695
\(489\) −1.99967e6 −0.378169
\(490\) 0 0
\(491\) −5.56781e6 −1.04227 −0.521136 0.853474i \(-0.674492\pi\)
−0.521136 + 0.853474i \(0.674492\pi\)
\(492\) 251266. 0.0467973
\(493\) 3.04941e6 0.565065
\(494\) −4.24697e6 −0.783000
\(495\) 0 0
\(496\) −1.87305e6 −0.341857
\(497\) −1.10780e7 −2.01174
\(498\) −2.81405e6 −0.508462
\(499\) −5.31330e6 −0.955241 −0.477621 0.878566i \(-0.658501\pi\)
−0.477621 + 0.878566i \(0.658501\pi\)
\(500\) 0 0
\(501\) −2.57535e6 −0.458398
\(502\) 2.60745e6 0.461804
\(503\) −1.03261e7 −1.81976 −0.909882 0.414867i \(-0.863828\pi\)
−0.909882 + 0.414867i \(0.863828\pi\)
\(504\) 2.11473e6 0.370832
\(505\) 0 0
\(506\) −423864. −0.0735954
\(507\) 2.64412e6 0.456837
\(508\) 8.76125e6 1.50628
\(509\) 1.37144e6 0.234629 0.117314 0.993095i \(-0.462571\pi\)
0.117314 + 0.993095i \(0.462571\pi\)
\(510\) 0 0
\(511\) −721191. −0.122179
\(512\) −4.43477e6 −0.747646
\(513\) −1.45515e6 −0.244126
\(514\) 579280. 0.0967121
\(515\) 0 0
\(516\) 4.58730e6 0.758460
\(517\) −1.10086e6 −0.181136
\(518\) 4.14939e6 0.679453
\(519\) −5.27064e6 −0.858904
\(520\) 0 0
\(521\) 524277. 0.0846187 0.0423094 0.999105i \(-0.486528\pi\)
0.0423094 + 0.999105i \(0.486528\pi\)
\(522\) −1.39184e6 −0.223570
\(523\) 1.42764e6 0.228225 0.114113 0.993468i \(-0.463598\pi\)
0.114113 + 0.993468i \(0.463598\pi\)
\(524\) 4.04321e6 0.643277
\(525\) 0 0
\(526\) −1.30871e6 −0.206243
\(527\) −2.08010e6 −0.326255
\(528\) 454010. 0.0708730
\(529\) −4.63349e6 −0.719895
\(530\) 0 0
\(531\) 2.37403e6 0.365384
\(532\) 8.79892e6 1.34788
\(533\) 903733. 0.137791
\(534\) −1.55690e6 −0.236270
\(535\) 0 0
\(536\) 9.61128e6 1.44500
\(537\) −4.60248e6 −0.688741
\(538\) 3.92757e6 0.585017
\(539\) 1.67066e6 0.247695
\(540\) 0 0
\(541\) 5.39626e6 0.792683 0.396341 0.918103i \(-0.370280\pi\)
0.396341 + 0.918103i \(0.370280\pi\)
\(542\) −2.44116e6 −0.356943
\(543\) −146496. −0.0213219
\(544\) −2.71429e6 −0.393241
\(545\) 0 0
\(546\) 3.35045e6 0.480974
\(547\) 7.34437e6 1.04951 0.524754 0.851254i \(-0.324157\pi\)
0.524754 + 0.851254i \(0.324157\pi\)
\(548\) −3.59249e6 −0.511027
\(549\) 2.62360e6 0.371507
\(550\) 0 0
\(551\) −1.31469e7 −1.84478
\(552\) 1.80315e6 0.251874
\(553\) −9.31417e6 −1.29518
\(554\) 2.78662e6 0.385747
\(555\) 0 0
\(556\) 3.12883e6 0.429235
\(557\) −7.05449e6 −0.963447 −0.481723 0.876323i \(-0.659989\pi\)
−0.481723 + 0.876323i \(0.659989\pi\)
\(558\) 949418. 0.129084
\(559\) 1.64992e7 2.23324
\(560\) 0 0
\(561\) 504197. 0.0676383
\(562\) −1.78318e6 −0.238152
\(563\) −126964. −0.0168814 −0.00844072 0.999964i \(-0.502687\pi\)
−0.00844072 + 0.999964i \(0.502687\pi\)
\(564\) 2.06290e6 0.273074
\(565\) 0 0
\(566\) 710638. 0.0932411
\(567\) 1.14797e6 0.149959
\(568\) −9.44736e6 −1.22868
\(569\) −1.36950e7 −1.77330 −0.886648 0.462445i \(-0.846972\pi\)
−0.886648 + 0.462445i \(0.846972\pi\)
\(570\) 0 0
\(571\) −484553. −0.0621944 −0.0310972 0.999516i \(-0.509900\pi\)
−0.0310972 + 0.999516i \(0.509900\pi\)
\(572\) 2.48606e6 0.317704
\(573\) −1.97178e6 −0.250883
\(574\) 505853. 0.0640833
\(575\) 0 0
\(576\) 158262. 0.0198756
\(577\) 4.31027e6 0.538970 0.269485 0.963005i \(-0.413147\pi\)
0.269485 + 0.963005i \(0.413147\pi\)
\(578\) 3.14505e6 0.391569
\(579\) 6.34054e6 0.786014
\(580\) 0 0
\(581\) 2.09696e7 2.57720
\(582\) −4.23976e6 −0.518841
\(583\) −4.63917e6 −0.565287
\(584\) −615032. −0.0746218
\(585\) 0 0
\(586\) 4.01551e6 0.483055
\(587\) 4.35136e6 0.521230 0.260615 0.965443i \(-0.416075\pi\)
0.260615 + 0.965443i \(0.416075\pi\)
\(588\) −3.13065e6 −0.373415
\(589\) 8.96790e6 1.06513
\(590\) 0 0
\(591\) 6.03024e6 0.710176
\(592\) −3.78965e6 −0.444422
\(593\) 2.85779e6 0.333729 0.166865 0.985980i \(-0.446636\pi\)
0.166865 + 0.985980i \(0.446636\pi\)
\(594\) −230131. −0.0267614
\(595\) 0 0
\(596\) 8.36492e6 0.964598
\(597\) −5.32015e6 −0.610926
\(598\) 2.85680e6 0.326683
\(599\) 1.62076e7 1.84566 0.922831 0.385206i \(-0.125870\pi\)
0.922831 + 0.385206i \(0.125870\pi\)
\(600\) 0 0
\(601\) −6.74322e6 −0.761520 −0.380760 0.924674i \(-0.624338\pi\)
−0.380760 + 0.924674i \(0.624338\pi\)
\(602\) 9.23524e6 1.03862
\(603\) 5.21745e6 0.584339
\(604\) 5.57065e6 0.621317
\(605\) 0 0
\(606\) 2.67237e6 0.295607
\(607\) 1.76156e6 0.194055 0.0970276 0.995282i \(-0.469067\pi\)
0.0970276 + 0.995282i \(0.469067\pi\)
\(608\) 1.17021e7 1.28382
\(609\) 1.03716e7 1.13319
\(610\) 0 0
\(611\) 7.41967e6 0.804047
\(612\) −944813. −0.101969
\(613\) 1.71643e7 1.84491 0.922456 0.386102i \(-0.126179\pi\)
0.922456 + 0.386102i \(0.126179\pi\)
\(614\) −2.08891e6 −0.223614
\(615\) 0 0
\(616\) 3.15904e6 0.335431
\(617\) −5.60639e6 −0.592885 −0.296443 0.955051i \(-0.595800\pi\)
−0.296443 + 0.955051i \(0.595800\pi\)
\(618\) 4.52601e6 0.476699
\(619\) 679689. 0.0712990 0.0356495 0.999364i \(-0.488650\pi\)
0.0356495 + 0.999364i \(0.488650\pi\)
\(620\) 0 0
\(621\) 978831. 0.101854
\(622\) −8.27637e6 −0.857757
\(623\) 1.16016e7 1.19756
\(624\) −3.05998e6 −0.314599
\(625\) 0 0
\(626\) 2.59953e6 0.265131
\(627\) −2.17374e6 −0.220820
\(628\) 4.50795e6 0.456121
\(629\) −4.20856e6 −0.424138
\(630\) 0 0
\(631\) −1.79404e6 −0.179374 −0.0896868 0.995970i \(-0.528587\pi\)
−0.0896868 + 0.995970i \(0.528587\pi\)
\(632\) −7.94312e6 −0.791040
\(633\) −9635.27 −0.000955773 0
\(634\) −4.53568e6 −0.448146
\(635\) 0 0
\(636\) 8.69333e6 0.852203
\(637\) −1.12601e7 −1.09949
\(638\) −2.07917e6 −0.202227
\(639\) −5.12846e6 −0.496861
\(640\) 0 0
\(641\) −1.87077e6 −0.179836 −0.0899179 0.995949i \(-0.528660\pi\)
−0.0899179 + 0.995949i \(0.528660\pi\)
\(642\) −3.26643e6 −0.312778
\(643\) 1.17131e7 1.11724 0.558618 0.829425i \(-0.311332\pi\)
0.558618 + 0.829425i \(0.311332\pi\)
\(644\) −5.91875e6 −0.562361
\(645\) 0 0
\(646\) 2.41109e6 0.227317
\(647\) 8.80680e6 0.827099 0.413550 0.910482i \(-0.364289\pi\)
0.413550 + 0.910482i \(0.364289\pi\)
\(648\) 978990. 0.0915884
\(649\) 3.54638e6 0.330502
\(650\) 0 0
\(651\) −7.07481e6 −0.654278
\(652\) 5.59763e6 0.515686
\(653\) −6.56212e6 −0.602229 −0.301114 0.953588i \(-0.597359\pi\)
−0.301114 + 0.953588i \(0.597359\pi\)
\(654\) −2.28604e6 −0.208997
\(655\) 0 0
\(656\) −461998. −0.0419161
\(657\) −333868. −0.0301760
\(658\) 4.15306e6 0.373941
\(659\) −1.31437e7 −1.17897 −0.589486 0.807778i \(-0.700670\pi\)
−0.589486 + 0.807778i \(0.700670\pi\)
\(660\) 0 0
\(661\) −1.73548e7 −1.54496 −0.772478 0.635041i \(-0.780983\pi\)
−0.772478 + 0.635041i \(0.780983\pi\)
\(662\) 1.38404e6 0.122745
\(663\) −3.39823e6 −0.300240
\(664\) 1.78828e7 1.57404
\(665\) 0 0
\(666\) 1.92091e6 0.167812
\(667\) 8.84350e6 0.769678
\(668\) 7.20913e6 0.625089
\(669\) −4.64000e6 −0.400823
\(670\) 0 0
\(671\) 3.91920e6 0.336041
\(672\) −9.23181e6 −0.788613
\(673\) 1.02596e7 0.873159 0.436580 0.899666i \(-0.356190\pi\)
0.436580 + 0.899666i \(0.356190\pi\)
\(674\) −2896.59 −0.000245605 0
\(675\) 0 0
\(676\) −7.40163e6 −0.622961
\(677\) −1.22085e6 −0.102375 −0.0511873 0.998689i \(-0.516301\pi\)
−0.0511873 + 0.998689i \(0.516301\pi\)
\(678\) −491973. −0.0411024
\(679\) 3.15936e7 2.62981
\(680\) 0 0
\(681\) −628868. −0.0519627
\(682\) 1.41827e6 0.116761
\(683\) −8.61333e6 −0.706511 −0.353256 0.935527i \(-0.614925\pi\)
−0.353256 + 0.935527i \(0.614925\pi\)
\(684\) 4.07336e6 0.332899
\(685\) 0 0
\(686\) 1.36939e6 0.111100
\(687\) 4.51904e6 0.365304
\(688\) −8.43458e6 −0.679348
\(689\) 3.12675e7 2.50926
\(690\) 0 0
\(691\) −1.53208e7 −1.22064 −0.610319 0.792156i \(-0.708959\pi\)
−0.610319 + 0.792156i \(0.708959\pi\)
\(692\) 1.47540e7 1.17124
\(693\) 1.71487e6 0.135643
\(694\) 2.83978e6 0.223814
\(695\) 0 0
\(696\) 8.84493e6 0.692104
\(697\) −513067. −0.0400030
\(698\) 1.54197e6 0.119795
\(699\) −1.37031e7 −1.06078
\(700\) 0 0
\(701\) 233349. 0.0179354 0.00896768 0.999960i \(-0.497145\pi\)
0.00896768 + 0.999960i \(0.497145\pi\)
\(702\) 1.55105e6 0.118791
\(703\) 1.81443e7 1.38469
\(704\) 236416. 0.0179781
\(705\) 0 0
\(706\) 2.55682e6 0.193058
\(707\) −1.99138e7 −1.49832
\(708\) −6.64556e6 −0.498252
\(709\) −3.15108e6 −0.235420 −0.117710 0.993048i \(-0.537555\pi\)
−0.117710 + 0.993048i \(0.537555\pi\)
\(710\) 0 0
\(711\) −4.31190e6 −0.319885
\(712\) 9.89387e6 0.731419
\(713\) −6.03242e6 −0.444393
\(714\) −1.90212e6 −0.139634
\(715\) 0 0
\(716\) 1.28836e7 0.939194
\(717\) −5.41778e6 −0.393571
\(718\) 7.50619e6 0.543386
\(719\) −2.38682e7 −1.72186 −0.860930 0.508723i \(-0.830118\pi\)
−0.860930 + 0.508723i \(0.830118\pi\)
\(720\) 0 0
\(721\) −3.37266e7 −2.41621
\(722\) −3.93494e6 −0.280928
\(723\) −8.64513e6 −0.615072
\(724\) 410083. 0.0290754
\(725\) 0 0
\(726\) −343775. −0.0242066
\(727\) −2.60780e7 −1.82995 −0.914974 0.403513i \(-0.867789\pi\)
−0.914974 + 0.403513i \(0.867789\pi\)
\(728\) −2.12916e7 −1.48895
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −9.36694e6 −0.648343
\(732\) −7.34419e6 −0.506601
\(733\) 822931. 0.0565723 0.0282861 0.999600i \(-0.490995\pi\)
0.0282861 + 0.999600i \(0.490995\pi\)
\(734\) 8.57021e6 0.587153
\(735\) 0 0
\(736\) −7.87161e6 −0.535635
\(737\) 7.79396e6 0.528554
\(738\) 234179. 0.0158273
\(739\) 1.58916e7 1.07043 0.535215 0.844716i \(-0.320231\pi\)
0.535215 + 0.844716i \(0.320231\pi\)
\(740\) 0 0
\(741\) 1.46508e7 0.980200
\(742\) 1.75016e7 1.16699
\(743\) 8.78449e6 0.583774 0.291887 0.956453i \(-0.405717\pi\)
0.291887 + 0.956453i \(0.405717\pi\)
\(744\) −6.03340e6 −0.399604
\(745\) 0 0
\(746\) 2.69715e6 0.177443
\(747\) 9.70763e6 0.636520
\(748\) −1.41139e6 −0.0922343
\(749\) 2.43406e7 1.58535
\(750\) 0 0
\(751\) −1.46203e7 −0.945926 −0.472963 0.881082i \(-0.656816\pi\)
−0.472963 + 0.881082i \(0.656816\pi\)
\(752\) −3.79301e6 −0.244590
\(753\) −8.99493e6 −0.578110
\(754\) 1.40134e7 0.897666
\(755\) 0 0
\(756\) −3.21349e6 −0.204490
\(757\) −7.20669e6 −0.457084 −0.228542 0.973534i \(-0.573396\pi\)
−0.228542 + 0.973534i \(0.573396\pi\)
\(758\) −8.90742e6 −0.563092
\(759\) 1.46220e6 0.0921305
\(760\) 0 0
\(761\) −2.92986e7 −1.83394 −0.916969 0.398958i \(-0.869372\pi\)
−0.916969 + 0.398958i \(0.869372\pi\)
\(762\) 8.16547e6 0.509441
\(763\) 1.70350e7 1.05933
\(764\) 5.51956e6 0.342114
\(765\) 0 0
\(766\) −4.76157e6 −0.293209
\(767\) −2.39022e7 −1.46707
\(768\) −4.84793e6 −0.296588
\(769\) −9.38411e6 −0.572239 −0.286119 0.958194i \(-0.592365\pi\)
−0.286119 + 0.958194i \(0.592365\pi\)
\(770\) 0 0
\(771\) −1.99834e6 −0.121069
\(772\) −1.77490e7 −1.07184
\(773\) −3.43323e6 −0.206659 −0.103329 0.994647i \(-0.532950\pi\)
−0.103329 + 0.994647i \(0.532950\pi\)
\(774\) 4.27535e6 0.256519
\(775\) 0 0
\(776\) 2.69430e7 1.60617
\(777\) −1.43141e7 −0.850575
\(778\) 3.42764e6 0.203024
\(779\) 2.21198e6 0.130598
\(780\) 0 0
\(781\) −7.66104e6 −0.449428
\(782\) −1.62186e6 −0.0948412
\(783\) 4.80144e6 0.279877
\(784\) 5.75627e6 0.334465
\(785\) 0 0
\(786\) 3.76827e6 0.217563
\(787\) 9.21027e6 0.530073 0.265036 0.964238i \(-0.414616\pi\)
0.265036 + 0.964238i \(0.414616\pi\)
\(788\) −1.68803e7 −0.968424
\(789\) 4.51465e6 0.258185
\(790\) 0 0
\(791\) 3.66605e6 0.208332
\(792\) 1.46244e6 0.0828449
\(793\) −2.64150e7 −1.49165
\(794\) −1.06313e7 −0.598462
\(795\) 0 0
\(796\) 1.48926e7 0.833082
\(797\) −1.76659e7 −0.985120 −0.492560 0.870279i \(-0.663939\pi\)
−0.492560 + 0.870279i \(0.663939\pi\)
\(798\) 8.20057e6 0.455866
\(799\) −4.21229e6 −0.233427
\(800\) 0 0
\(801\) 5.37085e6 0.295775
\(802\) −4.92705e6 −0.270490
\(803\) −498741. −0.0272952
\(804\) −1.46051e7 −0.796827
\(805\) 0 0
\(806\) −9.55896e6 −0.518290
\(807\) −1.35489e7 −0.732355
\(808\) −1.69825e7 −0.915108
\(809\) 5.29802e6 0.284605 0.142302 0.989823i \(-0.454549\pi\)
0.142302 + 0.989823i \(0.454549\pi\)
\(810\) 0 0
\(811\) 1.81272e7 0.967786 0.483893 0.875127i \(-0.339222\pi\)
0.483893 + 0.875127i \(0.339222\pi\)
\(812\) −2.90331e7 −1.54527
\(813\) 8.42128e6 0.446840
\(814\) 2.86951e6 0.151791
\(815\) 0 0
\(816\) 1.73721e6 0.0913329
\(817\) 4.03836e7 2.11665
\(818\) 7.23908e6 0.378268
\(819\) −1.15580e7 −0.602108
\(820\) 0 0
\(821\) 3.68262e7 1.90677 0.953387 0.301751i \(-0.0975710\pi\)
0.953387 + 0.301751i \(0.0975710\pi\)
\(822\) −3.34819e6 −0.172835
\(823\) −3.25146e7 −1.67332 −0.836660 0.547723i \(-0.815495\pi\)
−0.836660 + 0.547723i \(0.815495\pi\)
\(824\) −2.87621e7 −1.47571
\(825\) 0 0
\(826\) −1.33790e7 −0.682296
\(827\) 4.14035e6 0.210510 0.105255 0.994445i \(-0.466434\pi\)
0.105255 + 0.994445i \(0.466434\pi\)
\(828\) −2.74002e6 −0.138892
\(829\) −2.81888e7 −1.42459 −0.712296 0.701879i \(-0.752345\pi\)
−0.712296 + 0.701879i \(0.752345\pi\)
\(830\) 0 0
\(831\) −9.61298e6 −0.482898
\(832\) −1.59342e6 −0.0798033
\(833\) 6.39257e6 0.319200
\(834\) 2.91606e6 0.145172
\(835\) 0 0
\(836\) 6.08490e6 0.301119
\(837\) −3.27521e6 −0.161594
\(838\) −4.55763e6 −0.224197
\(839\) −470091. −0.0230556 −0.0115278 0.999934i \(-0.503669\pi\)
−0.0115278 + 0.999934i \(0.503669\pi\)
\(840\) 0 0
\(841\) 2.28686e7 1.11494
\(842\) −837911. −0.0407303
\(843\) 6.15142e6 0.298130
\(844\) 26971.8 0.00130333
\(845\) 0 0
\(846\) 1.92262e6 0.0923563
\(847\) 2.56172e6 0.122694
\(848\) −1.59843e7 −0.763314
\(849\) −2.45149e6 −0.116724
\(850\) 0 0
\(851\) −1.22051e7 −0.577721
\(852\) 1.43560e7 0.677540
\(853\) 3.80179e7 1.78902 0.894510 0.447047i \(-0.147524\pi\)
0.894510 + 0.447047i \(0.147524\pi\)
\(854\) −1.47855e7 −0.693730
\(855\) 0 0
\(856\) 2.07576e7 0.968263
\(857\) −2.72661e7 −1.26815 −0.634076 0.773271i \(-0.718619\pi\)
−0.634076 + 0.773271i \(0.718619\pi\)
\(858\) 2.31701e6 0.107451
\(859\) 3.07007e7 1.41960 0.709799 0.704404i \(-0.248786\pi\)
0.709799 + 0.704404i \(0.248786\pi\)
\(860\) 0 0
\(861\) −1.74504e6 −0.0802228
\(862\) −3.02892e6 −0.138841
\(863\) 2.20422e7 1.00746 0.503731 0.863861i \(-0.331960\pi\)
0.503731 + 0.863861i \(0.331960\pi\)
\(864\) −4.27377e6 −0.194772
\(865\) 0 0
\(866\) −4.32817e6 −0.196114
\(867\) −1.08495e7 −0.490186
\(868\) 1.98044e7 0.892199
\(869\) −6.44123e6 −0.289347
\(870\) 0 0
\(871\) −5.25305e7 −2.34620
\(872\) 1.45274e7 0.646990
\(873\) 1.46259e7 0.649512
\(874\) 6.99231e6 0.309630
\(875\) 0 0
\(876\) 934590. 0.0411491
\(877\) −6.53105e6 −0.286737 −0.143369 0.989669i \(-0.545793\pi\)
−0.143369 + 0.989669i \(0.545793\pi\)
\(878\) −1.90672e7 −0.834740
\(879\) −1.38523e7 −0.604713
\(880\) 0 0
\(881\) 2.27023e7 0.985439 0.492719 0.870188i \(-0.336003\pi\)
0.492719 + 0.870188i \(0.336003\pi\)
\(882\) −2.91776e6 −0.126293
\(883\) −2.91594e7 −1.25857 −0.629285 0.777174i \(-0.716652\pi\)
−0.629285 + 0.777174i \(0.716652\pi\)
\(884\) 9.51260e6 0.409420
\(885\) 0 0
\(886\) −8.73048e6 −0.373640
\(887\) 1.47884e7 0.631120 0.315560 0.948906i \(-0.397808\pi\)
0.315560 + 0.948906i \(0.397808\pi\)
\(888\) −1.22071e7 −0.519493
\(889\) −6.08468e7 −2.58216
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 1.29887e7 0.546578
\(893\) 1.81604e7 0.762073
\(894\) 7.79609e6 0.326237
\(895\) 0 0
\(896\) 3.19323e7 1.32880
\(897\) −9.85510e6 −0.408959
\(898\) 5.96454e6 0.246823
\(899\) −2.95907e7 −1.22111
\(900\) 0 0
\(901\) −1.77512e7 −0.728475
\(902\) 349823. 0.0143164
\(903\) −3.18588e7 −1.30020
\(904\) 3.12641e6 0.127240
\(905\) 0 0
\(906\) 5.19183e6 0.210136
\(907\) 6.96226e6 0.281017 0.140508 0.990080i \(-0.455126\pi\)
0.140508 + 0.990080i \(0.455126\pi\)
\(908\) 1.76038e6 0.0708584
\(909\) −9.21887e6 −0.370056
\(910\) 0 0
\(911\) 3.40287e7 1.35847 0.679234 0.733922i \(-0.262312\pi\)
0.679234 + 0.733922i \(0.262312\pi\)
\(912\) −7.48962e6 −0.298176
\(913\) 1.45015e7 0.575754
\(914\) −9.31869e6 −0.368968
\(915\) 0 0
\(916\) −1.26501e7 −0.498143
\(917\) −2.80801e7 −1.10275
\(918\) −880564. −0.0344869
\(919\) −2.25426e7 −0.880470 −0.440235 0.897883i \(-0.645105\pi\)
−0.440235 + 0.897883i \(0.645105\pi\)
\(920\) 0 0
\(921\) 7.20610e6 0.279931
\(922\) 2.07546e7 0.804059
\(923\) 5.16346e7 1.99497
\(924\) −4.80040e6 −0.184968
\(925\) 0 0
\(926\) −1.48749e7 −0.570069
\(927\) −1.56134e7 −0.596757
\(928\) −3.86124e7 −1.47183
\(929\) 3.13230e6 0.119076 0.0595381 0.998226i \(-0.481037\pi\)
0.0595381 + 0.998226i \(0.481037\pi\)
\(930\) 0 0
\(931\) −2.75602e7 −1.04210
\(932\) 3.83588e7 1.44652
\(933\) 2.85510e7 1.07378
\(934\) −1.33886e6 −0.0502191
\(935\) 0 0
\(936\) −9.85670e6 −0.367741
\(937\) 1.11644e7 0.415417 0.207709 0.978191i \(-0.433399\pi\)
0.207709 + 0.978191i \(0.433399\pi\)
\(938\) −2.94032e7 −1.09116
\(939\) −8.96761e6 −0.331904
\(940\) 0 0
\(941\) 2.20914e7 0.813296 0.406648 0.913585i \(-0.366698\pi\)
0.406648 + 0.913585i \(0.366698\pi\)
\(942\) 4.20140e6 0.154265
\(943\) −1.48793e6 −0.0544883
\(944\) 1.22191e7 0.446281
\(945\) 0 0
\(946\) 6.38664e6 0.232030
\(947\) −3.88629e7 −1.40819 −0.704093 0.710107i \(-0.748646\pi\)
−0.704093 + 0.710107i \(0.748646\pi\)
\(948\) 1.20702e7 0.436208
\(949\) 3.36146e6 0.121161
\(950\) 0 0
\(951\) 1.56467e7 0.561012
\(952\) 1.20876e7 0.432264
\(953\) 1.65779e6 0.0591287 0.0295643 0.999563i \(-0.490588\pi\)
0.0295643 + 0.999563i \(0.490588\pi\)
\(954\) 8.10216e6 0.288224
\(955\) 0 0
\(956\) 1.51659e7 0.536689
\(957\) 7.17252e6 0.253158
\(958\) −1.46003e7 −0.513981
\(959\) 2.49498e7 0.876033
\(960\) 0 0
\(961\) −8.44445e6 −0.294960
\(962\) −1.93402e7 −0.673789
\(963\) 1.12682e7 0.391551
\(964\) 2.42001e7 0.838736
\(965\) 0 0
\(966\) −5.51626e6 −0.190196
\(967\) 4.00243e7 1.37644 0.688221 0.725501i \(-0.258392\pi\)
0.688221 + 0.725501i \(0.258392\pi\)
\(968\) 2.18464e6 0.0749360
\(969\) −8.31753e6 −0.284567
\(970\) 0 0
\(971\) −3.50158e7 −1.19183 −0.595917 0.803046i \(-0.703211\pi\)
−0.595917 + 0.803046i \(0.703211\pi\)
\(972\) −1.48765e6 −0.0505052
\(973\) −2.17297e7 −0.735821
\(974\) −1.10969e6 −0.0374805
\(975\) 0 0
\(976\) 1.35036e7 0.453760
\(977\) 3.66007e7 1.22674 0.613370 0.789795i \(-0.289813\pi\)
0.613370 + 0.789795i \(0.289813\pi\)
\(978\) 5.21698e6 0.174410
\(979\) 8.02313e6 0.267539
\(980\) 0 0
\(981\) 7.88616e6 0.261633
\(982\) 1.45260e7 0.480692
\(983\) −4.10117e7 −1.35371 −0.676853 0.736119i \(-0.736657\pi\)
−0.676853 + 0.736119i \(0.736657\pi\)
\(984\) −1.48817e6 −0.0489965
\(985\) 0 0
\(986\) −7.95568e6 −0.260606
\(987\) −1.43268e7 −0.468119
\(988\) −4.10116e7 −1.33664
\(989\) −2.71648e7 −0.883111
\(990\) 0 0
\(991\) 3.49847e7 1.13160 0.565802 0.824541i \(-0.308567\pi\)
0.565802 + 0.824541i \(0.308567\pi\)
\(992\) 2.63387e7 0.849797
\(993\) −4.77451e6 −0.153658
\(994\) 2.89018e7 0.927809
\(995\) 0 0
\(996\) −2.71744e7 −0.867983
\(997\) 5.02983e7 1.60256 0.801282 0.598287i \(-0.204152\pi\)
0.801282 + 0.598287i \(0.204152\pi\)
\(998\) 1.38620e7 0.440554
\(999\) −6.62658e6 −0.210076
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.m.1.4 7
5.4 even 2 825.6.a.o.1.4 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.m.1.4 7 1.1 even 1 trivial
825.6.a.o.1.4 yes 7 5.4 even 2