Properties

Label 825.6.a.m.1.2
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 1.2
Root \(6.35371\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.35371 q^{2} +9.00000 q^{3} +22.0770 q^{4} -66.1834 q^{6} -251.987 q^{7} +72.9708 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-7.35371 q^{2} +9.00000 q^{3} +22.0770 q^{4} -66.1834 q^{6} -251.987 q^{7} +72.9708 q^{8} +81.0000 q^{9} +121.000 q^{11} +198.693 q^{12} +633.803 q^{13} +1853.04 q^{14} -1243.07 q^{16} -460.381 q^{17} -595.650 q^{18} -1693.64 q^{19} -2267.89 q^{21} -889.799 q^{22} +3139.26 q^{23} +656.737 q^{24} -4660.80 q^{26} +729.000 q^{27} -5563.13 q^{28} +223.138 q^{29} -7390.32 q^{31} +6806.11 q^{32} +1089.00 q^{33} +3385.51 q^{34} +1788.24 q^{36} -5715.38 q^{37} +12454.5 q^{38} +5704.23 q^{39} +19320.6 q^{41} +16677.4 q^{42} +2050.91 q^{43} +2671.32 q^{44} -23085.2 q^{46} +3367.25 q^{47} -11187.6 q^{48} +46690.7 q^{49} -4143.43 q^{51} +13992.5 q^{52} +20043.7 q^{53} -5360.85 q^{54} -18387.7 q^{56} -15242.7 q^{57} -1640.89 q^{58} -18420.6 q^{59} +21177.7 q^{61} +54346.3 q^{62} -20411.0 q^{63} -10271.9 q^{64} -8008.19 q^{66} -45127.5 q^{67} -10163.8 q^{68} +28253.4 q^{69} -54735.9 q^{71} +5910.64 q^{72} +64414.7 q^{73} +42029.2 q^{74} -37390.4 q^{76} -30490.5 q^{77} -41947.2 q^{78} -74351.1 q^{79} +6561.00 q^{81} -142078. q^{82} +118034. q^{83} -50068.2 q^{84} -15081.8 q^{86} +2008.24 q^{87} +8829.47 q^{88} -57632.9 q^{89} -159710. q^{91} +69305.5 q^{92} -66512.9 q^{93} -24761.8 q^{94} +61255.0 q^{96} +75016.9 q^{97} -343350. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.35371 −1.29996 −0.649982 0.759950i \(-0.725224\pi\)
−0.649982 + 0.759950i \(0.725224\pi\)
\(3\) 9.00000 0.577350
\(4\) 22.0770 0.689906
\(5\) 0 0
\(6\) −66.1834 −0.750535
\(7\) −251.987 −1.94372 −0.971860 0.235558i \(-0.924308\pi\)
−0.971860 + 0.235558i \(0.924308\pi\)
\(8\) 72.9708 0.403111
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 198.693 0.398318
\(13\) 633.803 1.04015 0.520075 0.854121i \(-0.325904\pi\)
0.520075 + 0.854121i \(0.325904\pi\)
\(14\) 1853.04 2.52677
\(15\) 0 0
\(16\) −1243.07 −1.21394
\(17\) −460.381 −0.386362 −0.193181 0.981163i \(-0.561881\pi\)
−0.193181 + 0.981163i \(0.561881\pi\)
\(18\) −595.650 −0.433321
\(19\) −1693.64 −1.07631 −0.538154 0.842847i \(-0.680878\pi\)
−0.538154 + 0.842847i \(0.680878\pi\)
\(20\) 0 0
\(21\) −2267.89 −1.12221
\(22\) −889.799 −0.391954
\(23\) 3139.26 1.23739 0.618697 0.785630i \(-0.287661\pi\)
0.618697 + 0.785630i \(0.287661\pi\)
\(24\) 656.737 0.232736
\(25\) 0 0
\(26\) −4660.80 −1.35216
\(27\) 729.000 0.192450
\(28\) −5563.13 −1.34099
\(29\) 223.138 0.0492695 0.0246348 0.999697i \(-0.492158\pi\)
0.0246348 + 0.999697i \(0.492158\pi\)
\(30\) 0 0
\(31\) −7390.32 −1.38121 −0.690604 0.723233i \(-0.742655\pi\)
−0.690604 + 0.723233i \(0.742655\pi\)
\(32\) 6806.11 1.17496
\(33\) 1089.00 0.174078
\(34\) 3385.51 0.502257
\(35\) 0 0
\(36\) 1788.24 0.229969
\(37\) −5715.38 −0.686342 −0.343171 0.939273i \(-0.611501\pi\)
−0.343171 + 0.939273i \(0.611501\pi\)
\(38\) 12454.5 1.39916
\(39\) 5704.23 0.600531
\(40\) 0 0
\(41\) 19320.6 1.79498 0.897492 0.441031i \(-0.145387\pi\)
0.897492 + 0.441031i \(0.145387\pi\)
\(42\) 16677.4 1.45883
\(43\) 2050.91 0.169151 0.0845755 0.996417i \(-0.473047\pi\)
0.0845755 + 0.996417i \(0.473047\pi\)
\(44\) 2671.32 0.208015
\(45\) 0 0
\(46\) −23085.2 −1.60857
\(47\) 3367.25 0.222347 0.111173 0.993801i \(-0.464539\pi\)
0.111173 + 0.993801i \(0.464539\pi\)
\(48\) −11187.6 −0.700866
\(49\) 46690.7 2.77805
\(50\) 0 0
\(51\) −4143.43 −0.223066
\(52\) 13992.5 0.717606
\(53\) 20043.7 0.980138 0.490069 0.871684i \(-0.336972\pi\)
0.490069 + 0.871684i \(0.336972\pi\)
\(54\) −5360.85 −0.250178
\(55\) 0 0
\(56\) −18387.7 −0.783534
\(57\) −15242.7 −0.621407
\(58\) −1640.89 −0.0640486
\(59\) −18420.6 −0.688928 −0.344464 0.938800i \(-0.611939\pi\)
−0.344464 + 0.938800i \(0.611939\pi\)
\(60\) 0 0
\(61\) 21177.7 0.728710 0.364355 0.931260i \(-0.381290\pi\)
0.364355 + 0.931260i \(0.381290\pi\)
\(62\) 54346.3 1.79552
\(63\) −20411.0 −0.647907
\(64\) −10271.9 −0.313473
\(65\) 0 0
\(66\) −8008.19 −0.226295
\(67\) −45127.5 −1.22816 −0.614080 0.789244i \(-0.710473\pi\)
−0.614080 + 0.789244i \(0.710473\pi\)
\(68\) −10163.8 −0.266554
\(69\) 28253.4 0.714410
\(70\) 0 0
\(71\) −54735.9 −1.28863 −0.644313 0.764762i \(-0.722856\pi\)
−0.644313 + 0.764762i \(0.722856\pi\)
\(72\) 5910.64 0.134370
\(73\) 64414.7 1.41474 0.707372 0.706842i \(-0.249881\pi\)
0.707372 + 0.706842i \(0.249881\pi\)
\(74\) 42029.2 0.892220
\(75\) 0 0
\(76\) −37390.4 −0.742552
\(77\) −30490.5 −0.586054
\(78\) −41947.2 −0.780668
\(79\) −74351.1 −1.34036 −0.670178 0.742201i \(-0.733782\pi\)
−0.670178 + 0.742201i \(0.733782\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −142078. −2.33341
\(83\) 118034. 1.88067 0.940333 0.340255i \(-0.110513\pi\)
0.940333 + 0.340255i \(0.110513\pi\)
\(84\) −50068.2 −0.774218
\(85\) 0 0
\(86\) −15081.8 −0.219890
\(87\) 2008.24 0.0284458
\(88\) 8829.47 0.121542
\(89\) −57632.9 −0.771250 −0.385625 0.922656i \(-0.626014\pi\)
−0.385625 + 0.922656i \(0.626014\pi\)
\(90\) 0 0
\(91\) −159710. −2.02176
\(92\) 69305.5 0.853686
\(93\) −66512.9 −0.797441
\(94\) −24761.8 −0.289043
\(95\) 0 0
\(96\) 61255.0 0.678365
\(97\) 75016.9 0.809524 0.404762 0.914422i \(-0.367354\pi\)
0.404762 + 0.914422i \(0.367354\pi\)
\(98\) −343350. −3.61137
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 32528.7 0.317295 0.158648 0.987335i \(-0.449287\pi\)
0.158648 + 0.987335i \(0.449287\pi\)
\(102\) 30469.5 0.289978
\(103\) −66562.8 −0.618213 −0.309107 0.951027i \(-0.600030\pi\)
−0.309107 + 0.951027i \(0.600030\pi\)
\(104\) 46249.1 0.419295
\(105\) 0 0
\(106\) −147395. −1.27414
\(107\) 64861.2 0.547678 0.273839 0.961776i \(-0.411706\pi\)
0.273839 + 0.961776i \(0.411706\pi\)
\(108\) 16094.1 0.132773
\(109\) −47344.4 −0.381682 −0.190841 0.981621i \(-0.561122\pi\)
−0.190841 + 0.981621i \(0.561122\pi\)
\(110\) 0 0
\(111\) −51438.4 −0.396260
\(112\) 313238. 2.35955
\(113\) −114489. −0.843469 −0.421734 0.906719i \(-0.638579\pi\)
−0.421734 + 0.906719i \(0.638579\pi\)
\(114\) 112091. 0.807806
\(115\) 0 0
\(116\) 4926.22 0.0339914
\(117\) 51338.0 0.346717
\(118\) 135460. 0.895581
\(119\) 116010. 0.750981
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −155735. −0.947296
\(123\) 173885. 1.03633
\(124\) −163156. −0.952905
\(125\) 0 0
\(126\) 150096. 0.842256
\(127\) 315900. 1.73796 0.868982 0.494843i \(-0.164775\pi\)
0.868982 + 0.494843i \(0.164775\pi\)
\(128\) −142259. −0.767459
\(129\) 18458.2 0.0976594
\(130\) 0 0
\(131\) 176456. 0.898378 0.449189 0.893437i \(-0.351713\pi\)
0.449189 + 0.893437i \(0.351713\pi\)
\(132\) 24041.9 0.120097
\(133\) 426775. 2.09204
\(134\) 331855. 1.59656
\(135\) 0 0
\(136\) −33594.4 −0.155747
\(137\) 345239. 1.57152 0.785758 0.618534i \(-0.212273\pi\)
0.785758 + 0.618534i \(0.212273\pi\)
\(138\) −207767. −0.928707
\(139\) −140830. −0.618239 −0.309120 0.951023i \(-0.600034\pi\)
−0.309120 + 0.951023i \(0.600034\pi\)
\(140\) 0 0
\(141\) 30305.2 0.128372
\(142\) 402512. 1.67517
\(143\) 76690.2 0.313617
\(144\) −100689. −0.404645
\(145\) 0 0
\(146\) −473687. −1.83912
\(147\) 420216. 1.60391
\(148\) −126178. −0.473512
\(149\) −31764.1 −0.117212 −0.0586059 0.998281i \(-0.518666\pi\)
−0.0586059 + 0.998281i \(0.518666\pi\)
\(150\) 0 0
\(151\) −106794. −0.381156 −0.190578 0.981672i \(-0.561036\pi\)
−0.190578 + 0.981672i \(0.561036\pi\)
\(152\) −123586. −0.433871
\(153\) −37290.8 −0.128787
\(154\) 224218. 0.761849
\(155\) 0 0
\(156\) 125932. 0.414310
\(157\) 55442.9 0.179513 0.0897566 0.995964i \(-0.471391\pi\)
0.0897566 + 0.995964i \(0.471391\pi\)
\(158\) 546757. 1.74241
\(159\) 180393. 0.565883
\(160\) 0 0
\(161\) −791055. −2.40515
\(162\) −48247.7 −0.144440
\(163\) −108411. −0.319598 −0.159799 0.987150i \(-0.551085\pi\)
−0.159799 + 0.987150i \(0.551085\pi\)
\(164\) 426540. 1.23837
\(165\) 0 0
\(166\) −867987. −2.44480
\(167\) −47112.2 −0.130720 −0.0653600 0.997862i \(-0.520820\pi\)
−0.0653600 + 0.997862i \(0.520820\pi\)
\(168\) −165490. −0.452374
\(169\) 30413.2 0.0819116
\(170\) 0 0
\(171\) −137185. −0.358769
\(172\) 45277.9 0.116698
\(173\) −496469. −1.26118 −0.630589 0.776117i \(-0.717187\pi\)
−0.630589 + 0.776117i \(0.717187\pi\)
\(174\) −14768.0 −0.0369785
\(175\) 0 0
\(176\) −150411. −0.366015
\(177\) −165785. −0.397753
\(178\) 423815. 1.00260
\(179\) −564791. −1.31751 −0.658757 0.752356i \(-0.728917\pi\)
−0.658757 + 0.752356i \(0.728917\pi\)
\(180\) 0 0
\(181\) 275289. 0.624585 0.312293 0.949986i \(-0.398903\pi\)
0.312293 + 0.949986i \(0.398903\pi\)
\(182\) 1.17446e6 2.62822
\(183\) 190599. 0.420721
\(184\) 229074. 0.498807
\(185\) 0 0
\(186\) 489116. 1.03664
\(187\) −55706.1 −0.116493
\(188\) 74338.7 0.153398
\(189\) −183699. −0.374069
\(190\) 0 0
\(191\) 365202. 0.724351 0.362176 0.932110i \(-0.382034\pi\)
0.362176 + 0.932110i \(0.382034\pi\)
\(192\) −92446.8 −0.180983
\(193\) −565527. −1.09285 −0.546425 0.837508i \(-0.684012\pi\)
−0.546425 + 0.837508i \(0.684012\pi\)
\(194\) −551652. −1.05235
\(195\) 0 0
\(196\) 1.03079e6 1.91659
\(197\) −365310. −0.670650 −0.335325 0.942103i \(-0.608846\pi\)
−0.335325 + 0.942103i \(0.608846\pi\)
\(198\) −72073.7 −0.130651
\(199\) −464819. −0.832053 −0.416026 0.909353i \(-0.636578\pi\)
−0.416026 + 0.909353i \(0.636578\pi\)
\(200\) 0 0
\(201\) −406148. −0.709078
\(202\) −239207. −0.412472
\(203\) −56228.0 −0.0957662
\(204\) −91474.5 −0.153895
\(205\) 0 0
\(206\) 489483. 0.803655
\(207\) 254280. 0.412465
\(208\) −787861. −1.26267
\(209\) −204930. −0.324519
\(210\) 0 0
\(211\) 632574. 0.978150 0.489075 0.872242i \(-0.337334\pi\)
0.489075 + 0.872242i \(0.337334\pi\)
\(212\) 442504. 0.676203
\(213\) −492623. −0.743988
\(214\) −476970. −0.711962
\(215\) 0 0
\(216\) 53195.7 0.0775787
\(217\) 1.86227e6 2.68468
\(218\) 348157. 0.496173
\(219\) 579732. 0.816802
\(220\) 0 0
\(221\) −291791. −0.401875
\(222\) 378263. 0.515124
\(223\) 625787. 0.842684 0.421342 0.906902i \(-0.361559\pi\)
0.421342 + 0.906902i \(0.361559\pi\)
\(224\) −1.71505e6 −2.28380
\(225\) 0 0
\(226\) 841921. 1.09648
\(227\) −1.02256e6 −1.31712 −0.658561 0.752527i \(-0.728835\pi\)
−0.658561 + 0.752527i \(0.728835\pi\)
\(228\) −336514. −0.428712
\(229\) −1.35027e6 −1.70150 −0.850748 0.525574i \(-0.823851\pi\)
−0.850748 + 0.525574i \(0.823851\pi\)
\(230\) 0 0
\(231\) −274414. −0.338358
\(232\) 16282.6 0.0198611
\(233\) 1.19472e6 1.44171 0.720853 0.693088i \(-0.243750\pi\)
0.720853 + 0.693088i \(0.243750\pi\)
\(234\) −377525. −0.450719
\(235\) 0 0
\(236\) −406672. −0.475296
\(237\) −669160. −0.773854
\(238\) −853105. −0.976248
\(239\) −1.26772e6 −1.43558 −0.717789 0.696260i \(-0.754846\pi\)
−0.717789 + 0.696260i \(0.754846\pi\)
\(240\) 0 0
\(241\) 1.43738e6 1.59415 0.797077 0.603878i \(-0.206379\pi\)
0.797077 + 0.603878i \(0.206379\pi\)
\(242\) −107666. −0.118179
\(243\) 59049.0 0.0641500
\(244\) 467540. 0.502741
\(245\) 0 0
\(246\) −1.27870e6 −1.34720
\(247\) −1.07343e6 −1.11952
\(248\) −539278. −0.556780
\(249\) 1.06231e6 1.08580
\(250\) 0 0
\(251\) −513175. −0.514139 −0.257070 0.966393i \(-0.582757\pi\)
−0.257070 + 0.966393i \(0.582757\pi\)
\(252\) −450613. −0.446995
\(253\) 379851. 0.373088
\(254\) −2.32304e6 −2.25929
\(255\) 0 0
\(256\) 1.37483e6 1.31114
\(257\) −907801. −0.857350 −0.428675 0.903459i \(-0.641019\pi\)
−0.428675 + 0.903459i \(0.641019\pi\)
\(258\) −135736. −0.126954
\(259\) 1.44020e6 1.33406
\(260\) 0 0
\(261\) 18074.2 0.0164232
\(262\) −1.29761e6 −1.16786
\(263\) −893422. −0.796466 −0.398233 0.917284i \(-0.630376\pi\)
−0.398233 + 0.917284i \(0.630376\pi\)
\(264\) 79465.2 0.0701725
\(265\) 0 0
\(266\) −3.13838e6 −2.71958
\(267\) −518696. −0.445281
\(268\) −996281. −0.847315
\(269\) −1.13167e6 −0.953541 −0.476771 0.879028i \(-0.658193\pi\)
−0.476771 + 0.879028i \(0.658193\pi\)
\(270\) 0 0
\(271\) −2.05455e6 −1.69939 −0.849694 0.527276i \(-0.823213\pi\)
−0.849694 + 0.527276i \(0.823213\pi\)
\(272\) 572286. 0.469019
\(273\) −1.43739e6 −1.16726
\(274\) −2.53879e6 −2.04292
\(275\) 0 0
\(276\) 623749. 0.492876
\(277\) 2.32890e6 1.82369 0.911845 0.410535i \(-0.134658\pi\)
0.911845 + 0.410535i \(0.134658\pi\)
\(278\) 1.03562e6 0.803689
\(279\) −598616. −0.460403
\(280\) 0 0
\(281\) −321019. −0.242530 −0.121265 0.992620i \(-0.538695\pi\)
−0.121265 + 0.992620i \(0.538695\pi\)
\(282\) −222856. −0.166879
\(283\) −1.18090e6 −0.876491 −0.438246 0.898855i \(-0.644400\pi\)
−0.438246 + 0.898855i \(0.644400\pi\)
\(284\) −1.20841e6 −0.889031
\(285\) 0 0
\(286\) −563957. −0.407691
\(287\) −4.86854e6 −3.48895
\(288\) 551295. 0.391654
\(289\) −1.20791e6 −0.850724
\(290\) 0 0
\(291\) 675152. 0.467379
\(292\) 1.42208e6 0.976040
\(293\) −423746. −0.288361 −0.144181 0.989551i \(-0.546055\pi\)
−0.144181 + 0.989551i \(0.546055\pi\)
\(294\) −3.09015e6 −2.08502
\(295\) 0 0
\(296\) −417056. −0.276672
\(297\) 88209.0 0.0580259
\(298\) 233584. 0.152371
\(299\) 1.98967e6 1.28707
\(300\) 0 0
\(301\) −516803. −0.328782
\(302\) 785329. 0.495489
\(303\) 292759. 0.183191
\(304\) 2.10531e6 1.30657
\(305\) 0 0
\(306\) 274226. 0.167419
\(307\) −247177. −0.149680 −0.0748398 0.997196i \(-0.523845\pi\)
−0.0748398 + 0.997196i \(0.523845\pi\)
\(308\) −673139. −0.404322
\(309\) −599065. −0.356926
\(310\) 0 0
\(311\) −1.65892e6 −0.972576 −0.486288 0.873799i \(-0.661649\pi\)
−0.486288 + 0.873799i \(0.661649\pi\)
\(312\) 416242. 0.242080
\(313\) −1.43394e6 −0.827311 −0.413656 0.910433i \(-0.635748\pi\)
−0.413656 + 0.910433i \(0.635748\pi\)
\(314\) −407711. −0.233361
\(315\) 0 0
\(316\) −1.64145e6 −0.924720
\(317\) −1.21845e6 −0.681018 −0.340509 0.940241i \(-0.610599\pi\)
−0.340509 + 0.940241i \(0.610599\pi\)
\(318\) −1.32656e6 −0.735627
\(319\) 26999.7 0.0148553
\(320\) 0 0
\(321\) 583751. 0.316202
\(322\) 5.81718e6 3.12661
\(323\) 779718. 0.415845
\(324\) 144847. 0.0766563
\(325\) 0 0
\(326\) 797222. 0.415466
\(327\) −426099. −0.220364
\(328\) 1.40984e6 0.723577
\(329\) −848504. −0.432180
\(330\) 0 0
\(331\) 2.72626e6 1.36772 0.683860 0.729613i \(-0.260300\pi\)
0.683860 + 0.729613i \(0.260300\pi\)
\(332\) 2.60584e6 1.29748
\(333\) −462946. −0.228781
\(334\) 346449. 0.169931
\(335\) 0 0
\(336\) 2.81914e6 1.36229
\(337\) −1.35050e6 −0.647769 −0.323884 0.946097i \(-0.604989\pi\)
−0.323884 + 0.946097i \(0.604989\pi\)
\(338\) −223650. −0.106482
\(339\) −1.03040e6 −0.486977
\(340\) 0 0
\(341\) −894229. −0.416450
\(342\) 1.00882e6 0.466387
\(343\) −7.53032e6 −3.45603
\(344\) 149656. 0.0681866
\(345\) 0 0
\(346\) 3.65089e6 1.63949
\(347\) −4.17183e6 −1.85996 −0.929978 0.367615i \(-0.880174\pi\)
−0.929978 + 0.367615i \(0.880174\pi\)
\(348\) 44335.9 0.0196249
\(349\) 1.92813e6 0.847371 0.423686 0.905809i \(-0.360736\pi\)
0.423686 + 0.905809i \(0.360736\pi\)
\(350\) 0 0
\(351\) 462042. 0.200177
\(352\) 823539. 0.354264
\(353\) −3.84016e6 −1.64026 −0.820129 0.572178i \(-0.806099\pi\)
−0.820129 + 0.572178i \(0.806099\pi\)
\(354\) 1.21914e6 0.517064
\(355\) 0 0
\(356\) −1.27236e6 −0.532090
\(357\) 1.04409e6 0.433579
\(358\) 4.15331e6 1.71272
\(359\) −2.65540e6 −1.08741 −0.543706 0.839276i \(-0.682979\pi\)
−0.543706 + 0.839276i \(0.682979\pi\)
\(360\) 0 0
\(361\) 392309. 0.158438
\(362\) −2.02439e6 −0.811939
\(363\) 131769. 0.0524864
\(364\) −3.52593e6 −1.39483
\(365\) 0 0
\(366\) −1.40161e6 −0.546922
\(367\) 969753. 0.375834 0.187917 0.982185i \(-0.439826\pi\)
0.187917 + 0.982185i \(0.439826\pi\)
\(368\) −3.90232e6 −1.50212
\(369\) 1.56497e6 0.598328
\(370\) 0 0
\(371\) −5.05075e6 −1.90511
\(372\) −1.46841e6 −0.550160
\(373\) 838185. 0.311938 0.155969 0.987762i \(-0.450150\pi\)
0.155969 + 0.987762i \(0.450150\pi\)
\(374\) 409646. 0.151436
\(375\) 0 0
\(376\) 245711. 0.0896303
\(377\) 141425. 0.0512477
\(378\) 1.35087e6 0.486277
\(379\) 9083.82 0.00324841 0.00162420 0.999999i \(-0.499483\pi\)
0.00162420 + 0.999999i \(0.499483\pi\)
\(380\) 0 0
\(381\) 2.84310e6 1.00341
\(382\) −2.68558e6 −0.941630
\(383\) 2.72358e6 0.948731 0.474365 0.880328i \(-0.342678\pi\)
0.474365 + 0.880328i \(0.342678\pi\)
\(384\) −1.28033e6 −0.443093
\(385\) 0 0
\(386\) 4.15872e6 1.42067
\(387\) 166123. 0.0563837
\(388\) 1.65615e6 0.558496
\(389\) 5.18178e6 1.73622 0.868110 0.496372i \(-0.165335\pi\)
0.868110 + 0.496372i \(0.165335\pi\)
\(390\) 0 0
\(391\) −1.44526e6 −0.478083
\(392\) 3.40706e6 1.11986
\(393\) 1.58811e6 0.518679
\(394\) 2.68638e6 0.871821
\(395\) 0 0
\(396\) 216377. 0.0693382
\(397\) 559564. 0.178186 0.0890929 0.996023i \(-0.471603\pi\)
0.0890929 + 0.996023i \(0.471603\pi\)
\(398\) 3.41814e6 1.08164
\(399\) 3.84098e6 1.20784
\(400\) 0 0
\(401\) −2.20014e6 −0.683265 −0.341633 0.939834i \(-0.610980\pi\)
−0.341633 + 0.939834i \(0.610980\pi\)
\(402\) 2.98669e6 0.921776
\(403\) −4.68401e6 −1.43666
\(404\) 718137. 0.218904
\(405\) 0 0
\(406\) 413484. 0.124493
\(407\) −691561. −0.206940
\(408\) −302349. −0.0899205
\(409\) −3.41580e6 −1.00968 −0.504840 0.863213i \(-0.668448\pi\)
−0.504840 + 0.863213i \(0.668448\pi\)
\(410\) 0 0
\(411\) 3.10715e6 0.907316
\(412\) −1.46951e6 −0.426509
\(413\) 4.64176e6 1.33908
\(414\) −1.86990e6 −0.536189
\(415\) 0 0
\(416\) 4.31373e6 1.22214
\(417\) −1.26747e6 −0.356941
\(418\) 1.50700e6 0.421863
\(419\) 1.12469e6 0.312966 0.156483 0.987681i \(-0.449984\pi\)
0.156483 + 0.987681i \(0.449984\pi\)
\(420\) 0 0
\(421\) −5.75430e6 −1.58230 −0.791148 0.611625i \(-0.790516\pi\)
−0.791148 + 0.611625i \(0.790516\pi\)
\(422\) −4.65177e6 −1.27156
\(423\) 272747. 0.0741155
\(424\) 1.46260e6 0.395104
\(425\) 0 0
\(426\) 3.62261e6 0.967158
\(427\) −5.33652e6 −1.41641
\(428\) 1.43194e6 0.377847
\(429\) 690211. 0.181067
\(430\) 0 0
\(431\) 1.30296e6 0.337861 0.168930 0.985628i \(-0.445969\pi\)
0.168930 + 0.985628i \(0.445969\pi\)
\(432\) −906198. −0.233622
\(433\) −6.66336e6 −1.70794 −0.853972 0.520320i \(-0.825813\pi\)
−0.853972 + 0.520320i \(0.825813\pi\)
\(434\) −1.36946e7 −3.48999
\(435\) 0 0
\(436\) −1.04522e6 −0.263325
\(437\) −5.31677e6 −1.33182
\(438\) −4.26318e6 −1.06181
\(439\) −6.12622e6 −1.51716 −0.758580 0.651580i \(-0.774106\pi\)
−0.758580 + 0.651580i \(0.774106\pi\)
\(440\) 0 0
\(441\) 3.78195e6 0.926017
\(442\) 2.14574e6 0.522423
\(443\) −6.57224e6 −1.59112 −0.795562 0.605872i \(-0.792825\pi\)
−0.795562 + 0.605872i \(0.792825\pi\)
\(444\) −1.13561e6 −0.273382
\(445\) 0 0
\(446\) −4.60186e6 −1.09546
\(447\) −285877. −0.0676722
\(448\) 2.58838e6 0.609303
\(449\) −1.83721e6 −0.430075 −0.215037 0.976606i \(-0.568987\pi\)
−0.215037 + 0.976606i \(0.568987\pi\)
\(450\) 0 0
\(451\) 2.33779e6 0.541208
\(452\) −2.52758e6 −0.581914
\(453\) −961142. −0.220061
\(454\) 7.51964e6 1.71221
\(455\) 0 0
\(456\) −1.11227e6 −0.250496
\(457\) 5.02595e6 1.12571 0.562857 0.826554i \(-0.309702\pi\)
0.562857 + 0.826554i \(0.309702\pi\)
\(458\) 9.92947e6 2.21188
\(459\) −335618. −0.0743555
\(460\) 0 0
\(461\) −3.24710e6 −0.711612 −0.355806 0.934560i \(-0.615794\pi\)
−0.355806 + 0.934560i \(0.615794\pi\)
\(462\) 2.01796e6 0.439854
\(463\) −8.24253e6 −1.78693 −0.893466 0.449131i \(-0.851734\pi\)
−0.893466 + 0.449131i \(0.851734\pi\)
\(464\) −277376. −0.0598100
\(465\) 0 0
\(466\) −8.78563e6 −1.87417
\(467\) 1.61860e6 0.343437 0.171718 0.985146i \(-0.445068\pi\)
0.171718 + 0.985146i \(0.445068\pi\)
\(468\) 1.13339e6 0.239202
\(469\) 1.13716e7 2.38720
\(470\) 0 0
\(471\) 498986. 0.103642
\(472\) −1.34417e6 −0.277714
\(473\) 248160. 0.0510010
\(474\) 4.92081e6 1.00598
\(475\) 0 0
\(476\) 2.56116e6 0.518106
\(477\) 1.62354e6 0.326713
\(478\) 9.32241e6 1.86620
\(479\) 107588. 0.0214252 0.0107126 0.999943i \(-0.496590\pi\)
0.0107126 + 0.999943i \(0.496590\pi\)
\(480\) 0 0
\(481\) −3.62243e6 −0.713899
\(482\) −1.05701e7 −2.07234
\(483\) −7.11949e6 −1.38861
\(484\) 323229. 0.0627188
\(485\) 0 0
\(486\) −434229. −0.0833927
\(487\) 8.33266e6 1.59207 0.796033 0.605253i \(-0.206928\pi\)
0.796033 + 0.605253i \(0.206928\pi\)
\(488\) 1.54536e6 0.293751
\(489\) −975698. −0.184520
\(490\) 0 0
\(491\) 9.57192e6 1.79182 0.895912 0.444232i \(-0.146523\pi\)
0.895912 + 0.444232i \(0.146523\pi\)
\(492\) 3.83886e6 0.714974
\(493\) −102728. −0.0190359
\(494\) 7.89371e6 1.45534
\(495\) 0 0
\(496\) 9.18669e6 1.67670
\(497\) 1.37928e7 2.50473
\(498\) −7.81189e6 −1.41151
\(499\) −815890. −0.146683 −0.0733416 0.997307i \(-0.523366\pi\)
−0.0733416 + 0.997307i \(0.523366\pi\)
\(500\) 0 0
\(501\) −424009. −0.0754712
\(502\) 3.77374e6 0.668363
\(503\) 5.19342e6 0.915238 0.457619 0.889148i \(-0.348702\pi\)
0.457619 + 0.889148i \(0.348702\pi\)
\(504\) −1.48941e6 −0.261178
\(505\) 0 0
\(506\) −2.79331e6 −0.485001
\(507\) 273719. 0.0472917
\(508\) 6.97413e6 1.19903
\(509\) 179583. 0.0307236 0.0153618 0.999882i \(-0.495110\pi\)
0.0153618 + 0.999882i \(0.495110\pi\)
\(510\) 0 0
\(511\) −1.62317e7 −2.74987
\(512\) −5.55782e6 −0.936978
\(513\) −1.23466e6 −0.207136
\(514\) 6.67571e6 1.11452
\(515\) 0 0
\(516\) 407501. 0.0673758
\(517\) 407437. 0.0670400
\(518\) −1.05908e7 −1.73423
\(519\) −4.46822e6 −0.728142
\(520\) 0 0
\(521\) 7.05065e6 1.13798 0.568991 0.822344i \(-0.307334\pi\)
0.568991 + 0.822344i \(0.307334\pi\)
\(522\) −132912. −0.0213495
\(523\) 4.20948e6 0.672936 0.336468 0.941695i \(-0.390768\pi\)
0.336468 + 0.941695i \(0.390768\pi\)
\(524\) 3.89563e6 0.619797
\(525\) 0 0
\(526\) 6.56996e6 1.03538
\(527\) 3.40236e6 0.533647
\(528\) −1.35370e6 −0.211319
\(529\) 3.41862e6 0.531143
\(530\) 0 0
\(531\) −1.49207e6 −0.229643
\(532\) 9.42192e6 1.44331
\(533\) 1.22454e7 1.86705
\(534\) 3.81434e6 0.578850
\(535\) 0 0
\(536\) −3.29299e6 −0.495084
\(537\) −5.08312e6 −0.760667
\(538\) 8.32197e6 1.23957
\(539\) 5.64957e6 0.837614
\(540\) 0 0
\(541\) −1.28203e7 −1.88324 −0.941618 0.336684i \(-0.890695\pi\)
−0.941618 + 0.336684i \(0.890695\pi\)
\(542\) 1.51085e7 2.20914
\(543\) 2.47760e6 0.360605
\(544\) −3.13340e6 −0.453961
\(545\) 0 0
\(546\) 1.05702e7 1.51740
\(547\) −4.35983e6 −0.623020 −0.311510 0.950243i \(-0.600835\pi\)
−0.311510 + 0.950243i \(0.600835\pi\)
\(548\) 7.62185e6 1.08420
\(549\) 1.71539e6 0.242903
\(550\) 0 0
\(551\) −377915. −0.0530292
\(552\) 2.06167e6 0.287986
\(553\) 1.87356e7 2.60528
\(554\) −1.71260e7 −2.37073
\(555\) 0 0
\(556\) −3.10909e6 −0.426527
\(557\) 2.40575e6 0.328558 0.164279 0.986414i \(-0.447470\pi\)
0.164279 + 0.986414i \(0.447470\pi\)
\(558\) 4.40205e6 0.598507
\(559\) 1.29987e6 0.175942
\(560\) 0 0
\(561\) −501355. −0.0672571
\(562\) 2.36068e6 0.315280
\(563\) 2.75282e6 0.366022 0.183011 0.983111i \(-0.441416\pi\)
0.183011 + 0.983111i \(0.441416\pi\)
\(564\) 669049. 0.0885646
\(565\) 0 0
\(566\) 8.68400e6 1.13941
\(567\) −1.65329e6 −0.215969
\(568\) −3.99413e6 −0.519459
\(569\) 1.50203e6 0.194491 0.0972454 0.995260i \(-0.468997\pi\)
0.0972454 + 0.995260i \(0.468997\pi\)
\(570\) 0 0
\(571\) 4.08404e6 0.524204 0.262102 0.965040i \(-0.415584\pi\)
0.262102 + 0.965040i \(0.415584\pi\)
\(572\) 1.69309e6 0.216366
\(573\) 3.28681e6 0.418204
\(574\) 3.58018e7 4.53551
\(575\) 0 0
\(576\) −832021. −0.104491
\(577\) −2.67916e6 −0.335011 −0.167506 0.985871i \(-0.553571\pi\)
−0.167506 + 0.985871i \(0.553571\pi\)
\(578\) 8.88259e6 1.10591
\(579\) −5.08975e6 −0.630957
\(580\) 0 0
\(581\) −2.97431e7 −3.65549
\(582\) −4.96487e6 −0.607575
\(583\) 2.42528e6 0.295523
\(584\) 4.70039e6 0.570298
\(585\) 0 0
\(586\) 3.11611e6 0.374859
\(587\) −3.66377e6 −0.438866 −0.219433 0.975628i \(-0.570421\pi\)
−0.219433 + 0.975628i \(0.570421\pi\)
\(588\) 9.27712e6 1.10655
\(589\) 1.25165e7 1.48661
\(590\) 0 0
\(591\) −3.28779e6 −0.387200
\(592\) 7.10462e6 0.833176
\(593\) −764508. −0.0892782 −0.0446391 0.999003i \(-0.514214\pi\)
−0.0446391 + 0.999003i \(0.514214\pi\)
\(594\) −648663. −0.0754316
\(595\) 0 0
\(596\) −701256. −0.0808651
\(597\) −4.18337e6 −0.480386
\(598\) −1.46315e7 −1.67315
\(599\) 8.57025e6 0.975947 0.487974 0.872858i \(-0.337736\pi\)
0.487974 + 0.872858i \(0.337736\pi\)
\(600\) 0 0
\(601\) −2.52254e6 −0.284873 −0.142436 0.989804i \(-0.545494\pi\)
−0.142436 + 0.989804i \(0.545494\pi\)
\(602\) 3.80042e6 0.427405
\(603\) −3.65533e6 −0.409386
\(604\) −2.35768e6 −0.262962
\(605\) 0 0
\(606\) −2.15286e6 −0.238141
\(607\) −9.32496e6 −1.02725 −0.513624 0.858015i \(-0.671697\pi\)
−0.513624 + 0.858015i \(0.671697\pi\)
\(608\) −1.15271e7 −1.26462
\(609\) −506052. −0.0552906
\(610\) 0 0
\(611\) 2.13417e6 0.231274
\(612\) −823270. −0.0888513
\(613\) 1.07420e7 1.15461 0.577303 0.816530i \(-0.304105\pi\)
0.577303 + 0.816530i \(0.304105\pi\)
\(614\) 1.81767e6 0.194578
\(615\) 0 0
\(616\) −2.22492e6 −0.236245
\(617\) −4.83417e6 −0.511221 −0.255611 0.966780i \(-0.582277\pi\)
−0.255611 + 0.966780i \(0.582277\pi\)
\(618\) 4.40535e6 0.463990
\(619\) −1.37750e7 −1.44499 −0.722495 0.691376i \(-0.757005\pi\)
−0.722495 + 0.691376i \(0.757005\pi\)
\(620\) 0 0
\(621\) 2.28852e6 0.238137
\(622\) 1.21992e7 1.26431
\(623\) 1.45228e7 1.49909
\(624\) −7.09075e6 −0.729006
\(625\) 0 0
\(626\) 1.05447e7 1.07547
\(627\) −1.84437e6 −0.187361
\(628\) 1.22401e6 0.123847
\(629\) 2.63125e6 0.265177
\(630\) 0 0
\(631\) −7.90778e6 −0.790644 −0.395322 0.918543i \(-0.629367\pi\)
−0.395322 + 0.918543i \(0.629367\pi\)
\(632\) −5.42546e6 −0.540311
\(633\) 5.69317e6 0.564735
\(634\) 8.96010e6 0.885299
\(635\) 0 0
\(636\) 3.98253e6 0.390406
\(637\) 2.95927e7 2.88959
\(638\) −198548. −0.0193114
\(639\) −4.43361e6 −0.429542
\(640\) 0 0
\(641\) −1.94986e7 −1.87438 −0.937190 0.348818i \(-0.886583\pi\)
−0.937190 + 0.348818i \(0.886583\pi\)
\(642\) −4.29273e6 −0.411051
\(643\) 2.57069e6 0.245201 0.122601 0.992456i \(-0.460877\pi\)
0.122601 + 0.992456i \(0.460877\pi\)
\(644\) −1.74641e7 −1.65933
\(645\) 0 0
\(646\) −5.73382e6 −0.540583
\(647\) −1.67753e7 −1.57547 −0.787735 0.616014i \(-0.788746\pi\)
−0.787735 + 0.616014i \(0.788746\pi\)
\(648\) 478761. 0.0447901
\(649\) −2.22889e6 −0.207720
\(650\) 0 0
\(651\) 1.67604e7 1.55000
\(652\) −2.39339e6 −0.220493
\(653\) −3.43799e6 −0.315517 −0.157758 0.987478i \(-0.550427\pi\)
−0.157758 + 0.987478i \(0.550427\pi\)
\(654\) 3.13341e6 0.286466
\(655\) 0 0
\(656\) −2.40168e7 −2.17899
\(657\) 5.21759e6 0.471581
\(658\) 6.23965e6 0.561818
\(659\) 1.55628e7 1.39596 0.697980 0.716117i \(-0.254083\pi\)
0.697980 + 0.716117i \(0.254083\pi\)
\(660\) 0 0
\(661\) 1.05335e7 0.937714 0.468857 0.883274i \(-0.344666\pi\)
0.468857 + 0.883274i \(0.344666\pi\)
\(662\) −2.00481e7 −1.77799
\(663\) −2.62612e6 −0.232023
\(664\) 8.61304e6 0.758117
\(665\) 0 0
\(666\) 3.40437e6 0.297407
\(667\) 700488. 0.0609658
\(668\) −1.04010e6 −0.0901845
\(669\) 5.63209e6 0.486524
\(670\) 0 0
\(671\) 2.56250e6 0.219714
\(672\) −1.54355e7 −1.31855
\(673\) −5.82081e6 −0.495389 −0.247694 0.968838i \(-0.579673\pi\)
−0.247694 + 0.968838i \(0.579673\pi\)
\(674\) 9.93118e6 0.842076
\(675\) 0 0
\(676\) 671432. 0.0565113
\(677\) −2.08364e7 −1.74723 −0.873616 0.486617i \(-0.838231\pi\)
−0.873616 + 0.486617i \(0.838231\pi\)
\(678\) 7.57729e6 0.633052
\(679\) −1.89033e7 −1.57349
\(680\) 0 0
\(681\) −9.20308e6 −0.760441
\(682\) 6.57590e6 0.541370
\(683\) −1.49980e6 −0.123022 −0.0615109 0.998106i \(-0.519592\pi\)
−0.0615109 + 0.998106i \(0.519592\pi\)
\(684\) −3.02863e6 −0.247517
\(685\) 0 0
\(686\) 5.53757e7 4.49272
\(687\) −1.21524e7 −0.982359
\(688\) −2.54942e6 −0.205338
\(689\) 1.27037e7 1.01949
\(690\) 0 0
\(691\) 9.02927e6 0.719378 0.359689 0.933072i \(-0.382883\pi\)
0.359689 + 0.933072i \(0.382883\pi\)
\(692\) −1.09605e7 −0.870095
\(693\) −2.46973e6 −0.195351
\(694\) 3.06784e7 2.41788
\(695\) 0 0
\(696\) 146543. 0.0114668
\(697\) −8.89482e6 −0.693514
\(698\) −1.41789e7 −1.10155
\(699\) 1.07525e7 0.832369
\(700\) 0 0
\(701\) 2.19167e7 1.68454 0.842268 0.539058i \(-0.181220\pi\)
0.842268 + 0.539058i \(0.181220\pi\)
\(702\) −3.39772e6 −0.260223
\(703\) 9.67978e6 0.738716
\(704\) −1.24290e6 −0.0945155
\(705\) 0 0
\(706\) 2.82394e7 2.13228
\(707\) −8.19683e6 −0.616733
\(708\) −3.66004e6 −0.274412
\(709\) 1.77866e7 1.32885 0.664425 0.747355i \(-0.268676\pi\)
0.664425 + 0.747355i \(0.268676\pi\)
\(710\) 0 0
\(711\) −6.02244e6 −0.446785
\(712\) −4.20552e6 −0.310899
\(713\) −2.32002e7 −1.70910
\(714\) −7.67795e6 −0.563637
\(715\) 0 0
\(716\) −1.24689e7 −0.908961
\(717\) −1.14094e7 −0.828832
\(718\) 1.95270e7 1.41360
\(719\) 1.51719e7 1.09451 0.547254 0.836967i \(-0.315673\pi\)
0.547254 + 0.836967i \(0.315673\pi\)
\(720\) 0 0
\(721\) 1.67730e7 1.20163
\(722\) −2.88492e6 −0.205964
\(723\) 1.29365e7 0.920385
\(724\) 6.07755e6 0.430905
\(725\) 0 0
\(726\) −968991. −0.0682304
\(727\) −1.83411e7 −1.28703 −0.643515 0.765434i \(-0.722524\pi\)
−0.643515 + 0.765434i \(0.722524\pi\)
\(728\) −1.16542e7 −0.814993
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −944198. −0.0653536
\(732\) 4.20786e6 0.290258
\(733\) −2.49198e7 −1.71311 −0.856555 0.516055i \(-0.827400\pi\)
−0.856555 + 0.516055i \(0.827400\pi\)
\(734\) −7.13128e6 −0.488571
\(735\) 0 0
\(736\) 2.13662e7 1.45389
\(737\) −5.46043e6 −0.370304
\(738\) −1.15083e7 −0.777805
\(739\) −2.11189e7 −1.42253 −0.711264 0.702924i \(-0.751877\pi\)
−0.711264 + 0.702924i \(0.751877\pi\)
\(740\) 0 0
\(741\) −9.66089e6 −0.646356
\(742\) 3.71417e7 2.47658
\(743\) 1.55948e6 0.103635 0.0518176 0.998657i \(-0.483499\pi\)
0.0518176 + 0.998657i \(0.483499\pi\)
\(744\) −4.85350e6 −0.321457
\(745\) 0 0
\(746\) −6.16377e6 −0.405508
\(747\) 9.56075e6 0.626889
\(748\) −1.22982e6 −0.0803690
\(749\) −1.63442e7 −1.06453
\(750\) 0 0
\(751\) 9.09412e6 0.588384 0.294192 0.955746i \(-0.404949\pi\)
0.294192 + 0.955746i \(0.404949\pi\)
\(752\) −4.18572e6 −0.269914
\(753\) −4.61857e6 −0.296838
\(754\) −1.04000e6 −0.0666201
\(755\) 0 0
\(756\) −4.05552e6 −0.258073
\(757\) −7.88031e6 −0.499808 −0.249904 0.968271i \(-0.580399\pi\)
−0.249904 + 0.968271i \(0.580399\pi\)
\(758\) −66799.8 −0.00422281
\(759\) 3.41866e6 0.215403
\(760\) 0 0
\(761\) 6.91680e6 0.432956 0.216478 0.976288i \(-0.430543\pi\)
0.216478 + 0.976288i \(0.430543\pi\)
\(762\) −2.09074e7 −1.30440
\(763\) 1.19302e7 0.741884
\(764\) 8.06256e6 0.499734
\(765\) 0 0
\(766\) −2.00284e7 −1.23332
\(767\) −1.16750e7 −0.716588
\(768\) 1.23735e7 0.756988
\(769\) 1.04635e7 0.638057 0.319029 0.947745i \(-0.396643\pi\)
0.319029 + 0.947745i \(0.396643\pi\)
\(770\) 0 0
\(771\) −8.17021e6 −0.494991
\(772\) −1.24852e7 −0.753964
\(773\) −8.77160e6 −0.527996 −0.263998 0.964523i \(-0.585041\pi\)
−0.263998 + 0.964523i \(0.585041\pi\)
\(774\) −1.22162e6 −0.0732968
\(775\) 0 0
\(776\) 5.47404e6 0.326328
\(777\) 1.29618e7 0.770219
\(778\) −3.81053e7 −2.25702
\(779\) −3.27221e7 −1.93195
\(780\) 0 0
\(781\) −6.62305e6 −0.388535
\(782\) 1.06280e7 0.621490
\(783\) 162668. 0.00948192
\(784\) −5.80398e7 −3.37237
\(785\) 0 0
\(786\) −1.16785e7 −0.674264
\(787\) 8.13351e6 0.468103 0.234051 0.972224i \(-0.424802\pi\)
0.234051 + 0.972224i \(0.424802\pi\)
\(788\) −8.06495e6 −0.462686
\(789\) −8.04080e6 −0.459840
\(790\) 0 0
\(791\) 2.88499e7 1.63947
\(792\) 715187. 0.0405141
\(793\) 1.34225e7 0.757967
\(794\) −4.11487e6 −0.231635
\(795\) 0 0
\(796\) −1.02618e7 −0.574038
\(797\) 2.51082e7 1.40014 0.700068 0.714076i \(-0.253153\pi\)
0.700068 + 0.714076i \(0.253153\pi\)
\(798\) −2.82454e7 −1.57015
\(799\) −1.55022e6 −0.0859064
\(800\) 0 0
\(801\) −4.66826e6 −0.257083
\(802\) 1.61792e7 0.888220
\(803\) 7.79417e6 0.426561
\(804\) −8.96653e6 −0.489198
\(805\) 0 0
\(806\) 3.44448e7 1.86761
\(807\) −1.01850e7 −0.550527
\(808\) 2.37365e6 0.127905
\(809\) 6.82106e6 0.366421 0.183211 0.983074i \(-0.441351\pi\)
0.183211 + 0.983074i \(0.441351\pi\)
\(810\) 0 0
\(811\) −1.80322e7 −0.962714 −0.481357 0.876525i \(-0.659856\pi\)
−0.481357 + 0.876525i \(0.659856\pi\)
\(812\) −1.24134e6 −0.0660697
\(813\) −1.84909e7 −0.981142
\(814\) 5.08554e6 0.269015
\(815\) 0 0
\(816\) 5.15057e6 0.270788
\(817\) −3.47349e6 −0.182059
\(818\) 2.51188e7 1.31255
\(819\) −1.29365e7 −0.673920
\(820\) 0 0
\(821\) −2.73897e7 −1.41817 −0.709086 0.705122i \(-0.750892\pi\)
−0.709086 + 0.705122i \(0.750892\pi\)
\(822\) −2.28491e7 −1.17948
\(823\) 1.92663e7 0.991513 0.495757 0.868462i \(-0.334891\pi\)
0.495757 + 0.868462i \(0.334891\pi\)
\(824\) −4.85714e6 −0.249208
\(825\) 0 0
\(826\) −3.41341e7 −1.74076
\(827\) −7.69161e6 −0.391069 −0.195534 0.980697i \(-0.562644\pi\)
−0.195534 + 0.980697i \(0.562644\pi\)
\(828\) 5.61374e6 0.284562
\(829\) 3.75125e7 1.89579 0.947894 0.318585i \(-0.103208\pi\)
0.947894 + 0.318585i \(0.103208\pi\)
\(830\) 0 0
\(831\) 2.09601e7 1.05291
\(832\) −6.51034e6 −0.326058
\(833\) −2.14955e7 −1.07333
\(834\) 9.32057e6 0.464010
\(835\) 0 0
\(836\) −4.52424e6 −0.223888
\(837\) −5.38755e6 −0.265814
\(838\) −8.27064e6 −0.406845
\(839\) 9.79411e6 0.480353 0.240176 0.970729i \(-0.422795\pi\)
0.240176 + 0.970729i \(0.422795\pi\)
\(840\) 0 0
\(841\) −2.04614e7 −0.997573
\(842\) 4.23155e7 2.05693
\(843\) −2.88917e6 −0.140025
\(844\) 1.39653e7 0.674832
\(845\) 0 0
\(846\) −2.00570e6 −0.0963475
\(847\) −3.68935e6 −0.176702
\(848\) −2.49157e7 −1.18982
\(849\) −1.06281e7 −0.506043
\(850\) 0 0
\(851\) −1.79421e7 −0.849276
\(852\) −1.08756e7 −0.513282
\(853\) −6.89811e6 −0.324607 −0.162303 0.986741i \(-0.551892\pi\)
−0.162303 + 0.986741i \(0.551892\pi\)
\(854\) 3.92432e7 1.84128
\(855\) 0 0
\(856\) 4.73297e6 0.220775
\(857\) −3.52694e6 −0.164039 −0.0820193 0.996631i \(-0.526137\pi\)
−0.0820193 + 0.996631i \(0.526137\pi\)
\(858\) −5.07561e6 −0.235380
\(859\) 1.95016e6 0.0901750 0.0450875 0.998983i \(-0.485643\pi\)
0.0450875 + 0.998983i \(0.485643\pi\)
\(860\) 0 0
\(861\) −4.38169e7 −2.01434
\(862\) −9.58158e6 −0.439207
\(863\) 1.22897e7 0.561712 0.280856 0.959750i \(-0.409382\pi\)
0.280856 + 0.959750i \(0.409382\pi\)
\(864\) 4.96165e6 0.226122
\(865\) 0 0
\(866\) 4.90004e7 2.22026
\(867\) −1.08712e7 −0.491166
\(868\) 4.11133e7 1.85218
\(869\) −8.99649e6 −0.404132
\(870\) 0 0
\(871\) −2.86020e7 −1.27747
\(872\) −3.45476e6 −0.153860
\(873\) 6.07637e6 0.269841
\(874\) 3.90980e7 1.73131
\(875\) 0 0
\(876\) 1.27987e7 0.563517
\(877\) 1.46545e7 0.643385 0.321693 0.946844i \(-0.395748\pi\)
0.321693 + 0.946844i \(0.395748\pi\)
\(878\) 4.50504e7 1.97225
\(879\) −3.81372e6 −0.166485
\(880\) 0 0
\(881\) 117098. 0.00508288 0.00254144 0.999997i \(-0.499191\pi\)
0.00254144 + 0.999997i \(0.499191\pi\)
\(882\) −2.78113e7 −1.20379
\(883\) −2.46470e7 −1.06381 −0.531903 0.846806i \(-0.678523\pi\)
−0.531903 + 0.846806i \(0.678523\pi\)
\(884\) −6.44186e6 −0.277256
\(885\) 0 0
\(886\) 4.83303e7 2.06840
\(887\) 9.09068e6 0.387961 0.193980 0.981005i \(-0.437860\pi\)
0.193980 + 0.981005i \(0.437860\pi\)
\(888\) −3.75350e6 −0.159737
\(889\) −7.96030e7 −3.37812
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 1.38155e7 0.581373
\(893\) −5.70290e6 −0.239313
\(894\) 2.10226e6 0.0879715
\(895\) 0 0
\(896\) 3.58475e7 1.49173
\(897\) 1.79071e7 0.743093
\(898\) 1.35103e7 0.559082
\(899\) −1.64906e6 −0.0680515
\(900\) 0 0
\(901\) −9.22771e6 −0.378688
\(902\) −1.71914e7 −0.703551
\(903\) −4.65123e6 −0.189823
\(904\) −8.35438e6 −0.340011
\(905\) 0 0
\(906\) 7.06796e6 0.286071
\(907\) 9.29801e6 0.375294 0.187647 0.982237i \(-0.439914\pi\)
0.187647 + 0.982237i \(0.439914\pi\)
\(908\) −2.25752e7 −0.908691
\(909\) 2.63483e6 0.105765
\(910\) 0 0
\(911\) 2.25087e7 0.898574 0.449287 0.893387i \(-0.351678\pi\)
0.449287 + 0.893387i \(0.351678\pi\)
\(912\) 1.89478e7 0.754348
\(913\) 1.42821e7 0.567042
\(914\) −3.69594e7 −1.46339
\(915\) 0 0
\(916\) −2.98099e7 −1.17387
\(917\) −4.44648e7 −1.74620
\(918\) 2.46803e6 0.0966595
\(919\) 2.05921e7 0.804288 0.402144 0.915576i \(-0.368265\pi\)
0.402144 + 0.915576i \(0.368265\pi\)
\(920\) 0 0
\(921\) −2.22459e6 −0.0864175
\(922\) 2.38782e7 0.925070
\(923\) −3.46918e7 −1.34036
\(924\) −6.05825e6 −0.233436
\(925\) 0 0
\(926\) 6.06131e7 2.32295
\(927\) −5.39158e6 −0.206071
\(928\) 1.51870e6 0.0578898
\(929\) 1.69947e7 0.646064 0.323032 0.946388i \(-0.395298\pi\)
0.323032 + 0.946388i \(0.395298\pi\)
\(930\) 0 0
\(931\) −7.90771e7 −2.99004
\(932\) 2.63759e7 0.994642
\(933\) −1.49303e7 −0.561517
\(934\) −1.19027e7 −0.446455
\(935\) 0 0
\(936\) 3.74618e6 0.139765
\(937\) −1.14605e7 −0.426438 −0.213219 0.977004i \(-0.568395\pi\)
−0.213219 + 0.977004i \(0.568395\pi\)
\(938\) −8.36232e7 −3.10327
\(939\) −1.29054e7 −0.477648
\(940\) 0 0
\(941\) 1.16402e7 0.428534 0.214267 0.976775i \(-0.431264\pi\)
0.214267 + 0.976775i \(0.431264\pi\)
\(942\) −3.66939e6 −0.134731
\(943\) 6.06523e7 2.22110
\(944\) 2.28981e7 0.836314
\(945\) 0 0
\(946\) −1.82489e6 −0.0662994
\(947\) 1.62911e7 0.590303 0.295151 0.955450i \(-0.404630\pi\)
0.295151 + 0.955450i \(0.404630\pi\)
\(948\) −1.47731e7 −0.533887
\(949\) 4.08262e7 1.47154
\(950\) 0 0
\(951\) −1.09660e7 −0.393186
\(952\) 8.46536e6 0.302728
\(953\) 1.77379e7 0.632660 0.316330 0.948649i \(-0.397549\pi\)
0.316330 + 0.948649i \(0.397549\pi\)
\(954\) −1.19390e7 −0.424715
\(955\) 0 0
\(956\) −2.79874e7 −0.990415
\(957\) 242997. 0.00857672
\(958\) −791172. −0.0278520
\(959\) −8.69960e7 −3.05459
\(960\) 0 0
\(961\) 2.59877e7 0.907737
\(962\) 2.66383e7 0.928043
\(963\) 5.25376e6 0.182559
\(964\) 3.17331e7 1.09982
\(965\) 0 0
\(966\) 5.23547e7 1.80515
\(967\) 4.62764e7 1.59145 0.795726 0.605657i \(-0.207089\pi\)
0.795726 + 0.605657i \(0.207089\pi\)
\(968\) 1.06837e6 0.0366464
\(969\) 7.01746e6 0.240088
\(970\) 0 0
\(971\) 3.08766e7 1.05095 0.525474 0.850810i \(-0.323888\pi\)
0.525474 + 0.850810i \(0.323888\pi\)
\(972\) 1.30362e6 0.0442575
\(973\) 3.54873e7 1.20168
\(974\) −6.12759e7 −2.06963
\(975\) 0 0
\(976\) −2.63254e7 −0.884607
\(977\) −3.37783e7 −1.13214 −0.566071 0.824357i \(-0.691537\pi\)
−0.566071 + 0.824357i \(0.691537\pi\)
\(978\) 7.17500e6 0.239869
\(979\) −6.97358e6 −0.232541
\(980\) 0 0
\(981\) −3.83489e6 −0.127227
\(982\) −7.03891e7 −2.32931
\(983\) −4.70641e7 −1.55348 −0.776740 0.629821i \(-0.783128\pi\)
−0.776740 + 0.629821i \(0.783128\pi\)
\(984\) 1.26885e7 0.417757
\(985\) 0 0
\(986\) 755435. 0.0247460
\(987\) −7.63654e6 −0.249519
\(988\) −2.36982e7 −0.772365
\(989\) 6.43833e6 0.209306
\(990\) 0 0
\(991\) −1.74875e7 −0.565644 −0.282822 0.959172i \(-0.591271\pi\)
−0.282822 + 0.959172i \(0.591271\pi\)
\(992\) −5.02993e7 −1.62287
\(993\) 2.45363e7 0.789654
\(994\) −1.01428e8 −3.25606
\(995\) 0 0
\(996\) 2.34525e7 0.749103
\(997\) 1.72629e7 0.550015 0.275008 0.961442i \(-0.411320\pi\)
0.275008 + 0.961442i \(0.411320\pi\)
\(998\) 5.99981e6 0.190683
\(999\) −4.16651e6 −0.132087
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.1.2 7
5.4 even 2 825.6.a.o.1.6 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.m.1.2 7 1.1 even 1 trivial
825.6.a.o.1.6 yes 7 5.4 even 2