Properties

Label 825.6.a.m.1.3
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.3
Root \(4.42881\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.42881 q^{2} +9.00000 q^{3} -2.52798 q^{4} -48.8593 q^{6} +29.0935 q^{7} +187.446 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-5.42881 q^{2} +9.00000 q^{3} -2.52798 q^{4} -48.8593 q^{6} +29.0935 q^{7} +187.446 q^{8} +81.0000 q^{9} +121.000 q^{11} -22.7518 q^{12} +433.055 q^{13} -157.943 q^{14} -936.714 q^{16} -681.634 q^{17} -439.734 q^{18} +1400.22 q^{19} +261.841 q^{21} -656.886 q^{22} -4762.16 q^{23} +1687.01 q^{24} -2350.97 q^{26} +729.000 q^{27} -73.5478 q^{28} -646.539 q^{29} +840.506 q^{31} -913.026 q^{32} +1089.00 q^{33} +3700.46 q^{34} -204.767 q^{36} -7782.45 q^{37} -7601.56 q^{38} +3897.49 q^{39} -15790.5 q^{41} -1421.49 q^{42} +857.270 q^{43} -305.886 q^{44} +25852.9 q^{46} +9406.46 q^{47} -8430.42 q^{48} -15960.6 q^{49} -6134.70 q^{51} -1094.75 q^{52} +38888.1 q^{53} -3957.61 q^{54} +5453.45 q^{56} +12602.0 q^{57} +3509.94 q^{58} +43344.6 q^{59} -35683.2 q^{61} -4562.95 q^{62} +2356.57 q^{63} +34931.5 q^{64} -5911.98 q^{66} +4283.17 q^{67} +1723.16 q^{68} -42859.4 q^{69} +790.935 q^{71} +15183.1 q^{72} -54354.4 q^{73} +42249.5 q^{74} -3539.74 q^{76} +3520.31 q^{77} -21158.8 q^{78} +24658.1 q^{79} +6561.00 q^{81} +85723.6 q^{82} -67785.3 q^{83} -661.930 q^{84} -4653.96 q^{86} -5818.85 q^{87} +22681.0 q^{88} -43023.8 q^{89} +12599.1 q^{91} +12038.6 q^{92} +7564.55 q^{93} -51065.9 q^{94} -8217.24 q^{96} +77298.9 q^{97} +86647.0 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.42881 −0.959688 −0.479844 0.877354i \(-0.659307\pi\)
−0.479844 + 0.877354i \(0.659307\pi\)
\(3\) 9.00000 0.577350
\(4\) −2.52798 −0.0789994
\(5\) 0 0
\(6\) −48.8593 −0.554076
\(7\) 29.0935 0.224414 0.112207 0.993685i \(-0.464208\pi\)
0.112207 + 0.993685i \(0.464208\pi\)
\(8\) 187.446 1.03550
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −22.7518 −0.0456103
\(13\) 433.055 0.710697 0.355348 0.934734i \(-0.384362\pi\)
0.355348 + 0.934734i \(0.384362\pi\)
\(14\) −157.943 −0.215368
\(15\) 0 0
\(16\) −936.714 −0.914760
\(17\) −681.634 −0.572043 −0.286021 0.958223i \(-0.592333\pi\)
−0.286021 + 0.958223i \(0.592333\pi\)
\(18\) −439.734 −0.319896
\(19\) 1400.22 0.889844 0.444922 0.895569i \(-0.353231\pi\)
0.444922 + 0.895569i \(0.353231\pi\)
\(20\) 0 0
\(21\) 261.841 0.129566
\(22\) −656.886 −0.289357
\(23\) −4762.16 −1.87709 −0.938543 0.345164i \(-0.887823\pi\)
−0.938543 + 0.345164i \(0.887823\pi\)
\(24\) 1687.01 0.597848
\(25\) 0 0
\(26\) −2350.97 −0.682047
\(27\) 729.000 0.192450
\(28\) −73.5478 −0.0177286
\(29\) −646.539 −0.142758 −0.0713789 0.997449i \(-0.522740\pi\)
−0.0713789 + 0.997449i \(0.522740\pi\)
\(30\) 0 0
\(31\) 840.506 0.157086 0.0785428 0.996911i \(-0.474973\pi\)
0.0785428 + 0.996911i \(0.474973\pi\)
\(32\) −913.026 −0.157619
\(33\) 1089.00 0.174078
\(34\) 3700.46 0.548983
\(35\) 0 0
\(36\) −204.767 −0.0263331
\(37\) −7782.45 −0.934571 −0.467285 0.884107i \(-0.654768\pi\)
−0.467285 + 0.884107i \(0.654768\pi\)
\(38\) −7601.56 −0.853972
\(39\) 3897.49 0.410321
\(40\) 0 0
\(41\) −15790.5 −1.46702 −0.733510 0.679679i \(-0.762119\pi\)
−0.733510 + 0.679679i \(0.762119\pi\)
\(42\) −1421.49 −0.124343
\(43\) 857.270 0.0707044 0.0353522 0.999375i \(-0.488745\pi\)
0.0353522 + 0.999375i \(0.488745\pi\)
\(44\) −305.886 −0.0238192
\(45\) 0 0
\(46\) 25852.9 1.80142
\(47\) 9406.46 0.621129 0.310564 0.950552i \(-0.399482\pi\)
0.310564 + 0.950552i \(0.399482\pi\)
\(48\) −8430.42 −0.528137
\(49\) −15960.6 −0.949638
\(50\) 0 0
\(51\) −6134.70 −0.330269
\(52\) −1094.75 −0.0561446
\(53\) 38888.1 1.90163 0.950817 0.309755i \(-0.100247\pi\)
0.950817 + 0.309755i \(0.100247\pi\)
\(54\) −3957.61 −0.184692
\(55\) 0 0
\(56\) 5453.45 0.232382
\(57\) 12602.0 0.513751
\(58\) 3509.94 0.137003
\(59\) 43344.6 1.62108 0.810540 0.585683i \(-0.199174\pi\)
0.810540 + 0.585683i \(0.199174\pi\)
\(60\) 0 0
\(61\) −35683.2 −1.22783 −0.613917 0.789371i \(-0.710407\pi\)
−0.613917 + 0.789371i \(0.710407\pi\)
\(62\) −4562.95 −0.150753
\(63\) 2356.57 0.0748047
\(64\) 34931.5 1.06602
\(65\) 0 0
\(66\) −5911.98 −0.167060
\(67\) 4283.17 0.116568 0.0582838 0.998300i \(-0.481437\pi\)
0.0582838 + 0.998300i \(0.481437\pi\)
\(68\) 1723.16 0.0451911
\(69\) −42859.4 −1.08374
\(70\) 0 0
\(71\) 790.935 0.0186207 0.00931033 0.999957i \(-0.497036\pi\)
0.00931033 + 0.999957i \(0.497036\pi\)
\(72\) 15183.1 0.345168
\(73\) −54354.4 −1.19379 −0.596894 0.802320i \(-0.703599\pi\)
−0.596894 + 0.802320i \(0.703599\pi\)
\(74\) 42249.5 0.896896
\(75\) 0 0
\(76\) −3539.74 −0.0702972
\(77\) 3520.31 0.0676634
\(78\) −21158.8 −0.393780
\(79\) 24658.1 0.444521 0.222261 0.974987i \(-0.428656\pi\)
0.222261 + 0.974987i \(0.428656\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 85723.6 1.40788
\(83\) −67785.3 −1.08004 −0.540021 0.841652i \(-0.681584\pi\)
−0.540021 + 0.841652i \(0.681584\pi\)
\(84\) −661.930 −0.0102356
\(85\) 0 0
\(86\) −4653.96 −0.0678542
\(87\) −5818.85 −0.0824212
\(88\) 22681.0 0.312216
\(89\) −43023.8 −0.575750 −0.287875 0.957668i \(-0.592949\pi\)
−0.287875 + 0.957668i \(0.592949\pi\)
\(90\) 0 0
\(91\) 12599.1 0.159490
\(92\) 12038.6 0.148289
\(93\) 7564.55 0.0906934
\(94\) −51065.9 −0.596090
\(95\) 0 0
\(96\) −8217.24 −0.0910013
\(97\) 77298.9 0.834150 0.417075 0.908872i \(-0.363055\pi\)
0.417075 + 0.908872i \(0.363055\pi\)
\(98\) 86647.0 0.911356
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −76235.3 −0.743623 −0.371811 0.928308i \(-0.621263\pi\)
−0.371811 + 0.928308i \(0.621263\pi\)
\(102\) 33304.2 0.316955
\(103\) −10863.8 −0.100899 −0.0504495 0.998727i \(-0.516065\pi\)
−0.0504495 + 0.998727i \(0.516065\pi\)
\(104\) 81174.3 0.735928
\(105\) 0 0
\(106\) −211116. −1.82497
\(107\) 68622.7 0.579440 0.289720 0.957111i \(-0.406438\pi\)
0.289720 + 0.957111i \(0.406438\pi\)
\(108\) −1842.90 −0.0152034
\(109\) −105784. −0.852813 −0.426406 0.904532i \(-0.640221\pi\)
−0.426406 + 0.904532i \(0.640221\pi\)
\(110\) 0 0
\(111\) −70042.1 −0.539575
\(112\) −27252.3 −0.205285
\(113\) 102154. 0.752594 0.376297 0.926499i \(-0.377197\pi\)
0.376297 + 0.926499i \(0.377197\pi\)
\(114\) −68414.0 −0.493041
\(115\) 0 0
\(116\) 1634.44 0.0112778
\(117\) 35077.4 0.236899
\(118\) −235309. −1.55573
\(119\) −19831.1 −0.128375
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 193718. 1.17834
\(123\) −142114. −0.846984
\(124\) −2124.78 −0.0124097
\(125\) 0 0
\(126\) −12793.4 −0.0717892
\(127\) 302945. 1.66669 0.833345 0.552754i \(-0.186423\pi\)
0.833345 + 0.552754i \(0.186423\pi\)
\(128\) −160420. −0.865432
\(129\) 7715.43 0.0408212
\(130\) 0 0
\(131\) −327544. −1.66760 −0.833800 0.552067i \(-0.813839\pi\)
−0.833800 + 0.552067i \(0.813839\pi\)
\(132\) −2752.97 −0.0137520
\(133\) 40737.4 0.199694
\(134\) −23252.5 −0.111869
\(135\) 0 0
\(136\) −127769. −0.592352
\(137\) −411208. −1.87180 −0.935902 0.352259i \(-0.885414\pi\)
−0.935902 + 0.352259i \(0.885414\pi\)
\(138\) 232676. 1.04005
\(139\) 384656. 1.68863 0.844317 0.535843i \(-0.180006\pi\)
0.844317 + 0.535843i \(0.180006\pi\)
\(140\) 0 0
\(141\) 84658.2 0.358609
\(142\) −4293.84 −0.0178700
\(143\) 52399.6 0.214283
\(144\) −75873.8 −0.304920
\(145\) 0 0
\(146\) 295080. 1.14566
\(147\) −143645. −0.548274
\(148\) 19673.9 0.0738306
\(149\) −61538.1 −0.227080 −0.113540 0.993533i \(-0.536219\pi\)
−0.113540 + 0.993533i \(0.536219\pi\)
\(150\) 0 0
\(151\) −337615. −1.20498 −0.602489 0.798127i \(-0.705824\pi\)
−0.602489 + 0.798127i \(0.705824\pi\)
\(152\) 262466. 0.921435
\(153\) −55212.3 −0.190681
\(154\) −19111.1 −0.0649358
\(155\) 0 0
\(156\) −9852.79 −0.0324151
\(157\) 455250. 1.47401 0.737006 0.675887i \(-0.236239\pi\)
0.737006 + 0.675887i \(0.236239\pi\)
\(158\) −133864. −0.426602
\(159\) 349993. 1.09791
\(160\) 0 0
\(161\) −138548. −0.421245
\(162\) −35618.4 −0.106632
\(163\) −36402.3 −0.107315 −0.0536575 0.998559i \(-0.517088\pi\)
−0.0536575 + 0.998559i \(0.517088\pi\)
\(164\) 39918.1 0.115894
\(165\) 0 0
\(166\) 367994. 1.03650
\(167\) 261013. 0.724220 0.362110 0.932135i \(-0.382056\pi\)
0.362110 + 0.932135i \(0.382056\pi\)
\(168\) 49081.1 0.134166
\(169\) −183757. −0.494910
\(170\) 0 0
\(171\) 113418. 0.296615
\(172\) −2167.16 −0.00558561
\(173\) 61304.1 0.155731 0.0778653 0.996964i \(-0.475190\pi\)
0.0778653 + 0.996964i \(0.475190\pi\)
\(174\) 31589.5 0.0790986
\(175\) 0 0
\(176\) −113342. −0.275810
\(177\) 390101. 0.935931
\(178\) 233568. 0.552540
\(179\) −766914. −1.78902 −0.894508 0.447052i \(-0.852474\pi\)
−0.894508 + 0.447052i \(0.852474\pi\)
\(180\) 0 0
\(181\) 116461. 0.264231 0.132115 0.991234i \(-0.457823\pi\)
0.132115 + 0.991234i \(0.457823\pi\)
\(182\) −68397.9 −0.153061
\(183\) −321149. −0.708890
\(184\) −892647. −1.94373
\(185\) 0 0
\(186\) −41066.5 −0.0870374
\(187\) −82477.7 −0.172477
\(188\) −23779.4 −0.0490688
\(189\) 21209.1 0.0431885
\(190\) 0 0
\(191\) −814613. −1.61573 −0.807863 0.589370i \(-0.799376\pi\)
−0.807863 + 0.589370i \(0.799376\pi\)
\(192\) 314383. 0.615470
\(193\) 81825.1 0.158122 0.0790612 0.996870i \(-0.474808\pi\)
0.0790612 + 0.996870i \(0.474808\pi\)
\(194\) −419641. −0.800523
\(195\) 0 0
\(196\) 40348.0 0.0750209
\(197\) 421809. 0.774372 0.387186 0.922002i \(-0.373447\pi\)
0.387186 + 0.922002i \(0.373447\pi\)
\(198\) −53207.8 −0.0964522
\(199\) 294642. 0.527426 0.263713 0.964601i \(-0.415053\pi\)
0.263713 + 0.964601i \(0.415053\pi\)
\(200\) 0 0
\(201\) 38548.5 0.0673003
\(202\) 413867. 0.713646
\(203\) −18810.1 −0.0320369
\(204\) 15508.4 0.0260911
\(205\) 0 0
\(206\) 58977.3 0.0968316
\(207\) −385735. −0.625695
\(208\) −405648. −0.650117
\(209\) 169427. 0.268298
\(210\) 0 0
\(211\) 792181. 1.22495 0.612475 0.790490i \(-0.290174\pi\)
0.612475 + 0.790490i \(0.290174\pi\)
\(212\) −98308.3 −0.150228
\(213\) 7118.42 0.0107506
\(214\) −372540. −0.556081
\(215\) 0 0
\(216\) 136648. 0.199283
\(217\) 24453.2 0.0352523
\(218\) 574282. 0.818434
\(219\) −489189. −0.689234
\(220\) 0 0
\(221\) −295185. −0.406549
\(222\) 380245. 0.517823
\(223\) −130498. −0.175728 −0.0878638 0.996132i \(-0.528004\pi\)
−0.0878638 + 0.996132i \(0.528004\pi\)
\(224\) −26563.1 −0.0353719
\(225\) 0 0
\(226\) −554577. −0.722255
\(227\) 615550. 0.792864 0.396432 0.918064i \(-0.370248\pi\)
0.396432 + 0.918064i \(0.370248\pi\)
\(228\) −31857.7 −0.0405861
\(229\) −698301. −0.879942 −0.439971 0.898012i \(-0.645011\pi\)
−0.439971 + 0.898012i \(0.645011\pi\)
\(230\) 0 0
\(231\) 31682.8 0.0390655
\(232\) −121191. −0.147826
\(233\) 630539. 0.760890 0.380445 0.924803i \(-0.375771\pi\)
0.380445 + 0.924803i \(0.375771\pi\)
\(234\) −190429. −0.227349
\(235\) 0 0
\(236\) −109574. −0.128064
\(237\) 221923. 0.256644
\(238\) 107659. 0.123200
\(239\) −880589. −0.997191 −0.498595 0.866835i \(-0.666151\pi\)
−0.498595 + 0.866835i \(0.666151\pi\)
\(240\) 0 0
\(241\) −97283.2 −0.107894 −0.0539468 0.998544i \(-0.517180\pi\)
−0.0539468 + 0.998544i \(0.517180\pi\)
\(242\) −79483.3 −0.0872443
\(243\) 59049.0 0.0641500
\(244\) 90206.6 0.0969982
\(245\) 0 0
\(246\) 771513. 0.812840
\(247\) 606374. 0.632409
\(248\) 157549. 0.162663
\(249\) −610068. −0.623562
\(250\) 0 0
\(251\) −35093.6 −0.0351595 −0.0175798 0.999845i \(-0.505596\pi\)
−0.0175798 + 0.999845i \(0.505596\pi\)
\(252\) −5957.37 −0.00590953
\(253\) −576221. −0.565962
\(254\) −1.64463e6 −1.59950
\(255\) 0 0
\(256\) −246919. −0.235480
\(257\) −1.45297e6 −1.37222 −0.686111 0.727497i \(-0.740683\pi\)
−0.686111 + 0.727497i \(0.740683\pi\)
\(258\) −41885.7 −0.0391756
\(259\) −226419. −0.209731
\(260\) 0 0
\(261\) −52369.7 −0.0475859
\(262\) 1.77818e6 1.60037
\(263\) −1.67823e6 −1.49610 −0.748051 0.663642i \(-0.769010\pi\)
−0.748051 + 0.663642i \(0.769010\pi\)
\(264\) 204129. 0.180258
\(265\) 0 0
\(266\) −221156. −0.191643
\(267\) −387215. −0.332410
\(268\) −10827.8 −0.00920878
\(269\) −2.08881e6 −1.76002 −0.880010 0.474955i \(-0.842464\pi\)
−0.880010 + 0.474955i \(0.842464\pi\)
\(270\) 0 0
\(271\) −59165.0 −0.0489375 −0.0244687 0.999701i \(-0.507789\pi\)
−0.0244687 + 0.999701i \(0.507789\pi\)
\(272\) 638496. 0.523282
\(273\) 113392. 0.0920818
\(274\) 2.23237e6 1.79635
\(275\) 0 0
\(276\) 108348. 0.0856145
\(277\) −1.37762e6 −1.07877 −0.539387 0.842058i \(-0.681344\pi\)
−0.539387 + 0.842058i \(0.681344\pi\)
\(278\) −2.08823e6 −1.62056
\(279\) 68081.0 0.0523619
\(280\) 0 0
\(281\) 348215. 0.263077 0.131538 0.991311i \(-0.458008\pi\)
0.131538 + 0.991311i \(0.458008\pi\)
\(282\) −459593. −0.344153
\(283\) −1.38544e6 −1.02831 −0.514154 0.857698i \(-0.671894\pi\)
−0.514154 + 0.857698i \(0.671894\pi\)
\(284\) −1999.47 −0.00147102
\(285\) 0 0
\(286\) −284468. −0.205645
\(287\) −459400. −0.329220
\(288\) −73955.1 −0.0525396
\(289\) −955233. −0.672767
\(290\) 0 0
\(291\) 695690. 0.481597
\(292\) 137407. 0.0943086
\(293\) −1.41063e6 −0.959937 −0.479969 0.877286i \(-0.659352\pi\)
−0.479969 + 0.877286i \(0.659352\pi\)
\(294\) 779823. 0.526172
\(295\) 0 0
\(296\) −1.45879e6 −0.967750
\(297\) 88209.0 0.0580259
\(298\) 334079. 0.217926
\(299\) −2.06227e6 −1.33404
\(300\) 0 0
\(301\) 24941.0 0.0158671
\(302\) 1.83285e6 1.15640
\(303\) −686118. −0.429331
\(304\) −1.31161e6 −0.813993
\(305\) 0 0
\(306\) 299737. 0.182994
\(307\) −1.99536e6 −1.20830 −0.604150 0.796871i \(-0.706487\pi\)
−0.604150 + 0.796871i \(0.706487\pi\)
\(308\) −8899.28 −0.00534537
\(309\) −97773.8 −0.0582541
\(310\) 0 0
\(311\) −1.98713e6 −1.16500 −0.582500 0.812831i \(-0.697925\pi\)
−0.582500 + 0.812831i \(0.697925\pi\)
\(312\) 730569. 0.424888
\(313\) −737295. −0.425383 −0.212692 0.977119i \(-0.568223\pi\)
−0.212692 + 0.977119i \(0.568223\pi\)
\(314\) −2.47147e6 −1.41459
\(315\) 0 0
\(316\) −62335.3 −0.0351169
\(317\) −2.58321e6 −1.44382 −0.721908 0.691989i \(-0.756734\pi\)
−0.721908 + 0.691989i \(0.756734\pi\)
\(318\) −1.90004e6 −1.05365
\(319\) −78231.2 −0.0430431
\(320\) 0 0
\(321\) 617604. 0.334540
\(322\) 752149. 0.404263
\(323\) −954440. −0.509029
\(324\) −16586.1 −0.00877772
\(325\) 0 0
\(326\) 197621. 0.102989
\(327\) −952056. −0.492372
\(328\) −2.95986e6 −1.51910
\(329\) 273667. 0.139390
\(330\) 0 0
\(331\) −263417. −0.132152 −0.0660759 0.997815i \(-0.521048\pi\)
−0.0660759 + 0.997815i \(0.521048\pi\)
\(332\) 171360. 0.0853227
\(333\) −630379. −0.311524
\(334\) −1.41699e6 −0.695025
\(335\) 0 0
\(336\) −245270. −0.118521
\(337\) 1.74494e6 0.836963 0.418482 0.908225i \(-0.362562\pi\)
0.418482 + 0.908225i \(0.362562\pi\)
\(338\) 997581. 0.474959
\(339\) 919389. 0.434510
\(340\) 0 0
\(341\) 101701. 0.0473631
\(342\) −615726. −0.284657
\(343\) −953322. −0.437527
\(344\) 160692. 0.0732146
\(345\) 0 0
\(346\) −332808. −0.149453
\(347\) −3.21441e6 −1.43310 −0.716551 0.697535i \(-0.754280\pi\)
−0.716551 + 0.697535i \(0.754280\pi\)
\(348\) 14709.9 0.00651123
\(349\) 287220. 0.126227 0.0631133 0.998006i \(-0.479897\pi\)
0.0631133 + 0.998006i \(0.479897\pi\)
\(350\) 0 0
\(351\) 315697. 0.136774
\(352\) −110476. −0.0475239
\(353\) 1.23752e6 0.528587 0.264293 0.964442i \(-0.414861\pi\)
0.264293 + 0.964442i \(0.414861\pi\)
\(354\) −2.11779e6 −0.898202
\(355\) 0 0
\(356\) 108763. 0.0454839
\(357\) −178480. −0.0741171
\(358\) 4.16344e6 1.71690
\(359\) 4.49015e6 1.83876 0.919379 0.393373i \(-0.128692\pi\)
0.919379 + 0.393373i \(0.128692\pi\)
\(360\) 0 0
\(361\) −515470. −0.208178
\(362\) −632245. −0.253579
\(363\) 131769. 0.0524864
\(364\) −31850.2 −0.0125997
\(365\) 0 0
\(366\) 1.74346e6 0.680313
\(367\) 2.56230e6 0.993035 0.496517 0.868027i \(-0.334612\pi\)
0.496517 + 0.868027i \(0.334612\pi\)
\(368\) 4.46078e6 1.71708
\(369\) −1.27903e6 −0.489007
\(370\) 0 0
\(371\) 1.13139e6 0.426754
\(372\) −19123.1 −0.00716473
\(373\) −1.44404e6 −0.537411 −0.268706 0.963222i \(-0.586596\pi\)
−0.268706 + 0.963222i \(0.586596\pi\)
\(374\) 447756. 0.165524
\(375\) 0 0
\(376\) 1.76320e6 0.643181
\(377\) −279987. −0.101457
\(378\) −115140. −0.0414475
\(379\) −3.07135e6 −1.09833 −0.549163 0.835715i \(-0.685053\pi\)
−0.549163 + 0.835715i \(0.685053\pi\)
\(380\) 0 0
\(381\) 2.72651e6 0.962263
\(382\) 4.42238e6 1.55059
\(383\) 490553. 0.170879 0.0854395 0.996343i \(-0.472771\pi\)
0.0854395 + 0.996343i \(0.472771\pi\)
\(384\) −1.44378e6 −0.499657
\(385\) 0 0
\(386\) −444213. −0.151748
\(387\) 69438.9 0.0235681
\(388\) −195410. −0.0658974
\(389\) −2.10754e6 −0.706157 −0.353079 0.935594i \(-0.614865\pi\)
−0.353079 + 0.935594i \(0.614865\pi\)
\(390\) 0 0
\(391\) 3.24604e6 1.07377
\(392\) −2.99174e6 −0.983353
\(393\) −2.94790e6 −0.962789
\(394\) −2.28992e6 −0.743156
\(395\) 0 0
\(396\) −24776.8 −0.00793974
\(397\) 1.76935e6 0.563425 0.281713 0.959499i \(-0.409098\pi\)
0.281713 + 0.959499i \(0.409098\pi\)
\(398\) −1.59956e6 −0.506164
\(399\) 366637. 0.115293
\(400\) 0 0
\(401\) −2.75581e6 −0.855831 −0.427916 0.903819i \(-0.640752\pi\)
−0.427916 + 0.903819i \(0.640752\pi\)
\(402\) −209273. −0.0645873
\(403\) 363985. 0.111640
\(404\) 192722. 0.0587458
\(405\) 0 0
\(406\) 102116. 0.0307454
\(407\) −941677. −0.281784
\(408\) −1.14993e6 −0.341995
\(409\) 2.37262e6 0.701326 0.350663 0.936502i \(-0.385956\pi\)
0.350663 + 0.936502i \(0.385956\pi\)
\(410\) 0 0
\(411\) −3.70088e6 −1.08069
\(412\) 27463.4 0.00797097
\(413\) 1.26104e6 0.363793
\(414\) 2.09408e6 0.600472
\(415\) 0 0
\(416\) −395390. −0.112019
\(417\) 3.46191e6 0.974934
\(418\) −919789. −0.257482
\(419\) −1.55244e6 −0.431996 −0.215998 0.976394i \(-0.569300\pi\)
−0.215998 + 0.976394i \(0.569300\pi\)
\(420\) 0 0
\(421\) −1.71814e6 −0.472446 −0.236223 0.971699i \(-0.575910\pi\)
−0.236223 + 0.971699i \(0.575910\pi\)
\(422\) −4.30061e6 −1.17557
\(423\) 761923. 0.207043
\(424\) 7.28941e6 1.96915
\(425\) 0 0
\(426\) −38644.6 −0.0103173
\(427\) −1.03815e6 −0.275543
\(428\) −173477. −0.0457754
\(429\) 471596. 0.123716
\(430\) 0 0
\(431\) −2.93876e6 −0.762029 −0.381015 0.924569i \(-0.624425\pi\)
−0.381015 + 0.924569i \(0.624425\pi\)
\(432\) −682864. −0.176046
\(433\) −6.64391e6 −1.70296 −0.851480 0.524388i \(-0.824294\pi\)
−0.851480 + 0.524388i \(0.824294\pi\)
\(434\) −132752. −0.0338312
\(435\) 0 0
\(436\) 267420. 0.0673717
\(437\) −6.66809e6 −1.67031
\(438\) 2.65572e6 0.661449
\(439\) 1.90964e6 0.472922 0.236461 0.971641i \(-0.424012\pi\)
0.236461 + 0.971641i \(0.424012\pi\)
\(440\) 0 0
\(441\) −1.29281e6 −0.316546
\(442\) 1.60250e6 0.390160
\(443\) 4.69600e6 1.13689 0.568445 0.822721i \(-0.307545\pi\)
0.568445 + 0.822721i \(0.307545\pi\)
\(444\) 177065. 0.0426261
\(445\) 0 0
\(446\) 708447. 0.168644
\(447\) −553843. −0.131105
\(448\) 1.01628e6 0.239231
\(449\) 4.56839e6 1.06942 0.534708 0.845037i \(-0.320421\pi\)
0.534708 + 0.845037i \(0.320421\pi\)
\(450\) 0 0
\(451\) −1.91065e6 −0.442323
\(452\) −258244. −0.0594545
\(453\) −3.03853e6 −0.695694
\(454\) −3.34171e6 −0.760902
\(455\) 0 0
\(456\) 2.36220e6 0.531991
\(457\) −3.14110e6 −0.703545 −0.351772 0.936086i \(-0.614421\pi\)
−0.351772 + 0.936086i \(0.614421\pi\)
\(458\) 3.79095e6 0.844470
\(459\) −496911. −0.110090
\(460\) 0 0
\(461\) 1.26633e6 0.277521 0.138760 0.990326i \(-0.455688\pi\)
0.138760 + 0.990326i \(0.455688\pi\)
\(462\) −172000. −0.0374907
\(463\) 1.84439e6 0.399852 0.199926 0.979811i \(-0.435930\pi\)
0.199926 + 0.979811i \(0.435930\pi\)
\(464\) 605622. 0.130589
\(465\) 0 0
\(466\) −3.42308e6 −0.730217
\(467\) 3.10954e6 0.659787 0.329894 0.944018i \(-0.392987\pi\)
0.329894 + 0.944018i \(0.392987\pi\)
\(468\) −88675.1 −0.0187149
\(469\) 124612. 0.0261594
\(470\) 0 0
\(471\) 4.09725e6 0.851021
\(472\) 8.12476e6 1.67863
\(473\) 103730. 0.0213182
\(474\) −1.20478e6 −0.246299
\(475\) 0 0
\(476\) 50132.6 0.0101415
\(477\) 3.14993e6 0.633878
\(478\) 4.78055e6 0.956992
\(479\) −2.08110e6 −0.414433 −0.207216 0.978295i \(-0.566440\pi\)
−0.207216 + 0.978295i \(0.566440\pi\)
\(480\) 0 0
\(481\) −3.37023e6 −0.664196
\(482\) 528132. 0.103544
\(483\) −1.24693e6 −0.243206
\(484\) −37012.2 −0.00718177
\(485\) 0 0
\(486\) −320566. −0.0615640
\(487\) 4.11669e6 0.786549 0.393274 0.919421i \(-0.371342\pi\)
0.393274 + 0.919421i \(0.371342\pi\)
\(488\) −6.68868e6 −1.27142
\(489\) −327621. −0.0619583
\(490\) 0 0
\(491\) −8.82019e6 −1.65110 −0.825552 0.564326i \(-0.809136\pi\)
−0.825552 + 0.564326i \(0.809136\pi\)
\(492\) 359263. 0.0669113
\(493\) 440703. 0.0816636
\(494\) −3.29189e6 −0.606915
\(495\) 0 0
\(496\) −787313. −0.143696
\(497\) 23011.1 0.00417874
\(498\) 3.31194e6 0.598425
\(499\) 6.69775e6 1.20414 0.602071 0.798442i \(-0.294342\pi\)
0.602071 + 0.798442i \(0.294342\pi\)
\(500\) 0 0
\(501\) 2.34911e6 0.418129
\(502\) 190516. 0.0337422
\(503\) 3.07340e6 0.541626 0.270813 0.962632i \(-0.412707\pi\)
0.270813 + 0.962632i \(0.412707\pi\)
\(504\) 441730. 0.0774605
\(505\) 0 0
\(506\) 3.12820e6 0.543147
\(507\) −1.65381e6 −0.285737
\(508\) −765840. −0.131668
\(509\) −1.01798e6 −0.174158 −0.0870789 0.996201i \(-0.527753\pi\)
−0.0870789 + 0.996201i \(0.527753\pi\)
\(510\) 0 0
\(511\) −1.58136e6 −0.267903
\(512\) 6.47391e6 1.09142
\(513\) 1.02076e6 0.171250
\(514\) 7.88791e6 1.31690
\(515\) 0 0
\(516\) −19504.5 −0.00322485
\(517\) 1.13818e6 0.187277
\(518\) 1.22918e6 0.201276
\(519\) 551737. 0.0899112
\(520\) 0 0
\(521\) −1.06152e6 −0.171331 −0.0856653 0.996324i \(-0.527302\pi\)
−0.0856653 + 0.996324i \(0.527302\pi\)
\(522\) 284305. 0.0456676
\(523\) 5.52884e6 0.883852 0.441926 0.897051i \(-0.354295\pi\)
0.441926 + 0.897051i \(0.354295\pi\)
\(524\) 828026. 0.131739
\(525\) 0 0
\(526\) 9.11077e6 1.43579
\(527\) −572917. −0.0898597
\(528\) −1.02008e6 −0.159239
\(529\) 1.62418e7 2.52345
\(530\) 0 0
\(531\) 3.51091e6 0.540360
\(532\) −102983. −0.0157757
\(533\) −6.83814e6 −1.04261
\(534\) 2.10212e6 0.319009
\(535\) 0 0
\(536\) 802862. 0.120706
\(537\) −6.90223e6 −1.03289
\(538\) 1.13397e7 1.68907
\(539\) −1.93123e6 −0.286327
\(540\) 0 0
\(541\) 2.20998e6 0.324634 0.162317 0.986739i \(-0.448103\pi\)
0.162317 + 0.986739i \(0.448103\pi\)
\(542\) 321196. 0.0469647
\(543\) 1.04815e6 0.152554
\(544\) 622349. 0.0901648
\(545\) 0 0
\(546\) −615581. −0.0883698
\(547\) 3.95818e6 0.565624 0.282812 0.959175i \(-0.408733\pi\)
0.282812 + 0.959175i \(0.408733\pi\)
\(548\) 1.03953e6 0.147872
\(549\) −2.89034e6 −0.409278
\(550\) 0 0
\(551\) −905300. −0.127032
\(552\) −8.03382e6 −1.12221
\(553\) 717391. 0.0997569
\(554\) 7.47885e6 1.03529
\(555\) 0 0
\(556\) −972404. −0.133401
\(557\) 2.04981e6 0.279947 0.139974 0.990155i \(-0.455298\pi\)
0.139974 + 0.990155i \(0.455298\pi\)
\(558\) −369599. −0.0502511
\(559\) 371245. 0.0502494
\(560\) 0 0
\(561\) −742299. −0.0995799
\(562\) −1.89040e6 −0.252471
\(563\) 2.43373e6 0.323595 0.161798 0.986824i \(-0.448271\pi\)
0.161798 + 0.986824i \(0.448271\pi\)
\(564\) −214014. −0.0283299
\(565\) 0 0
\(566\) 7.52132e6 0.986854
\(567\) 190882. 0.0249349
\(568\) 148258. 0.0192817
\(569\) −2.17438e6 −0.281549 −0.140775 0.990042i \(-0.544959\pi\)
−0.140775 + 0.990042i \(0.544959\pi\)
\(570\) 0 0
\(571\) −9.30560e6 −1.19441 −0.597206 0.802088i \(-0.703722\pi\)
−0.597206 + 0.802088i \(0.703722\pi\)
\(572\) −132465. −0.0169282
\(573\) −7.33152e6 −0.932840
\(574\) 2.49400e6 0.315949
\(575\) 0 0
\(576\) 2.82945e6 0.355342
\(577\) 1.43013e7 1.78828 0.894138 0.447791i \(-0.147789\pi\)
0.894138 + 0.447791i \(0.147789\pi\)
\(578\) 5.18578e6 0.645646
\(579\) 736426. 0.0912920
\(580\) 0 0
\(581\) −1.97211e6 −0.242377
\(582\) −3.77677e6 −0.462182
\(583\) 4.70546e6 0.573364
\(584\) −1.01885e7 −1.23617
\(585\) 0 0
\(586\) 7.65803e6 0.921240
\(587\) −1.27072e7 −1.52214 −0.761068 0.648672i \(-0.775325\pi\)
−0.761068 + 0.648672i \(0.775325\pi\)
\(588\) 363132. 0.0433133
\(589\) 1.17690e6 0.139782
\(590\) 0 0
\(591\) 3.79628e6 0.447084
\(592\) 7.28993e6 0.854908
\(593\) 1.24734e7 1.45662 0.728311 0.685246i \(-0.240305\pi\)
0.728311 + 0.685246i \(0.240305\pi\)
\(594\) −478870. −0.0556867
\(595\) 0 0
\(596\) 155567. 0.0179392
\(597\) 2.65178e6 0.304510
\(598\) 1.11957e7 1.28026
\(599\) −2.89494e6 −0.329665 −0.164832 0.986322i \(-0.552708\pi\)
−0.164832 + 0.986322i \(0.552708\pi\)
\(600\) 0 0
\(601\) −1.26362e7 −1.42702 −0.713508 0.700647i \(-0.752895\pi\)
−0.713508 + 0.700647i \(0.752895\pi\)
\(602\) −135400. −0.0152274
\(603\) 346936. 0.0388559
\(604\) 853484. 0.0951925
\(605\) 0 0
\(606\) 3.72481e6 0.412024
\(607\) −4.63540e6 −0.510641 −0.255320 0.966857i \(-0.582181\pi\)
−0.255320 + 0.966857i \(0.582181\pi\)
\(608\) −1.27844e6 −0.140256
\(609\) −169291. −0.0184965
\(610\) 0 0
\(611\) 4.07351e6 0.441434
\(612\) 139576. 0.0150637
\(613\) 1.22937e6 0.132139 0.0660696 0.997815i \(-0.478954\pi\)
0.0660696 + 0.997815i \(0.478954\pi\)
\(614\) 1.08324e7 1.15959
\(615\) 0 0
\(616\) 659868. 0.0700657
\(617\) −2.09847e6 −0.221917 −0.110959 0.993825i \(-0.535392\pi\)
−0.110959 + 0.993825i \(0.535392\pi\)
\(618\) 530796. 0.0559057
\(619\) 4.56327e6 0.478684 0.239342 0.970935i \(-0.423068\pi\)
0.239342 + 0.970935i \(0.423068\pi\)
\(620\) 0 0
\(621\) −3.47161e6 −0.361245
\(622\) 1.07878e7 1.11804
\(623\) −1.25171e6 −0.129207
\(624\) −3.65083e6 −0.375345
\(625\) 0 0
\(626\) 4.00264e6 0.408235
\(627\) 1.52484e6 0.154902
\(628\) −1.15086e6 −0.116446
\(629\) 5.30478e6 0.534615
\(630\) 0 0
\(631\) 4.71323e6 0.471243 0.235622 0.971845i \(-0.424287\pi\)
0.235622 + 0.971845i \(0.424287\pi\)
\(632\) 4.62207e6 0.460303
\(633\) 7.12963e6 0.707225
\(634\) 1.40238e7 1.38561
\(635\) 0 0
\(636\) −884775. −0.0867341
\(637\) −6.91180e6 −0.674905
\(638\) 424703. 0.0413079
\(639\) 64065.8 0.00620689
\(640\) 0 0
\(641\) −3.81979e6 −0.367193 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(642\) −3.35286e6 −0.321054
\(643\) 1.69452e7 1.61629 0.808143 0.588986i \(-0.200473\pi\)
0.808143 + 0.588986i \(0.200473\pi\)
\(644\) 350246. 0.0332781
\(645\) 0 0
\(646\) 5.18148e6 0.488509
\(647\) −9.06068e6 −0.850942 −0.425471 0.904972i \(-0.639892\pi\)
−0.425471 + 0.904972i \(0.639892\pi\)
\(648\) 1.22983e6 0.115056
\(649\) 5.24469e6 0.488774
\(650\) 0 0
\(651\) 220079. 0.0203529
\(652\) 92024.4 0.00847782
\(653\) 1.19377e7 1.09556 0.547781 0.836622i \(-0.315473\pi\)
0.547781 + 0.836622i \(0.315473\pi\)
\(654\) 5.16853e6 0.472523
\(655\) 0 0
\(656\) 1.47912e7 1.34197
\(657\) −4.40270e6 −0.397929
\(658\) −1.48569e6 −0.133771
\(659\) 1.37103e7 1.22979 0.614897 0.788607i \(-0.289197\pi\)
0.614897 + 0.788607i \(0.289197\pi\)
\(660\) 0 0
\(661\) −249304. −0.0221935 −0.0110967 0.999938i \(-0.503532\pi\)
−0.0110967 + 0.999938i \(0.503532\pi\)
\(662\) 1.43004e6 0.126824
\(663\) −2.65666e6 −0.234721
\(664\) −1.27061e7 −1.11839
\(665\) 0 0
\(666\) 3.42221e6 0.298965
\(667\) 3.07892e6 0.267968
\(668\) −659835. −0.0572130
\(669\) −1.17448e6 −0.101456
\(670\) 0 0
\(671\) −4.31767e6 −0.370206
\(672\) −239068. −0.0204220
\(673\) 1.03766e7 0.883116 0.441558 0.897233i \(-0.354426\pi\)
0.441558 + 0.897233i \(0.354426\pi\)
\(674\) −9.47297e6 −0.803223
\(675\) 0 0
\(676\) 464534. 0.0390976
\(677\) −2.08995e7 −1.75253 −0.876263 0.481834i \(-0.839971\pi\)
−0.876263 + 0.481834i \(0.839971\pi\)
\(678\) −4.99119e6 −0.416994
\(679\) 2.24889e6 0.187195
\(680\) 0 0
\(681\) 5.53995e6 0.457760
\(682\) −552117. −0.0454538
\(683\) −1.51871e7 −1.24573 −0.622864 0.782330i \(-0.714031\pi\)
−0.622864 + 0.782330i \(0.714031\pi\)
\(684\) −286719. −0.0234324
\(685\) 0 0
\(686\) 5.17541e6 0.419889
\(687\) −6.28471e6 −0.508035
\(688\) −803017. −0.0646776
\(689\) 1.68407e7 1.35148
\(690\) 0 0
\(691\) −1.73343e7 −1.38105 −0.690527 0.723307i \(-0.742621\pi\)
−0.690527 + 0.723307i \(0.742621\pi\)
\(692\) −154976. −0.0123026
\(693\) 285145. 0.0225545
\(694\) 1.74504e7 1.37533
\(695\) 0 0
\(696\) −1.09072e6 −0.0853474
\(697\) 1.07633e7 0.839198
\(698\) −1.55926e6 −0.121138
\(699\) 5.67485e6 0.439300
\(700\) 0 0
\(701\) −7.14097e6 −0.548861 −0.274430 0.961607i \(-0.588489\pi\)
−0.274430 + 0.961607i \(0.588489\pi\)
\(702\) −1.71386e6 −0.131260
\(703\) −1.08972e7 −0.831622
\(704\) 4.22671e6 0.321419
\(705\) 0 0
\(706\) −6.71828e6 −0.507278
\(707\) −2.21795e6 −0.166880
\(708\) −986168. −0.0739380
\(709\) 2.01962e7 1.50887 0.754437 0.656372i \(-0.227910\pi\)
0.754437 + 0.656372i \(0.227910\pi\)
\(710\) 0 0
\(711\) 1.99731e6 0.148174
\(712\) −8.06465e6 −0.596191
\(713\) −4.00262e6 −0.294863
\(714\) 968933. 0.0711293
\(715\) 0 0
\(716\) 1.93875e6 0.141331
\(717\) −7.92530e6 −0.575728
\(718\) −2.43762e7 −1.76463
\(719\) 1.32950e6 0.0959106 0.0479553 0.998849i \(-0.484729\pi\)
0.0479553 + 0.998849i \(0.484729\pi\)
\(720\) 0 0
\(721\) −316064. −0.0226432
\(722\) 2.79839e6 0.199786
\(723\) −875549. −0.0622924
\(724\) −294411. −0.0208741
\(725\) 0 0
\(726\) −715349. −0.0503705
\(727\) −1.28863e7 −0.904256 −0.452128 0.891953i \(-0.649335\pi\)
−0.452128 + 0.891953i \(0.649335\pi\)
\(728\) 2.36164e6 0.165153
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −584344. −0.0404460
\(732\) 811859. 0.0560019
\(733\) 1.88150e7 1.29344 0.646718 0.762729i \(-0.276141\pi\)
0.646718 + 0.762729i \(0.276141\pi\)
\(734\) −1.39102e7 −0.953003
\(735\) 0 0
\(736\) 4.34797e6 0.295864
\(737\) 518263. 0.0351465
\(738\) 6.94361e6 0.469294
\(739\) 3.60917e6 0.243106 0.121553 0.992585i \(-0.461213\pi\)
0.121553 + 0.992585i \(0.461213\pi\)
\(740\) 0 0
\(741\) 5.45736e6 0.365121
\(742\) −6.14210e6 −0.409550
\(743\) −9.49631e6 −0.631078 −0.315539 0.948913i \(-0.602185\pi\)
−0.315539 + 0.948913i \(0.602185\pi\)
\(744\) 1.41794e6 0.0939133
\(745\) 0 0
\(746\) 7.83941e6 0.515747
\(747\) −5.49061e6 −0.360014
\(748\) 208502. 0.0136256
\(749\) 1.99647e6 0.130035
\(750\) 0 0
\(751\) −2.77857e7 −1.79772 −0.898858 0.438241i \(-0.855602\pi\)
−0.898858 + 0.438241i \(0.855602\pi\)
\(752\) −8.81116e6 −0.568184
\(753\) −315842. −0.0202994
\(754\) 1.52000e6 0.0973675
\(755\) 0 0
\(756\) −53616.3 −0.00341187
\(757\) −1.28586e7 −0.815557 −0.407778 0.913081i \(-0.633696\pi\)
−0.407778 + 0.913081i \(0.633696\pi\)
\(758\) 1.66738e7 1.05405
\(759\) −5.18599e6 −0.326759
\(760\) 0 0
\(761\) 2.18383e7 1.36696 0.683481 0.729968i \(-0.260465\pi\)
0.683481 + 0.729968i \(0.260465\pi\)
\(762\) −1.48017e7 −0.923472
\(763\) −3.07762e6 −0.191383
\(764\) 2.05933e6 0.127642
\(765\) 0 0
\(766\) −2.66312e6 −0.163991
\(767\) 1.87706e7 1.15210
\(768\) −2.22227e6 −0.135955
\(769\) −2.24376e7 −1.36824 −0.684119 0.729371i \(-0.739813\pi\)
−0.684119 + 0.729371i \(0.739813\pi\)
\(770\) 0 0
\(771\) −1.30767e7 −0.792252
\(772\) −206852. −0.0124916
\(773\) −2.89760e7 −1.74417 −0.872085 0.489354i \(-0.837233\pi\)
−0.872085 + 0.489354i \(0.837233\pi\)
\(774\) −376971. −0.0226181
\(775\) 0 0
\(776\) 1.44894e7 0.863764
\(777\) −2.03777e6 −0.121088
\(778\) 1.14414e7 0.677690
\(779\) −2.21102e7 −1.30542
\(780\) 0 0
\(781\) 95703.2 0.00561434
\(782\) −1.76222e7 −1.03049
\(783\) −471327. −0.0274737
\(784\) 1.49505e7 0.868691
\(785\) 0 0
\(786\) 1.60036e7 0.923977
\(787\) 1.92570e7 1.10829 0.554143 0.832421i \(-0.313046\pi\)
0.554143 + 0.832421i \(0.313046\pi\)
\(788\) −1.06632e6 −0.0611750
\(789\) −1.51040e7 −0.863774
\(790\) 0 0
\(791\) 2.97203e6 0.168893
\(792\) 1.83716e6 0.104072
\(793\) −1.54528e7 −0.872617
\(794\) −9.60544e6 −0.540712
\(795\) 0 0
\(796\) −744849. −0.0416664
\(797\) −1.99816e7 −1.11425 −0.557127 0.830427i \(-0.688097\pi\)
−0.557127 + 0.830427i \(0.688097\pi\)
\(798\) −1.99040e6 −0.110645
\(799\) −6.41176e6 −0.355312
\(800\) 0 0
\(801\) −3.48493e6 −0.191917
\(802\) 1.49608e7 0.821331
\(803\) −6.57688e6 −0.359941
\(804\) −97449.9 −0.00531669
\(805\) 0 0
\(806\) −1.97601e6 −0.107140
\(807\) −1.87993e7 −1.01615
\(808\) −1.42900e7 −0.770023
\(809\) 6.88532e6 0.369873 0.184936 0.982750i \(-0.440792\pi\)
0.184936 + 0.982750i \(0.440792\pi\)
\(810\) 0 0
\(811\) 1.40242e7 0.748729 0.374365 0.927282i \(-0.377861\pi\)
0.374365 + 0.927282i \(0.377861\pi\)
\(812\) 47551.5 0.00253089
\(813\) −532485. −0.0282541
\(814\) 5.11219e6 0.270424
\(815\) 0 0
\(816\) 5.74646e6 0.302117
\(817\) 1.20037e6 0.0629159
\(818\) −1.28805e7 −0.673054
\(819\) 1.02052e6 0.0531635
\(820\) 0 0
\(821\) −1.51193e7 −0.782843 −0.391421 0.920212i \(-0.628016\pi\)
−0.391421 + 0.920212i \(0.628016\pi\)
\(822\) 2.00914e7 1.03712
\(823\) 1.05051e7 0.540633 0.270316 0.962772i \(-0.412872\pi\)
0.270316 + 0.962772i \(0.412872\pi\)
\(824\) −2.03637e6 −0.104481
\(825\) 0 0
\(826\) −6.84597e6 −0.349128
\(827\) −2.78766e7 −1.41735 −0.708673 0.705537i \(-0.750706\pi\)
−0.708673 + 0.705537i \(0.750706\pi\)
\(828\) 975130. 0.0494296
\(829\) −1.72518e7 −0.871865 −0.435933 0.899979i \(-0.643581\pi\)
−0.435933 + 0.899979i \(0.643581\pi\)
\(830\) 0 0
\(831\) −1.23986e7 −0.622831
\(832\) 1.51272e7 0.757620
\(833\) 1.08793e7 0.543234
\(834\) −1.87940e7 −0.935632
\(835\) 0 0
\(836\) −428309. −0.0211954
\(837\) 612729. 0.0302311
\(838\) 8.42791e6 0.414581
\(839\) 7.21252e6 0.353738 0.176869 0.984234i \(-0.443403\pi\)
0.176869 + 0.984234i \(0.443403\pi\)
\(840\) 0 0
\(841\) −2.00931e7 −0.979620
\(842\) 9.32744e6 0.453401
\(843\) 3.13394e6 0.151887
\(844\) −2.00262e6 −0.0967704
\(845\) 0 0
\(846\) −4.13634e6 −0.198697
\(847\) 425957. 0.0204013
\(848\) −3.64270e7 −1.73954
\(849\) −1.24690e7 −0.593693
\(850\) 0 0
\(851\) 3.70612e7 1.75427
\(852\) −17995.2 −0.000849295 0
\(853\) 807294. 0.0379891 0.0189946 0.999820i \(-0.493953\pi\)
0.0189946 + 0.999820i \(0.493953\pi\)
\(854\) 5.63592e6 0.264436
\(855\) 0 0
\(856\) 1.28630e7 0.600011
\(857\) 1.65206e7 0.768376 0.384188 0.923255i \(-0.374481\pi\)
0.384188 + 0.923255i \(0.374481\pi\)
\(858\) −2.56021e6 −0.118729
\(859\) −1.50645e7 −0.696580 −0.348290 0.937387i \(-0.613237\pi\)
−0.348290 + 0.937387i \(0.613237\pi\)
\(860\) 0 0
\(861\) −4.13460e6 −0.190075
\(862\) 1.59540e7 0.731310
\(863\) −3.63009e6 −0.165917 −0.0829584 0.996553i \(-0.526437\pi\)
−0.0829584 + 0.996553i \(0.526437\pi\)
\(864\) −665596. −0.0303338
\(865\) 0 0
\(866\) 3.60686e7 1.63431
\(867\) −8.59709e6 −0.388422
\(868\) −61817.3 −0.00278491
\(869\) 2.98363e6 0.134028
\(870\) 0 0
\(871\) 1.85484e6 0.0828442
\(872\) −1.98288e7 −0.883090
\(873\) 6.26121e6 0.278050
\(874\) 3.61998e7 1.60298
\(875\) 0 0
\(876\) 1.23666e6 0.0544491
\(877\) −3.86808e7 −1.69823 −0.849116 0.528207i \(-0.822864\pi\)
−0.849116 + 0.528207i \(0.822864\pi\)
\(878\) −1.03671e7 −0.453858
\(879\) −1.26956e7 −0.554220
\(880\) 0 0
\(881\) 8.53508e6 0.370483 0.185241 0.982693i \(-0.440693\pi\)
0.185241 + 0.982693i \(0.440693\pi\)
\(882\) 7.01840e6 0.303785
\(883\) 2.01181e7 0.868331 0.434165 0.900833i \(-0.357043\pi\)
0.434165 + 0.900833i \(0.357043\pi\)
\(884\) 746221. 0.0321171
\(885\) 0 0
\(886\) −2.54937e7 −1.09106
\(887\) 1.42523e7 0.608243 0.304122 0.952633i \(-0.401637\pi\)
0.304122 + 0.952633i \(0.401637\pi\)
\(888\) −1.31291e7 −0.558731
\(889\) 8.81372e6 0.374029
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 329895. 0.0138824
\(893\) 1.31712e7 0.552708
\(894\) 3.00671e6 0.125819
\(895\) 0 0
\(896\) −4.66717e6 −0.194215
\(897\) −1.85605e7 −0.770207
\(898\) −2.48009e7 −1.02631
\(899\) −543420. −0.0224252
\(900\) 0 0
\(901\) −2.65074e7 −1.08782
\(902\) 1.03726e7 0.424492
\(903\) 224469. 0.00916086
\(904\) 1.91484e7 0.779313
\(905\) 0 0
\(906\) 1.64956e7 0.667649
\(907\) 2.45446e7 0.990689 0.495345 0.868697i \(-0.335042\pi\)
0.495345 + 0.868697i \(0.335042\pi\)
\(908\) −1.55610e6 −0.0626358
\(909\) −6.17506e6 −0.247874
\(910\) 0 0
\(911\) 4.04542e7 1.61498 0.807491 0.589880i \(-0.200825\pi\)
0.807491 + 0.589880i \(0.200825\pi\)
\(912\) −1.18045e7 −0.469959
\(913\) −8.20202e6 −0.325645
\(914\) 1.70525e7 0.675183
\(915\) 0 0
\(916\) 1.76529e6 0.0695149
\(917\) −9.52940e6 −0.374233
\(918\) 2.69764e6 0.105652
\(919\) −3.79964e7 −1.48407 −0.742033 0.670363i \(-0.766138\pi\)
−0.742033 + 0.670363i \(0.766138\pi\)
\(920\) 0 0
\(921\) −1.79582e7 −0.697612
\(922\) −6.87468e6 −0.266333
\(923\) 342518. 0.0132336
\(924\) −80093.5 −0.00308615
\(925\) 0 0
\(926\) −1.00128e7 −0.383733
\(927\) −879964. −0.0336330
\(928\) 590307. 0.0225013
\(929\) −4.20556e6 −0.159876 −0.0799382 0.996800i \(-0.525472\pi\)
−0.0799382 + 0.996800i \(0.525472\pi\)
\(930\) 0 0
\(931\) −2.23484e7 −0.845030
\(932\) −1.59399e6 −0.0601099
\(933\) −1.78842e7 −0.672613
\(934\) −1.68811e7 −0.633190
\(935\) 0 0
\(936\) 6.57512e6 0.245309
\(937\) −3.39406e6 −0.126291 −0.0631453 0.998004i \(-0.520113\pi\)
−0.0631453 + 0.998004i \(0.520113\pi\)
\(938\) −676496. −0.0251049
\(939\) −6.63566e6 −0.245595
\(940\) 0 0
\(941\) 5.89822e6 0.217144 0.108572 0.994089i \(-0.465372\pi\)
0.108572 + 0.994089i \(0.465372\pi\)
\(942\) −2.22432e7 −0.816714
\(943\) 7.51968e7 2.75372
\(944\) −4.06014e7 −1.48290
\(945\) 0 0
\(946\) −563129. −0.0204588
\(947\) −2.92768e7 −1.06084 −0.530419 0.847735i \(-0.677966\pi\)
−0.530419 + 0.847735i \(0.677966\pi\)
\(948\) −561018. −0.0202748
\(949\) −2.35384e7 −0.848421
\(950\) 0 0
\(951\) −2.32489e7 −0.833587
\(952\) −3.71726e6 −0.132932
\(953\) −7.23007e6 −0.257876 −0.128938 0.991653i \(-0.541157\pi\)
−0.128938 + 0.991653i \(0.541157\pi\)
\(954\) −1.71004e7 −0.608325
\(955\) 0 0
\(956\) 2.22611e6 0.0787775
\(957\) −704081. −0.0248509
\(958\) 1.12979e7 0.397726
\(959\) −1.19635e7 −0.420060
\(960\) 0 0
\(961\) −2.79227e7 −0.975324
\(962\) 1.82963e7 0.637421
\(963\) 5.55844e6 0.193147
\(964\) 245930. 0.00852353
\(965\) 0 0
\(966\) 6.76934e6 0.233402
\(967\) 5.69785e7 1.95950 0.979748 0.200233i \(-0.0641700\pi\)
0.979748 + 0.200233i \(0.0641700\pi\)
\(968\) 2.74440e6 0.0941366
\(969\) −8.58996e6 −0.293888
\(970\) 0 0
\(971\) 6.34775e6 0.216059 0.108029 0.994148i \(-0.465546\pi\)
0.108029 + 0.994148i \(0.465546\pi\)
\(972\) −149275. −0.00506782
\(973\) 1.11910e7 0.378954
\(974\) −2.23487e7 −0.754841
\(975\) 0 0
\(976\) 3.34250e7 1.12317
\(977\) −2.53017e7 −0.848034 −0.424017 0.905654i \(-0.639380\pi\)
−0.424017 + 0.905654i \(0.639380\pi\)
\(978\) 1.77859e6 0.0594606
\(979\) −5.20588e6 −0.173595
\(980\) 0 0
\(981\) −8.56850e6 −0.284271
\(982\) 4.78832e7 1.58454
\(983\) 5.27528e7 1.74125 0.870627 0.491944i \(-0.163713\pi\)
0.870627 + 0.491944i \(0.163713\pi\)
\(984\) −2.66388e7 −0.877054
\(985\) 0 0
\(986\) −2.39249e6 −0.0783715
\(987\) 2.46300e6 0.0804769
\(988\) −1.53290e6 −0.0499599
\(989\) −4.08245e6 −0.132718
\(990\) 0 0
\(991\) −2.05136e6 −0.0663526 −0.0331763 0.999450i \(-0.510562\pi\)
−0.0331763 + 0.999450i \(0.510562\pi\)
\(992\) −767404. −0.0247597
\(993\) −2.37075e6 −0.0762979
\(994\) −124923. −0.00401029
\(995\) 0 0
\(996\) 1.54224e6 0.0492611
\(997\) 4.46568e7 1.42282 0.711409 0.702778i \(-0.248057\pi\)
0.711409 + 0.702778i \(0.248057\pi\)
\(998\) −3.63609e7 −1.15560
\(999\) −5.67341e6 −0.179858
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.1.3 7
5.4 even 2 825.6.a.o.1.5 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.m.1.3 7 1.1 even 1 trivial
825.6.a.o.1.5 yes 7 5.4 even 2