Properties

Label 43.6.a.a.1.6
Level $43$
Weight $6$
Character 43.1
Self dual yes
Analytic conductor $6.897$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.89650425196\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(5.65705\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.65705 q^{2} +7.84314 q^{3} -18.6260 q^{4} -107.102 q^{5} +28.6827 q^{6} -25.5214 q^{7} -185.142 q^{8} -181.485 q^{9} +O(q^{10})\) \(q+3.65705 q^{2} +7.84314 q^{3} -18.6260 q^{4} -107.102 q^{5} +28.6827 q^{6} -25.5214 q^{7} -185.142 q^{8} -181.485 q^{9} -391.677 q^{10} +512.073 q^{11} -146.086 q^{12} +862.516 q^{13} -93.3331 q^{14} -840.016 q^{15} -81.0406 q^{16} -1521.49 q^{17} -663.700 q^{18} -1543.11 q^{19} +1994.88 q^{20} -200.168 q^{21} +1872.67 q^{22} -3126.31 q^{23} -1452.09 q^{24} +8345.85 q^{25} +3154.26 q^{26} -3329.30 q^{27} +475.362 q^{28} -947.120 q^{29} -3071.98 q^{30} +339.499 q^{31} +5628.17 q^{32} +4016.26 q^{33} -5564.17 q^{34} +2733.40 q^{35} +3380.34 q^{36} -7448.67 q^{37} -5643.25 q^{38} +6764.84 q^{39} +19829.1 q^{40} +5116.45 q^{41} -732.024 q^{42} -1849.00 q^{43} -9537.86 q^{44} +19437.4 q^{45} -11433.1 q^{46} -17159.9 q^{47} -635.613 q^{48} -16155.7 q^{49} +30521.2 q^{50} -11933.3 q^{51} -16065.2 q^{52} +18090.4 q^{53} -12175.4 q^{54} -54844.0 q^{55} +4725.08 q^{56} -12102.9 q^{57} -3463.66 q^{58} +17031.6 q^{59} +15646.1 q^{60} +10664.9 q^{61} +1241.56 q^{62} +4631.76 q^{63} +23175.8 q^{64} -92377.3 q^{65} +14687.6 q^{66} -8799.60 q^{67} +28339.3 q^{68} -24520.1 q^{69} +9996.16 q^{70} -77057.3 q^{71} +33600.5 q^{72} -7964.75 q^{73} -27240.1 q^{74} +65457.6 q^{75} +28742.0 q^{76} -13068.8 q^{77} +24739.3 q^{78} +68997.0 q^{79} +8679.62 q^{80} +17988.8 q^{81} +18711.1 q^{82} +40813.4 q^{83} +3728.33 q^{84} +162955. q^{85} -6761.88 q^{86} -7428.39 q^{87} -94806.0 q^{88} -83692.1 q^{89} +71083.6 q^{90} -22012.6 q^{91} +58230.7 q^{92} +2662.74 q^{93} -62754.7 q^{94} +165271. q^{95} +44142.5 q^{96} +159824. q^{97} -59082.0 q^{98} -92933.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} + O(q^{10}) \) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} - 617 q^{10} - 532 q^{11} - 4195 q^{12} - 2492 q^{13} - 4240 q^{14} - 1780 q^{15} + 1882 q^{16} - 2534 q^{17} - 3711 q^{18} - 1678 q^{19} - 2607 q^{20} - 2256 q^{21} + 11502 q^{22} - 2488 q^{23} + 19953 q^{24} + 4378 q^{25} + 4586 q^{26} - 8960 q^{27} + 18640 q^{28} - 4360 q^{29} + 25092 q^{30} + 5704 q^{31} - 18294 q^{32} - 12852 q^{33} + 30007 q^{34} + 5640 q^{35} + 67969 q^{36} - 3772 q^{37} - 6559 q^{38} + 11120 q^{39} + 14869 q^{40} - 10698 q^{41} + 78698 q^{42} - 14792 q^{43} - 356 q^{44} - 44912 q^{45} - 19389 q^{46} - 77864 q^{47} - 118727 q^{48} + 7188 q^{49} + 26877 q^{50} - 80246 q^{51} - 60736 q^{52} - 62352 q^{53} + 61026 q^{54} - 49552 q^{55} - 144528 q^{56} - 808 q^{57} + 52951 q^{58} - 26224 q^{59} + 101500 q^{60} - 82540 q^{61} - 9023 q^{62} - 61768 q^{63} + 153858 q^{64} - 5000 q^{65} - 48516 q^{66} + 27784 q^{67} + 40507 q^{68} - 93776 q^{69} + 185910 q^{70} - 9504 q^{71} - 186687 q^{72} + 14260 q^{73} - 15239 q^{74} + 167420 q^{75} + 1279 q^{76} - 218140 q^{77} + 264170 q^{78} + 160248 q^{79} - 1291 q^{80} + 161076 q^{81} - 47781 q^{82} - 77176 q^{83} + 16382 q^{84} + 141096 q^{85} + 22188 q^{86} + 268136 q^{87} + 129544 q^{88} - 265692 q^{89} + 48990 q^{90} + 401148 q^{91} + 190391 q^{92} - 123860 q^{93} + 248737 q^{94} + 135884 q^{95} + 950817 q^{96} + 144742 q^{97} - 292244 q^{98} + 239516 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.65705 0.646481 0.323241 0.946317i \(-0.395228\pi\)
0.323241 + 0.946317i \(0.395228\pi\)
\(3\) 7.84314 0.503138 0.251569 0.967839i \(-0.419054\pi\)
0.251569 + 0.967839i \(0.419054\pi\)
\(4\) −18.6260 −0.582062
\(5\) −107.102 −1.91590 −0.957950 0.286936i \(-0.907363\pi\)
−0.957950 + 0.286936i \(0.907363\pi\)
\(6\) 28.6827 0.325269
\(7\) −25.5214 −0.196861 −0.0984305 0.995144i \(-0.531382\pi\)
−0.0984305 + 0.995144i \(0.531382\pi\)
\(8\) −185.142 −1.02277
\(9\) −181.485 −0.746853
\(10\) −391.677 −1.23859
\(11\) 512.073 1.27600 0.637999 0.770037i \(-0.279762\pi\)
0.637999 + 0.770037i \(0.279762\pi\)
\(12\) −146.086 −0.292857
\(13\) 862.516 1.41550 0.707749 0.706464i \(-0.249711\pi\)
0.707749 + 0.706464i \(0.249711\pi\)
\(14\) −93.3331 −0.127267
\(15\) −840.016 −0.963961
\(16\) −81.0406 −0.0791413
\(17\) −1521.49 −1.27687 −0.638435 0.769675i \(-0.720418\pi\)
−0.638435 + 0.769675i \(0.720418\pi\)
\(18\) −663.700 −0.482826
\(19\) −1543.11 −0.980650 −0.490325 0.871540i \(-0.663122\pi\)
−0.490325 + 0.871540i \(0.663122\pi\)
\(20\) 1994.88 1.11517
\(21\) −200.168 −0.0990481
\(22\) 1872.67 0.824908
\(23\) −3126.31 −1.23229 −0.616145 0.787633i \(-0.711306\pi\)
−0.616145 + 0.787633i \(0.711306\pi\)
\(24\) −1452.09 −0.514596
\(25\) 8345.85 2.67067
\(26\) 3154.26 0.915092
\(27\) −3329.30 −0.878907
\(28\) 475.362 0.114585
\(29\) −947.120 −0.209127 −0.104563 0.994518i \(-0.533345\pi\)
−0.104563 + 0.994518i \(0.533345\pi\)
\(30\) −3071.98 −0.623183
\(31\) 339.499 0.0634504 0.0317252 0.999497i \(-0.489900\pi\)
0.0317252 + 0.999497i \(0.489900\pi\)
\(32\) 5628.17 0.971610
\(33\) 4016.26 0.642002
\(34\) −5564.17 −0.825473
\(35\) 2733.40 0.377166
\(36\) 3380.34 0.434715
\(37\) −7448.67 −0.894488 −0.447244 0.894412i \(-0.647594\pi\)
−0.447244 + 0.894412i \(0.647594\pi\)
\(38\) −5643.25 −0.633972
\(39\) 6764.84 0.712190
\(40\) 19829.1 1.95953
\(41\) 5116.45 0.475345 0.237672 0.971345i \(-0.423616\pi\)
0.237672 + 0.971345i \(0.423616\pi\)
\(42\) −732.024 −0.0640328
\(43\) −1849.00 −0.152499
\(44\) −9537.86 −0.742710
\(45\) 19437.4 1.43089
\(46\) −11433.1 −0.796652
\(47\) −17159.9 −1.13311 −0.566553 0.824025i \(-0.691724\pi\)
−0.566553 + 0.824025i \(0.691724\pi\)
\(48\) −635.613 −0.0398189
\(49\) −16155.7 −0.961246
\(50\) 30521.2 1.72654
\(51\) −11933.3 −0.642442
\(52\) −16065.2 −0.823907
\(53\) 18090.4 0.884623 0.442312 0.896861i \(-0.354159\pi\)
0.442312 + 0.896861i \(0.354159\pi\)
\(54\) −12175.4 −0.568197
\(55\) −54844.0 −2.44468
\(56\) 4725.08 0.201344
\(57\) −12102.9 −0.493402
\(58\) −3463.66 −0.135197
\(59\) 17031.6 0.636980 0.318490 0.947926i \(-0.396824\pi\)
0.318490 + 0.947926i \(0.396824\pi\)
\(60\) 15646.1 0.561085
\(61\) 10664.9 0.366972 0.183486 0.983022i \(-0.441262\pi\)
0.183486 + 0.983022i \(0.441262\pi\)
\(62\) 1241.56 0.0410195
\(63\) 4631.76 0.147026
\(64\) 23175.8 0.707269
\(65\) −92377.3 −2.71195
\(66\) 14687.6 0.415042
\(67\) −8799.60 −0.239484 −0.119742 0.992805i \(-0.538207\pi\)
−0.119742 + 0.992805i \(0.538207\pi\)
\(68\) 28339.3 0.743218
\(69\) −24520.1 −0.620011
\(70\) 9996.16 0.243831
\(71\) −77057.3 −1.81413 −0.907064 0.420992i \(-0.861682\pi\)
−0.907064 + 0.420992i \(0.861682\pi\)
\(72\) 33600.5 0.763861
\(73\) −7964.75 −0.174930 −0.0874652 0.996168i \(-0.527877\pi\)
−0.0874652 + 0.996168i \(0.527877\pi\)
\(74\) −27240.1 −0.578269
\(75\) 65457.6 1.34371
\(76\) 28742.0 0.570800
\(77\) −13068.8 −0.251194
\(78\) 24739.3 0.460417
\(79\) 68997.0 1.24383 0.621917 0.783083i \(-0.286354\pi\)
0.621917 + 0.783083i \(0.286354\pi\)
\(80\) 8679.62 0.151627
\(81\) 17988.8 0.304641
\(82\) 18711.1 0.307301
\(83\) 40813.4 0.650290 0.325145 0.945664i \(-0.394587\pi\)
0.325145 + 0.945664i \(0.394587\pi\)
\(84\) 3728.33 0.0576522
\(85\) 162955. 2.44636
\(86\) −6761.88 −0.0985874
\(87\) −7428.39 −0.105220
\(88\) −94806.0 −1.30506
\(89\) −83692.1 −1.11998 −0.559989 0.828500i \(-0.689195\pi\)
−0.559989 + 0.828500i \(0.689195\pi\)
\(90\) 71083.6 0.925046
\(91\) −22012.6 −0.278656
\(92\) 58230.7 0.717269
\(93\) 2662.74 0.0319243
\(94\) −62754.7 −0.732532
\(95\) 165271. 1.87883
\(96\) 44142.5 0.488853
\(97\) 159824. 1.72469 0.862346 0.506320i \(-0.168994\pi\)
0.862346 + 0.506320i \(0.168994\pi\)
\(98\) −59082.0 −0.621427
\(99\) −92933.6 −0.952982
\(100\) −155450. −1.55450
\(101\) −168968. −1.64817 −0.824085 0.566466i \(-0.808310\pi\)
−0.824085 + 0.566466i \(0.808310\pi\)
\(102\) −43640.5 −0.415326
\(103\) −145084. −1.34749 −0.673747 0.738962i \(-0.735316\pi\)
−0.673747 + 0.738962i \(0.735316\pi\)
\(104\) −159688. −1.44773
\(105\) 21438.4 0.189766
\(106\) 66157.5 0.571892
\(107\) 72693.1 0.613810 0.306905 0.951740i \(-0.400707\pi\)
0.306905 + 0.951740i \(0.400707\pi\)
\(108\) 62011.4 0.511579
\(109\) −3254.39 −0.0262364 −0.0131182 0.999914i \(-0.504176\pi\)
−0.0131182 + 0.999914i \(0.504176\pi\)
\(110\) −200567. −1.58044
\(111\) −58420.9 −0.450050
\(112\) 2068.27 0.0155798
\(113\) 116394. 0.857503 0.428752 0.903422i \(-0.358954\pi\)
0.428752 + 0.903422i \(0.358954\pi\)
\(114\) −44260.8 −0.318975
\(115\) 334834. 2.36094
\(116\) 17641.0 0.121725
\(117\) −156534. −1.05717
\(118\) 62285.4 0.411795
\(119\) 38830.6 0.251366
\(120\) 155522. 0.985914
\(121\) 101167. 0.628170
\(122\) 39002.1 0.237240
\(123\) 40129.0 0.239164
\(124\) −6323.50 −0.0369321
\(125\) −559163. −3.20084
\(126\) 16938.6 0.0950496
\(127\) −6451.01 −0.0354910 −0.0177455 0.999843i \(-0.505649\pi\)
−0.0177455 + 0.999843i \(0.505649\pi\)
\(128\) −95346.3 −0.514374
\(129\) −14502.0 −0.0767278
\(130\) −337828. −1.75322
\(131\) −48048.9 −0.244628 −0.122314 0.992491i \(-0.539031\pi\)
−0.122314 + 0.992491i \(0.539031\pi\)
\(132\) −74806.8 −0.373685
\(133\) 39382.5 0.193052
\(134\) −32180.6 −0.154822
\(135\) 356574. 1.68390
\(136\) 281691. 1.30595
\(137\) −103084. −0.469233 −0.234616 0.972088i \(-0.575383\pi\)
−0.234616 + 0.972088i \(0.575383\pi\)
\(138\) −89671.2 −0.400826
\(139\) 219626. 0.964156 0.482078 0.876128i \(-0.339882\pi\)
0.482078 + 0.876128i \(0.339882\pi\)
\(140\) −50912.2 −0.219534
\(141\) −134588. −0.570109
\(142\) −281802. −1.17280
\(143\) 441671. 1.80617
\(144\) 14707.7 0.0591069
\(145\) 101438. 0.400666
\(146\) −29127.5 −0.113089
\(147\) −126711. −0.483639
\(148\) 138739. 0.520648
\(149\) −335627. −1.23849 −0.619243 0.785200i \(-0.712560\pi\)
−0.619243 + 0.785200i \(0.712560\pi\)
\(150\) 239382. 0.868686
\(151\) −84920.3 −0.303088 −0.151544 0.988450i \(-0.548425\pi\)
−0.151544 + 0.988450i \(0.548425\pi\)
\(152\) 285695. 1.00298
\(153\) 276128. 0.953634
\(154\) −47793.3 −0.162392
\(155\) −36361.0 −0.121565
\(156\) −126002. −0.414539
\(157\) 313767. 1.01592 0.507958 0.861382i \(-0.330401\pi\)
0.507958 + 0.861382i \(0.330401\pi\)
\(158\) 252325. 0.804115
\(159\) 141885. 0.445087
\(160\) −602788. −1.86151
\(161\) 79787.9 0.242590
\(162\) 65785.8 0.196945
\(163\) 239642. 0.706469 0.353235 0.935535i \(-0.385082\pi\)
0.353235 + 0.935535i \(0.385082\pi\)
\(164\) −95298.9 −0.276680
\(165\) −430149. −1.23001
\(166\) 149257. 0.420400
\(167\) −506675. −1.40585 −0.702923 0.711266i \(-0.748122\pi\)
−0.702923 + 0.711266i \(0.748122\pi\)
\(168\) 37059.4 0.101304
\(169\) 372641. 1.00363
\(170\) 595933. 1.58152
\(171\) 280052. 0.732401
\(172\) 34439.5 0.0887637
\(173\) −427174. −1.08515 −0.542575 0.840007i \(-0.682551\pi\)
−0.542575 + 0.840007i \(0.682551\pi\)
\(174\) −27166.0 −0.0680225
\(175\) −212998. −0.525751
\(176\) −41498.7 −0.100984
\(177\) 133581. 0.320489
\(178\) −306066. −0.724044
\(179\) 11674.9 0.0272345 0.0136172 0.999907i \(-0.495665\pi\)
0.0136172 + 0.999907i \(0.495665\pi\)
\(180\) −362041. −0.832870
\(181\) 71691.3 0.162656 0.0813280 0.996687i \(-0.474084\pi\)
0.0813280 + 0.996687i \(0.474084\pi\)
\(182\) −80501.3 −0.180146
\(183\) 83646.4 0.184637
\(184\) 578811. 1.26035
\(185\) 797768. 1.71375
\(186\) 9737.76 0.0206384
\(187\) −779114. −1.62928
\(188\) 319621. 0.659539
\(189\) 84968.3 0.173023
\(190\) 604403. 1.21463
\(191\) 212639. 0.421755 0.210877 0.977513i \(-0.432368\pi\)
0.210877 + 0.977513i \(0.432368\pi\)
\(192\) 181771. 0.355853
\(193\) −137469. −0.265651 −0.132825 0.991139i \(-0.542405\pi\)
−0.132825 + 0.991139i \(0.542405\pi\)
\(194\) 584483. 1.11498
\(195\) −724528. −1.36448
\(196\) 300915. 0.559505
\(197\) −28517.3 −0.0523531 −0.0261766 0.999657i \(-0.508333\pi\)
−0.0261766 + 0.999657i \(0.508333\pi\)
\(198\) −339863. −0.616085
\(199\) −353575. −0.632921 −0.316460 0.948606i \(-0.602494\pi\)
−0.316460 + 0.948606i \(0.602494\pi\)
\(200\) −1.54516e6 −2.73149
\(201\) −69016.5 −0.120493
\(202\) −617926. −1.06551
\(203\) 24171.8 0.0411689
\(204\) 222269. 0.373941
\(205\) −547982. −0.910713
\(206\) −530580. −0.871130
\(207\) 567379. 0.920339
\(208\) −69898.9 −0.112024
\(209\) −790187. −1.25131
\(210\) 78401.3 0.122680
\(211\) 327840. 0.506939 0.253469 0.967343i \(-0.418428\pi\)
0.253469 + 0.967343i \(0.418428\pi\)
\(212\) −336952. −0.514906
\(213\) −604371. −0.912756
\(214\) 265842. 0.396816
\(215\) 198032. 0.292172
\(216\) 616392. 0.898923
\(217\) −8664.49 −0.0124909
\(218\) −11901.5 −0.0169613
\(219\) −62468.7 −0.0880140
\(220\) 1.02152e6 1.42296
\(221\) −1.31231e6 −1.80741
\(222\) −213648. −0.290949
\(223\) 247951. 0.333890 0.166945 0.985966i \(-0.446610\pi\)
0.166945 + 0.985966i \(0.446610\pi\)
\(224\) −143639. −0.191272
\(225\) −1.51465e6 −1.99460
\(226\) 425660. 0.554359
\(227\) −868910. −1.11921 −0.559603 0.828761i \(-0.689046\pi\)
−0.559603 + 0.828761i \(0.689046\pi\)
\(228\) 225428. 0.287191
\(229\) 238721. 0.300817 0.150408 0.988624i \(-0.451941\pi\)
0.150408 + 0.988624i \(0.451941\pi\)
\(230\) 1.22451e6 1.52631
\(231\) −102501. −0.126385
\(232\) 175351. 0.213889
\(233\) 507726. 0.612688 0.306344 0.951921i \(-0.400894\pi\)
0.306344 + 0.951921i \(0.400894\pi\)
\(234\) −572452. −0.683439
\(235\) 1.83786e6 2.17092
\(236\) −317231. −0.370762
\(237\) 541153. 0.625819
\(238\) 142005. 0.162503
\(239\) 143880. 0.162932 0.0814660 0.996676i \(-0.474040\pi\)
0.0814660 + 0.996676i \(0.474040\pi\)
\(240\) 68075.4 0.0762891
\(241\) 398237. 0.441671 0.220835 0.975311i \(-0.429122\pi\)
0.220835 + 0.975311i \(0.429122\pi\)
\(242\) 369974. 0.406100
\(243\) 950107. 1.03218
\(244\) −198645. −0.213600
\(245\) 1.73030e6 1.84165
\(246\) 146754. 0.154615
\(247\) −1.33096e6 −1.38811
\(248\) −62855.4 −0.0648953
\(249\) 320105. 0.327185
\(250\) −2.04489e6 −2.06928
\(251\) 1.65625e6 1.65937 0.829684 0.558233i \(-0.188520\pi\)
0.829684 + 0.558233i \(0.188520\pi\)
\(252\) −86271.1 −0.0855784
\(253\) −1.60090e6 −1.57240
\(254\) −23591.7 −0.0229443
\(255\) 1.27808e6 1.23085
\(256\) −1.09031e6 −1.03980
\(257\) 32691.4 0.0308746 0.0154373 0.999881i \(-0.495086\pi\)
0.0154373 + 0.999881i \(0.495086\pi\)
\(258\) −53034.4 −0.0496030
\(259\) 190101. 0.176090
\(260\) 1.72062e6 1.57852
\(261\) 171888. 0.156187
\(262\) −175717. −0.158147
\(263\) 1.02727e6 0.915791 0.457895 0.889006i \(-0.348603\pi\)
0.457895 + 0.889006i \(0.348603\pi\)
\(264\) −743577. −0.656623
\(265\) −1.93752e6 −1.69485
\(266\) 144024. 0.124804
\(267\) −656409. −0.563503
\(268\) 163901. 0.139394
\(269\) −1.88917e6 −1.59180 −0.795902 0.605426i \(-0.793003\pi\)
−0.795902 + 0.605426i \(0.793003\pi\)
\(270\) 1.30401e6 1.08861
\(271\) 605259. 0.500631 0.250316 0.968164i \(-0.419466\pi\)
0.250316 + 0.968164i \(0.419466\pi\)
\(272\) 123303. 0.101053
\(273\) −172648. −0.140202
\(274\) −376982. −0.303350
\(275\) 4.27368e6 3.40777
\(276\) 456711. 0.360885
\(277\) −1.16555e6 −0.912710 −0.456355 0.889798i \(-0.650845\pi\)
−0.456355 + 0.889798i \(0.650845\pi\)
\(278\) 803184. 0.623309
\(279\) −61614.0 −0.0473881
\(280\) −506066. −0.385755
\(281\) 938873. 0.709318 0.354659 0.934996i \(-0.384597\pi\)
0.354659 + 0.934996i \(0.384597\pi\)
\(282\) −492194. −0.368564
\(283\) −2.54537e6 −1.88923 −0.944616 0.328179i \(-0.893565\pi\)
−0.944616 + 0.328179i \(0.893565\pi\)
\(284\) 1.43527e6 1.05594
\(285\) 1.29624e6 0.945309
\(286\) 1.61521e6 1.16766
\(287\) −130579. −0.0935768
\(288\) −1.02143e6 −0.725649
\(289\) 895076. 0.630399
\(290\) 370965. 0.259023
\(291\) 1.25352e6 0.867757
\(292\) 148351. 0.101820
\(293\) −1.61560e6 −1.09942 −0.549712 0.835354i \(-0.685263\pi\)
−0.549712 + 0.835354i \(0.685263\pi\)
\(294\) −463389. −0.312663
\(295\) −1.82412e6 −1.22039
\(296\) 1.37906e6 0.914858
\(297\) −1.70484e6 −1.12148
\(298\) −1.22740e6 −0.800657
\(299\) −2.69650e6 −1.74430
\(300\) −1.21921e6 −0.782126
\(301\) 47189.1 0.0300210
\(302\) −310558. −0.195941
\(303\) −1.32524e6 −0.829257
\(304\) 125055. 0.0776099
\(305\) −1.14223e6 −0.703081
\(306\) 1.00981e6 0.616506
\(307\) −1.49055e6 −0.902613 −0.451307 0.892369i \(-0.649042\pi\)
−0.451307 + 0.892369i \(0.649042\pi\)
\(308\) 243420. 0.146211
\(309\) −1.13792e6 −0.677975
\(310\) −132974. −0.0785892
\(311\) −754959. −0.442611 −0.221305 0.975205i \(-0.571032\pi\)
−0.221305 + 0.975205i \(0.571032\pi\)
\(312\) −1.25245e6 −0.728409
\(313\) 1.31681e6 0.759735 0.379867 0.925041i \(-0.375970\pi\)
0.379867 + 0.925041i \(0.375970\pi\)
\(314\) 1.14746e6 0.656770
\(315\) −496071. −0.281687
\(316\) −1.28514e6 −0.723988
\(317\) −2.19577e6 −1.22727 −0.613633 0.789591i \(-0.710293\pi\)
−0.613633 + 0.789591i \(0.710293\pi\)
\(318\) 518882. 0.287740
\(319\) −484994. −0.266845
\(320\) −2.48217e6 −1.35506
\(321\) 570142. 0.308831
\(322\) 291788. 0.156830
\(323\) 2.34783e6 1.25216
\(324\) −335059. −0.177320
\(325\) 7.19843e6 3.78033
\(326\) 876381. 0.456719
\(327\) −25524.7 −0.0132005
\(328\) −947268. −0.486170
\(329\) 437946. 0.223065
\(330\) −1.57308e6 −0.795179
\(331\) 727999. 0.365225 0.182613 0.983185i \(-0.441545\pi\)
0.182613 + 0.983185i \(0.441545\pi\)
\(332\) −760190. −0.378509
\(333\) 1.35182e6 0.668050
\(334\) −1.85293e6 −0.908853
\(335\) 942455. 0.458827
\(336\) 16221.7 0.00783879
\(337\) 626937. 0.300711 0.150355 0.988632i \(-0.451958\pi\)
0.150355 + 0.988632i \(0.451958\pi\)
\(338\) 1.36277e6 0.648829
\(339\) 912897. 0.431442
\(340\) −3.03519e6 −1.42393
\(341\) 173848. 0.0809625
\(342\) 1.02417e6 0.473484
\(343\) 841254. 0.386093
\(344\) 342327. 0.155971
\(345\) 2.62615e6 1.18788
\(346\) −1.56220e6 −0.701529
\(347\) −3.30393e6 −1.47302 −0.736508 0.676428i \(-0.763527\pi\)
−0.736508 + 0.676428i \(0.763527\pi\)
\(348\) 138361. 0.0612444
\(349\) −2.93486e6 −1.28980 −0.644902 0.764265i \(-0.723102\pi\)
−0.644902 + 0.764265i \(0.723102\pi\)
\(350\) −778943. −0.339888
\(351\) −2.87157e6 −1.24409
\(352\) 2.88203e6 1.23977
\(353\) 633691. 0.270670 0.135335 0.990800i \(-0.456789\pi\)
0.135335 + 0.990800i \(0.456789\pi\)
\(354\) 488513. 0.207190
\(355\) 8.25300e6 3.47569
\(356\) 1.55885e6 0.651897
\(357\) 304554. 0.126472
\(358\) 42695.6 0.0176066
\(359\) 3.08062e6 1.26154 0.630771 0.775969i \(-0.282739\pi\)
0.630771 + 0.775969i \(0.282739\pi\)
\(360\) −3.59868e6 −1.46348
\(361\) −94896.2 −0.0383249
\(362\) 262179. 0.105154
\(363\) 793470. 0.316056
\(364\) 410007. 0.162195
\(365\) 853042. 0.335149
\(366\) 305899. 0.119364
\(367\) −1.32881e6 −0.514988 −0.257494 0.966280i \(-0.582897\pi\)
−0.257494 + 0.966280i \(0.582897\pi\)
\(368\) 253358. 0.0975250
\(369\) −928559. −0.355012
\(370\) 2.91748e6 1.10791
\(371\) −461692. −0.174148
\(372\) −49596.1 −0.0185819
\(373\) −1.91319e6 −0.712011 −0.356006 0.934484i \(-0.615862\pi\)
−0.356006 + 0.934484i \(0.615862\pi\)
\(374\) −2.84926e6 −1.05330
\(375\) −4.38560e6 −1.61046
\(376\) 3.17702e6 1.15891
\(377\) −816906. −0.296019
\(378\) 310733. 0.111856
\(379\) 703691. 0.251642 0.125821 0.992053i \(-0.459843\pi\)
0.125821 + 0.992053i \(0.459843\pi\)
\(380\) −3.07833e6 −1.09359
\(381\) −50596.2 −0.0178569
\(382\) 777632. 0.272656
\(383\) 1.73748e6 0.605234 0.302617 0.953112i \(-0.402140\pi\)
0.302617 + 0.953112i \(0.402140\pi\)
\(384\) −747815. −0.258801
\(385\) 1.39970e6 0.481263
\(386\) −502731. −0.171738
\(387\) 335566. 0.113894
\(388\) −2.97687e6 −1.00388
\(389\) 3.93096e6 1.31712 0.658559 0.752529i \(-0.271166\pi\)
0.658559 + 0.752529i \(0.271166\pi\)
\(390\) −2.64963e6 −0.882113
\(391\) 4.75666e6 1.57347
\(392\) 2.99109e6 0.983136
\(393\) −376854. −0.123081
\(394\) −104289. −0.0338453
\(395\) −7.38971e6 −2.38306
\(396\) 1.73098e6 0.554695
\(397\) 3.99784e6 1.27306 0.636530 0.771252i \(-0.280369\pi\)
0.636530 + 0.771252i \(0.280369\pi\)
\(398\) −1.29304e6 −0.409171
\(399\) 308882. 0.0971316
\(400\) −676353. −0.211360
\(401\) 4.39985e6 1.36640 0.683198 0.730233i \(-0.260588\pi\)
0.683198 + 0.730233i \(0.260588\pi\)
\(402\) −252397. −0.0778966
\(403\) 292823. 0.0898138
\(404\) 3.14721e6 0.959338
\(405\) −1.92663e6 −0.583662
\(406\) 88397.6 0.0266149
\(407\) −3.81426e6 −1.14136
\(408\) 2.20934e6 0.657072
\(409\) −1.64907e6 −0.487451 −0.243726 0.969844i \(-0.578370\pi\)
−0.243726 + 0.969844i \(0.578370\pi\)
\(410\) −2.00400e6 −0.588759
\(411\) −808499. −0.236089
\(412\) 2.70234e6 0.784326
\(413\) −434671. −0.125396
\(414\) 2.07493e6 0.594982
\(415\) −4.37120e6 −1.24589
\(416\) 4.85439e6 1.37531
\(417\) 1.72256e6 0.485103
\(418\) −2.88975e6 −0.808947
\(419\) −5.93011e6 −1.65017 −0.825083 0.565012i \(-0.808871\pi\)
−0.825083 + 0.565012i \(0.808871\pi\)
\(420\) −399311. −0.110456
\(421\) 4.85544e6 1.33513 0.667565 0.744552i \(-0.267337\pi\)
0.667565 + 0.744552i \(0.267337\pi\)
\(422\) 1.19893e6 0.327726
\(423\) 3.11427e6 0.846264
\(424\) −3.34929e6 −0.904769
\(425\) −1.26981e7 −3.41010
\(426\) −2.21022e6 −0.590080
\(427\) −272184. −0.0722424
\(428\) −1.35398e6 −0.357275
\(429\) 3.46409e6 0.908752
\(430\) 724212. 0.188884
\(431\) 2.59168e6 0.672029 0.336015 0.941857i \(-0.390921\pi\)
0.336015 + 0.941857i \(0.390921\pi\)
\(432\) 269808. 0.0695578
\(433\) −5.08691e6 −1.30387 −0.651934 0.758275i \(-0.726042\pi\)
−0.651934 + 0.758275i \(0.726042\pi\)
\(434\) −31686.5 −0.00807513
\(435\) 795596. 0.201590
\(436\) 60616.3 0.0152712
\(437\) 4.82426e6 1.20845
\(438\) −228451. −0.0568994
\(439\) −5.71552e6 −1.41545 −0.707725 0.706488i \(-0.750278\pi\)
−0.707725 + 0.706488i \(0.750278\pi\)
\(440\) 1.01539e7 2.50036
\(441\) 2.93201e6 0.717909
\(442\) −4.79918e6 −1.16845
\(443\) −4.81790e6 −1.16640 −0.583201 0.812328i \(-0.698200\pi\)
−0.583201 + 0.812328i \(0.698200\pi\)
\(444\) 1.08815e6 0.261957
\(445\) 8.96359e6 2.14576
\(446\) 906769. 0.215854
\(447\) −2.63237e6 −0.623128
\(448\) −591479. −0.139234
\(449\) −2.36680e6 −0.554046 −0.277023 0.960863i \(-0.589348\pi\)
−0.277023 + 0.960863i \(0.589348\pi\)
\(450\) −5.53914e6 −1.28947
\(451\) 2.61999e6 0.606539
\(452\) −2.16796e6 −0.499120
\(453\) −666042. −0.152495
\(454\) −3.17765e6 −0.723546
\(455\) 2.35760e6 0.533877
\(456\) 2.24074e6 0.504638
\(457\) 1.78378e6 0.399531 0.199766 0.979844i \(-0.435982\pi\)
0.199766 + 0.979844i \(0.435982\pi\)
\(458\) 873015. 0.194472
\(459\) 5.06549e6 1.12225
\(460\) −6.23662e6 −1.37422
\(461\) −4.26575e6 −0.934852 −0.467426 0.884032i \(-0.654819\pi\)
−0.467426 + 0.884032i \(0.654819\pi\)
\(462\) −374850. −0.0817056
\(463\) −2.33897e6 −0.507074 −0.253537 0.967326i \(-0.581594\pi\)
−0.253537 + 0.967326i \(0.581594\pi\)
\(464\) 76755.2 0.0165506
\(465\) −285185. −0.0611637
\(466\) 1.85678e6 0.396091
\(467\) −6.84641e6 −1.45268 −0.726342 0.687334i \(-0.758781\pi\)
−0.726342 + 0.687334i \(0.758781\pi\)
\(468\) 2.91560e6 0.615337
\(469\) 224578. 0.0471450
\(470\) 6.72116e6 1.40346
\(471\) 2.46091e6 0.511145
\(472\) −3.15326e6 −0.651486
\(473\) −946822. −0.194588
\(474\) 1.97902e6 0.404580
\(475\) −1.28786e7 −2.61899
\(476\) −723258. −0.146311
\(477\) −3.28314e6 −0.660683
\(478\) 526177. 0.105332
\(479\) 9.12451e6 1.81707 0.908533 0.417814i \(-0.137203\pi\)
0.908533 + 0.417814i \(0.137203\pi\)
\(480\) −4.72775e6 −0.936594
\(481\) −6.42460e6 −1.26614
\(482\) 1.45637e6 0.285532
\(483\) 625788. 0.122056
\(484\) −1.88434e6 −0.365634
\(485\) −1.71174e7 −3.30434
\(486\) 3.47459e6 0.667287
\(487\) 2.68151e6 0.512338 0.256169 0.966632i \(-0.417540\pi\)
0.256169 + 0.966632i \(0.417540\pi\)
\(488\) −1.97452e6 −0.375329
\(489\) 1.87954e6 0.355451
\(490\) 6.32781e6 1.19059
\(491\) 4.83155e6 0.904446 0.452223 0.891905i \(-0.350631\pi\)
0.452223 + 0.891905i \(0.350631\pi\)
\(492\) −747442. −0.139208
\(493\) 1.44103e6 0.267028
\(494\) −4.86739e6 −0.897385
\(495\) 9.95338e6 1.82582
\(496\) −27513.2 −0.00502154
\(497\) 1.96661e6 0.357131
\(498\) 1.17064e6 0.211519
\(499\) 8.26537e6 1.48597 0.742987 0.669306i \(-0.233409\pi\)
0.742987 + 0.669306i \(0.233409\pi\)
\(500\) 1.04150e7 1.86309
\(501\) −3.97392e6 −0.707334
\(502\) 6.05701e6 1.07275
\(503\) −1.06078e7 −1.86941 −0.934703 0.355429i \(-0.884335\pi\)
−0.934703 + 0.355429i \(0.884335\pi\)
\(504\) −857532. −0.150374
\(505\) 1.80969e7 3.15773
\(506\) −5.85457e6 −1.01653
\(507\) 2.92268e6 0.504965
\(508\) 120157. 0.0206580
\(509\) −1.89756e6 −0.324639 −0.162320 0.986738i \(-0.551898\pi\)
−0.162320 + 0.986738i \(0.551898\pi\)
\(510\) 4.67399e6 0.795724
\(511\) 203272. 0.0344370
\(512\) −936238. −0.157838
\(513\) 5.13749e6 0.861901
\(514\) 119554. 0.0199598
\(515\) 1.55388e7 2.58166
\(516\) 270113. 0.0446603
\(517\) −8.78713e6 −1.44584
\(518\) 695207. 0.113839
\(519\) −3.35039e6 −0.545980
\(520\) 1.71029e7 2.77371
\(521\) −1.76155e6 −0.284316 −0.142158 0.989844i \(-0.545404\pi\)
−0.142158 + 0.989844i \(0.545404\pi\)
\(522\) 628604. 0.100972
\(523\) −7.12167e6 −1.13849 −0.569243 0.822170i \(-0.692764\pi\)
−0.569243 + 0.822170i \(0.692764\pi\)
\(524\) 894959. 0.142388
\(525\) −1.67057e6 −0.264525
\(526\) 3.75679e6 0.592041
\(527\) −516544. −0.0810179
\(528\) −325480. −0.0508089
\(529\) 3.33749e6 0.518538
\(530\) −7.08560e6 −1.09569
\(531\) −3.09099e6 −0.475730
\(532\) −733538. −0.112368
\(533\) 4.41302e6 0.672849
\(534\) −2.40052e6 −0.364294
\(535\) −7.78558e6 −1.17600
\(536\) 1.62917e6 0.244937
\(537\) 91567.6 0.0137027
\(538\) −6.90877e6 −1.02907
\(539\) −8.27287e6 −1.22655
\(540\) −6.64155e6 −0.980133
\(541\) 6.26443e6 0.920214 0.460107 0.887864i \(-0.347811\pi\)
0.460107 + 0.887864i \(0.347811\pi\)
\(542\) 2.21346e6 0.323649
\(543\) 562285. 0.0818383
\(544\) −8.56320e6 −1.24062
\(545\) 348552. 0.0502663
\(546\) −631383. −0.0906382
\(547\) −3.19734e6 −0.456899 −0.228449 0.973556i \(-0.573366\pi\)
−0.228449 + 0.973556i \(0.573366\pi\)
\(548\) 1.92003e6 0.273123
\(549\) −1.93552e6 −0.274074
\(550\) 1.56291e7 2.20306
\(551\) 1.46151e6 0.205080
\(552\) 4.53969e6 0.634131
\(553\) −1.76090e6 −0.244862
\(554\) −4.26248e6 −0.590049
\(555\) 6.25700e6 0.862251
\(556\) −4.09076e6 −0.561199
\(557\) 1.13299e7 1.54735 0.773677 0.633581i \(-0.218416\pi\)
0.773677 + 0.633581i \(0.218416\pi\)
\(558\) −225326. −0.0306355
\(559\) −1.59479e6 −0.215861
\(560\) −221516. −0.0298494
\(561\) −6.11070e6 −0.819754
\(562\) 3.43350e6 0.458561
\(563\) −4.20790e6 −0.559492 −0.279746 0.960074i \(-0.590250\pi\)
−0.279746 + 0.960074i \(0.590250\pi\)
\(564\) 2.50683e6 0.331839
\(565\) −1.24661e7 −1.64289
\(566\) −9.30855e6 −1.22135
\(567\) −459099. −0.0599720
\(568\) 1.42665e7 1.85544
\(569\) −3.28836e6 −0.425794 −0.212897 0.977075i \(-0.568290\pi\)
−0.212897 + 0.977075i \(0.568290\pi\)
\(570\) 4.74042e6 0.611124
\(571\) −2.54135e6 −0.326192 −0.163096 0.986610i \(-0.552148\pi\)
−0.163096 + 0.986610i \(0.552148\pi\)
\(572\) −8.22656e6 −1.05130
\(573\) 1.66776e6 0.212201
\(574\) −477534. −0.0604956
\(575\) −2.60917e7 −3.29104
\(576\) −4.20606e6 −0.528225
\(577\) −1.69533e6 −0.211990 −0.105995 0.994367i \(-0.533803\pi\)
−0.105995 + 0.994367i \(0.533803\pi\)
\(578\) 3.27334e6 0.407541
\(579\) −1.07819e6 −0.133659
\(580\) −1.88939e6 −0.233213
\(581\) −1.04161e6 −0.128017
\(582\) 4.58418e6 0.560988
\(583\) 9.26360e6 1.12878
\(584\) 1.47461e6 0.178914
\(585\) 1.67651e7 2.02543
\(586\) −5.90834e6 −0.710757
\(587\) 6.20781e6 0.743607 0.371803 0.928311i \(-0.378740\pi\)
0.371803 + 0.928311i \(0.378740\pi\)
\(588\) 2.36012e6 0.281508
\(589\) −523886. −0.0622226
\(590\) −6.67090e6 −0.788959
\(591\) −223665. −0.0263408
\(592\) 603645. 0.0707909
\(593\) 3.01403e6 0.351974 0.175987 0.984393i \(-0.443688\pi\)
0.175987 + 0.984393i \(0.443688\pi\)
\(594\) −6.23469e6 −0.725018
\(595\) −4.15884e6 −0.481592
\(596\) 6.25138e6 0.720876
\(597\) −2.77314e6 −0.318446
\(598\) −9.86122e6 −1.12766
\(599\) 1.21463e7 1.38317 0.691586 0.722294i \(-0.256912\pi\)
0.691586 + 0.722294i \(0.256912\pi\)
\(600\) −1.21189e7 −1.37432
\(601\) 1.68363e6 0.190134 0.0950671 0.995471i \(-0.469693\pi\)
0.0950671 + 0.995471i \(0.469693\pi\)
\(602\) 172573. 0.0194080
\(603\) 1.59700e6 0.178859
\(604\) 1.58173e6 0.176416
\(605\) −1.08352e7 −1.20351
\(606\) −4.84648e6 −0.536099
\(607\) −5.11718e6 −0.563714 −0.281857 0.959456i \(-0.590950\pi\)
−0.281857 + 0.959456i \(0.590950\pi\)
\(608\) −8.68491e6 −0.952810
\(609\) 189583. 0.0207136
\(610\) −4.17720e6 −0.454528
\(611\) −1.48007e7 −1.60391
\(612\) −5.14316e6 −0.555075
\(613\) 4.10152e6 0.440853 0.220426 0.975404i \(-0.429255\pi\)
0.220426 + 0.975404i \(0.429255\pi\)
\(614\) −5.45103e6 −0.583522
\(615\) −4.29790e6 −0.458214
\(616\) 2.41958e6 0.256915
\(617\) 1.52311e7 1.61071 0.805355 0.592792i \(-0.201975\pi\)
0.805355 + 0.592792i \(0.201975\pi\)
\(618\) −4.16141e6 −0.438298
\(619\) 5.27961e6 0.553829 0.276914 0.960895i \(-0.410688\pi\)
0.276914 + 0.960895i \(0.410688\pi\)
\(620\) 677260. 0.0707581
\(621\) 1.04084e7 1.08307
\(622\) −2.76092e6 −0.286140
\(623\) 2.13594e6 0.220480
\(624\) −548227. −0.0563636
\(625\) 3.38068e7 3.46181
\(626\) 4.81564e6 0.491154
\(627\) −6.19754e6 −0.629580
\(628\) −5.84421e6 −0.591326
\(629\) 1.13331e7 1.14215
\(630\) −1.81416e6 −0.182105
\(631\) −1.84888e7 −1.84857 −0.924283 0.381709i \(-0.875336\pi\)
−0.924283 + 0.381709i \(0.875336\pi\)
\(632\) −1.27742e7 −1.27216
\(633\) 2.57129e6 0.255060
\(634\) −8.03004e6 −0.793404
\(635\) 690917. 0.0679973
\(636\) −2.64276e6 −0.259068
\(637\) −1.39345e7 −1.36064
\(638\) −1.77365e6 −0.172511
\(639\) 1.39848e7 1.35489
\(640\) 1.02118e7 0.985489
\(641\) −3.87212e6 −0.372223 −0.186112 0.982529i \(-0.559589\pi\)
−0.186112 + 0.982529i \(0.559589\pi\)
\(642\) 2.08504e6 0.199653
\(643\) 3.24879e6 0.309880 0.154940 0.987924i \(-0.450482\pi\)
0.154940 + 0.987924i \(0.450482\pi\)
\(644\) −1.48613e6 −0.141202
\(645\) 1.55319e6 0.147003
\(646\) 8.58615e6 0.809500
\(647\) 1.85942e7 1.74629 0.873147 0.487456i \(-0.162075\pi\)
0.873147 + 0.487456i \(0.162075\pi\)
\(648\) −3.33047e6 −0.311579
\(649\) 8.72142e6 0.812785
\(650\) 2.63250e7 2.44391
\(651\) −67956.8 −0.00628464
\(652\) −4.46356e6 −0.411209
\(653\) 9.77909e6 0.897461 0.448730 0.893667i \(-0.351876\pi\)
0.448730 + 0.893667i \(0.351876\pi\)
\(654\) −93345.0 −0.00853388
\(655\) 5.14614e6 0.468682
\(656\) −414640. −0.0376194
\(657\) 1.44548e6 0.130647
\(658\) 1.60159e6 0.144207
\(659\) 1.92292e7 1.72484 0.862418 0.506196i \(-0.168949\pi\)
0.862418 + 0.506196i \(0.168949\pi\)
\(660\) 8.01196e6 0.715944
\(661\) −1.39614e7 −1.24287 −0.621434 0.783467i \(-0.713449\pi\)
−0.621434 + 0.783467i \(0.713449\pi\)
\(662\) 2.66233e6 0.236111
\(663\) −1.02926e7 −0.909374
\(664\) −7.55626e6 −0.665099
\(665\) −4.21794e6 −0.369868
\(666\) 4.94368e6 0.431882
\(667\) 2.96099e6 0.257705
\(668\) 9.43732e6 0.818290
\(669\) 1.94471e6 0.167993
\(670\) 3.44660e6 0.296623
\(671\) 5.46121e6 0.468255
\(672\) −1.12658e6 −0.0962362
\(673\) 1.13333e7 0.964537 0.482269 0.876023i \(-0.339813\pi\)
0.482269 + 0.876023i \(0.339813\pi\)
\(674\) 2.29274e6 0.194404
\(675\) −2.77858e7 −2.34727
\(676\) −6.94082e6 −0.584176
\(677\) −9.84009e6 −0.825140 −0.412570 0.910926i \(-0.635369\pi\)
−0.412570 + 0.910926i \(0.635369\pi\)
\(678\) 3.33851e6 0.278919
\(679\) −4.07892e6 −0.339524
\(680\) −3.01697e7 −2.50207
\(681\) −6.81498e6 −0.563115
\(682\) 635771. 0.0523407
\(683\) 4282.68 0.000351288 0 0.000175644 1.00000i \(-0.499944\pi\)
0.000175644 1.00000i \(0.499944\pi\)
\(684\) −5.21625e6 −0.426303
\(685\) 1.10405e7 0.899003
\(686\) 3.07651e6 0.249602
\(687\) 1.87232e6 0.151352
\(688\) 149844. 0.0120689
\(689\) 1.56033e7 1.25218
\(690\) 9.60397e6 0.767941
\(691\) 5.24701e6 0.418039 0.209019 0.977911i \(-0.432973\pi\)
0.209019 + 0.977911i \(0.432973\pi\)
\(692\) 7.95655e6 0.631625
\(693\) 2.37180e6 0.187605
\(694\) −1.20826e7 −0.952277
\(695\) −2.35224e7 −1.84723
\(696\) 1.37531e6 0.107616
\(697\) −7.78462e6 −0.606954
\(698\) −1.07329e7 −0.833834
\(699\) 3.98217e6 0.308267
\(700\) 3.96730e6 0.306020
\(701\) −3.73379e6 −0.286982 −0.143491 0.989652i \(-0.545833\pi\)
−0.143491 + 0.989652i \(0.545833\pi\)
\(702\) −1.05015e7 −0.804281
\(703\) 1.14942e7 0.877180
\(704\) 1.18677e7 0.902473
\(705\) 1.44146e7 1.09227
\(706\) 2.31744e6 0.174983
\(707\) 4.31231e6 0.324461
\(708\) −2.48808e6 −0.186544
\(709\) −1.21242e7 −0.905810 −0.452905 0.891559i \(-0.649612\pi\)
−0.452905 + 0.891559i \(0.649612\pi\)
\(710\) 3.01816e7 2.24697
\(711\) −1.25219e7 −0.928960
\(712\) 1.54949e7 1.14548
\(713\) −1.06138e6 −0.0781892
\(714\) 1.11377e6 0.0817615
\(715\) −4.73039e7 −3.46044
\(716\) −217456. −0.0158522
\(717\) 1.12847e6 0.0819772
\(718\) 1.12660e7 0.815563
\(719\) 1.59176e7 1.14830 0.574149 0.818751i \(-0.305333\pi\)
0.574149 + 0.818751i \(0.305333\pi\)
\(720\) −1.57522e6 −0.113243
\(721\) 3.70275e6 0.265269
\(722\) −347040. −0.0247763
\(723\) 3.12342e6 0.222221
\(724\) −1.33532e6 −0.0946759
\(725\) −7.90452e6 −0.558509
\(726\) 2.90176e6 0.204324
\(727\) −633198. −0.0444328 −0.0222164 0.999753i \(-0.507072\pi\)
−0.0222164 + 0.999753i \(0.507072\pi\)
\(728\) 4.07546e6 0.285002
\(729\) 3.08055e6 0.214689
\(730\) 3.11961e6 0.216668
\(731\) 2.81324e6 0.194721
\(732\) −1.55800e6 −0.107470
\(733\) −2.64288e7 −1.81684 −0.908421 0.418057i \(-0.862711\pi\)
−0.908421 + 0.418057i \(0.862711\pi\)
\(734\) −4.85952e6 −0.332930
\(735\) 1.35710e7 0.926603
\(736\) −1.75954e7 −1.19730
\(737\) −4.50603e6 −0.305581
\(738\) −3.39579e6 −0.229509
\(739\) 2.15407e6 0.145094 0.0725470 0.997365i \(-0.476887\pi\)
0.0725470 + 0.997365i \(0.476887\pi\)
\(740\) −1.48592e7 −0.997508
\(741\) −1.04389e7 −0.698409
\(742\) −1.68843e6 −0.112583
\(743\) 1.76951e7 1.17593 0.587964 0.808887i \(-0.299930\pi\)
0.587964 + 0.808887i \(0.299930\pi\)
\(744\) −492984. −0.0326513
\(745\) 3.59463e7 2.37281
\(746\) −6.99664e6 −0.460302
\(747\) −7.40702e6 −0.485671
\(748\) 1.45118e7 0.948345
\(749\) −1.85523e6 −0.120835
\(750\) −1.60383e7 −1.04113
\(751\) 2.58345e7 1.67148 0.835739 0.549127i \(-0.185040\pi\)
0.835739 + 0.549127i \(0.185040\pi\)
\(752\) 1.39065e6 0.0896755
\(753\) 1.29902e7 0.834891
\(754\) −2.98747e6 −0.191370
\(755\) 9.09514e6 0.580687
\(756\) −1.58262e6 −0.100710
\(757\) −2.47558e7 −1.57013 −0.785067 0.619411i \(-0.787371\pi\)
−0.785067 + 0.619411i \(0.787371\pi\)
\(758\) 2.57343e6 0.162682
\(759\) −1.25561e7 −0.791133
\(760\) −3.05985e7 −1.92161
\(761\) −1.12818e7 −0.706183 −0.353092 0.935589i \(-0.614870\pi\)
−0.353092 + 0.935589i \(0.614870\pi\)
\(762\) −185033. −0.0115441
\(763\) 83056.7 0.00516492
\(764\) −3.96062e6 −0.245488
\(765\) −2.95739e7 −1.82707
\(766\) 6.35406e6 0.391272
\(767\) 1.46900e7 0.901643
\(768\) −8.55146e6 −0.523163
\(769\) −1.45315e7 −0.886124 −0.443062 0.896491i \(-0.646108\pi\)
−0.443062 + 0.896491i \(0.646108\pi\)
\(770\) 5.11876e6 0.311127
\(771\) 256403. 0.0155341
\(772\) 2.56050e6 0.154625
\(773\) −5.49667e6 −0.330865 −0.165433 0.986221i \(-0.552902\pi\)
−0.165433 + 0.986221i \(0.552902\pi\)
\(774\) 1.22718e6 0.0736303
\(775\) 2.83341e6 0.169455
\(776\) −2.95900e7 −1.76397
\(777\) 1.49099e6 0.0885974
\(778\) 1.43757e7 0.851492
\(779\) −7.89526e6 −0.466147
\(780\) 1.34950e7 0.794215
\(781\) −3.94590e7 −2.31482
\(782\) 1.73953e7 1.01722
\(783\) 3.15324e6 0.183803
\(784\) 1.30926e6 0.0760742
\(785\) −3.36050e7 −1.94639
\(786\) −1.37817e6 −0.0795697
\(787\) 1.62860e7 0.937297 0.468649 0.883385i \(-0.344741\pi\)
0.468649 + 0.883385i \(0.344741\pi\)
\(788\) 531163. 0.0304728
\(789\) 8.05704e6 0.460769
\(790\) −2.70245e7 −1.54060
\(791\) −2.97055e6 −0.168809
\(792\) 1.72059e7 0.974685
\(793\) 9.19866e6 0.519447
\(794\) 1.46203e7 0.823009
\(795\) −1.51962e7 −0.852742
\(796\) 6.58569e6 0.368399
\(797\) 1.65610e7 0.923506 0.461753 0.887009i \(-0.347221\pi\)
0.461753 + 0.887009i \(0.347221\pi\)
\(798\) 1.12960e6 0.0627937
\(799\) 2.61087e7 1.44683
\(800\) 4.69718e7 2.59485
\(801\) 1.51889e7 0.836458
\(802\) 1.60905e7 0.883350
\(803\) −4.07853e6 −0.223211
\(804\) 1.28550e6 0.0701346
\(805\) −8.54545e6 −0.464778
\(806\) 1.07087e6 0.0580629
\(807\) −1.48170e7 −0.800896
\(808\) 3.12831e7 1.68571
\(809\) −1.93881e7 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(810\) −7.04580e6 −0.377327
\(811\) 1.99327e7 1.06418 0.532089 0.846688i \(-0.321407\pi\)
0.532089 + 0.846688i \(0.321407\pi\)
\(812\) −450224. −0.0239629
\(813\) 4.74713e6 0.251886
\(814\) −1.39489e7 −0.737870
\(815\) −2.56661e7 −1.35352
\(816\) 967079. 0.0508436
\(817\) 2.85322e6 0.149548
\(818\) −6.03074e6 −0.315128
\(819\) 3.99497e6 0.208115
\(820\) 1.02067e7 0.530092
\(821\) −2.97106e7 −1.53834 −0.769172 0.639041i \(-0.779331\pi\)
−0.769172 + 0.639041i \(0.779331\pi\)
\(822\) −2.95672e6 −0.152627
\(823\) −3.70742e6 −0.190797 −0.0953987 0.995439i \(-0.530413\pi\)
−0.0953987 + 0.995439i \(0.530413\pi\)
\(824\) 2.68611e7 1.37818
\(825\) 3.35191e7 1.71458
\(826\) −1.58961e6 −0.0810664
\(827\) −5.42093e6 −0.275620 −0.137810 0.990459i \(-0.544006\pi\)
−0.137810 + 0.990459i \(0.544006\pi\)
\(828\) −1.05680e7 −0.535694
\(829\) −3.24596e7 −1.64043 −0.820214 0.572056i \(-0.806146\pi\)
−0.820214 + 0.572056i \(0.806146\pi\)
\(830\) −1.59857e7 −0.805445
\(831\) −9.14159e6 −0.459218
\(832\) 1.99895e7 1.00114
\(833\) 2.45807e7 1.22739
\(834\) 6.29948e6 0.313610
\(835\) 5.42659e7 2.69346
\(836\) 1.47180e7 0.728339
\(837\) −1.13029e6 −0.0557670
\(838\) −2.16867e7 −1.06680
\(839\) −6.99699e6 −0.343168 −0.171584 0.985170i \(-0.554888\pi\)
−0.171584 + 0.985170i \(0.554888\pi\)
\(840\) −3.96914e6 −0.194088
\(841\) −1.96141e7 −0.956266
\(842\) 1.77566e7 0.863136
\(843\) 7.36371e6 0.356885
\(844\) −6.10634e6 −0.295070
\(845\) −3.99107e7 −1.92286
\(846\) 1.13891e7 0.547094
\(847\) −2.58194e6 −0.123662
\(848\) −1.46606e6 −0.0700102
\(849\) −1.99637e7 −0.950543
\(850\) −4.64377e7 −2.20457
\(851\) 2.32869e7 1.10227
\(852\) 1.12570e7 0.531281
\(853\) −5.68399e6 −0.267474 −0.133737 0.991017i \(-0.542698\pi\)
−0.133737 + 0.991017i \(0.542698\pi\)
\(854\) −995389. −0.0467033
\(855\) −2.99942e7 −1.40321
\(856\) −1.34585e7 −0.627788
\(857\) −8.71203e6 −0.405198 −0.202599 0.979262i \(-0.564939\pi\)
−0.202599 + 0.979262i \(0.564939\pi\)
\(858\) 1.26683e7 0.587491
\(859\) 1.41568e7 0.654609 0.327304 0.944919i \(-0.393860\pi\)
0.327304 + 0.944919i \(0.393860\pi\)
\(860\) −3.68854e6 −0.170062
\(861\) −1.02415e6 −0.0470820
\(862\) 9.47790e6 0.434454
\(863\) 9.82195e6 0.448922 0.224461 0.974483i \(-0.427938\pi\)
0.224461 + 0.974483i \(0.427938\pi\)
\(864\) −1.87378e7 −0.853955
\(865\) 4.57513e7 2.07904
\(866\) −1.86031e7 −0.842927
\(867\) 7.02021e6 0.317177
\(868\) 161385. 0.00727048
\(869\) 3.53315e7 1.58713
\(870\) 2.90953e6 0.130324
\(871\) −7.58980e6 −0.338988
\(872\) 602524. 0.0268339
\(873\) −2.90056e7 −1.28809
\(874\) 1.76426e7 0.781237
\(875\) 1.42706e7 0.630120
\(876\) 1.16354e6 0.0512297
\(877\) 2.13034e7 0.935298 0.467649 0.883914i \(-0.345101\pi\)
0.467649 + 0.883914i \(0.345101\pi\)
\(878\) −2.09019e7 −0.915061
\(879\) −1.26714e7 −0.553162
\(880\) 4.44460e6 0.193475
\(881\) 3.10088e7 1.34600 0.673000 0.739642i \(-0.265005\pi\)
0.673000 + 0.739642i \(0.265005\pi\)
\(882\) 1.07225e7 0.464114
\(883\) −2.21207e7 −0.954766 −0.477383 0.878695i \(-0.658415\pi\)
−0.477383 + 0.878695i \(0.658415\pi\)
\(884\) 2.44431e7 1.05202
\(885\) −1.43068e7 −0.614024
\(886\) −1.76193e7 −0.754057
\(887\) 4.29991e7 1.83506 0.917530 0.397666i \(-0.130180\pi\)
0.917530 + 0.397666i \(0.130180\pi\)
\(888\) 1.08162e7 0.460299
\(889\) 164639. 0.00698680
\(890\) 3.27803e7 1.38720
\(891\) 9.21156e6 0.388722
\(892\) −4.61833e6 −0.194345
\(893\) 2.64797e7 1.11118
\(894\) −9.62669e6 −0.402841
\(895\) −1.25040e6 −0.0521786
\(896\) 2.43337e6 0.101260
\(897\) −2.11490e7 −0.877624
\(898\) −8.65551e6 −0.358181
\(899\) −321546. −0.0132692
\(900\) 2.82118e7 1.16098
\(901\) −2.75244e7 −1.12955
\(902\) 9.58144e6 0.392116
\(903\) 370111. 0.0151047
\(904\) −2.15494e7 −0.877031
\(905\) −7.67828e6 −0.311632
\(906\) −2.43575e6 −0.0985852
\(907\) 3.34186e7 1.34887 0.674435 0.738334i \(-0.264387\pi\)
0.674435 + 0.738334i \(0.264387\pi\)
\(908\) 1.61843e7 0.651448
\(909\) 3.06653e7 1.23094
\(910\) 8.62185e6 0.345141
\(911\) −3.77821e7 −1.50831 −0.754155 0.656697i \(-0.771953\pi\)
−0.754155 + 0.656697i \(0.771953\pi\)
\(912\) 980824. 0.0390485
\(913\) 2.08994e7 0.829769
\(914\) 6.52337e6 0.258289
\(915\) −8.95869e6 −0.353746
\(916\) −4.44642e6 −0.175094
\(917\) 1.22628e6 0.0481576
\(918\) 1.85248e7 0.725514
\(919\) 3.71228e7 1.44995 0.724973 0.688777i \(-0.241852\pi\)
0.724973 + 0.688777i \(0.241852\pi\)
\(920\) −6.19918e7 −2.41471
\(921\) −1.16906e7 −0.454139
\(922\) −1.56000e7 −0.604364
\(923\) −6.64632e7 −2.56789
\(924\) 1.90917e6 0.0735641
\(925\) −6.21655e7 −2.38888
\(926\) −8.55371e6 −0.327814
\(927\) 2.63306e7 1.00638
\(928\) −5.33055e6 −0.203190
\(929\) 4.73143e6 0.179868 0.0899339 0.995948i \(-0.471334\pi\)
0.0899339 + 0.995948i \(0.471334\pi\)
\(930\) −1.04293e6 −0.0395412
\(931\) 2.49300e7 0.942646
\(932\) −9.45690e6 −0.356623
\(933\) −5.92124e6 −0.222694
\(934\) −2.50377e7 −0.939132
\(935\) 8.34447e7 3.12154
\(936\) 2.89810e7 1.08124
\(937\) −1.74328e7 −0.648660 −0.324330 0.945944i \(-0.605139\pi\)
−0.324330 + 0.945944i \(0.605139\pi\)
\(938\) 821293. 0.0304783
\(939\) 1.03279e7 0.382251
\(940\) −3.42320e7 −1.26361
\(941\) 3.82735e7 1.40904 0.704521 0.709683i \(-0.251162\pi\)
0.704521 + 0.709683i \(0.251162\pi\)
\(942\) 8.99968e6 0.330446
\(943\) −1.59956e7 −0.585762
\(944\) −1.38025e6 −0.0504114
\(945\) −9.10028e6 −0.331494
\(946\) −3.46258e6 −0.125797
\(947\) −1.65041e7 −0.598021 −0.299011 0.954250i \(-0.596657\pi\)
−0.299011 + 0.954250i \(0.596657\pi\)
\(948\) −1.00795e7 −0.364266
\(949\) −6.86973e6 −0.247613
\(950\) −4.70977e7 −1.69313
\(951\) −1.72217e7 −0.617484
\(952\) −7.18916e6 −0.257090
\(953\) −2.15575e7 −0.768894 −0.384447 0.923147i \(-0.625608\pi\)
−0.384447 + 0.923147i \(0.625608\pi\)
\(954\) −1.20066e7 −0.427119
\(955\) −2.27741e7 −0.808040
\(956\) −2.67991e6 −0.0948365
\(957\) −3.80388e6 −0.134260
\(958\) 3.33688e7 1.17470
\(959\) 2.63084e6 0.0923736
\(960\) −1.94680e7 −0.681779
\(961\) −2.85139e7 −0.995974
\(962\) −2.34951e7 −0.818539
\(963\) −1.31927e7 −0.458425
\(964\) −7.41755e6 −0.257080
\(965\) 1.47232e7 0.508960
\(966\) 2.28854e6 0.0789069
\(967\) −3.38528e7 −1.16420 −0.582101 0.813116i \(-0.697769\pi\)
−0.582101 + 0.813116i \(0.697769\pi\)
\(968\) −1.87303e7 −0.642476
\(969\) 1.84144e7 0.630011
\(970\) −6.25993e7 −2.13619
\(971\) 1.12966e7 0.384502 0.192251 0.981346i \(-0.438421\pi\)
0.192251 + 0.981346i \(0.438421\pi\)
\(972\) −1.76967e7 −0.600795
\(973\) −5.60517e6 −0.189805
\(974\) 9.80641e6 0.331217
\(975\) 5.64583e7 1.90202
\(976\) −864291. −0.0290426
\(977\) −2.28369e7 −0.765423 −0.382711 0.923868i \(-0.625010\pi\)
−0.382711 + 0.923868i \(0.625010\pi\)
\(978\) 6.87358e6 0.229793
\(979\) −4.28564e7 −1.42909
\(980\) −3.22286e7 −1.07196
\(981\) 590624. 0.0195947
\(982\) 1.76692e7 0.584707
\(983\) 1.60143e7 0.528598 0.264299 0.964441i \(-0.414860\pi\)
0.264299 + 0.964441i \(0.414860\pi\)
\(984\) −7.42955e6 −0.244610
\(985\) 3.05426e6 0.100303
\(986\) 5.26993e6 0.172629
\(987\) 3.43487e6 0.112232
\(988\) 2.47905e7 0.807965
\(989\) 5.78055e6 0.187922
\(990\) 3.64000e7 1.18036
\(991\) 1.70581e7 0.551755 0.275878 0.961193i \(-0.411032\pi\)
0.275878 + 0.961193i \(0.411032\pi\)
\(992\) 1.91076e6 0.0616490
\(993\) 5.70980e6 0.183759
\(994\) 7.19200e6 0.230879
\(995\) 3.78687e7 1.21261
\(996\) −5.96227e6 −0.190442
\(997\) −2.65006e7 −0.844341 −0.422171 0.906516i \(-0.638732\pi\)
−0.422171 + 0.906516i \(0.638732\pi\)
\(998\) 3.02269e7 0.960653
\(999\) 2.47988e7 0.786172
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 43.6.a.a.1.6 8
3.2 odd 2 387.6.a.c.1.3 8
4.3 odd 2 688.6.a.e.1.3 8
5.4 even 2 1075.6.a.a.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.6 8 1.1 even 1 trivial
387.6.a.c.1.3 8 3.2 odd 2
688.6.a.e.1.3 8 4.3 odd 2
1075.6.a.a.1.3 8 5.4 even 2