Properties

Label 29.6.a.b.1.2
Level $29$
Weight $6$
Character 29.1
Self dual yes
Analytic conductor $4.651$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.65113077458\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(7.83842\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q-6.83842 q^{2} -24.5656 q^{3} +14.7640 q^{4} -54.3066 q^{5} +167.990 q^{6} -37.6697 q^{7} +117.867 q^{8} +360.469 q^{9} +O(q^{10})\) \(q-6.83842 q^{2} -24.5656 q^{3} +14.7640 q^{4} -54.3066 q^{5} +167.990 q^{6} -37.6697 q^{7} +117.867 q^{8} +360.469 q^{9} +371.372 q^{10} -146.903 q^{11} -362.686 q^{12} +162.581 q^{13} +257.601 q^{14} +1334.08 q^{15} -1278.47 q^{16} -2162.70 q^{17} -2465.04 q^{18} +2494.06 q^{19} -801.781 q^{20} +925.378 q^{21} +1004.59 q^{22} -122.601 q^{23} -2895.48 q^{24} -175.788 q^{25} -1111.79 q^{26} -2885.69 q^{27} -556.154 q^{28} +841.000 q^{29} -9122.96 q^{30} +8674.26 q^{31} +4970.98 q^{32} +3608.77 q^{33} +14789.4 q^{34} +2045.71 q^{35} +5321.95 q^{36} +11752.8 q^{37} -17055.4 q^{38} -3993.89 q^{39} -6400.97 q^{40} +9519.28 q^{41} -6328.12 q^{42} -14834.7 q^{43} -2168.88 q^{44} -19575.8 q^{45} +838.394 q^{46} -3827.35 q^{47} +31406.4 q^{48} -15388.0 q^{49} +1202.11 q^{50} +53127.9 q^{51} +2400.34 q^{52} -33207.9 q^{53} +19733.5 q^{54} +7977.83 q^{55} -4440.02 q^{56} -61268.2 q^{57} -5751.11 q^{58} +19608.0 q^{59} +19696.2 q^{60} +24024.9 q^{61} -59318.2 q^{62} -13578.7 q^{63} +6917.49 q^{64} -8829.21 q^{65} -24678.3 q^{66} +55181.2 q^{67} -31930.0 q^{68} +3011.76 q^{69} -13989.4 q^{70} -41626.8 q^{71} +42487.4 q^{72} -15707.6 q^{73} -80370.5 q^{74} +4318.34 q^{75} +36822.3 q^{76} +5533.80 q^{77} +27311.9 q^{78} +56772.8 q^{79} +69429.5 q^{80} -16705.2 q^{81} -65096.8 q^{82} +107406. q^{83} +13662.2 q^{84} +117449. q^{85} +101446. q^{86} -20659.7 q^{87} -17315.1 q^{88} +37615.4 q^{89} +133868. q^{90} -6124.36 q^{91} -1810.07 q^{92} -213088. q^{93} +26173.0 q^{94} -135444. q^{95} -122115. q^{96} -43944.4 q^{97} +105230. q^{98} -52954.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} + 26 q^{3} + 154 q^{4} + 32 q^{5} + 22 q^{6} + 184 q^{7} + 942 q^{8} + 1005 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 4 q^{2} + 26 q^{3} + 154 q^{4} + 32 q^{5} + 22 q^{6} + 184 q^{7} + 942 q^{8} + 1005 q^{9} + 922 q^{10} + 1106 q^{11} + 214 q^{12} + 408 q^{13} - 2008 q^{14} - 614 q^{15} + 242 q^{16} - 874 q^{17} - 5598 q^{18} + 4288 q^{19} - 6350 q^{20} - 4200 q^{21} - 6114 q^{22} - 4532 q^{23} - 4318 q^{24} + 5527 q^{25} - 19806 q^{26} + 5942 q^{27} - 496 q^{28} + 5887 q^{29} - 16734 q^{30} + 7794 q^{31} + 7898 q^{32} + 34410 q^{33} + 20840 q^{34} + 7088 q^{35} - 572 q^{36} + 5086 q^{37} + 23732 q^{38} + 33394 q^{39} + 22906 q^{40} + 19826 q^{41} - 55440 q^{42} + 19498 q^{43} - 6074 q^{44} + 7854 q^{45} - 12404 q^{46} + 14278 q^{47} - 16406 q^{48} + 38431 q^{49} - 41066 q^{50} + 23892 q^{51} - 34302 q^{52} - 58644 q^{53} - 31194 q^{54} - 25574 q^{55} - 79560 q^{56} - 88540 q^{57} + 3364 q^{58} + 12888 q^{59} - 180822 q^{60} + 102866 q^{61} - 42654 q^{62} - 88632 q^{63} - 10170 q^{64} - 149206 q^{65} + 7710 q^{66} + 102996 q^{67} + 85100 q^{68} - 107244 q^{69} + 349480 q^{70} - 51596 q^{71} + 135568 q^{72} - 17566 q^{73} + 12132 q^{74} + 39356 q^{75} + 360740 q^{76} - 94104 q^{77} + 46386 q^{78} + 212058 q^{79} + 142510 q^{80} - 128285 q^{81} + 201924 q^{82} - 122928 q^{83} - 12328 q^{84} - 109336 q^{85} - 63290 q^{86} + 21866 q^{87} + 136666 q^{88} - 66510 q^{89} + 56084 q^{90} + 194368 q^{91} - 110108 q^{92} - 474274 q^{93} + 438926 q^{94} - 131676 q^{95} - 117018 q^{96} - 118182 q^{97} - 29132 q^{98} + 300668 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.83842 −1.20887 −0.604436 0.796653i \(-0.706602\pi\)
−0.604436 + 0.796653i \(0.706602\pi\)
\(3\) −24.5656 −1.57588 −0.787942 0.615750i \(-0.788853\pi\)
−0.787942 + 0.615750i \(0.788853\pi\)
\(4\) 14.7640 0.461374
\(5\) −54.3066 −0.971467 −0.485733 0.874107i \(-0.661447\pi\)
−0.485733 + 0.874107i \(0.661447\pi\)
\(6\) 167.990 1.90504
\(7\) −37.6697 −0.290567 −0.145284 0.989390i \(-0.546409\pi\)
−0.145284 + 0.989390i \(0.546409\pi\)
\(8\) 117.867 0.651131
\(9\) 360.469 1.48341
\(10\) 371.372 1.17438
\(11\) −146.903 −0.366058 −0.183029 0.983107i \(-0.558590\pi\)
−0.183029 + 0.983107i \(0.558590\pi\)
\(12\) −362.686 −0.727072
\(13\) 162.581 0.266815 0.133408 0.991061i \(-0.457408\pi\)
0.133408 + 0.991061i \(0.457408\pi\)
\(14\) 257.601 0.351259
\(15\) 1334.08 1.53092
\(16\) −1278.47 −1.24851
\(17\) −2162.70 −1.81499 −0.907493 0.420068i \(-0.862006\pi\)
−0.907493 + 0.420068i \(0.862006\pi\)
\(18\) −2465.04 −1.79325
\(19\) 2494.06 1.58498 0.792489 0.609886i \(-0.208785\pi\)
0.792489 + 0.609886i \(0.208785\pi\)
\(20\) −801.781 −0.448209
\(21\) 925.378 0.457900
\(22\) 1004.59 0.442518
\(23\) −122.601 −0.0483251 −0.0241626 0.999708i \(-0.507692\pi\)
−0.0241626 + 0.999708i \(0.507692\pi\)
\(24\) −2895.48 −1.02611
\(25\) −175.788 −0.0562522
\(26\) −1111.79 −0.322546
\(27\) −2885.69 −0.761798
\(28\) −556.154 −0.134060
\(29\) 841.000 0.185695
\(30\) −9122.96 −1.85069
\(31\) 8674.26 1.62117 0.810584 0.585622i \(-0.199150\pi\)
0.810584 + 0.585622i \(0.199150\pi\)
\(32\) 4970.98 0.858157
\(33\) 3608.77 0.576865
\(34\) 14789.4 2.19409
\(35\) 2045.71 0.282276
\(36\) 5321.95 0.684407
\(37\) 11752.8 1.41136 0.705679 0.708532i \(-0.250642\pi\)
0.705679 + 0.708532i \(0.250642\pi\)
\(38\) −17055.4 −1.91604
\(39\) −3993.89 −0.420470
\(40\) −6400.97 −0.632552
\(41\) 9519.28 0.884391 0.442195 0.896919i \(-0.354200\pi\)
0.442195 + 0.896919i \(0.354200\pi\)
\(42\) −6328.12 −0.553543
\(43\) −14834.7 −1.22351 −0.611756 0.791046i \(-0.709537\pi\)
−0.611756 + 0.791046i \(0.709537\pi\)
\(44\) −2168.88 −0.168890
\(45\) −19575.8 −1.44108
\(46\) 838.394 0.0584189
\(47\) −3827.35 −0.252728 −0.126364 0.991984i \(-0.540331\pi\)
−0.126364 + 0.991984i \(0.540331\pi\)
\(48\) 31406.4 1.96750
\(49\) −15388.0 −0.915571
\(50\) 1202.11 0.0680017
\(51\) 53127.9 2.86021
\(52\) 2400.34 0.123102
\(53\) −33207.9 −1.62387 −0.811936 0.583746i \(-0.801586\pi\)
−0.811936 + 0.583746i \(0.801586\pi\)
\(54\) 19733.5 0.920917
\(55\) 7977.83 0.355613
\(56\) −4440.02 −0.189197
\(57\) −61268.2 −2.49774
\(58\) −5751.11 −0.224482
\(59\) 19608.0 0.733336 0.366668 0.930352i \(-0.380498\pi\)
0.366668 + 0.930352i \(0.380498\pi\)
\(60\) 19696.2 0.706326
\(61\) 24024.9 0.826679 0.413340 0.910577i \(-0.364362\pi\)
0.413340 + 0.910577i \(0.364362\pi\)
\(62\) −59318.2 −1.95979
\(63\) −13578.7 −0.431030
\(64\) 6917.49 0.211105
\(65\) −8829.21 −0.259202
\(66\) −24678.3 −0.697357
\(67\) 55181.2 1.50177 0.750886 0.660432i \(-0.229627\pi\)
0.750886 + 0.660432i \(0.229627\pi\)
\(68\) −31930.0 −0.837387
\(69\) 3011.76 0.0761548
\(70\) −13989.4 −0.341236
\(71\) −41626.8 −0.980002 −0.490001 0.871722i \(-0.663004\pi\)
−0.490001 + 0.871722i \(0.663004\pi\)
\(72\) 42487.4 0.965894
\(73\) −15707.6 −0.344987 −0.172493 0.985011i \(-0.555182\pi\)
−0.172493 + 0.985011i \(0.555182\pi\)
\(74\) −80370.5 −1.70615
\(75\) 4318.34 0.0886469
\(76\) 36822.3 0.731268
\(77\) 5533.80 0.106365
\(78\) 27311.9 0.508295
\(79\) 56772.8 1.02346 0.511732 0.859145i \(-0.329004\pi\)
0.511732 + 0.859145i \(0.329004\pi\)
\(80\) 69429.5 1.21288
\(81\) −16705.2 −0.282905
\(82\) −65096.8 −1.06912
\(83\) 107406. 1.71133 0.855664 0.517532i \(-0.173149\pi\)
0.855664 + 0.517532i \(0.173149\pi\)
\(84\) 13662.2 0.211263
\(85\) 117449. 1.76320
\(86\) 101446. 1.47907
\(87\) −20659.7 −0.292634
\(88\) −17315.1 −0.238352
\(89\) 37615.4 0.503374 0.251687 0.967809i \(-0.419015\pi\)
0.251687 + 0.967809i \(0.419015\pi\)
\(90\) 133868. 1.74209
\(91\) −6124.36 −0.0775278
\(92\) −1810.07 −0.0222960
\(93\) −213088. −2.55477
\(94\) 26173.0 0.305516
\(95\) −135444. −1.53975
\(96\) −122115. −1.35236
\(97\) −43944.4 −0.474214 −0.237107 0.971484i \(-0.576199\pi\)
−0.237107 + 0.971484i \(0.576199\pi\)
\(98\) 105230. 1.10681
\(99\) −52954.1 −0.543014
\(100\) −2595.33 −0.0259533
\(101\) 71165.6 0.694171 0.347086 0.937833i \(-0.387171\pi\)
0.347086 + 0.937833i \(0.387171\pi\)
\(102\) −363311. −3.45763
\(103\) −36173.8 −0.335970 −0.167985 0.985790i \(-0.553726\pi\)
−0.167985 + 0.985790i \(0.553726\pi\)
\(104\) 19162.9 0.173732
\(105\) −50254.2 −0.444835
\(106\) 227090. 1.96306
\(107\) 44407.3 0.374969 0.187484 0.982268i \(-0.439967\pi\)
0.187484 + 0.982268i \(0.439967\pi\)
\(108\) −42604.2 −0.351474
\(109\) −142245. −1.14676 −0.573379 0.819290i \(-0.694368\pi\)
−0.573379 + 0.819290i \(0.694368\pi\)
\(110\) −54555.8 −0.429891
\(111\) −288715. −2.22414
\(112\) 48159.6 0.362776
\(113\) 29934.7 0.220536 0.110268 0.993902i \(-0.464829\pi\)
0.110268 + 0.993902i \(0.464829\pi\)
\(114\) 418977. 3.01945
\(115\) 6658.03 0.0469463
\(116\) 12416.5 0.0856750
\(117\) 58605.2 0.395796
\(118\) −134088. −0.886510
\(119\) 81468.0 0.527375
\(120\) 157244. 0.996828
\(121\) −139470. −0.866001
\(122\) −164292. −0.999350
\(123\) −233847. −1.39370
\(124\) 128066. 0.747965
\(125\) 179255. 1.02611
\(126\) 92857.1 0.521061
\(127\) 219282. 1.20641 0.603204 0.797587i \(-0.293890\pi\)
0.603204 + 0.797587i \(0.293890\pi\)
\(128\) −206376. −1.11336
\(129\) 364424. 1.92811
\(130\) 60377.8 0.313342
\(131\) 207170. 1.05475 0.527374 0.849633i \(-0.323177\pi\)
0.527374 + 0.849633i \(0.323177\pi\)
\(132\) 53279.8 0.266151
\(133\) −93950.5 −0.460543
\(134\) −377352. −1.81545
\(135\) 156712. 0.740062
\(136\) −254911. −1.18179
\(137\) 11815.8 0.0537852 0.0268926 0.999638i \(-0.491439\pi\)
0.0268926 + 0.999638i \(0.491439\pi\)
\(138\) −20595.7 −0.0920615
\(139\) −51342.5 −0.225393 −0.112696 0.993629i \(-0.535949\pi\)
−0.112696 + 0.993629i \(0.535949\pi\)
\(140\) 30202.8 0.130235
\(141\) 94021.2 0.398270
\(142\) 284661. 1.18470
\(143\) −23883.7 −0.0976699
\(144\) −460849. −1.85205
\(145\) −45671.9 −0.180397
\(146\) 107415. 0.417045
\(147\) 378015. 1.44283
\(148\) 173518. 0.651163
\(149\) 159538. 0.588707 0.294353 0.955697i \(-0.404896\pi\)
0.294353 + 0.955697i \(0.404896\pi\)
\(150\) −29530.6 −0.107163
\(151\) 64145.0 0.228939 0.114470 0.993427i \(-0.463483\pi\)
0.114470 + 0.993427i \(0.463483\pi\)
\(152\) 293968. 1.03203
\(153\) −779584. −2.69237
\(154\) −37842.5 −0.128581
\(155\) −471070. −1.57491
\(156\) −58965.7 −0.193994
\(157\) 556965. 1.80335 0.901673 0.432419i \(-0.142340\pi\)
0.901673 + 0.432419i \(0.142340\pi\)
\(158\) −388236. −1.23724
\(159\) 815772. 2.55903
\(160\) −269957. −0.833671
\(161\) 4618.32 0.0140417
\(162\) 114237. 0.341996
\(163\) −175236. −0.516599 −0.258299 0.966065i \(-0.583162\pi\)
−0.258299 + 0.966065i \(0.583162\pi\)
\(164\) 140542. 0.408035
\(165\) −195980. −0.560406
\(166\) −734487. −2.06878
\(167\) 464707. 1.28940 0.644700 0.764436i \(-0.276982\pi\)
0.644700 + 0.764436i \(0.276982\pi\)
\(168\) 109072. 0.298153
\(169\) −344861. −0.928810
\(170\) −803164. −2.13148
\(171\) 899031. 2.35117
\(172\) −219019. −0.564497
\(173\) −92110.9 −0.233989 −0.116995 0.993133i \(-0.537326\pi\)
−0.116995 + 0.993133i \(0.537326\pi\)
\(174\) 141279. 0.353758
\(175\) 6621.88 0.0163450
\(176\) 187812. 0.457027
\(177\) −481682. −1.15565
\(178\) −257230. −0.608516
\(179\) 46322.2 0.108058 0.0540290 0.998539i \(-0.482794\pi\)
0.0540290 + 0.998539i \(0.482794\pi\)
\(180\) −289017. −0.664878
\(181\) 118183. 0.268139 0.134069 0.990972i \(-0.457196\pi\)
0.134069 + 0.990972i \(0.457196\pi\)
\(182\) 41880.9 0.0937212
\(183\) −590186. −1.30275
\(184\) −14450.6 −0.0314660
\(185\) −638255. −1.37109
\(186\) 1.45719e6 3.08840
\(187\) 317707. 0.664390
\(188\) −56506.9 −0.116602
\(189\) 108703. 0.221354
\(190\) 926224. 1.86137
\(191\) −349094. −0.692404 −0.346202 0.938160i \(-0.612529\pi\)
−0.346202 + 0.938160i \(0.612529\pi\)
\(192\) −169932. −0.332677
\(193\) −128726. −0.248756 −0.124378 0.992235i \(-0.539694\pi\)
−0.124378 + 0.992235i \(0.539694\pi\)
\(194\) 300510. 0.573264
\(195\) 216895. 0.408472
\(196\) −227188. −0.422420
\(197\) 615954. 1.13079 0.565396 0.824820i \(-0.308724\pi\)
0.565396 + 0.824820i \(0.308724\pi\)
\(198\) 362122. 0.656436
\(199\) −333500. −0.596985 −0.298492 0.954412i \(-0.596484\pi\)
−0.298492 + 0.954412i \(0.596484\pi\)
\(200\) −20719.7 −0.0366275
\(201\) −1.35556e6 −2.36662
\(202\) −486660. −0.839165
\(203\) −31680.2 −0.0539570
\(204\) 784379. 1.31962
\(205\) −516960. −0.859156
\(206\) 247371. 0.406145
\(207\) −44193.7 −0.0716860
\(208\) −207855. −0.333121
\(209\) −366386. −0.580195
\(210\) 343659. 0.537749
\(211\) 246416. 0.381033 0.190516 0.981684i \(-0.438984\pi\)
0.190516 + 0.981684i \(0.438984\pi\)
\(212\) −490280. −0.749212
\(213\) 1.02259e6 1.54437
\(214\) −303676. −0.453290
\(215\) 805624. 1.18860
\(216\) −340128. −0.496030
\(217\) −326756. −0.471058
\(218\) 972733. 1.38628
\(219\) 385866. 0.543659
\(220\) 117784. 0.164071
\(221\) −351613. −0.484266
\(222\) 1.97435e6 2.68870
\(223\) −275147. −0.370512 −0.185256 0.982690i \(-0.559311\pi\)
−0.185256 + 0.982690i \(0.559311\pi\)
\(224\) −187255. −0.249352
\(225\) −63366.1 −0.0834451
\(226\) −204706. −0.266600
\(227\) −461856. −0.594897 −0.297448 0.954738i \(-0.596136\pi\)
−0.297448 + 0.954738i \(0.596136\pi\)
\(228\) −904561. −1.15239
\(229\) −311105. −0.392029 −0.196015 0.980601i \(-0.562800\pi\)
−0.196015 + 0.980601i \(0.562800\pi\)
\(230\) −45530.4 −0.0567521
\(231\) −135941. −0.167618
\(232\) 99126.3 0.120912
\(233\) 1.52969e6 1.84592 0.922962 0.384891i \(-0.125761\pi\)
0.922962 + 0.384891i \(0.125761\pi\)
\(234\) −400767. −0.478467
\(235\) 207851. 0.245517
\(236\) 289492. 0.338342
\(237\) −1.39466e6 −1.61286
\(238\) −557113. −0.637530
\(239\) 1.21977e6 1.38129 0.690644 0.723195i \(-0.257327\pi\)
0.690644 + 0.723195i \(0.257327\pi\)
\(240\) −1.70558e6 −1.91136
\(241\) 563284. 0.624719 0.312360 0.949964i \(-0.398881\pi\)
0.312360 + 0.949964i \(0.398881\pi\)
\(242\) 953757. 1.04689
\(243\) 1.11160e6 1.20762
\(244\) 354703. 0.381408
\(245\) 835670. 0.889447
\(246\) 1.59914e6 1.68480
\(247\) 405486. 0.422896
\(248\) 1.02241e6 1.05559
\(249\) −2.63849e6 −2.69685
\(250\) −1.22582e6 −1.24044
\(251\) −1.52639e6 −1.52926 −0.764629 0.644470i \(-0.777078\pi\)
−0.764629 + 0.644470i \(0.777078\pi\)
\(252\) −200476. −0.198866
\(253\) 18010.4 0.0176898
\(254\) −1.49954e6 −1.45839
\(255\) −2.88520e6 −2.77860
\(256\) 1.18993e6 1.13480
\(257\) 197606. 0.186624 0.0933121 0.995637i \(-0.470255\pi\)
0.0933121 + 0.995637i \(0.470255\pi\)
\(258\) −2.49208e6 −2.33084
\(259\) −442724. −0.410094
\(260\) −130354. −0.119589
\(261\) 303154. 0.275462
\(262\) −1.41671e6 −1.27506
\(263\) −2.03558e6 −1.81468 −0.907339 0.420400i \(-0.861890\pi\)
−0.907339 + 0.420400i \(0.861890\pi\)
\(264\) 425356. 0.375615
\(265\) 1.80341e6 1.57754
\(266\) 642473. 0.556738
\(267\) −924046. −0.793260
\(268\) 814693. 0.692878
\(269\) −728283. −0.613648 −0.306824 0.951766i \(-0.599266\pi\)
−0.306824 + 0.951766i \(0.599266\pi\)
\(270\) −1.07166e6 −0.894640
\(271\) −878997. −0.727049 −0.363525 0.931585i \(-0.618427\pi\)
−0.363525 + 0.931585i \(0.618427\pi\)
\(272\) 2.76495e6 2.26602
\(273\) 150449. 0.122175
\(274\) −80801.6 −0.0650195
\(275\) 25823.9 0.0205916
\(276\) 44465.5 0.0351358
\(277\) −981143. −0.768304 −0.384152 0.923270i \(-0.625506\pi\)
−0.384152 + 0.923270i \(0.625506\pi\)
\(278\) 351102. 0.272471
\(279\) 3.12680e6 2.40486
\(280\) 241123. 0.183799
\(281\) 1.45423e6 1.09867 0.549333 0.835603i \(-0.314882\pi\)
0.549333 + 0.835603i \(0.314882\pi\)
\(282\) −642956. −0.481458
\(283\) 1.81952e6 1.35049 0.675243 0.737595i \(-0.264039\pi\)
0.675243 + 0.737595i \(0.264039\pi\)
\(284\) −614576. −0.452147
\(285\) 3.32727e6 2.42647
\(286\) 163326. 0.118070
\(287\) −358588. −0.256975
\(288\) 1.79188e6 1.27300
\(289\) 3.25740e6 2.29417
\(290\) 312323. 0.218077
\(291\) 1.07952e6 0.747306
\(292\) −231906. −0.159168
\(293\) 279641. 0.190297 0.0951484 0.995463i \(-0.469667\pi\)
0.0951484 + 0.995463i \(0.469667\pi\)
\(294\) −2.58503e6 −1.74420
\(295\) −1.06484e6 −0.712412
\(296\) 1.38527e6 0.918978
\(297\) 423917. 0.278862
\(298\) −1.09099e6 −0.711672
\(299\) −19932.5 −0.0128939
\(300\) 63755.8 0.0408994
\(301\) 558819. 0.355513
\(302\) −438650. −0.276759
\(303\) −1.74823e6 −1.09393
\(304\) −3.18859e6 −1.97886
\(305\) −1.30471e6 −0.803091
\(306\) 5.33112e6 3.25473
\(307\) −1.65629e6 −1.00297 −0.501487 0.865165i \(-0.667214\pi\)
−0.501487 + 0.865165i \(0.667214\pi\)
\(308\) 81700.9 0.0490738
\(309\) 888630. 0.529450
\(310\) 3.22137e6 1.90387
\(311\) −1.33517e6 −0.782771 −0.391386 0.920227i \(-0.628004\pi\)
−0.391386 + 0.920227i \(0.628004\pi\)
\(312\) −470749. −0.273781
\(313\) −2.25773e6 −1.30260 −0.651300 0.758821i \(-0.725776\pi\)
−0.651300 + 0.758821i \(0.725776\pi\)
\(314\) −3.80876e6 −2.18002
\(315\) 737416. 0.418732
\(316\) 838191. 0.472199
\(317\) 2.76702e6 1.54655 0.773274 0.634072i \(-0.218618\pi\)
0.773274 + 0.634072i \(0.218618\pi\)
\(318\) −5.57859e6 −3.09355
\(319\) −123546. −0.0679753
\(320\) −375666. −0.205082
\(321\) −1.09089e6 −0.590907
\(322\) −31582.0 −0.0169746
\(323\) −5.39390e6 −2.87671
\(324\) −246636. −0.130525
\(325\) −28579.7 −0.0150089
\(326\) 1.19833e6 0.624503
\(327\) 3.49434e6 1.80716
\(328\) 1.12201e6 0.575854
\(329\) 144175. 0.0734346
\(330\) 1.34019e6 0.677459
\(331\) 1.50762e6 0.756350 0.378175 0.925734i \(-0.376552\pi\)
0.378175 + 0.925734i \(0.376552\pi\)
\(332\) 1.58574e6 0.789562
\(333\) 4.23652e6 2.09362
\(334\) −3.17786e6 −1.55872
\(335\) −2.99670e6 −1.45892
\(336\) −1.18307e6 −0.571692
\(337\) 2.65301e6 1.27252 0.636259 0.771476i \(-0.280481\pi\)
0.636259 + 0.771476i \(0.280481\pi\)
\(338\) 2.35830e6 1.12281
\(339\) −735364. −0.347539
\(340\) 1.73401e6 0.813493
\(341\) −1.27428e6 −0.593442
\(342\) −6.14795e6 −2.84227
\(343\) 1.21277e6 0.556602
\(344\) −1.74853e6 −0.796666
\(345\) −163558. −0.0739818
\(346\) 629893. 0.282863
\(347\) −1.30221e6 −0.580574 −0.290287 0.956940i \(-0.593751\pi\)
−0.290287 + 0.956940i \(0.593751\pi\)
\(348\) −305019. −0.135014
\(349\) −1.20351e6 −0.528914 −0.264457 0.964397i \(-0.585193\pi\)
−0.264457 + 0.964397i \(0.585193\pi\)
\(350\) −45283.2 −0.0197591
\(351\) −469157. −0.203259
\(352\) −730253. −0.314135
\(353\) −1.97691e6 −0.844404 −0.422202 0.906502i \(-0.638743\pi\)
−0.422202 + 0.906502i \(0.638743\pi\)
\(354\) 3.29394e6 1.39704
\(355\) 2.26061e6 0.952039
\(356\) 555353. 0.232244
\(357\) −2.00131e6 −0.831082
\(358\) −316771. −0.130628
\(359\) −2.29106e6 −0.938209 −0.469105 0.883143i \(-0.655423\pi\)
−0.469105 + 0.883143i \(0.655423\pi\)
\(360\) −2.30735e6 −0.938334
\(361\) 3.74425e6 1.51216
\(362\) −808187. −0.324146
\(363\) 3.42617e6 1.36472
\(364\) −90419.8 −0.0357693
\(365\) 853027. 0.335143
\(366\) 4.03594e6 1.57486
\(367\) 755682. 0.292869 0.146435 0.989220i \(-0.453220\pi\)
0.146435 + 0.989220i \(0.453220\pi\)
\(368\) 156741. 0.0603343
\(369\) 3.43140e6 1.31191
\(370\) 4.36466e6 1.65747
\(371\) 1.25093e6 0.471844
\(372\) −3.14603e6 −1.17871
\(373\) −4.45947e6 −1.65963 −0.829815 0.558038i \(-0.811554\pi\)
−0.829815 + 0.558038i \(0.811554\pi\)
\(374\) −2.17262e6 −0.803164
\(375\) −4.40350e6 −1.61704
\(376\) −451119. −0.164559
\(377\) 136730. 0.0495463
\(378\) −743356. −0.267588
\(379\) 4.02987e6 1.44109 0.720547 0.693406i \(-0.243891\pi\)
0.720547 + 0.693406i \(0.243891\pi\)
\(380\) −1.99969e6 −0.710402
\(381\) −5.38680e6 −1.90116
\(382\) 2.38725e6 0.837028
\(383\) 4.12660e6 1.43746 0.718729 0.695291i \(-0.244724\pi\)
0.718729 + 0.695291i \(0.244724\pi\)
\(384\) 5.06975e6 1.75452
\(385\) −300522. −0.103330
\(386\) 880284. 0.300715
\(387\) −5.34745e6 −1.81497
\(388\) −648793. −0.218790
\(389\) −1.37522e6 −0.460784 −0.230392 0.973098i \(-0.574001\pi\)
−0.230392 + 0.973098i \(0.574001\pi\)
\(390\) −1.48322e6 −0.493791
\(391\) 265148. 0.0877094
\(392\) −1.81374e6 −0.596156
\(393\) −5.08925e6 −1.66216
\(394\) −4.21215e6 −1.36698
\(395\) −3.08314e6 −0.994261
\(396\) −781812. −0.250533
\(397\) 3.95058e6 1.25801 0.629006 0.777400i \(-0.283462\pi\)
0.629006 + 0.777400i \(0.283462\pi\)
\(398\) 2.28061e6 0.721679
\(399\) 2.30795e6 0.725762
\(400\) 224740. 0.0702313
\(401\) 601236. 0.186717 0.0933585 0.995633i \(-0.470240\pi\)
0.0933585 + 0.995633i \(0.470240\pi\)
\(402\) 9.26988e6 2.86094
\(403\) 1.41027e6 0.432552
\(404\) 1.05069e6 0.320272
\(405\) 907205. 0.274832
\(406\) 216642. 0.0652272
\(407\) −1.72653e6 −0.516639
\(408\) 6.26204e6 1.86237
\(409\) 6.38666e6 1.88784 0.943921 0.330171i \(-0.107106\pi\)
0.943921 + 0.330171i \(0.107106\pi\)
\(410\) 3.53519e6 1.03861
\(411\) −290263. −0.0847593
\(412\) −534068. −0.155008
\(413\) −738627. −0.213084
\(414\) 302215. 0.0866592
\(415\) −5.83286e6 −1.66250
\(416\) 808185. 0.228969
\(417\) 1.26126e6 0.355193
\(418\) 2.50550e6 0.701381
\(419\) 4.44705e6 1.23748 0.618738 0.785597i \(-0.287644\pi\)
0.618738 + 0.785597i \(0.287644\pi\)
\(420\) −741951. −0.205235
\(421\) 413525. 0.113710 0.0568548 0.998382i \(-0.481893\pi\)
0.0568548 + 0.998382i \(0.481893\pi\)
\(422\) −1.68509e6 −0.460620
\(423\) −1.37964e6 −0.374900
\(424\) −3.91412e6 −1.05735
\(425\) 380176. 0.102097
\(426\) −6.99288e6 −1.86695
\(427\) −905010. −0.240206
\(428\) 655628. 0.173001
\(429\) 586716. 0.153916
\(430\) −5.50920e6 −1.43687
\(431\) −3.32051e6 −0.861016 −0.430508 0.902587i \(-0.641666\pi\)
−0.430508 + 0.902587i \(0.641666\pi\)
\(432\) 3.68927e6 0.951111
\(433\) 1.14444e6 0.293341 0.146670 0.989185i \(-0.453144\pi\)
0.146670 + 0.989185i \(0.453144\pi\)
\(434\) 2.23450e6 0.569450
\(435\) 1.12196e6 0.284285
\(436\) −2.10011e6 −0.529084
\(437\) −305774. −0.0765943
\(438\) −2.63872e6 −0.657215
\(439\) 900181. 0.222930 0.111465 0.993768i \(-0.464446\pi\)
0.111465 + 0.993768i \(0.464446\pi\)
\(440\) 940325. 0.231551
\(441\) −5.54689e6 −1.35817
\(442\) 2.40447e6 0.585416
\(443\) 3.62823e6 0.878386 0.439193 0.898393i \(-0.355264\pi\)
0.439193 + 0.898393i \(0.355264\pi\)
\(444\) −4.26257e6 −1.02616
\(445\) −2.04277e6 −0.489012
\(446\) 1.88157e6 0.447902
\(447\) −3.91915e6 −0.927733
\(448\) −260580. −0.0613403
\(449\) −2.10851e6 −0.493583 −0.246791 0.969069i \(-0.579376\pi\)
−0.246791 + 0.969069i \(0.579376\pi\)
\(450\) 433324. 0.100874
\(451\) −1.39841e6 −0.323739
\(452\) 441955. 0.101749
\(453\) −1.57576e6 −0.360782
\(454\) 3.15836e6 0.719155
\(455\) 332593. 0.0753157
\(456\) −7.22151e6 −1.62636
\(457\) 157338. 0.0352407 0.0176203 0.999845i \(-0.494391\pi\)
0.0176203 + 0.999845i \(0.494391\pi\)
\(458\) 2.12747e6 0.473914
\(459\) 6.24086e6 1.38265
\(460\) 98298.9 0.0216598
\(461\) −79527.0 −0.0174286 −0.00871429 0.999962i \(-0.502774\pi\)
−0.00871429 + 0.999962i \(0.502774\pi\)
\(462\) 929623. 0.202629
\(463\) 421922. 0.0914703 0.0457352 0.998954i \(-0.485437\pi\)
0.0457352 + 0.998954i \(0.485437\pi\)
\(464\) −1.07520e6 −0.231842
\(465\) 1.15721e7 2.48188
\(466\) −1.04607e7 −2.23149
\(467\) 3.56062e6 0.755498 0.377749 0.925908i \(-0.376698\pi\)
0.377749 + 0.925908i \(0.376698\pi\)
\(468\) 865246. 0.182610
\(469\) −2.07866e6 −0.436366
\(470\) −1.42137e6 −0.296799
\(471\) −1.36822e7 −2.84186
\(472\) 2.31114e6 0.477498
\(473\) 2.17927e6 0.447877
\(474\) 9.53725e6 1.94974
\(475\) −438427. −0.0891585
\(476\) 1.20279e6 0.243317
\(477\) −1.19704e7 −2.40887
\(478\) −8.34132e6 −1.66980
\(479\) 3.58247e6 0.713417 0.356708 0.934216i \(-0.383899\pi\)
0.356708 + 0.934216i \(0.383899\pi\)
\(480\) 6.63166e6 1.31377
\(481\) 1.91078e6 0.376572
\(482\) −3.85197e6 −0.755206
\(483\) −113452. −0.0221281
\(484\) −2.05914e6 −0.399550
\(485\) 2.38647e6 0.460683
\(486\) −7.60156e6 −1.45986
\(487\) 8.65020e6 1.65274 0.826368 0.563130i \(-0.190403\pi\)
0.826368 + 0.563130i \(0.190403\pi\)
\(488\) 2.83175e6 0.538276
\(489\) 4.30477e6 0.814100
\(490\) −5.71466e6 −1.07523
\(491\) 146402. 0.0274059 0.0137030 0.999906i \(-0.495638\pi\)
0.0137030 + 0.999906i \(0.495638\pi\)
\(492\) −3.45250e6 −0.643016
\(493\) −1.81883e6 −0.337034
\(494\) −2.77289e6 −0.511228
\(495\) 2.87576e6 0.527521
\(496\) −1.10898e7 −2.02404
\(497\) 1.56807e6 0.284756
\(498\) 1.80431e7 3.26015
\(499\) −4.24620e6 −0.763394 −0.381697 0.924287i \(-0.624660\pi\)
−0.381697 + 0.924287i \(0.624660\pi\)
\(500\) 2.64651e6 0.473422
\(501\) −1.14158e7 −2.03194
\(502\) 1.04381e7 1.84868
\(503\) 2.20106e6 0.387892 0.193946 0.981012i \(-0.437871\pi\)
0.193946 + 0.981012i \(0.437871\pi\)
\(504\) −1.60049e6 −0.280657
\(505\) −3.86476e6 −0.674364
\(506\) −123163. −0.0213847
\(507\) 8.47171e6 1.46370
\(508\) 3.23748e6 0.556605
\(509\) −1.03089e6 −0.176367 −0.0881834 0.996104i \(-0.528106\pi\)
−0.0881834 + 0.996104i \(0.528106\pi\)
\(510\) 1.97302e7 3.35897
\(511\) 591700. 0.100242
\(512\) −1.53318e6 −0.258474
\(513\) −7.19709e6 −1.20743
\(514\) −1.35131e6 −0.225605
\(515\) 1.96448e6 0.326384
\(516\) 5.38034e6 0.889581
\(517\) 562251. 0.0925133
\(518\) 3.02753e6 0.495752
\(519\) 2.26276e6 0.368740
\(520\) −1.04067e6 −0.168774
\(521\) 1.09737e7 1.77117 0.885583 0.464480i \(-0.153759\pi\)
0.885583 + 0.464480i \(0.153759\pi\)
\(522\) −2.07309e6 −0.332999
\(523\) −1.91358e6 −0.305909 −0.152954 0.988233i \(-0.548879\pi\)
−0.152954 + 0.988233i \(0.548879\pi\)
\(524\) 3.05865e6 0.486633
\(525\) −162670. −0.0257579
\(526\) 1.39202e7 2.19372
\(527\) −1.87598e7 −2.94240
\(528\) −4.61371e6 −0.720221
\(529\) −6.42131e6 −0.997665
\(530\) −1.23325e7 −1.90704
\(531\) 7.06807e6 1.08784
\(532\) −1.38708e6 −0.212482
\(533\) 1.54765e6 0.235969
\(534\) 6.31901e6 0.958950
\(535\) −2.41161e6 −0.364270
\(536\) 6.50405e6 0.977849
\(537\) −1.13793e6 −0.170287
\(538\) 4.98030e6 0.741822
\(539\) 2.26055e6 0.335152
\(540\) 2.31369e6 0.341445
\(541\) −3.63173e6 −0.533483 −0.266741 0.963768i \(-0.585947\pi\)
−0.266741 + 0.963768i \(0.585947\pi\)
\(542\) 6.01095e6 0.878910
\(543\) −2.90324e6 −0.422556
\(544\) −1.07507e7 −1.55754
\(545\) 7.72487e6 1.11404
\(546\) −1.02883e6 −0.147694
\(547\) −7.98387e6 −1.14089 −0.570447 0.821335i \(-0.693230\pi\)
−0.570447 + 0.821335i \(0.693230\pi\)
\(548\) 174449. 0.0248151
\(549\) 8.66022e6 1.22630
\(550\) −176594. −0.0248926
\(551\) 2.09751e6 0.294323
\(552\) 354987. 0.0495867
\(553\) −2.13861e6 −0.297385
\(554\) 6.70947e6 0.928782
\(555\) 1.56791e7 2.16067
\(556\) −758019. −0.103990
\(557\) −4.40511e6 −0.601614 −0.300807 0.953685i \(-0.597256\pi\)
−0.300807 + 0.953685i \(0.597256\pi\)
\(558\) −2.13823e7 −2.90717
\(559\) −2.41184e6 −0.326452
\(560\) −2.61539e6 −0.352424
\(561\) −7.80467e6 −1.04700
\(562\) −9.94460e6 −1.32815
\(563\) 8.26618e6 1.09909 0.549545 0.835464i \(-0.314801\pi\)
0.549545 + 0.835464i \(0.314801\pi\)
\(564\) 1.38813e6 0.183752
\(565\) −1.62565e6 −0.214243
\(566\) −1.24426e7 −1.63257
\(567\) 629281. 0.0822028
\(568\) −4.90643e6 −0.638109
\(569\) −1.12505e7 −1.45677 −0.728385 0.685168i \(-0.759729\pi\)
−0.728385 + 0.685168i \(0.759729\pi\)
\(570\) −2.27532e7 −2.93330
\(571\) 2.47253e6 0.317360 0.158680 0.987330i \(-0.449276\pi\)
0.158680 + 0.987330i \(0.449276\pi\)
\(572\) −352617. −0.0450623
\(573\) 8.57571e6 1.09115
\(574\) 2.45217e6 0.310650
\(575\) 21551.7 0.00271839
\(576\) 2.49354e6 0.313156
\(577\) 1.24000e7 1.55054 0.775270 0.631630i \(-0.217614\pi\)
0.775270 + 0.631630i \(0.217614\pi\)
\(578\) −2.22754e7 −2.77336
\(579\) 3.16224e6 0.392011
\(580\) −674298. −0.0832304
\(581\) −4.04595e6 −0.497256
\(582\) −7.38221e6 −0.903398
\(583\) 4.87836e6 0.594432
\(584\) −1.85141e6 −0.224631
\(585\) −3.18265e6 −0.384503
\(586\) −1.91230e6 −0.230045
\(587\) 5.73588e6 0.687076 0.343538 0.939139i \(-0.388375\pi\)
0.343538 + 0.939139i \(0.388375\pi\)
\(588\) 5.58100e6 0.665685
\(589\) 2.16341e7 2.56952
\(590\) 7.28185e6 0.861215
\(591\) −1.51313e7 −1.78200
\(592\) −1.50256e7 −1.76209
\(593\) −8.23346e6 −0.961492 −0.480746 0.876860i \(-0.659634\pi\)
−0.480746 + 0.876860i \(0.659634\pi\)
\(594\) −2.89892e6 −0.337109
\(595\) −4.42426e6 −0.512328
\(596\) 2.35542e6 0.271614
\(597\) 8.19263e6 0.940779
\(598\) 136307. 0.0155871
\(599\) −2.06935e6 −0.235650 −0.117825 0.993034i \(-0.537592\pi\)
−0.117825 + 0.993034i \(0.537592\pi\)
\(600\) 508991. 0.0577207
\(601\) 2.18287e6 0.246514 0.123257 0.992375i \(-0.460666\pi\)
0.123257 + 0.992375i \(0.460666\pi\)
\(602\) −3.82144e6 −0.429770
\(603\) 1.98911e7 2.22774
\(604\) 947034. 0.105627
\(605\) 7.57417e6 0.841292
\(606\) 1.19551e7 1.32243
\(607\) −7.87116e6 −0.867095 −0.433548 0.901131i \(-0.642738\pi\)
−0.433548 + 0.901131i \(0.642738\pi\)
\(608\) 1.23979e7 1.36016
\(609\) 778243. 0.0850300
\(610\) 8.92217e6 0.970836
\(611\) −622253. −0.0674317
\(612\) −1.15097e7 −1.24219
\(613\) 1.01857e7 1.09481 0.547406 0.836867i \(-0.315615\pi\)
0.547406 + 0.836867i \(0.315615\pi\)
\(614\) 1.13264e7 1.21247
\(615\) 1.26994e7 1.35393
\(616\) 652254. 0.0692572
\(617\) 1.28805e7 1.36213 0.681067 0.732221i \(-0.261516\pi\)
0.681067 + 0.732221i \(0.261516\pi\)
\(618\) −6.07682e6 −0.640038
\(619\) 2.22093e6 0.232974 0.116487 0.993192i \(-0.462837\pi\)
0.116487 + 0.993192i \(0.462837\pi\)
\(620\) −6.95486e6 −0.726623
\(621\) 353787. 0.0368140
\(622\) 9.13043e6 0.946271
\(623\) −1.41696e6 −0.146264
\(624\) 5.10608e6 0.524960
\(625\) −9.18539e6 −0.940584
\(626\) 1.54393e7 1.57468
\(627\) 9.00050e6 0.914319
\(628\) 8.22301e6 0.832017
\(629\) −2.54177e7 −2.56159
\(630\) −5.04276e6 −0.506194
\(631\) −1.12269e7 −1.12250 −0.561252 0.827645i \(-0.689680\pi\)
−0.561252 + 0.827645i \(0.689680\pi\)
\(632\) 6.69165e6 0.666408
\(633\) −6.05335e6 −0.600464
\(634\) −1.89220e7 −1.86958
\(635\) −1.19085e7 −1.17199
\(636\) 1.20440e7 1.18067
\(637\) −2.50179e6 −0.244288
\(638\) 844858. 0.0821735
\(639\) −1.50051e7 −1.45374
\(640\) 1.12076e7 1.08159
\(641\) 1.73945e6 0.167212 0.0836060 0.996499i \(-0.473356\pi\)
0.0836060 + 0.996499i \(0.473356\pi\)
\(642\) 7.45998e6 0.714332
\(643\) 6.08165e6 0.580088 0.290044 0.957013i \(-0.406330\pi\)
0.290044 + 0.957013i \(0.406330\pi\)
\(644\) 68184.8 0.00647847
\(645\) −1.97906e7 −1.87310
\(646\) 3.68857e7 3.47758
\(647\) −2.01823e6 −0.189544 −0.0947721 0.995499i \(-0.530212\pi\)
−0.0947721 + 0.995499i \(0.530212\pi\)
\(648\) −1.96900e6 −0.184208
\(649\) −2.88048e6 −0.268444
\(650\) 195440. 0.0181439
\(651\) 8.02697e6 0.742333
\(652\) −2.58717e6 −0.238345
\(653\) 5.21279e6 0.478396 0.239198 0.970971i \(-0.423116\pi\)
0.239198 + 0.970971i \(0.423116\pi\)
\(654\) −2.38958e7 −2.18462
\(655\) −1.12507e7 −1.02465
\(656\) −1.21701e7 −1.10417
\(657\) −5.66209e6 −0.511757
\(658\) −985930. −0.0887731
\(659\) 7.29296e6 0.654170 0.327085 0.944995i \(-0.393934\pi\)
0.327085 + 0.944995i \(0.393934\pi\)
\(660\) −2.89344e6 −0.258556
\(661\) −1.08067e7 −0.962034 −0.481017 0.876711i \(-0.659732\pi\)
−0.481017 + 0.876711i \(0.659732\pi\)
\(662\) −1.03098e7 −0.914331
\(663\) 8.63757e6 0.763146
\(664\) 1.26596e7 1.11430
\(665\) 5.10214e6 0.447402
\(666\) −2.89711e7 −2.53092
\(667\) −103107. −0.00897375
\(668\) 6.86091e6 0.594895
\(669\) 6.75915e6 0.583884
\(670\) 2.04927e7 1.76365
\(671\) −3.52934e6 −0.302613
\(672\) 4.60003e6 0.392950
\(673\) 3.23697e6 0.275487 0.137743 0.990468i \(-0.456015\pi\)
0.137743 + 0.990468i \(0.456015\pi\)
\(674\) −1.81424e7 −1.53831
\(675\) 507269. 0.0428528
\(676\) −5.09151e6 −0.428529
\(677\) −6.00931e6 −0.503910 −0.251955 0.967739i \(-0.581073\pi\)
−0.251955 + 0.967739i \(0.581073\pi\)
\(678\) 5.02873e6 0.420130
\(679\) 1.65537e6 0.137791
\(680\) 1.38434e7 1.14807
\(681\) 1.13458e7 0.937488
\(682\) 8.71405e6 0.717396
\(683\) −7.39622e6 −0.606678 −0.303339 0.952883i \(-0.598101\pi\)
−0.303339 + 0.952883i \(0.598101\pi\)
\(684\) 1.32733e7 1.08477
\(685\) −641678. −0.0522506
\(686\) −8.29346e6 −0.672861
\(687\) 7.64249e6 0.617793
\(688\) 1.89658e7 1.52757
\(689\) −5.39896e6 −0.433274
\(690\) 1.11848e6 0.0894347
\(691\) −1.10125e7 −0.877383 −0.438692 0.898638i \(-0.644558\pi\)
−0.438692 + 0.898638i \(0.644558\pi\)
\(692\) −1.35992e6 −0.107956
\(693\) 1.99476e6 0.157782
\(694\) 8.90506e6 0.701840
\(695\) 2.78824e6 0.218962
\(696\) −2.43510e6 −0.190543
\(697\) −2.05873e7 −1.60516
\(698\) 8.23008e6 0.639390
\(699\) −3.75778e7 −2.90896
\(700\) 97765.2 0.00754118
\(701\) 1.97899e7 1.52106 0.760532 0.649300i \(-0.224938\pi\)
0.760532 + 0.649300i \(0.224938\pi\)
\(702\) 3.20829e6 0.245715
\(703\) 2.93122e7 2.23697
\(704\) −1.01620e6 −0.0772768
\(705\) −5.10598e6 −0.386906
\(706\) 1.35189e7 1.02078
\(707\) −2.68078e6 −0.201703
\(708\) −7.11154e6 −0.533188
\(709\) −1.20373e7 −0.899317 −0.449658 0.893201i \(-0.648454\pi\)
−0.449658 + 0.893201i \(0.648454\pi\)
\(710\) −1.54590e7 −1.15089
\(711\) 2.04648e7 1.51822
\(712\) 4.43363e6 0.327762
\(713\) −1.06347e6 −0.0783431
\(714\) 1.36858e7 1.00467
\(715\) 1.29704e6 0.0948831
\(716\) 683900. 0.0498551
\(717\) −2.99645e7 −2.17675
\(718\) 1.56672e7 1.13418
\(719\) −1.01004e7 −0.728643 −0.364322 0.931273i \(-0.618699\pi\)
−0.364322 + 0.931273i \(0.618699\pi\)
\(720\) 2.50272e7 1.79920
\(721\) 1.36265e6 0.0976219
\(722\) −2.56048e7 −1.82801
\(723\) −1.38374e7 −0.984485
\(724\) 1.74485e6 0.123712
\(725\) −147838. −0.0104458
\(726\) −2.34296e7 −1.64977
\(727\) 2.01441e7 1.41356 0.706778 0.707436i \(-0.250148\pi\)
0.706778 + 0.707436i \(0.250148\pi\)
\(728\) −721861. −0.0504807
\(729\) −2.32477e7 −1.62017
\(730\) −5.83335e6 −0.405146
\(731\) 3.20830e7 2.22066
\(732\) −8.71349e6 −0.601055
\(733\) −2.42377e7 −1.66622 −0.833110 0.553108i \(-0.813442\pi\)
−0.833110 + 0.553108i \(0.813442\pi\)
\(734\) −5.16767e6 −0.354042
\(735\) −2.05287e7 −1.40166
\(736\) −609445. −0.0414705
\(737\) −8.10630e6 −0.549736
\(738\) −2.34654e7 −1.58594
\(739\) −6.78569e6 −0.457070 −0.228535 0.973536i \(-0.573394\pi\)
−0.228535 + 0.973536i \(0.573394\pi\)
\(740\) −9.42317e6 −0.632584
\(741\) −9.96102e6 −0.666436
\(742\) −8.55439e6 −0.570400
\(743\) 4.71605e6 0.313406 0.156703 0.987646i \(-0.449914\pi\)
0.156703 + 0.987646i \(0.449914\pi\)
\(744\) −2.51161e7 −1.66349
\(745\) −8.66398e6 −0.571909
\(746\) 3.04957e7 2.00628
\(747\) 3.87165e7 2.53860
\(748\) 4.69062e6 0.306532
\(749\) −1.67281e6 −0.108954
\(750\) 3.01130e7 1.95479
\(751\) −1.69859e7 −1.09898 −0.549490 0.835501i \(-0.685178\pi\)
−0.549490 + 0.835501i \(0.685178\pi\)
\(752\) 4.89316e6 0.315533
\(753\) 3.74967e7 2.40993
\(754\) −935019. −0.0598952
\(755\) −3.48350e6 −0.222407
\(756\) 1.60489e6 0.102127
\(757\) −2.07327e7 −1.31497 −0.657486 0.753467i \(-0.728380\pi\)
−0.657486 + 0.753467i \(0.728380\pi\)
\(758\) −2.75579e7 −1.74210
\(759\) −442437. −0.0278771
\(760\) −1.59644e7 −1.00258
\(761\) −52400.3 −0.00327999 −0.00163999 0.999999i \(-0.500522\pi\)
−0.00163999 + 0.999999i \(0.500522\pi\)
\(762\) 3.68372e7 2.29826
\(763\) 5.35834e6 0.333210
\(764\) −5.15402e6 −0.319457
\(765\) 4.23366e7 2.61555
\(766\) −2.82194e7 −1.73770
\(767\) 3.18788e6 0.195665
\(768\) −2.92312e7 −1.78832
\(769\) −2.09357e7 −1.27665 −0.638325 0.769767i \(-0.720373\pi\)
−0.638325 + 0.769767i \(0.720373\pi\)
\(770\) 2.05510e6 0.124912
\(771\) −4.85432e6 −0.294098
\(772\) −1.90051e6 −0.114770
\(773\) 2.78692e7 1.67755 0.838777 0.544476i \(-0.183271\pi\)
0.838777 + 0.544476i \(0.183271\pi\)
\(774\) 3.65681e7 2.19407
\(775\) −1.52483e6 −0.0911942
\(776\) −5.17960e6 −0.308775
\(777\) 1.08758e7 0.646261
\(778\) 9.40431e6 0.557029
\(779\) 2.37417e7 1.40174
\(780\) 3.20223e6 0.188459
\(781\) 6.11512e6 0.358738
\(782\) −1.81319e6 −0.106030
\(783\) −2.42686e6 −0.141462
\(784\) 1.96731e7 1.14310
\(785\) −3.02469e7 −1.75189
\(786\) 3.48024e7 2.00934
\(787\) 1.87393e7 1.07849 0.539245 0.842149i \(-0.318710\pi\)
0.539245 + 0.842149i \(0.318710\pi\)
\(788\) 9.09392e6 0.521717
\(789\) 5.00053e7 2.85972
\(790\) 2.10838e7 1.20194
\(791\) −1.12763e6 −0.0640805
\(792\) −6.24155e6 −0.353573
\(793\) 3.90598e6 0.220571
\(794\) −2.70157e7 −1.52078
\(795\) −4.43019e7 −2.48602
\(796\) −4.92378e6 −0.275433
\(797\) −2.79661e6 −0.155950 −0.0779751 0.996955i \(-0.524845\pi\)
−0.0779751 + 0.996955i \(0.524845\pi\)
\(798\) −1.57827e7 −0.877354
\(799\) 8.27740e6 0.458698
\(800\) −873838. −0.0482732
\(801\) 1.35592e7 0.746711
\(802\) −4.11150e6 −0.225717
\(803\) 2.30750e6 0.126285
\(804\) −2.00134e7 −1.09190
\(805\) −250806. −0.0136410
\(806\) −9.64399e6 −0.522901
\(807\) 1.78907e7 0.967038
\(808\) 8.38809e6 0.451996
\(809\) 2.31444e7 1.24330 0.621648 0.783296i \(-0.286463\pi\)
0.621648 + 0.783296i \(0.286463\pi\)
\(810\) −6.20385e6 −0.332238
\(811\) 1.44543e7 0.771694 0.385847 0.922563i \(-0.373909\pi\)
0.385847 + 0.922563i \(0.373909\pi\)
\(812\) −467725. −0.0248943
\(813\) 2.15931e7 1.14575
\(814\) 1.18067e7 0.624551
\(815\) 9.51646e6 0.501859
\(816\) −6.79226e7 −3.57099
\(817\) −3.69987e7 −1.93924
\(818\) −4.36747e7 −2.28216
\(819\) −2.20764e6 −0.115005
\(820\) −7.63238e6 −0.396392
\(821\) 1.80178e7 0.932920 0.466460 0.884542i \(-0.345529\pi\)
0.466460 + 0.884542i \(0.345529\pi\)
\(822\) 1.98494e6 0.102463
\(823\) 6.65519e6 0.342500 0.171250 0.985228i \(-0.445219\pi\)
0.171250 + 0.985228i \(0.445219\pi\)
\(824\) −4.26370e6 −0.218760
\(825\) −634379. −0.0324499
\(826\) 5.05104e6 0.257591
\(827\) 5.27683e6 0.268293 0.134146 0.990962i \(-0.457171\pi\)
0.134146 + 0.990962i \(0.457171\pi\)
\(828\) −652474. −0.0330740
\(829\) −724962. −0.0366378 −0.0183189 0.999832i \(-0.505831\pi\)
−0.0183189 + 0.999832i \(0.505831\pi\)
\(830\) 3.98875e7 2.00975
\(831\) 2.41024e7 1.21076
\(832\) 1.12465e6 0.0563261
\(833\) 3.32796e7 1.66175
\(834\) −8.62503e6 −0.429383
\(835\) −2.52367e7 −1.25261
\(836\) −5.40932e6 −0.267687
\(837\) −2.50312e7 −1.23500
\(838\) −3.04108e7 −1.49595
\(839\) 1.71120e7 0.839256 0.419628 0.907696i \(-0.362160\pi\)
0.419628 + 0.907696i \(0.362160\pi\)
\(840\) −5.92332e6 −0.289646
\(841\) 707281. 0.0344828
\(842\) −2.82786e6 −0.137460
\(843\) −3.57239e7 −1.73137
\(844\) 3.63808e6 0.175799
\(845\) 1.87282e7 0.902308
\(846\) 9.43456e6 0.453206
\(847\) 5.25380e6 0.251632
\(848\) 4.24554e7 2.02742
\(849\) −4.46975e7 −2.12821
\(850\) −2.59980e6 −0.123422
\(851\) −1.44090e6 −0.0682040
\(852\) 1.50974e7 0.712532
\(853\) −3.96032e6 −0.186362 −0.0931810 0.995649i \(-0.529704\pi\)
−0.0931810 + 0.995649i \(0.529704\pi\)
\(854\) 6.18884e6 0.290378
\(855\) −4.88234e7 −2.28409
\(856\) 5.23417e6 0.244154
\(857\) 1.78958e7 0.832338 0.416169 0.909287i \(-0.363373\pi\)
0.416169 + 0.909287i \(0.363373\pi\)
\(858\) −4.01221e6 −0.186065
\(859\) 3.52591e7 1.63038 0.815188 0.579197i \(-0.196634\pi\)
0.815188 + 0.579197i \(0.196634\pi\)
\(860\) 1.18942e7 0.548390
\(861\) 8.80893e6 0.404963
\(862\) 2.27070e7 1.04086
\(863\) −4.23888e7 −1.93742 −0.968711 0.248192i \(-0.920163\pi\)
−0.968711 + 0.248192i \(0.920163\pi\)
\(864\) −1.43447e7 −0.653742
\(865\) 5.00223e6 0.227313
\(866\) −7.82614e6 −0.354612
\(867\) −8.00199e7 −3.61535
\(868\) −4.82422e6 −0.217334
\(869\) −8.34011e6 −0.374647
\(870\) −7.67241e6 −0.343664
\(871\) 8.97139e6 0.400695
\(872\) −1.67661e7 −0.746689
\(873\) −1.58406e7 −0.703453
\(874\) 2.09101e6 0.0925928
\(875\) −6.75247e6 −0.298155
\(876\) 5.69692e6 0.250830
\(877\) 3.53937e6 0.155391 0.0776956 0.996977i \(-0.475244\pi\)
0.0776956 + 0.996977i \(0.475244\pi\)
\(878\) −6.15582e6 −0.269494
\(879\) −6.86954e6 −0.299886
\(880\) −1.01994e7 −0.443986
\(881\) −1.82655e7 −0.792851 −0.396426 0.918067i \(-0.629750\pi\)
−0.396426 + 0.918067i \(0.629750\pi\)
\(882\) 3.79320e7 1.64185
\(883\) −2.56061e7 −1.10520 −0.552602 0.833446i \(-0.686365\pi\)
−0.552602 + 0.833446i \(0.686365\pi\)
\(884\) −5.19119e6 −0.223428
\(885\) 2.61585e7 1.12268
\(886\) −2.48114e7 −1.06186
\(887\) −4.46206e6 −0.190426 −0.0952130 0.995457i \(-0.530353\pi\)
−0.0952130 + 0.995457i \(0.530353\pi\)
\(888\) −3.40300e7 −1.44820
\(889\) −8.26030e6 −0.350543
\(890\) 1.39693e7 0.591153
\(891\) 2.45406e6 0.103560
\(892\) −4.06226e6 −0.170945
\(893\) −9.54566e6 −0.400569
\(894\) 2.68008e7 1.12151
\(895\) −2.51561e6 −0.104975
\(896\) 7.77412e6 0.323505
\(897\) 489653. 0.0203193
\(898\) 1.44189e7 0.596679
\(899\) 7.29505e6 0.301043
\(900\) −935535. −0.0384994
\(901\) 7.18186e7 2.94730
\(902\) 9.56294e6 0.391359
\(903\) −1.37277e7 −0.560247
\(904\) 3.52832e6 0.143598
\(905\) −6.41814e6 −0.260488
\(906\) 1.07757e7 0.436139
\(907\) 4.62490e7 1.86674 0.933371 0.358914i \(-0.116853\pi\)
0.933371 + 0.358914i \(0.116853\pi\)
\(908\) −6.81882e6 −0.274470
\(909\) 2.56530e7 1.02974
\(910\) −2.27441e6 −0.0910471
\(911\) −3.42662e7 −1.36795 −0.683974 0.729507i \(-0.739750\pi\)
−0.683974 + 0.729507i \(0.739750\pi\)
\(912\) 7.83296e7 3.11845
\(913\) −1.57783e7 −0.626446
\(914\) −1.07595e6 −0.0426015
\(915\) 3.20510e7 1.26558
\(916\) −4.59315e6 −0.180872
\(917\) −7.80402e6 −0.306475
\(918\) −4.26776e7 −1.67145
\(919\) 1.31785e7 0.514728 0.257364 0.966315i \(-0.417146\pi\)
0.257364 + 0.966315i \(0.417146\pi\)
\(920\) 784763. 0.0305681
\(921\) 4.06877e7 1.58057
\(922\) 543839. 0.0210690
\(923\) −6.76771e6 −0.261479
\(924\) −2.00703e6 −0.0773347
\(925\) −2.06600e6 −0.0793919
\(926\) −2.88528e6 −0.110576
\(927\) −1.30395e7 −0.498381
\(928\) 4.18059e6 0.159356
\(929\) −1.62539e7 −0.617899 −0.308949 0.951078i \(-0.599977\pi\)
−0.308949 + 0.951078i \(0.599977\pi\)
\(930\) −7.91349e7 −3.00027
\(931\) −3.83786e7 −1.45116
\(932\) 2.25843e7 0.851661
\(933\) 3.27992e7 1.23356
\(934\) −2.43490e7 −0.913301
\(935\) −1.72536e7 −0.645433
\(936\) 6.90764e6 0.257715
\(937\) −2.53494e7 −0.943233 −0.471617 0.881804i \(-0.656329\pi\)
−0.471617 + 0.881804i \(0.656329\pi\)
\(938\) 1.42147e7 0.527511
\(939\) 5.54625e7 2.05275
\(940\) 3.06870e6 0.113275
\(941\) 9.02296e6 0.332181 0.166091 0.986110i \(-0.446886\pi\)
0.166091 + 0.986110i \(0.446886\pi\)
\(942\) 9.35645e7 3.43545
\(943\) −1.16707e6 −0.0427383
\(944\) −2.50683e7 −0.915576
\(945\) −5.90329e6 −0.215038
\(946\) −1.49028e7 −0.541426
\(947\) −338514. −0.0122660 −0.00613298 0.999981i \(-0.501952\pi\)
−0.00613298 + 0.999981i \(0.501952\pi\)
\(948\) −2.05907e7 −0.744131
\(949\) −2.55375e6 −0.0920477
\(950\) 2.99814e6 0.107781
\(951\) −6.79734e7 −2.43718
\(952\) 9.60241e6 0.343390
\(953\) −1.25549e7 −0.447797 −0.223899 0.974612i \(-0.571878\pi\)
−0.223899 + 0.974612i \(0.571878\pi\)
\(954\) 8.18587e7 2.91202
\(955\) 1.89581e7 0.672647
\(956\) 1.80087e7 0.637291
\(957\) 3.03498e6 0.107121
\(958\) −2.44984e7 −0.862430
\(959\) −445099. −0.0156282
\(960\) 9.22846e6 0.323185
\(961\) 4.66136e7 1.62819
\(962\) −1.30667e7 −0.455227
\(963\) 1.60074e7 0.556232
\(964\) 8.31630e6 0.288229
\(965\) 6.99069e6 0.241658
\(966\) 775832. 0.0267501
\(967\) −1.51562e6 −0.0521224 −0.0260612 0.999660i \(-0.508296\pi\)
−0.0260612 + 0.999660i \(0.508296\pi\)
\(968\) −1.64390e7 −0.563880
\(969\) 1.32504e8 4.53337
\(970\) −1.63197e7 −0.556907
\(971\) 3.97841e7 1.35413 0.677067 0.735921i \(-0.263251\pi\)
0.677067 + 0.735921i \(0.263251\pi\)
\(972\) 1.64116e7 0.557166
\(973\) 1.93406e6 0.0654918
\(974\) −5.91537e7 −1.99795
\(975\) 702079. 0.0236523
\(976\) −3.07152e7 −1.03212
\(977\) −1.08895e7 −0.364982 −0.182491 0.983208i \(-0.558416\pi\)
−0.182491 + 0.983208i \(0.558416\pi\)
\(978\) −2.94378e7 −0.984143
\(979\) −5.52584e6 −0.184264
\(980\) 1.23378e7 0.410367
\(981\) −5.12750e7 −1.70111
\(982\) −1.00116e6 −0.0331303
\(983\) 1.51586e7 0.500353 0.250177 0.968200i \(-0.419511\pi\)
0.250177 + 0.968200i \(0.419511\pi\)
\(984\) −2.75629e7 −0.907479
\(985\) −3.34504e7 −1.09853
\(986\) 1.24379e7 0.407432
\(987\) −3.54175e6 −0.115724
\(988\) 5.98659e6 0.195113
\(989\) 1.81875e6 0.0591264
\(990\) −1.96656e7 −0.637705
\(991\) −4.43710e7 −1.43521 −0.717605 0.696451i \(-0.754762\pi\)
−0.717605 + 0.696451i \(0.754762\pi\)
\(992\) 4.31195e7 1.39122
\(993\) −3.70357e7 −1.19192
\(994\) −1.07231e7 −0.344234
\(995\) 1.81113e7 0.579951
\(996\) −3.89546e7 −1.24426
\(997\) −4.48040e7 −1.42751 −0.713754 0.700397i \(-0.753006\pi\)
−0.713754 + 0.700397i \(0.753006\pi\)
\(998\) 2.90373e7 0.922847
\(999\) −3.39149e7 −1.07517
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 29.6.a.b.1.2 7
3.2 odd 2 261.6.a.e.1.6 7
4.3 odd 2 464.6.a.k.1.7 7
5.4 even 2 725.6.a.b.1.6 7
29.28 even 2 841.6.a.b.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.6.a.b.1.2 7 1.1 even 1 trivial
261.6.a.e.1.6 7 3.2 odd 2
464.6.a.k.1.7 7 4.3 odd 2
725.6.a.b.1.6 7 5.4 even 2
841.6.a.b.1.6 7 29.28 even 2