Properties

Label 825.6.a.v.1.2
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-9.04603\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-10.0460 q^{2} -9.00000 q^{3} +68.9227 q^{4} +90.4143 q^{6} +29.0524 q^{7} -370.927 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-10.0460 q^{2} -9.00000 q^{3} +68.9227 q^{4} +90.4143 q^{6} +29.0524 q^{7} -370.927 q^{8} +81.0000 q^{9} +121.000 q^{11} -620.305 q^{12} -1023.51 q^{13} -291.862 q^{14} +1520.82 q^{16} +1509.69 q^{17} -813.728 q^{18} +1643.04 q^{19} -261.472 q^{21} -1215.57 q^{22} -1478.11 q^{23} +3338.34 q^{24} +10282.2 q^{26} -729.000 q^{27} +2002.37 q^{28} -4572.33 q^{29} +7531.62 q^{31} -3408.50 q^{32} -1089.00 q^{33} -15166.4 q^{34} +5582.74 q^{36} -4408.40 q^{37} -16506.0 q^{38} +9211.62 q^{39} +5629.62 q^{41} +2626.75 q^{42} +2283.11 q^{43} +8339.65 q^{44} +14849.1 q^{46} -5980.48 q^{47} -13687.3 q^{48} -15963.0 q^{49} -13587.2 q^{51} -70543.3 q^{52} -28498.0 q^{53} +7323.56 q^{54} -10776.3 q^{56} -14787.3 q^{57} +45933.7 q^{58} +25942.8 q^{59} -51360.7 q^{61} -75662.8 q^{62} +2353.25 q^{63} -14424.2 q^{64} +10940.1 q^{66} +39180.2 q^{67} +104052. q^{68} +13303.0 q^{69} +37560.3 q^{71} -30045.1 q^{72} -58687.6 q^{73} +44286.9 q^{74} +113242. q^{76} +3515.34 q^{77} -92540.2 q^{78} -8274.06 q^{79} +6561.00 q^{81} -56555.3 q^{82} +21355.4 q^{83} -18021.4 q^{84} -22936.2 q^{86} +41150.9 q^{87} -44882.1 q^{88} +78385.8 q^{89} -29735.5 q^{91} -101875. q^{92} -67784.5 q^{93} +60080.1 q^{94} +30676.5 q^{96} -64574.7 q^{97} +160364. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.0460 −1.77590 −0.887952 0.459936i \(-0.847872\pi\)
−0.887952 + 0.459936i \(0.847872\pi\)
\(3\) −9.00000 −0.577350
\(4\) 68.9227 2.15384
\(5\) 0 0
\(6\) 90.4143 1.02532
\(7\) 29.0524 0.224098 0.112049 0.993703i \(-0.464259\pi\)
0.112049 + 0.993703i \(0.464259\pi\)
\(8\) −370.927 −2.04910
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −620.305 −1.24352
\(13\) −1023.51 −1.67971 −0.839856 0.542809i \(-0.817361\pi\)
−0.839856 + 0.542809i \(0.817361\pi\)
\(14\) −291.862 −0.397976
\(15\) 0 0
\(16\) 1520.82 1.48517
\(17\) 1509.69 1.26696 0.633482 0.773757i \(-0.281625\pi\)
0.633482 + 0.773757i \(0.281625\pi\)
\(18\) −813.728 −0.591968
\(19\) 1643.04 1.04415 0.522075 0.852900i \(-0.325158\pi\)
0.522075 + 0.852900i \(0.325158\pi\)
\(20\) 0 0
\(21\) −261.472 −0.129383
\(22\) −1215.57 −0.535455
\(23\) −1478.11 −0.582622 −0.291311 0.956628i \(-0.594092\pi\)
−0.291311 + 0.956628i \(0.594092\pi\)
\(24\) 3338.34 1.18305
\(25\) 0 0
\(26\) 10282.2 2.98301
\(27\) −729.000 −0.192450
\(28\) 2002.37 0.482669
\(29\) −4572.33 −1.00958 −0.504792 0.863241i \(-0.668431\pi\)
−0.504792 + 0.863241i \(0.668431\pi\)
\(30\) 0 0
\(31\) 7531.62 1.40762 0.703808 0.710391i \(-0.251482\pi\)
0.703808 + 0.710391i \(0.251482\pi\)
\(32\) −3408.50 −0.588421
\(33\) −1089.00 −0.174078
\(34\) −15166.4 −2.25001
\(35\) 0 0
\(36\) 5582.74 0.717945
\(37\) −4408.40 −0.529391 −0.264695 0.964332i \(-0.585271\pi\)
−0.264695 + 0.964332i \(0.585271\pi\)
\(38\) −16506.0 −1.85431
\(39\) 9211.62 0.969783
\(40\) 0 0
\(41\) 5629.62 0.523021 0.261511 0.965201i \(-0.415779\pi\)
0.261511 + 0.965201i \(0.415779\pi\)
\(42\) 2626.75 0.229771
\(43\) 2283.11 0.188302 0.0941510 0.995558i \(-0.469986\pi\)
0.0941510 + 0.995558i \(0.469986\pi\)
\(44\) 8339.65 0.649406
\(45\) 0 0
\(46\) 14849.1 1.03468
\(47\) −5980.48 −0.394904 −0.197452 0.980313i \(-0.563267\pi\)
−0.197452 + 0.980313i \(0.563267\pi\)
\(48\) −13687.3 −0.857464
\(49\) −15963.0 −0.949780
\(50\) 0 0
\(51\) −13587.2 −0.731482
\(52\) −70543.3 −3.61782
\(53\) −28498.0 −1.39356 −0.696778 0.717287i \(-0.745384\pi\)
−0.696778 + 0.717287i \(0.745384\pi\)
\(54\) 7323.56 0.341773
\(55\) 0 0
\(56\) −10776.3 −0.459199
\(57\) −14787.3 −0.602840
\(58\) 45933.7 1.79292
\(59\) 25942.8 0.970258 0.485129 0.874443i \(-0.338773\pi\)
0.485129 + 0.874443i \(0.338773\pi\)
\(60\) 0 0
\(61\) −51360.7 −1.76728 −0.883642 0.468163i \(-0.844916\pi\)
−0.883642 + 0.468163i \(0.844916\pi\)
\(62\) −75662.8 −2.49979
\(63\) 2353.25 0.0746992
\(64\) −14424.2 −0.440193
\(65\) 0 0
\(66\) 10940.1 0.309145
\(67\) 39180.2 1.06630 0.533151 0.846020i \(-0.321008\pi\)
0.533151 + 0.846020i \(0.321008\pi\)
\(68\) 104052. 2.72883
\(69\) 13303.0 0.336377
\(70\) 0 0
\(71\) 37560.3 0.884267 0.442133 0.896949i \(-0.354222\pi\)
0.442133 + 0.896949i \(0.354222\pi\)
\(72\) −30045.1 −0.683034
\(73\) −58687.6 −1.28896 −0.644480 0.764621i \(-0.722926\pi\)
−0.644480 + 0.764621i \(0.722926\pi\)
\(74\) 44286.9 0.940147
\(75\) 0 0
\(76\) 113242. 2.24893
\(77\) 3515.34 0.0675680
\(78\) −92540.2 −1.72224
\(79\) −8274.06 −0.149160 −0.0745798 0.997215i \(-0.523762\pi\)
−0.0745798 + 0.997215i \(0.523762\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −56555.3 −0.928836
\(83\) 21355.4 0.340261 0.170130 0.985422i \(-0.445581\pi\)
0.170130 + 0.985422i \(0.445581\pi\)
\(84\) −18021.4 −0.278669
\(85\) 0 0
\(86\) −22936.2 −0.334406
\(87\) 41150.9 0.582883
\(88\) −44882.1 −0.617827
\(89\) 78385.8 1.04897 0.524484 0.851420i \(-0.324258\pi\)
0.524484 + 0.851420i \(0.324258\pi\)
\(90\) 0 0
\(91\) −29735.5 −0.376420
\(92\) −101875. −1.25487
\(93\) −67784.5 −0.812687
\(94\) 60080.1 0.701312
\(95\) 0 0
\(96\) 30676.5 0.339725
\(97\) −64574.7 −0.696840 −0.348420 0.937339i \(-0.613282\pi\)
−0.348420 + 0.937339i \(0.613282\pi\)
\(98\) 160364. 1.68672
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 48364.4 0.471762 0.235881 0.971782i \(-0.424203\pi\)
0.235881 + 0.971782i \(0.424203\pi\)
\(102\) 136497. 1.29904
\(103\) −121923. −1.13238 −0.566192 0.824273i \(-0.691584\pi\)
−0.566192 + 0.824273i \(0.691584\pi\)
\(104\) 379648. 3.44190
\(105\) 0 0
\(106\) 286291. 2.47482
\(107\) 191883. 1.62023 0.810117 0.586269i \(-0.199404\pi\)
0.810117 + 0.586269i \(0.199404\pi\)
\(108\) −50244.7 −0.414506
\(109\) −15660.0 −0.126248 −0.0631240 0.998006i \(-0.520106\pi\)
−0.0631240 + 0.998006i \(0.520106\pi\)
\(110\) 0 0
\(111\) 39675.6 0.305644
\(112\) 44183.4 0.332823
\(113\) 15119.6 0.111389 0.0556947 0.998448i \(-0.482263\pi\)
0.0556947 + 0.998448i \(0.482263\pi\)
\(114\) 148554. 1.07059
\(115\) 0 0
\(116\) −315137. −2.17448
\(117\) −82904.6 −0.559904
\(118\) −260622. −1.72309
\(119\) 43860.0 0.283924
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 515971. 3.13853
\(123\) −50666.6 −0.301966
\(124\) 519099. 3.03177
\(125\) 0 0
\(126\) −23640.8 −0.132659
\(127\) 223751. 1.23099 0.615496 0.788140i \(-0.288956\pi\)
0.615496 + 0.788140i \(0.288956\pi\)
\(128\) 253978. 1.37016
\(129\) −20548.0 −0.108716
\(130\) 0 0
\(131\) 269255. 1.37084 0.685418 0.728149i \(-0.259619\pi\)
0.685418 + 0.728149i \(0.259619\pi\)
\(132\) −75056.9 −0.374935
\(133\) 47734.1 0.233991
\(134\) −393606. −1.89365
\(135\) 0 0
\(136\) −559983. −2.59614
\(137\) −172187. −0.783790 −0.391895 0.920010i \(-0.628180\pi\)
−0.391895 + 0.920010i \(0.628180\pi\)
\(138\) −133642. −0.597374
\(139\) −350209. −1.53741 −0.768707 0.639601i \(-0.779099\pi\)
−0.768707 + 0.639601i \(0.779099\pi\)
\(140\) 0 0
\(141\) 53824.4 0.227998
\(142\) −377332. −1.57037
\(143\) −123845. −0.506452
\(144\) 123186. 0.495057
\(145\) 0 0
\(146\) 589578. 2.28907
\(147\) 143667. 0.548356
\(148\) −303839. −1.14022
\(149\) 519184. 1.91582 0.957912 0.287062i \(-0.0926785\pi\)
0.957912 + 0.287062i \(0.0926785\pi\)
\(150\) 0 0
\(151\) −230203. −0.821617 −0.410808 0.911722i \(-0.634753\pi\)
−0.410808 + 0.911722i \(0.634753\pi\)
\(152\) −609446. −2.13957
\(153\) 122285. 0.422321
\(154\) −35315.2 −0.119994
\(155\) 0 0
\(156\) 634890. 2.08875
\(157\) 328864. 1.06480 0.532399 0.846494i \(-0.321291\pi\)
0.532399 + 0.846494i \(0.321291\pi\)
\(158\) 83121.5 0.264893
\(159\) 256482. 0.804569
\(160\) 0 0
\(161\) −42942.7 −0.130564
\(162\) −65912.0 −0.197323
\(163\) 246382. 0.726341 0.363171 0.931723i \(-0.381694\pi\)
0.363171 + 0.931723i \(0.381694\pi\)
\(164\) 388009. 1.12650
\(165\) 0 0
\(166\) −214537. −0.604270
\(167\) 581413. 1.61322 0.806610 0.591084i \(-0.201300\pi\)
0.806610 + 0.591084i \(0.201300\pi\)
\(168\) 96986.9 0.265118
\(169\) 676286. 1.82143
\(170\) 0 0
\(171\) 133086. 0.348050
\(172\) 157358. 0.405572
\(173\) 264650. 0.672290 0.336145 0.941810i \(-0.390877\pi\)
0.336145 + 0.941810i \(0.390877\pi\)
\(174\) −413404. −1.03515
\(175\) 0 0
\(176\) 184019. 0.447796
\(177\) −233485. −0.560179
\(178\) −787466. −1.86287
\(179\) −13886.5 −0.0323937 −0.0161969 0.999869i \(-0.505156\pi\)
−0.0161969 + 0.999869i \(0.505156\pi\)
\(180\) 0 0
\(181\) −752593. −1.70751 −0.853756 0.520673i \(-0.825681\pi\)
−0.853756 + 0.520673i \(0.825681\pi\)
\(182\) 298724. 0.668485
\(183\) 462246. 1.02034
\(184\) 548271. 1.19385
\(185\) 0 0
\(186\) 680966. 1.44325
\(187\) 182672. 0.382004
\(188\) −412191. −0.850558
\(189\) −21179.2 −0.0431276
\(190\) 0 0
\(191\) −25416.1 −0.0504110 −0.0252055 0.999682i \(-0.508024\pi\)
−0.0252055 + 0.999682i \(0.508024\pi\)
\(192\) 129818. 0.254145
\(193\) 489822. 0.946553 0.473277 0.880914i \(-0.343071\pi\)
0.473277 + 0.880914i \(0.343071\pi\)
\(194\) 648719. 1.23752
\(195\) 0 0
\(196\) −1.10021e6 −2.04567
\(197\) 324827. 0.596329 0.298164 0.954515i \(-0.403626\pi\)
0.298164 + 0.954515i \(0.403626\pi\)
\(198\) −98461.1 −0.178485
\(199\) −398516. −0.713368 −0.356684 0.934225i \(-0.616093\pi\)
−0.356684 + 0.934225i \(0.616093\pi\)
\(200\) 0 0
\(201\) −352622. −0.615629
\(202\) −485870. −0.837803
\(203\) −132837. −0.226245
\(204\) −936465. −1.57549
\(205\) 0 0
\(206\) 1.22485e6 2.01101
\(207\) −119727. −0.194207
\(208\) −1.55657e6 −2.49466
\(209\) 198807. 0.314823
\(210\) 0 0
\(211\) −868645. −1.34319 −0.671593 0.740920i \(-0.734390\pi\)
−0.671593 + 0.740920i \(0.734390\pi\)
\(212\) −1.96416e6 −3.00149
\(213\) −338043. −0.510532
\(214\) −1.92766e6 −2.87738
\(215\) 0 0
\(216\) 270406. 0.394350
\(217\) 218812. 0.315443
\(218\) 157320. 0.224204
\(219\) 528189. 0.744181
\(220\) 0 0
\(221\) −1.54518e6 −2.12814
\(222\) −398582. −0.542794
\(223\) 48571.9 0.0654068 0.0327034 0.999465i \(-0.489588\pi\)
0.0327034 + 0.999465i \(0.489588\pi\)
\(224\) −99025.1 −0.131864
\(225\) 0 0
\(226\) −151892. −0.197817
\(227\) −78669.6 −0.101331 −0.0506655 0.998716i \(-0.516134\pi\)
−0.0506655 + 0.998716i \(0.516134\pi\)
\(228\) −1.01918e6 −1.29842
\(229\) −324090. −0.408391 −0.204196 0.978930i \(-0.565458\pi\)
−0.204196 + 0.978930i \(0.565458\pi\)
\(230\) 0 0
\(231\) −31638.1 −0.0390104
\(232\) 1.69600e6 2.06874
\(233\) −1.26828e6 −1.53048 −0.765238 0.643747i \(-0.777379\pi\)
−0.765238 + 0.643747i \(0.777379\pi\)
\(234\) 832862. 0.994336
\(235\) 0 0
\(236\) 1.78805e6 2.08978
\(237\) 74466.6 0.0861173
\(238\) −440619. −0.504221
\(239\) −1.35381e6 −1.53307 −0.766535 0.642202i \(-0.778021\pi\)
−0.766535 + 0.642202i \(0.778021\pi\)
\(240\) 0 0
\(241\) 1.14198e6 1.26653 0.633264 0.773936i \(-0.281715\pi\)
0.633264 + 0.773936i \(0.281715\pi\)
\(242\) −147084. −0.161446
\(243\) −59049.0 −0.0641500
\(244\) −3.53992e6 −3.80644
\(245\) 0 0
\(246\) 508998. 0.536264
\(247\) −1.68167e6 −1.75387
\(248\) −2.79368e6 −2.88434
\(249\) −192198. −0.196449
\(250\) 0 0
\(251\) −321578. −0.322183 −0.161091 0.986940i \(-0.551501\pi\)
−0.161091 + 0.986940i \(0.551501\pi\)
\(252\) 162192. 0.160890
\(253\) −178851. −0.175667
\(254\) −2.24781e6 −2.18612
\(255\) 0 0
\(256\) −2.08990e6 −1.99308
\(257\) −1.44275e6 −1.36257 −0.681286 0.732018i \(-0.738579\pi\)
−0.681286 + 0.732018i \(0.738579\pi\)
\(258\) 206425. 0.193070
\(259\) −128075. −0.118635
\(260\) 0 0
\(261\) −370359. −0.336528
\(262\) −2.70495e6 −2.43447
\(263\) −838618. −0.747610 −0.373805 0.927507i \(-0.621947\pi\)
−0.373805 + 0.927507i \(0.621947\pi\)
\(264\) 403939. 0.356703
\(265\) 0 0
\(266\) −479539. −0.415546
\(267\) −705472. −0.605622
\(268\) 2.70041e6 2.29664
\(269\) −415262. −0.349898 −0.174949 0.984577i \(-0.555976\pi\)
−0.174949 + 0.984577i \(0.555976\pi\)
\(270\) 0 0
\(271\) 647209. 0.535330 0.267665 0.963512i \(-0.413748\pi\)
0.267665 + 0.963512i \(0.413748\pi\)
\(272\) 2.29595e6 1.88166
\(273\) 267620. 0.217326
\(274\) 1.72980e6 1.39193
\(275\) 0 0
\(276\) 916878. 0.724501
\(277\) −679011. −0.531713 −0.265856 0.964013i \(-0.585655\pi\)
−0.265856 + 0.964013i \(0.585655\pi\)
\(278\) 3.51821e6 2.73030
\(279\) 610061. 0.469205
\(280\) 0 0
\(281\) −1.78685e6 −1.34997 −0.674983 0.737834i \(-0.735849\pi\)
−0.674983 + 0.737834i \(0.735849\pi\)
\(282\) −540721. −0.404903
\(283\) 872555. 0.647630 0.323815 0.946120i \(-0.395034\pi\)
0.323815 + 0.946120i \(0.395034\pi\)
\(284\) 2.58876e6 1.90457
\(285\) 0 0
\(286\) 1.24415e6 0.899411
\(287\) 163554. 0.117208
\(288\) −276088. −0.196140
\(289\) 859295. 0.605198
\(290\) 0 0
\(291\) 581172. 0.402321
\(292\) −4.04491e6 −2.77621
\(293\) −653130. −0.444458 −0.222229 0.974995i \(-0.571333\pi\)
−0.222229 + 0.974995i \(0.571333\pi\)
\(294\) −1.44328e6 −0.973827
\(295\) 0 0
\(296\) 1.63519e6 1.08477
\(297\) −88209.0 −0.0580259
\(298\) −5.21574e6 −3.40232
\(299\) 1.51286e6 0.978638
\(300\) 0 0
\(301\) 66329.8 0.0421980
\(302\) 2.31263e6 1.45911
\(303\) −435280. −0.272372
\(304\) 2.49875e6 1.55074
\(305\) 0 0
\(306\) −1.22847e6 −0.750002
\(307\) 1.87392e6 1.13476 0.567381 0.823456i \(-0.307957\pi\)
0.567381 + 0.823456i \(0.307957\pi\)
\(308\) 242287. 0.145530
\(309\) 1.09731e6 0.653783
\(310\) 0 0
\(311\) −1.60626e6 −0.941705 −0.470853 0.882212i \(-0.656054\pi\)
−0.470853 + 0.882212i \(0.656054\pi\)
\(312\) −3.41684e6 −1.98718
\(313\) 2.58202e6 1.48970 0.744849 0.667234i \(-0.232522\pi\)
0.744849 + 0.667234i \(0.232522\pi\)
\(314\) −3.30377e6 −1.89098
\(315\) 0 0
\(316\) −570271. −0.321265
\(317\) 1.34588e6 0.752242 0.376121 0.926571i \(-0.377258\pi\)
0.376121 + 0.926571i \(0.377258\pi\)
\(318\) −2.57662e6 −1.42884
\(319\) −553252. −0.304401
\(320\) 0 0
\(321\) −1.72695e6 −0.935442
\(322\) 431403. 0.231870
\(323\) 2.48047e6 1.32290
\(324\) 452202. 0.239315
\(325\) 0 0
\(326\) −2.47516e6 −1.28991
\(327\) 140940. 0.0728893
\(328\) −2.08818e6 −1.07172
\(329\) −173748. −0.0884971
\(330\) 0 0
\(331\) 2.74357e6 1.37641 0.688203 0.725518i \(-0.258400\pi\)
0.688203 + 0.725518i \(0.258400\pi\)
\(332\) 1.47187e6 0.732865
\(333\) −357080. −0.176464
\(334\) −5.84089e6 −2.86492
\(335\) 0 0
\(336\) −397650. −0.192156
\(337\) −3.70365e6 −1.77646 −0.888229 0.459400i \(-0.848064\pi\)
−0.888229 + 0.459400i \(0.848064\pi\)
\(338\) −6.79399e6 −3.23469
\(339\) −136076. −0.0643107
\(340\) 0 0
\(341\) 911325. 0.424412
\(342\) −1.33698e6 −0.618103
\(343\) −952047. −0.436941
\(344\) −846866. −0.385850
\(345\) 0 0
\(346\) −2.65868e6 −1.19392
\(347\) 1.17221e6 0.522616 0.261308 0.965255i \(-0.415846\pi\)
0.261308 + 0.965255i \(0.415846\pi\)
\(348\) 2.83624e6 1.25543
\(349\) −360082. −0.158248 −0.0791240 0.996865i \(-0.525212\pi\)
−0.0791240 + 0.996865i \(0.525212\pi\)
\(350\) 0 0
\(351\) 746141. 0.323261
\(352\) −412428. −0.177415
\(353\) −998723. −0.426588 −0.213294 0.976988i \(-0.568419\pi\)
−0.213294 + 0.976988i \(0.568419\pi\)
\(354\) 2.34560e6 0.994824
\(355\) 0 0
\(356\) 5.40256e6 2.25930
\(357\) −394740. −0.163923
\(358\) 139504. 0.0575282
\(359\) 1.47706e6 0.604871 0.302436 0.953170i \(-0.402200\pi\)
0.302436 + 0.953170i \(0.402200\pi\)
\(360\) 0 0
\(361\) 223465. 0.0902489
\(362\) 7.56057e6 3.03238
\(363\) −131769. −0.0524864
\(364\) −2.04945e6 −0.810746
\(365\) 0 0
\(366\) −4.64374e6 −1.81203
\(367\) −3.21905e6 −1.24756 −0.623781 0.781599i \(-0.714404\pi\)
−0.623781 + 0.781599i \(0.714404\pi\)
\(368\) −2.24793e6 −0.865294
\(369\) 455999. 0.174340
\(370\) 0 0
\(371\) −827935. −0.312292
\(372\) −4.67190e6 −1.75039
\(373\) 4.65790e6 1.73348 0.866739 0.498762i \(-0.166212\pi\)
0.866739 + 0.498762i \(0.166212\pi\)
\(374\) −1.83513e6 −0.678403
\(375\) 0 0
\(376\) 2.21832e6 0.809198
\(377\) 4.67984e6 1.69581
\(378\) 212767. 0.0765905
\(379\) −4.04644e6 −1.44702 −0.723510 0.690314i \(-0.757472\pi\)
−0.723510 + 0.690314i \(0.757472\pi\)
\(380\) 0 0
\(381\) −2.01376e6 −0.710714
\(382\) 255331. 0.0895252
\(383\) −824148. −0.287084 −0.143542 0.989644i \(-0.545849\pi\)
−0.143542 + 0.989644i \(0.545849\pi\)
\(384\) −2.28580e6 −0.791062
\(385\) 0 0
\(386\) −4.92077e6 −1.68099
\(387\) 184932. 0.0627674
\(388\) −4.45066e6 −1.50088
\(389\) −2.54462e6 −0.852607 −0.426303 0.904580i \(-0.640184\pi\)
−0.426303 + 0.904580i \(0.640184\pi\)
\(390\) 0 0
\(391\) −2.23148e6 −0.738162
\(392\) 5.92109e6 1.94620
\(393\) −2.42330e6 −0.791453
\(394\) −3.26322e6 −1.05902
\(395\) 0 0
\(396\) 675512. 0.216469
\(397\) 5.17776e6 1.64879 0.824395 0.566015i \(-0.191516\pi\)
0.824395 + 0.566015i \(0.191516\pi\)
\(398\) 4.00351e6 1.26687
\(399\) −429607. −0.135095
\(400\) 0 0
\(401\) −4.82239e6 −1.49762 −0.748809 0.662785i \(-0.769374\pi\)
−0.748809 + 0.662785i \(0.769374\pi\)
\(402\) 3.54245e6 1.09330
\(403\) −7.70871e6 −2.36439
\(404\) 3.33341e6 1.01610
\(405\) 0 0
\(406\) 1.33449e6 0.401790
\(407\) −533416. −0.159617
\(408\) 5.03985e6 1.49888
\(409\) −3.68626e6 −1.08963 −0.544813 0.838558i \(-0.683399\pi\)
−0.544813 + 0.838558i \(0.683399\pi\)
\(410\) 0 0
\(411\) 1.54968e6 0.452521
\(412\) −8.40329e6 −2.43897
\(413\) 753702. 0.217433
\(414\) 1.20278e6 0.344894
\(415\) 0 0
\(416\) 3.48864e6 0.988377
\(417\) 3.15188e6 0.887626
\(418\) −1.99722e6 −0.559095
\(419\) 1.31904e6 0.367049 0.183525 0.983015i \(-0.441249\pi\)
0.183525 + 0.983015i \(0.441249\pi\)
\(420\) 0 0
\(421\) −3.29518e6 −0.906095 −0.453048 0.891486i \(-0.649663\pi\)
−0.453048 + 0.891486i \(0.649663\pi\)
\(422\) 8.72643e6 2.38537
\(423\) −484419. −0.131635
\(424\) 1.05707e7 2.85553
\(425\) 0 0
\(426\) 3.39599e6 0.906655
\(427\) −1.49215e6 −0.396044
\(428\) 1.32251e7 3.48972
\(429\) 1.11461e6 0.292400
\(430\) 0 0
\(431\) 1.87137e6 0.485251 0.242625 0.970120i \(-0.421991\pi\)
0.242625 + 0.970120i \(0.421991\pi\)
\(432\) −1.10867e6 −0.285821
\(433\) −4.31194e6 −1.10523 −0.552615 0.833437i \(-0.686370\pi\)
−0.552615 + 0.833437i \(0.686370\pi\)
\(434\) −2.19819e6 −0.560197
\(435\) 0 0
\(436\) −1.07933e6 −0.271917
\(437\) −2.42859e6 −0.608345
\(438\) −5.30620e6 −1.32159
\(439\) 1.13612e6 0.281360 0.140680 0.990055i \(-0.455071\pi\)
0.140680 + 0.990055i \(0.455071\pi\)
\(440\) 0 0
\(441\) −1.29300e6 −0.316593
\(442\) 1.55230e7 3.77936
\(443\) 875875. 0.212047 0.106024 0.994364i \(-0.466188\pi\)
0.106024 + 0.994364i \(0.466188\pi\)
\(444\) 2.73455e6 0.658306
\(445\) 0 0
\(446\) −487955. −0.116156
\(447\) −4.67266e6 −1.10610
\(448\) −419059. −0.0986461
\(449\) 4.08806e6 0.956977 0.478489 0.878094i \(-0.341185\pi\)
0.478489 + 0.878094i \(0.341185\pi\)
\(450\) 0 0
\(451\) 681184. 0.157697
\(452\) 1.04208e6 0.239915
\(453\) 2.07183e6 0.474361
\(454\) 790317. 0.179954
\(455\) 0 0
\(456\) 5.48501e6 1.23528
\(457\) −2.85279e6 −0.638968 −0.319484 0.947592i \(-0.603510\pi\)
−0.319484 + 0.947592i \(0.603510\pi\)
\(458\) 3.25581e6 0.725263
\(459\) −1.10056e6 −0.243827
\(460\) 0 0
\(461\) 6.16387e6 1.35083 0.675416 0.737437i \(-0.263964\pi\)
0.675416 + 0.737437i \(0.263964\pi\)
\(462\) 317837. 0.0692787
\(463\) 4.52412e6 0.980804 0.490402 0.871496i \(-0.336850\pi\)
0.490402 + 0.871496i \(0.336850\pi\)
\(464\) −6.95366e6 −1.49940
\(465\) 0 0
\(466\) 1.27412e7 2.71798
\(467\) −2.23084e6 −0.473344 −0.236672 0.971590i \(-0.576057\pi\)
−0.236672 + 0.971590i \(0.576057\pi\)
\(468\) −5.71401e6 −1.20594
\(469\) 1.13828e6 0.238956
\(470\) 0 0
\(471\) −2.95977e6 −0.614761
\(472\) −9.62289e6 −1.98816
\(473\) 276256. 0.0567752
\(474\) −748093. −0.152936
\(475\) 0 0
\(476\) 3.02295e6 0.611525
\(477\) −2.30834e6 −0.464518
\(478\) 1.36004e7 2.72259
\(479\) −6.62215e6 −1.31874 −0.659372 0.751817i \(-0.729178\pi\)
−0.659372 + 0.751817i \(0.729178\pi\)
\(480\) 0 0
\(481\) 4.51205e6 0.889224
\(482\) −1.14723e7 −2.24923
\(483\) 386484. 0.0753813
\(484\) 1.00910e6 0.195803
\(485\) 0 0
\(486\) 593208. 0.113924
\(487\) −2.17583e6 −0.415721 −0.207860 0.978158i \(-0.566650\pi\)
−0.207860 + 0.978158i \(0.566650\pi\)
\(488\) 1.90511e7 3.62134
\(489\) −2.21744e6 −0.419353
\(490\) 0 0
\(491\) 7.67198e6 1.43616 0.718081 0.695959i \(-0.245021\pi\)
0.718081 + 0.695959i \(0.245021\pi\)
\(492\) −3.49208e6 −0.650386
\(493\) −6.90278e6 −1.27911
\(494\) 1.68941e7 3.11471
\(495\) 0 0
\(496\) 1.14542e7 2.09055
\(497\) 1.09122e6 0.198162
\(498\) 1.93083e6 0.348875
\(499\) 6.65009e6 1.19557 0.597787 0.801655i \(-0.296047\pi\)
0.597787 + 0.801655i \(0.296047\pi\)
\(500\) 0 0
\(501\) −5.23272e6 −0.931393
\(502\) 3.23058e6 0.572165
\(503\) −4.18197e6 −0.736989 −0.368495 0.929630i \(-0.620127\pi\)
−0.368495 + 0.929630i \(0.620127\pi\)
\(504\) −872882. −0.153066
\(505\) 0 0
\(506\) 1.79675e6 0.311968
\(507\) −6.08657e6 −1.05161
\(508\) 1.54215e7 2.65135
\(509\) −1.01322e7 −1.73345 −0.866725 0.498787i \(-0.833779\pi\)
−0.866725 + 0.498787i \(0.833779\pi\)
\(510\) 0 0
\(511\) −1.70502e6 −0.288853
\(512\) 1.28679e7 2.16936
\(513\) −1.19777e6 −0.200947
\(514\) 1.44939e7 2.41980
\(515\) 0 0
\(516\) −1.41622e6 −0.234157
\(517\) −723639. −0.119068
\(518\) 1.28664e6 0.210685
\(519\) −2.38185e6 −0.388147
\(520\) 0 0
\(521\) −6.42546e6 −1.03707 −0.518537 0.855055i \(-0.673523\pi\)
−0.518537 + 0.855055i \(0.673523\pi\)
\(522\) 3.72063e6 0.597641
\(523\) 1.17398e6 0.187674 0.0938372 0.995588i \(-0.470087\pi\)
0.0938372 + 0.995588i \(0.470087\pi\)
\(524\) 1.85578e7 2.95256
\(525\) 0 0
\(526\) 8.42478e6 1.32768
\(527\) 1.13704e7 1.78340
\(528\) −1.65617e6 −0.258535
\(529\) −4.25153e6 −0.660551
\(530\) 0 0
\(531\) 2.10137e6 0.323419
\(532\) 3.28997e6 0.503979
\(533\) −5.76199e6 −0.878525
\(534\) 7.08719e6 1.07553
\(535\) 0 0
\(536\) −1.45330e7 −2.18496
\(537\) 124979. 0.0187025
\(538\) 4.17174e6 0.621386
\(539\) −1.93152e6 −0.286370
\(540\) 0 0
\(541\) −5.25283e6 −0.771614 −0.385807 0.922580i \(-0.626077\pi\)
−0.385807 + 0.922580i \(0.626077\pi\)
\(542\) −6.50188e6 −0.950694
\(543\) 6.77334e6 0.985833
\(544\) −5.14576e6 −0.745508
\(545\) 0 0
\(546\) −2.68852e6 −0.385950
\(547\) −1.13849e7 −1.62689 −0.813447 0.581640i \(-0.802411\pi\)
−0.813447 + 0.581640i \(0.802411\pi\)
\(548\) −1.18676e7 −1.68815
\(549\) −4.16022e6 −0.589095
\(550\) 0 0
\(551\) −7.51249e6 −1.05416
\(552\) −4.93444e6 −0.689271
\(553\) −240382. −0.0334263
\(554\) 6.82136e6 0.944271
\(555\) 0 0
\(556\) −2.41374e7 −3.31134
\(557\) −4.54355e6 −0.620522 −0.310261 0.950651i \(-0.600416\pi\)
−0.310261 + 0.950651i \(0.600416\pi\)
\(558\) −6.12869e6 −0.833263
\(559\) −2.33679e6 −0.316293
\(560\) 0 0
\(561\) −1.64405e6 −0.220550
\(562\) 1.79508e7 2.39741
\(563\) 1.22077e7 1.62317 0.811583 0.584237i \(-0.198606\pi\)
0.811583 + 0.584237i \(0.198606\pi\)
\(564\) 3.70972e6 0.491070
\(565\) 0 0
\(566\) −8.76572e6 −1.15013
\(567\) 190613. 0.0248997
\(568\) −1.39321e7 −1.81195
\(569\) −1.58948e6 −0.205814 −0.102907 0.994691i \(-0.532814\pi\)
−0.102907 + 0.994691i \(0.532814\pi\)
\(570\) 0 0
\(571\) −692853. −0.0889306 −0.0444653 0.999011i \(-0.514158\pi\)
−0.0444653 + 0.999011i \(0.514158\pi\)
\(572\) −8.53574e6 −1.09082
\(573\) 228745. 0.0291048
\(574\) −1.64307e6 −0.208150
\(575\) 0 0
\(576\) −1.16836e6 −0.146731
\(577\) −1.24815e7 −1.56073 −0.780365 0.625324i \(-0.784967\pi\)
−0.780365 + 0.625324i \(0.784967\pi\)
\(578\) −8.63250e6 −1.07477
\(579\) −4.40840e6 −0.546493
\(580\) 0 0
\(581\) 620425. 0.0762516
\(582\) −5.83847e6 −0.714483
\(583\) −3.44825e6 −0.420173
\(584\) 2.17688e7 2.64121
\(585\) 0 0
\(586\) 6.56136e6 0.789314
\(587\) 9.13291e6 1.09399 0.546996 0.837135i \(-0.315771\pi\)
0.546996 + 0.837135i \(0.315771\pi\)
\(588\) 9.90189e6 1.18107
\(589\) 1.23747e7 1.46976
\(590\) 0 0
\(591\) −2.92344e6 −0.344291
\(592\) −6.70436e6 −0.786236
\(593\) −1.08272e7 −1.26438 −0.632191 0.774813i \(-0.717844\pi\)
−0.632191 + 0.774813i \(0.717844\pi\)
\(594\) 886150. 0.103048
\(595\) 0 0
\(596\) 3.57836e7 4.12637
\(597\) 3.58665e6 0.411863
\(598\) −1.51983e7 −1.73797
\(599\) 7.47502e6 0.851227 0.425614 0.904905i \(-0.360058\pi\)
0.425614 + 0.904905i \(0.360058\pi\)
\(600\) 0 0
\(601\) −1.02516e6 −0.115772 −0.0578861 0.998323i \(-0.518436\pi\)
−0.0578861 + 0.998323i \(0.518436\pi\)
\(602\) −666351. −0.0749397
\(603\) 3.17360e6 0.355434
\(604\) −1.58662e7 −1.76963
\(605\) 0 0
\(606\) 4.37283e6 0.483706
\(607\) −1.33264e7 −1.46805 −0.734024 0.679123i \(-0.762360\pi\)
−0.734024 + 0.679123i \(0.762360\pi\)
\(608\) −5.60028e6 −0.614399
\(609\) 1.19553e6 0.130623
\(610\) 0 0
\(611\) 6.12110e6 0.663325
\(612\) 8.42819e6 0.909611
\(613\) 1.57435e7 1.69220 0.846098 0.533027i \(-0.178946\pi\)
0.846098 + 0.533027i \(0.178946\pi\)
\(614\) −1.88254e7 −2.01523
\(615\) 0 0
\(616\) −1.30394e6 −0.138454
\(617\) 5.87960e6 0.621777 0.310888 0.950446i \(-0.399373\pi\)
0.310888 + 0.950446i \(0.399373\pi\)
\(618\) −1.10236e7 −1.16106
\(619\) −1.81138e6 −0.190013 −0.0950063 0.995477i \(-0.530287\pi\)
−0.0950063 + 0.995477i \(0.530287\pi\)
\(620\) 0 0
\(621\) 1.07754e6 0.112126
\(622\) 1.61365e7 1.67238
\(623\) 2.27730e6 0.235071
\(624\) 1.40092e7 1.44029
\(625\) 0 0
\(626\) −2.59390e7 −2.64556
\(627\) −1.78927e6 −0.181763
\(628\) 2.26662e7 2.29340
\(629\) −6.65529e6 −0.670719
\(630\) 0 0
\(631\) −1.43835e7 −1.43811 −0.719054 0.694954i \(-0.755425\pi\)
−0.719054 + 0.694954i \(0.755425\pi\)
\(632\) 3.06907e6 0.305643
\(633\) 7.81780e6 0.775489
\(634\) −1.35207e7 −1.33591
\(635\) 0 0
\(636\) 1.76774e7 1.73291
\(637\) 1.63383e7 1.59536
\(638\) 5.55798e6 0.540587
\(639\) 3.04239e6 0.294756
\(640\) 0 0
\(641\) −3.80867e6 −0.366124 −0.183062 0.983101i \(-0.558601\pi\)
−0.183062 + 0.983101i \(0.558601\pi\)
\(642\) 1.73490e7 1.66126
\(643\) −483920. −0.0461579 −0.0230789 0.999734i \(-0.507347\pi\)
−0.0230789 + 0.999734i \(0.507347\pi\)
\(644\) −2.95973e6 −0.281214
\(645\) 0 0
\(646\) −2.49188e7 −2.34934
\(647\) −1.80652e7 −1.69661 −0.848306 0.529506i \(-0.822377\pi\)
−0.848306 + 0.529506i \(0.822377\pi\)
\(648\) −2.43365e6 −0.227678
\(649\) 3.13908e6 0.292544
\(650\) 0 0
\(651\) −1.96931e6 −0.182121
\(652\) 1.69813e7 1.56442
\(653\) −2.03744e7 −1.86983 −0.934915 0.354871i \(-0.884525\pi\)
−0.934915 + 0.354871i \(0.884525\pi\)
\(654\) −1.41588e6 −0.129444
\(655\) 0 0
\(656\) 8.56161e6 0.776776
\(657\) −4.75370e6 −0.429653
\(658\) 1.74547e6 0.157162
\(659\) −1.37586e7 −1.23413 −0.617066 0.786912i \(-0.711679\pi\)
−0.617066 + 0.786912i \(0.711679\pi\)
\(660\) 0 0
\(661\) −1.34803e7 −1.20004 −0.600019 0.799985i \(-0.704840\pi\)
−0.600019 + 0.799985i \(0.704840\pi\)
\(662\) −2.75620e7 −2.44437
\(663\) 1.39066e7 1.22868
\(664\) −7.92127e6 −0.697228
\(665\) 0 0
\(666\) 3.58724e6 0.313382
\(667\) 6.75840e6 0.588206
\(668\) 4.00726e7 3.47461
\(669\) −437147. −0.0377627
\(670\) 0 0
\(671\) −6.21465e6 −0.532856
\(672\) 891226. 0.0761315
\(673\) 680925. 0.0579511 0.0289755 0.999580i \(-0.490776\pi\)
0.0289755 + 0.999580i \(0.490776\pi\)
\(674\) 3.72070e7 3.15482
\(675\) 0 0
\(676\) 4.66115e7 3.92307
\(677\) −1.36705e7 −1.14634 −0.573171 0.819436i \(-0.694287\pi\)
−0.573171 + 0.819436i \(0.694287\pi\)
\(678\) 1.36703e6 0.114210
\(679\) −1.87605e6 −0.156160
\(680\) 0 0
\(681\) 708026. 0.0585034
\(682\) −9.15520e6 −0.753715
\(683\) −6.79648e6 −0.557484 −0.278742 0.960366i \(-0.589917\pi\)
−0.278742 + 0.960366i \(0.589917\pi\)
\(684\) 9.17264e6 0.749642
\(685\) 0 0
\(686\) 9.56429e6 0.775966
\(687\) 2.91681e6 0.235785
\(688\) 3.47218e6 0.279661
\(689\) 2.91680e7 2.34077
\(690\) 0 0
\(691\) 9.11491e6 0.726202 0.363101 0.931750i \(-0.381718\pi\)
0.363101 + 0.931750i \(0.381718\pi\)
\(692\) 1.82404e7 1.44800
\(693\) 284743. 0.0225227
\(694\) −1.17761e7 −0.928116
\(695\) 0 0
\(696\) −1.52640e7 −1.19439
\(697\) 8.49896e6 0.662649
\(698\) 3.61740e6 0.281033
\(699\) 1.14146e7 0.883621
\(700\) 0 0
\(701\) −1.03729e7 −0.797269 −0.398634 0.917110i \(-0.630516\pi\)
−0.398634 + 0.917110i \(0.630516\pi\)
\(702\) −7.49575e6 −0.574080
\(703\) −7.24315e6 −0.552763
\(704\) −1.74533e6 −0.132723
\(705\) 0 0
\(706\) 1.00332e7 0.757579
\(707\) 1.40510e6 0.105721
\(708\) −1.60925e7 −1.20653
\(709\) 1.78828e7 1.33604 0.668022 0.744142i \(-0.267141\pi\)
0.668022 + 0.744142i \(0.267141\pi\)
\(710\) 0 0
\(711\) −670199. −0.0497199
\(712\) −2.90754e7 −2.14944
\(713\) −1.11326e7 −0.820108
\(714\) 3.96557e6 0.291112
\(715\) 0 0
\(716\) −957097. −0.0697708
\(717\) 1.21843e7 0.885119
\(718\) −1.48386e7 −1.07419
\(719\) −2.62904e7 −1.89660 −0.948298 0.317382i \(-0.897196\pi\)
−0.948298 + 0.317382i \(0.897196\pi\)
\(720\) 0 0
\(721\) −3.54217e6 −0.253765
\(722\) −2.24494e6 −0.160273
\(723\) −1.02778e7 −0.731231
\(724\) −5.18708e7 −3.67770
\(725\) 0 0
\(726\) 1.32376e6 0.0932108
\(727\) −1.40763e7 −0.987763 −0.493882 0.869529i \(-0.664422\pi\)
−0.493882 + 0.869529i \(0.664422\pi\)
\(728\) 1.10297e7 0.771322
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 3.44677e6 0.238572
\(732\) 3.18593e7 2.19765
\(733\) 5.60463e6 0.385290 0.192645 0.981269i \(-0.438293\pi\)
0.192645 + 0.981269i \(0.438293\pi\)
\(734\) 3.23387e7 2.21555
\(735\) 0 0
\(736\) 5.03813e6 0.342827
\(737\) 4.74081e6 0.321502
\(738\) −4.58098e6 −0.309612
\(739\) −1.92933e6 −0.129956 −0.0649778 0.997887i \(-0.520698\pi\)
−0.0649778 + 0.997887i \(0.520698\pi\)
\(740\) 0 0
\(741\) 1.51350e7 1.01260
\(742\) 8.31746e6 0.554601
\(743\) 4.43790e6 0.294921 0.147460 0.989068i \(-0.452890\pi\)
0.147460 + 0.989068i \(0.452890\pi\)
\(744\) 2.51431e7 1.66528
\(745\) 0 0
\(746\) −4.67934e7 −3.07849
\(747\) 1.72978e6 0.113420
\(748\) 1.25903e7 0.822774
\(749\) 5.57467e6 0.363090
\(750\) 0 0
\(751\) −1.11577e7 −0.721897 −0.360949 0.932586i \(-0.617547\pi\)
−0.360949 + 0.932586i \(0.617547\pi\)
\(752\) −9.09521e6 −0.586500
\(753\) 2.89420e6 0.186012
\(754\) −4.70138e7 −3.01160
\(755\) 0 0
\(756\) −1.45973e6 −0.0928898
\(757\) −1.16077e7 −0.736216 −0.368108 0.929783i \(-0.619994\pi\)
−0.368108 + 0.929783i \(0.619994\pi\)
\(758\) 4.06506e7 2.56977
\(759\) 1.60966e6 0.101422
\(760\) 0 0
\(761\) −1.38952e6 −0.0869770 −0.0434885 0.999054i \(-0.513847\pi\)
−0.0434885 + 0.999054i \(0.513847\pi\)
\(762\) 2.02303e7 1.26216
\(763\) −454960. −0.0282919
\(764\) −1.75175e6 −0.108577
\(765\) 0 0
\(766\) 8.27942e6 0.509833
\(767\) −2.65528e7 −1.62975
\(768\) 1.88091e7 1.15071
\(769\) −2.19019e7 −1.33557 −0.667784 0.744355i \(-0.732757\pi\)
−0.667784 + 0.744355i \(0.732757\pi\)
\(770\) 0 0
\(771\) 1.29848e7 0.786681
\(772\) 3.37599e7 2.03872
\(773\) 9.21665e6 0.554784 0.277392 0.960757i \(-0.410530\pi\)
0.277392 + 0.960757i \(0.410530\pi\)
\(774\) −1.85783e6 −0.111469
\(775\) 0 0
\(776\) 2.39525e7 1.42789
\(777\) 1.15267e6 0.0684941
\(778\) 2.55633e7 1.51415
\(779\) 9.24966e6 0.546113
\(780\) 0 0
\(781\) 4.54480e6 0.266617
\(782\) 2.24175e7 1.31090
\(783\) 3.33323e6 0.194294
\(784\) −2.42767e7 −1.41059
\(785\) 0 0
\(786\) 2.43445e7 1.40554
\(787\) 2.18581e7 1.25798 0.628992 0.777412i \(-0.283468\pi\)
0.628992 + 0.777412i \(0.283468\pi\)
\(788\) 2.23879e7 1.28439
\(789\) 7.54756e6 0.431633
\(790\) 0 0
\(791\) 439261. 0.0249621
\(792\) −3.63545e6 −0.205942
\(793\) 5.25684e7 2.96853
\(794\) −5.20159e7 −2.92809
\(795\) 0 0
\(796\) −2.74668e7 −1.53648
\(797\) −2.79333e7 −1.55767 −0.778837 0.627226i \(-0.784190\pi\)
−0.778837 + 0.627226i \(0.784190\pi\)
\(798\) 4.31585e6 0.239916
\(799\) −9.02865e6 −0.500329
\(800\) 0 0
\(801\) 6.34925e6 0.349656
\(802\) 4.84459e7 2.65963
\(803\) −7.10120e6 −0.388636
\(804\) −2.43037e7 −1.32596
\(805\) 0 0
\(806\) 7.74419e7 4.19893
\(807\) 3.73736e6 0.202014
\(808\) −1.79397e7 −0.966687
\(809\) 1.07778e7 0.578974 0.289487 0.957182i \(-0.406515\pi\)
0.289487 + 0.957182i \(0.406515\pi\)
\(810\) 0 0
\(811\) −2.71394e7 −1.44893 −0.724467 0.689309i \(-0.757914\pi\)
−0.724467 + 0.689309i \(0.757914\pi\)
\(812\) −9.15550e6 −0.487295
\(813\) −5.82488e6 −0.309073
\(814\) 5.35871e6 0.283465
\(815\) 0 0
\(816\) −2.06636e7 −1.08638
\(817\) 3.75122e6 0.196616
\(818\) 3.70322e7 1.93507
\(819\) −2.40858e6 −0.125473
\(820\) 0 0
\(821\) 6.84095e6 0.354208 0.177104 0.984192i \(-0.443327\pi\)
0.177104 + 0.984192i \(0.443327\pi\)
\(822\) −1.55682e7 −0.803634
\(823\) −1.95930e7 −1.00833 −0.504163 0.863609i \(-0.668199\pi\)
−0.504163 + 0.863609i \(0.668199\pi\)
\(824\) 4.52247e7 2.32037
\(825\) 0 0
\(826\) −7.57171e6 −0.386139
\(827\) 1.86828e7 0.949902 0.474951 0.880012i \(-0.342466\pi\)
0.474951 + 0.880012i \(0.342466\pi\)
\(828\) −8.25191e6 −0.418291
\(829\) −1.58057e7 −0.798779 −0.399390 0.916781i \(-0.630778\pi\)
−0.399390 + 0.916781i \(0.630778\pi\)
\(830\) 0 0
\(831\) 6.11109e6 0.306985
\(832\) 1.47634e7 0.739397
\(833\) −2.40991e7 −1.20334
\(834\) −3.16639e7 −1.57634
\(835\) 0 0
\(836\) 1.37023e7 0.678077
\(837\) −5.49055e6 −0.270896
\(838\) −1.32512e7 −0.651844
\(839\) −3.69461e7 −1.81202 −0.906011 0.423255i \(-0.860888\pi\)
−0.906011 + 0.423255i \(0.860888\pi\)
\(840\) 0 0
\(841\) 395030. 0.0192593
\(842\) 3.31035e7 1.60914
\(843\) 1.60817e7 0.779403
\(844\) −5.98694e7 −2.89300
\(845\) 0 0
\(846\) 4.86649e6 0.233771
\(847\) 425357. 0.0203725
\(848\) −4.33401e7 −2.06967
\(849\) −7.85300e6 −0.373909
\(850\) 0 0
\(851\) 6.51609e6 0.308435
\(852\) −2.32988e7 −1.09960
\(853\) −2.39478e7 −1.12692 −0.563459 0.826144i \(-0.690530\pi\)
−0.563459 + 0.826144i \(0.690530\pi\)
\(854\) 1.49902e7 0.703337
\(855\) 0 0
\(856\) −7.11746e7 −3.32002
\(857\) 4.09428e6 0.190426 0.0952129 0.995457i \(-0.469647\pi\)
0.0952129 + 0.995457i \(0.469647\pi\)
\(858\) −1.11974e7 −0.519275
\(859\) −6.51782e6 −0.301384 −0.150692 0.988581i \(-0.548150\pi\)
−0.150692 + 0.988581i \(0.548150\pi\)
\(860\) 0 0
\(861\) −1.47199e6 −0.0676700
\(862\) −1.87998e7 −0.861759
\(863\) −8.54609e6 −0.390607 −0.195304 0.980743i \(-0.562569\pi\)
−0.195304 + 0.980743i \(0.562569\pi\)
\(864\) 2.48479e6 0.113242
\(865\) 0 0
\(866\) 4.33178e7 1.96278
\(867\) −7.73365e6 −0.349411
\(868\) 1.50811e7 0.679413
\(869\) −1.00116e6 −0.0449733
\(870\) 0 0
\(871\) −4.01015e7 −1.79108
\(872\) 5.80870e6 0.258695
\(873\) −5.23055e6 −0.232280
\(874\) 2.43977e7 1.08036
\(875\) 0 0
\(876\) 3.64042e7 1.60284
\(877\) 3.63188e7 1.59453 0.797266 0.603629i \(-0.206279\pi\)
0.797266 + 0.603629i \(0.206279\pi\)
\(878\) −1.14135e7 −0.499668
\(879\) 5.87817e6 0.256608
\(880\) 0 0
\(881\) 1.57431e7 0.683364 0.341682 0.939816i \(-0.389004\pi\)
0.341682 + 0.939816i \(0.389004\pi\)
\(882\) 1.29895e7 0.562240
\(883\) −1.30059e7 −0.561354 −0.280677 0.959802i \(-0.590559\pi\)
−0.280677 + 0.959802i \(0.590559\pi\)
\(884\) −1.06498e8 −4.58365
\(885\) 0 0
\(886\) −8.79907e6 −0.376576
\(887\) 3.12186e7 1.33231 0.666154 0.745814i \(-0.267939\pi\)
0.666154 + 0.745814i \(0.267939\pi\)
\(888\) −1.47167e7 −0.626295
\(889\) 6.50050e6 0.275862
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 3.34771e6 0.140876
\(893\) −9.82614e6 −0.412339
\(894\) 4.69416e7 1.96433
\(895\) 0 0
\(896\) 7.37868e6 0.307050
\(897\) −1.36158e7 −0.565017
\(898\) −4.10688e7 −1.69950
\(899\) −3.44370e7 −1.42111
\(900\) 0 0
\(901\) −4.30230e7 −1.76558
\(902\) −6.84319e6 −0.280054
\(903\) −596968. −0.0243631
\(904\) −5.60827e6 −0.228248
\(905\) 0 0
\(906\) −2.08137e7 −0.842419
\(907\) −3.59253e7 −1.45005 −0.725023 0.688725i \(-0.758171\pi\)
−0.725023 + 0.688725i \(0.758171\pi\)
\(908\) −5.42212e6 −0.218250
\(909\) 3.91752e6 0.157254
\(910\) 0 0
\(911\) −1.47634e7 −0.589372 −0.294686 0.955594i \(-0.595215\pi\)
−0.294686 + 0.955594i \(0.595215\pi\)
\(912\) −2.24888e7 −0.895321
\(913\) 2.58400e6 0.102592
\(914\) 2.86592e7 1.13475
\(915\) 0 0
\(916\) −2.23371e7 −0.879607
\(917\) 7.82251e6 0.307201
\(918\) 1.10563e7 0.433014
\(919\) −1.26772e7 −0.495146 −0.247573 0.968869i \(-0.579633\pi\)
−0.247573 + 0.968869i \(0.579633\pi\)
\(920\) 0 0
\(921\) −1.68653e7 −0.655155
\(922\) −6.19224e7 −2.39895
\(923\) −3.84435e7 −1.48531
\(924\) −2.18058e6 −0.0840220
\(925\) 0 0
\(926\) −4.54495e7 −1.74181
\(927\) −9.87580e6 −0.377462
\(928\) 1.55848e7 0.594060
\(929\) 1.20341e7 0.457482 0.228741 0.973487i \(-0.426539\pi\)
0.228741 + 0.973487i \(0.426539\pi\)
\(930\) 0 0
\(931\) −2.62277e7 −0.991713
\(932\) −8.74136e7 −3.29639
\(933\) 1.44563e7 0.543694
\(934\) 2.24111e7 0.840614
\(935\) 0 0
\(936\) 3.07515e7 1.14730
\(937\) 4.51344e7 1.67942 0.839708 0.543038i \(-0.182726\pi\)
0.839708 + 0.543038i \(0.182726\pi\)
\(938\) −1.14352e7 −0.424362
\(939\) −2.32381e7 −0.860077
\(940\) 0 0
\(941\) 8.44095e6 0.310754 0.155377 0.987855i \(-0.450341\pi\)
0.155377 + 0.987855i \(0.450341\pi\)
\(942\) 2.97340e7 1.09176
\(943\) −8.32120e6 −0.304724
\(944\) 3.94543e7 1.44100
\(945\) 0 0
\(946\) −2.77528e6 −0.100827
\(947\) −2.68109e7 −0.971485 −0.485743 0.874102i \(-0.661451\pi\)
−0.485743 + 0.874102i \(0.661451\pi\)
\(948\) 5.13244e6 0.185483
\(949\) 6.00675e7 2.16508
\(950\) 0 0
\(951\) −1.21129e7 −0.434307
\(952\) −1.62689e7 −0.581788
\(953\) −1.42183e7 −0.507127 −0.253563 0.967319i \(-0.581603\pi\)
−0.253563 + 0.967319i \(0.581603\pi\)
\(954\) 2.31896e7 0.824940
\(955\) 0 0
\(956\) −9.33081e7 −3.30198
\(957\) 4.97926e6 0.175746
\(958\) 6.65263e7 2.34196
\(959\) −5.00246e6 −0.175645
\(960\) 0 0
\(961\) 2.80961e7 0.981380
\(962\) −4.53282e7 −1.57918
\(963\) 1.55425e7 0.540078
\(964\) 7.87082e7 2.72789
\(965\) 0 0
\(966\) −3.88263e6 −0.133870
\(967\) −277112. −0.00952993 −0.00476496 0.999989i \(-0.501517\pi\)
−0.00476496 + 0.999989i \(0.501517\pi\)
\(968\) −5.43074e6 −0.186282
\(969\) −2.23242e7 −0.763777
\(970\) 0 0
\(971\) 7.77514e6 0.264643 0.132321 0.991207i \(-0.457757\pi\)
0.132321 + 0.991207i \(0.457757\pi\)
\(972\) −4.06982e6 −0.138169
\(973\) −1.01744e7 −0.344531
\(974\) 2.18584e7 0.738280
\(975\) 0 0
\(976\) −7.81102e7 −2.62472
\(977\) −4.83351e6 −0.162004 −0.0810020 0.996714i \(-0.525812\pi\)
−0.0810020 + 0.996714i \(0.525812\pi\)
\(978\) 2.22765e7 0.744731
\(979\) 9.48468e6 0.316276
\(980\) 0 0
\(981\) −1.26846e6 −0.0420827
\(982\) −7.70729e7 −2.55049
\(983\) 9.75876e6 0.322115 0.161057 0.986945i \(-0.448510\pi\)
0.161057 + 0.986945i \(0.448510\pi\)
\(984\) 1.87936e7 0.618760
\(985\) 0 0
\(986\) 6.93455e7 2.27157
\(987\) 1.56373e6 0.0510938
\(988\) −1.15905e8 −3.77755
\(989\) −3.37468e6 −0.109709
\(990\) 0 0
\(991\) 2.72969e7 0.882935 0.441468 0.897277i \(-0.354458\pi\)
0.441468 + 0.897277i \(0.354458\pi\)
\(992\) −2.56715e7 −0.828270
\(993\) −2.46922e7 −0.794668
\(994\) −1.09624e7 −0.351917
\(995\) 0 0
\(996\) −1.32468e7 −0.423120
\(997\) −4.85951e7 −1.54830 −0.774149 0.633004i \(-0.781822\pi\)
−0.774149 + 0.633004i \(0.781822\pi\)
\(998\) −6.68070e7 −2.12322
\(999\) 3.21372e6 0.101881
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.v.1.2 13
5.2 odd 4 165.6.c.b.34.3 26
5.3 odd 4 165.6.c.b.34.24 yes 26
5.4 even 2 825.6.a.y.1.12 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.3 26 5.2 odd 4
165.6.c.b.34.24 yes 26 5.3 odd 4
825.6.a.v.1.2 13 1.1 even 1 trivial
825.6.a.y.1.12 13 5.4 even 2