Properties

Label 23.6.a.a.1.1
Level $23$
Weight $6$
Character 23.1
Self dual yes
Analytic conductor $3.689$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.68882785570\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7925.1
Defining polynomial: \( x^{3} - x^{2} - 13x + 12 \) Copy content Toggle raw display
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 1.1
Root \(-3.57511\) of defining polynomial
Character \(\chi\) \(=\) 23.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.15022 q^{2} +9.09413 q^{3} +51.7266 q^{4} +3.86330 q^{5} -83.2134 q^{6} -226.379 q^{7} -180.503 q^{8} -160.297 q^{9} +O(q^{10})\) \(q-9.15022 q^{2} +9.09413 q^{3} +51.7266 q^{4} +3.86330 q^{5} -83.2134 q^{6} -226.379 q^{7} -180.503 q^{8} -160.297 q^{9} -35.3501 q^{10} +18.3261 q^{11} +470.409 q^{12} -711.185 q^{13} +2071.42 q^{14} +35.1334 q^{15} -3.60999 q^{16} +165.441 q^{17} +1466.75 q^{18} +213.254 q^{19} +199.835 q^{20} -2058.72 q^{21} -167.688 q^{22} -529.000 q^{23} -1641.52 q^{24} -3110.07 q^{25} +6507.50 q^{26} -3667.63 q^{27} -11709.8 q^{28} +5475.82 q^{29} -321.478 q^{30} +6077.13 q^{31} +5809.12 q^{32} +166.660 q^{33} -1513.82 q^{34} -874.570 q^{35} -8291.60 q^{36} +11499.7 q^{37} -1951.32 q^{38} -6467.61 q^{39} -697.337 q^{40} -13183.0 q^{41} +18837.8 q^{42} -21455.2 q^{43} +947.948 q^{44} -619.274 q^{45} +4840.47 q^{46} +12960.8 q^{47} -32.8298 q^{48} +34440.5 q^{49} +28457.9 q^{50} +1504.54 q^{51} -36787.2 q^{52} -29103.4 q^{53} +33559.7 q^{54} +70.7993 q^{55} +40862.1 q^{56} +1939.36 q^{57} -50105.0 q^{58} +10506.2 q^{59} +1817.33 q^{60} -34103.3 q^{61} -55607.1 q^{62} +36287.8 q^{63} -53039.3 q^{64} -2747.52 q^{65} -1524.98 q^{66} +12988.2 q^{67} +8557.70 q^{68} -4810.80 q^{69} +8002.51 q^{70} +45243.0 q^{71} +28934.0 q^{72} -52589.2 q^{73} -105225. q^{74} -28283.4 q^{75} +11030.9 q^{76} -4148.65 q^{77} +59180.1 q^{78} +57603.3 q^{79} -13.9465 q^{80} +5598.14 q^{81} +120627. q^{82} -40816.8 q^{83} -106491. q^{84} +639.148 q^{85} +196320. q^{86} +49797.8 q^{87} -3307.92 q^{88} +54699.9 q^{89} +5666.50 q^{90} +160997. q^{91} -27363.4 q^{92} +55266.2 q^{93} -118594. q^{94} +823.864 q^{95} +52828.9 q^{96} -117497. q^{97} -315138. q^{98} -2937.62 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} - 20 q^{3} + 16 q^{4} - 58 q^{5} - 230 q^{6} - 282 q^{7} - 360 q^{8} + 121 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{2} - 20 q^{3} + 16 q^{4} - 58 q^{5} - 230 q^{6} - 282 q^{7} - 360 q^{8} + 121 q^{9} - 156 q^{10} + 136 q^{11} + 620 q^{12} - 1116 q^{13} + 2016 q^{14} + 750 q^{15} + 128 q^{16} - 896 q^{17} + 4282 q^{18} + 1654 q^{19} + 1504 q^{20} - 1670 q^{21} + 1352 q^{22} - 1587 q^{23} + 3600 q^{24} - 7347 q^{25} + 1998 q^{26} - 10700 q^{27} - 10264 q^{28} - 844 q^{29} + 3180 q^{30} - 3020 q^{31} + 6656 q^{32} - 7370 q^{33} - 11212 q^{34} + 1072 q^{35} - 2548 q^{36} + 8938 q^{37} + 10728 q^{38} + 16020 q^{39} + 3440 q^{40} - 12792 q^{41} + 20560 q^{42} - 16730 q^{43} + 5112 q^{44} - 3936 q^{45} + 2116 q^{46} + 22500 q^{47} + 23120 q^{48} + 2887 q^{49} + 15156 q^{50} + 50290 q^{51} - 47412 q^{52} + 17108 q^{53} - 7610 q^{54} - 436 q^{55} + 42640 q^{56} - 61960 q^{57} - 55678 q^{58} + 54176 q^{59} - 2400 q^{60} - 71324 q^{61} - 72710 q^{62} + 40696 q^{63} - 49984 q^{64} + 846 q^{65} - 42860 q^{66} - 62960 q^{67} - 8352 q^{68} + 10580 q^{69} + 9224 q^{70} + 98400 q^{71} - 72840 q^{72} - 81772 q^{73} - 59044 q^{74} + 44800 q^{75} + 31488 q^{76} - 304 q^{77} + 182070 q^{78} + 58224 q^{79} - 17568 q^{80} + 149947 q^{81} + 61926 q^{82} + 9892 q^{83} - 109720 q^{84} + 15536 q^{85} + 191140 q^{86} + 90500 q^{87} - 58400 q^{88} + 27542 q^{89} - 62112 q^{90} + 151974 q^{91} - 8464 q^{92} + 157330 q^{93} - 146990 q^{94} - 20644 q^{95} + 16160 q^{96} - 273672 q^{97} - 401276 q^{98} + 183082 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.15022 −1.61755 −0.808773 0.588121i \(-0.799868\pi\)
−0.808773 + 0.588121i \(0.799868\pi\)
\(3\) 9.09413 0.583389 0.291694 0.956512i \(-0.405781\pi\)
0.291694 + 0.956512i \(0.405781\pi\)
\(4\) 51.7266 1.61646
\(5\) 3.86330 0.0691088 0.0345544 0.999403i \(-0.488999\pi\)
0.0345544 + 0.999403i \(0.488999\pi\)
\(6\) −83.2134 −0.943659
\(7\) −226.379 −1.74619 −0.873094 0.487551i \(-0.837890\pi\)
−0.873094 + 0.487551i \(0.837890\pi\)
\(8\) −180.503 −0.997147
\(9\) −160.297 −0.659657
\(10\) −35.3501 −0.111787
\(11\) 18.3261 0.0456656 0.0228328 0.999739i \(-0.492731\pi\)
0.0228328 + 0.999739i \(0.492731\pi\)
\(12\) 470.409 0.943023
\(13\) −711.185 −1.16714 −0.583572 0.812062i \(-0.698345\pi\)
−0.583572 + 0.812062i \(0.698345\pi\)
\(14\) 2071.42 2.82454
\(15\) 35.1334 0.0403173
\(16\) −3.60999 −0.00352539
\(17\) 165.441 0.138842 0.0694210 0.997587i \(-0.477885\pi\)
0.0694210 + 0.997587i \(0.477885\pi\)
\(18\) 1466.75 1.06703
\(19\) 213.254 0.135523 0.0677615 0.997702i \(-0.478414\pi\)
0.0677615 + 0.997702i \(0.478414\pi\)
\(20\) 199.835 0.111711
\(21\) −2058.72 −1.01871
\(22\) −167.688 −0.0738662
\(23\) −529.000 −0.208514
\(24\) −1641.52 −0.581724
\(25\) −3110.07 −0.995224
\(26\) 6507.50 1.88791
\(27\) −3667.63 −0.968226
\(28\) −11709.8 −2.82264
\(29\) 5475.82 1.20908 0.604539 0.796576i \(-0.293357\pi\)
0.604539 + 0.796576i \(0.293357\pi\)
\(30\) −321.478 −0.0652151
\(31\) 6077.13 1.13578 0.567890 0.823104i \(-0.307760\pi\)
0.567890 + 0.823104i \(0.307760\pi\)
\(32\) 5809.12 1.00285
\(33\) 166.660 0.0266408
\(34\) −1513.82 −0.224583
\(35\) −874.570 −0.120677
\(36\) −8291.60 −1.06631
\(37\) 11499.7 1.38096 0.690481 0.723350i \(-0.257399\pi\)
0.690481 + 0.723350i \(0.257399\pi\)
\(38\) −1951.32 −0.219215
\(39\) −6467.61 −0.680899
\(40\) −697.337 −0.0689116
\(41\) −13183.0 −1.22477 −0.612384 0.790561i \(-0.709789\pi\)
−0.612384 + 0.790561i \(0.709789\pi\)
\(42\) 18837.8 1.64781
\(43\) −21455.2 −1.76955 −0.884773 0.466022i \(-0.845687\pi\)
−0.884773 + 0.466022i \(0.845687\pi\)
\(44\) 947.948 0.0738164
\(45\) −619.274 −0.0455881
\(46\) 4840.47 0.337282
\(47\) 12960.8 0.855830 0.427915 0.903819i \(-0.359248\pi\)
0.427915 + 0.903819i \(0.359248\pi\)
\(48\) −32.8298 −0.00205667
\(49\) 34440.5 2.04917
\(50\) 28457.9 1.60982
\(51\) 1504.54 0.0809988
\(52\) −36787.2 −1.88664
\(53\) −29103.4 −1.42316 −0.711581 0.702604i \(-0.752021\pi\)
−0.711581 + 0.702604i \(0.752021\pi\)
\(54\) 33559.7 1.56615
\(55\) 70.7993 0.00315589
\(56\) 40862.1 1.74121
\(57\) 1939.36 0.0790626
\(58\) −50105.0 −1.95574
\(59\) 10506.2 0.392930 0.196465 0.980511i \(-0.437054\pi\)
0.196465 + 0.980511i \(0.437054\pi\)
\(60\) 1817.33 0.0651712
\(61\) −34103.3 −1.17347 −0.586734 0.809780i \(-0.699587\pi\)
−0.586734 + 0.809780i \(0.699587\pi\)
\(62\) −55607.1 −1.83718
\(63\) 36287.8 1.15189
\(64\) −53039.3 −1.61863
\(65\) −2747.52 −0.0806599
\(66\) −1524.98 −0.0430927
\(67\) 12988.2 0.353476 0.176738 0.984258i \(-0.443445\pi\)
0.176738 + 0.984258i \(0.443445\pi\)
\(68\) 8557.70 0.224432
\(69\) −4810.80 −0.121645
\(70\) 8002.51 0.195201
\(71\) 45243.0 1.06514 0.532569 0.846387i \(-0.321227\pi\)
0.532569 + 0.846387i \(0.321227\pi\)
\(72\) 28934.0 0.657775
\(73\) −52589.2 −1.15502 −0.577510 0.816384i \(-0.695976\pi\)
−0.577510 + 0.816384i \(0.695976\pi\)
\(74\) −105225. −2.23377
\(75\) −28283.4 −0.580603
\(76\) 11030.9 0.219067
\(77\) −4148.65 −0.0797407
\(78\) 59180.1 1.10139
\(79\) 57603.3 1.03844 0.519218 0.854642i \(-0.326223\pi\)
0.519218 + 0.854642i \(0.326223\pi\)
\(80\) −13.9465 −0.000243635 0
\(81\) 5598.14 0.0948051
\(82\) 120627. 1.98112
\(83\) −40816.8 −0.650345 −0.325173 0.945655i \(-0.605422\pi\)
−0.325173 + 0.945655i \(0.605422\pi\)
\(84\) −106491. −1.64670
\(85\) 639.148 0.00959520
\(86\) 196320. 2.86232
\(87\) 49797.8 0.705362
\(88\) −3307.92 −0.0455353
\(89\) 54699.9 0.732000 0.366000 0.930615i \(-0.380727\pi\)
0.366000 + 0.930615i \(0.380727\pi\)
\(90\) 5666.50 0.0737409
\(91\) 160997. 2.03805
\(92\) −27363.4 −0.337054
\(93\) 55266.2 0.662602
\(94\) −118594. −1.38434
\(95\) 823.864 0.00936583
\(96\) 52828.9 0.585051
\(97\) −117497. −1.26794 −0.633968 0.773360i \(-0.718575\pi\)
−0.633968 + 0.773360i \(0.718575\pi\)
\(98\) −315138. −3.31464
\(99\) −2937.62 −0.0301236
\(100\) −160874. −1.60874
\(101\) −141707. −1.38226 −0.691128 0.722733i \(-0.742886\pi\)
−0.691128 + 0.722733i \(0.742886\pi\)
\(102\) −13766.9 −0.131019
\(103\) 60185.0 0.558979 0.279489 0.960149i \(-0.409835\pi\)
0.279489 + 0.960149i \(0.409835\pi\)
\(104\) 128371. 1.16381
\(105\) −7953.46 −0.0704017
\(106\) 266303. 2.30203
\(107\) −186999. −1.57899 −0.789495 0.613757i \(-0.789658\pi\)
−0.789495 + 0.613757i \(0.789658\pi\)
\(108\) −189714. −1.56509
\(109\) −63597.2 −0.512710 −0.256355 0.966583i \(-0.582522\pi\)
−0.256355 + 0.966583i \(0.582522\pi\)
\(110\) −647.830 −0.00510480
\(111\) 104580. 0.805638
\(112\) 817.227 0.00615599
\(113\) 42281.3 0.311496 0.155748 0.987797i \(-0.450221\pi\)
0.155748 + 0.987797i \(0.450221\pi\)
\(114\) −17745.6 −0.127887
\(115\) −2043.69 −0.0144102
\(116\) 283246. 1.95442
\(117\) 114001. 0.769915
\(118\) −96134.0 −0.635583
\(119\) −37452.4 −0.242444
\(120\) −6341.67 −0.0402023
\(121\) −160715. −0.997915
\(122\) 312052. 1.89814
\(123\) −119888. −0.714516
\(124\) 314349. 1.83594
\(125\) −24088.0 −0.137888
\(126\) −332042. −1.86323
\(127\) −119942. −0.659877 −0.329938 0.944002i \(-0.607028\pi\)
−0.329938 + 0.944002i \(0.607028\pi\)
\(128\) 299429. 1.61536
\(129\) −195117. −1.03233
\(130\) 25140.4 0.130471
\(131\) 188708. 0.960753 0.480376 0.877062i \(-0.340500\pi\)
0.480376 + 0.877062i \(0.340500\pi\)
\(132\) 8620.77 0.0430637
\(133\) −48276.2 −0.236649
\(134\) −118844. −0.571764
\(135\) −14169.2 −0.0669129
\(136\) −29862.6 −0.138446
\(137\) −320875. −1.46061 −0.730305 0.683121i \(-0.760622\pi\)
−0.730305 + 0.683121i \(0.760622\pi\)
\(138\) 44019.9 0.196766
\(139\) 174260. 0.765000 0.382500 0.923955i \(-0.375063\pi\)
0.382500 + 0.923955i \(0.375063\pi\)
\(140\) −45238.6 −0.195069
\(141\) 117867. 0.499282
\(142\) −413984. −1.72291
\(143\) −13033.3 −0.0532983
\(144\) 578.670 0.00232555
\(145\) 21154.7 0.0835579
\(146\) 481203. 1.86830
\(147\) 313206. 1.19547
\(148\) 594840. 2.23227
\(149\) −144356. −0.532682 −0.266341 0.963879i \(-0.585815\pi\)
−0.266341 + 0.963879i \(0.585815\pi\)
\(150\) 258800. 0.939152
\(151\) −122240. −0.436286 −0.218143 0.975917i \(-0.570000\pi\)
−0.218143 + 0.975917i \(0.570000\pi\)
\(152\) −38492.9 −0.135136
\(153\) −26519.6 −0.0915881
\(154\) 37961.1 0.128984
\(155\) 23477.8 0.0784924
\(156\) −334548. −1.10064
\(157\) −115272. −0.373227 −0.186614 0.982433i \(-0.559751\pi\)
−0.186614 + 0.982433i \(0.559751\pi\)
\(158\) −527083. −1.67972
\(159\) −264670. −0.830257
\(160\) 22442.4 0.0693057
\(161\) 119755. 0.364106
\(162\) −51224.3 −0.153352
\(163\) 120609. 0.355557 0.177778 0.984071i \(-0.443109\pi\)
0.177778 + 0.984071i \(0.443109\pi\)
\(164\) −681910. −1.97978
\(165\) 643.858 0.00184111
\(166\) 373483. 1.05196
\(167\) 476969. 1.32342 0.661711 0.749759i \(-0.269830\pi\)
0.661711 + 0.749759i \(0.269830\pi\)
\(168\) 371605. 1.01580
\(169\) 134491. 0.362224
\(170\) −5848.35 −0.0155207
\(171\) −34183.9 −0.0893987
\(172\) −1.10981e6 −2.86039
\(173\) 341220. 0.866801 0.433400 0.901202i \(-0.357314\pi\)
0.433400 + 0.901202i \(0.357314\pi\)
\(174\) −455661. −1.14096
\(175\) 704056. 1.73785
\(176\) −66.1572 −0.000160989 0
\(177\) 95544.7 0.229231
\(178\) −500516. −1.18404
\(179\) 157948. 0.368452 0.184226 0.982884i \(-0.441022\pi\)
0.184226 + 0.982884i \(0.441022\pi\)
\(180\) −32033.0 −0.0736912
\(181\) −133395. −0.302652 −0.151326 0.988484i \(-0.548354\pi\)
−0.151326 + 0.988484i \(0.548354\pi\)
\(182\) −1.47316e6 −3.29665
\(183\) −310140. −0.684588
\(184\) 95486.0 0.207919
\(185\) 44426.8 0.0954367
\(186\) −505698. −1.07179
\(187\) 3031.89 0.00634030
\(188\) 670418. 1.38341
\(189\) 830276. 1.69070
\(190\) −7538.54 −0.0151497
\(191\) −819880. −1.62617 −0.813087 0.582142i \(-0.802215\pi\)
−0.813087 + 0.582142i \(0.802215\pi\)
\(192\) −482346. −0.944291
\(193\) 389602. 0.752884 0.376442 0.926440i \(-0.377147\pi\)
0.376442 + 0.926440i \(0.377147\pi\)
\(194\) 1.07512e6 2.05094
\(195\) −24986.3 −0.0470561
\(196\) 1.78149e6 3.31240
\(197\) 251073. 0.460930 0.230465 0.973081i \(-0.425975\pi\)
0.230465 + 0.973081i \(0.425975\pi\)
\(198\) 26879.9 0.0487264
\(199\) 487551. 0.872745 0.436373 0.899766i \(-0.356263\pi\)
0.436373 + 0.899766i \(0.356263\pi\)
\(200\) 561377. 0.992384
\(201\) 118116. 0.206214
\(202\) 1.29665e6 2.23586
\(203\) −1.23961e6 −2.11128
\(204\) 77824.8 0.130931
\(205\) −50929.8 −0.0846423
\(206\) −550707. −0.904174
\(207\) 84797.0 0.137548
\(208\) 2567.37 0.00411463
\(209\) 3908.12 0.00618873
\(210\) 72775.9 0.113878
\(211\) −274245. −0.424064 −0.212032 0.977263i \(-0.568008\pi\)
−0.212032 + 0.977263i \(0.568008\pi\)
\(212\) −1.50542e6 −2.30048
\(213\) 411446. 0.621390
\(214\) 1.71108e6 2.55409
\(215\) −82888.0 −0.122291
\(216\) 662018. 0.965463
\(217\) −1.37573e6 −1.98329
\(218\) 581928. 0.829332
\(219\) −478253. −0.673826
\(220\) 3662.21 0.00510136
\(221\) −117659. −0.162048
\(222\) −956928. −1.30316
\(223\) −215015. −0.289539 −0.144769 0.989465i \(-0.546244\pi\)
−0.144769 + 0.989465i \(0.546244\pi\)
\(224\) −1.31506e6 −1.75116
\(225\) 498535. 0.656507
\(226\) −386884. −0.503859
\(227\) 837285. 1.07847 0.539236 0.842155i \(-0.318713\pi\)
0.539236 + 0.842155i \(0.318713\pi\)
\(228\) 100316. 0.127801
\(229\) −1.33914e6 −1.68748 −0.843740 0.536753i \(-0.819651\pi\)
−0.843740 + 0.536753i \(0.819651\pi\)
\(230\) 18700.2 0.0233091
\(231\) −37728.4 −0.0465198
\(232\) −988401. −1.20563
\(233\) −320747. −0.387055 −0.193528 0.981095i \(-0.561993\pi\)
−0.193528 + 0.981095i \(0.561993\pi\)
\(234\) −1.04313e6 −1.24537
\(235\) 50071.5 0.0591454
\(236\) 543450. 0.635154
\(237\) 523852. 0.605812
\(238\) 342697. 0.392165
\(239\) −278679. −0.315580 −0.157790 0.987473i \(-0.550437\pi\)
−0.157790 + 0.987473i \(0.550437\pi\)
\(240\) −126.831 −0.000142134 0
\(241\) −680862. −0.755120 −0.377560 0.925985i \(-0.623237\pi\)
−0.377560 + 0.925985i \(0.623237\pi\)
\(242\) 1.47058e6 1.61417
\(243\) 942145. 1.02353
\(244\) −1.76405e6 −1.89686
\(245\) 133054. 0.141616
\(246\) 1.09700e6 1.15576
\(247\) −151663. −0.158175
\(248\) −1.09694e6 −1.13254
\(249\) −371194. −0.379404
\(250\) 220410. 0.223040
\(251\) 410308. 0.411079 0.205540 0.978649i \(-0.434105\pi\)
0.205540 + 0.978649i \(0.434105\pi\)
\(252\) 1.87705e6 1.86197
\(253\) −9694.52 −0.00952193
\(254\) 1.09750e6 1.06738
\(255\) 5812.50 0.00559773
\(256\) −1.04259e6 −0.994289
\(257\) 1.69725e6 1.60292 0.801460 0.598048i \(-0.204057\pi\)
0.801460 + 0.598048i \(0.204057\pi\)
\(258\) 1.78536e6 1.66985
\(259\) −2.60329e6 −2.41142
\(260\) −142120. −0.130383
\(261\) −877756. −0.797577
\(262\) −1.72672e6 −1.55406
\(263\) 1.11082e6 0.990269 0.495135 0.868816i \(-0.335119\pi\)
0.495135 + 0.868816i \(0.335119\pi\)
\(264\) −30082.6 −0.0265648
\(265\) −112435. −0.0983531
\(266\) 441738. 0.382790
\(267\) 497448. 0.427041
\(268\) 671833. 0.571379
\(269\) 133948. 0.112864 0.0564322 0.998406i \(-0.482028\pi\)
0.0564322 + 0.998406i \(0.482028\pi\)
\(270\) 129651. 0.108235
\(271\) 93482.4 0.0773226 0.0386613 0.999252i \(-0.487691\pi\)
0.0386613 + 0.999252i \(0.487691\pi\)
\(272\) −597.241 −0.000489471 0
\(273\) 1.46413e6 1.18898
\(274\) 2.93608e6 2.36261
\(275\) −56995.6 −0.0454475
\(276\) −248846. −0.196634
\(277\) 576551. 0.451479 0.225740 0.974188i \(-0.427520\pi\)
0.225740 + 0.974188i \(0.427520\pi\)
\(278\) −1.59452e6 −1.23742
\(279\) −974144. −0.749226
\(280\) 157862. 0.120333
\(281\) −434678. −0.328399 −0.164200 0.986427i \(-0.552504\pi\)
−0.164200 + 0.986427i \(0.552504\pi\)
\(282\) −1.07851e6 −0.807611
\(283\) −1.52198e6 −1.12965 −0.564823 0.825212i \(-0.691056\pi\)
−0.564823 + 0.825212i \(0.691056\pi\)
\(284\) 2.34027e6 1.72175
\(285\) 7492.33 0.00546392
\(286\) 119257. 0.0862124
\(287\) 2.98435e6 2.13868
\(288\) −931183. −0.661537
\(289\) −1.39249e6 −0.980723
\(290\) −193571. −0.135159
\(291\) −1.06853e6 −0.739699
\(292\) −2.72026e6 −1.86704
\(293\) 1.36342e6 0.927813 0.463907 0.885884i \(-0.346447\pi\)
0.463907 + 0.885884i \(0.346447\pi\)
\(294\) −2.86591e6 −1.93372
\(295\) 40588.6 0.0271549
\(296\) −2.07573e6 −1.37702
\(297\) −67213.5 −0.0442146
\(298\) 1.32089e6 0.861638
\(299\) 376217. 0.243366
\(300\) −1.46301e6 −0.938519
\(301\) 4.85701e6 3.08996
\(302\) 1.11852e6 0.705713
\(303\) −1.28870e6 −0.806392
\(304\) −769.845 −0.000477771 0
\(305\) −131751. −0.0810970
\(306\) 242661. 0.148148
\(307\) −3.03847e6 −1.83996 −0.919980 0.391966i \(-0.871795\pi\)
−0.919980 + 0.391966i \(0.871795\pi\)
\(308\) −214596. −0.128897
\(309\) 547331. 0.326102
\(310\) −214827. −0.126965
\(311\) −3.32689e6 −1.95046 −0.975232 0.221186i \(-0.929007\pi\)
−0.975232 + 0.221186i \(0.929007\pi\)
\(312\) 1.16742e6 0.678956
\(313\) −134964. −0.0778677 −0.0389339 0.999242i \(-0.512396\pi\)
−0.0389339 + 0.999242i \(0.512396\pi\)
\(314\) 1.05476e6 0.603712
\(315\) 140191. 0.0796055
\(316\) 2.97962e6 1.67859
\(317\) 2.83907e6 1.58682 0.793411 0.608687i \(-0.208303\pi\)
0.793411 + 0.608687i \(0.208303\pi\)
\(318\) 2.42179e6 1.34298
\(319\) 100351. 0.0552132
\(320\) −204907. −0.111862
\(321\) −1.70059e6 −0.921166
\(322\) −1.09578e6 −0.588958
\(323\) 35280.9 0.0188163
\(324\) 289573. 0.153248
\(325\) 2.21184e6 1.16157
\(326\) −1.10360e6 −0.575130
\(327\) −578361. −0.299109
\(328\) 2.37956e6 1.22127
\(329\) −2.93406e6 −1.49444
\(330\) −5891.45 −0.00297809
\(331\) 2.26789e6 1.13777 0.568883 0.822419i \(-0.307376\pi\)
0.568883 + 0.822419i \(0.307376\pi\)
\(332\) −2.11132e6 −1.05125
\(333\) −1.84336e6 −0.910962
\(334\) −4.36437e6 −2.14070
\(335\) 50177.1 0.0244283
\(336\) 7431.97 0.00359134
\(337\) −1.16585e6 −0.559199 −0.279599 0.960117i \(-0.590202\pi\)
−0.279599 + 0.960117i \(0.590202\pi\)
\(338\) −1.23063e6 −0.585915
\(339\) 384512. 0.181723
\(340\) 33061.0 0.0155102
\(341\) 111370. 0.0518660
\(342\) 312790. 0.144607
\(343\) −3.99185e6 −1.83206
\(344\) 3.87273e6 1.76450
\(345\) −18585.6 −0.00840674
\(346\) −3.12224e6 −1.40209
\(347\) 247150. 0.110189 0.0550944 0.998481i \(-0.482454\pi\)
0.0550944 + 0.998481i \(0.482454\pi\)
\(348\) 2.57587e6 1.14019
\(349\) 869941. 0.382319 0.191160 0.981559i \(-0.438775\pi\)
0.191160 + 0.981559i \(0.438775\pi\)
\(350\) −6.44227e6 −2.81105
\(351\) 2.60837e6 1.13006
\(352\) 106459. 0.0457957
\(353\) 3.16976e6 1.35391 0.676954 0.736025i \(-0.263299\pi\)
0.676954 + 0.736025i \(0.263299\pi\)
\(354\) −874256. −0.370792
\(355\) 174787. 0.0736104
\(356\) 2.82944e6 1.18325
\(357\) −340597. −0.141439
\(358\) −1.44526e6 −0.595988
\(359\) 1.48523e6 0.608215 0.304108 0.952638i \(-0.401642\pi\)
0.304108 + 0.952638i \(0.401642\pi\)
\(360\) 111781. 0.0454581
\(361\) −2.43062e6 −0.981634
\(362\) 1.22060e6 0.489554
\(363\) −1.46157e6 −0.582172
\(364\) 8.32785e6 3.29442
\(365\) −203168. −0.0798221
\(366\) 2.83785e6 1.10735
\(367\) 2.27000e6 0.879752 0.439876 0.898059i \(-0.355022\pi\)
0.439876 + 0.898059i \(0.355022\pi\)
\(368\) 1909.69 0.000735094 0
\(369\) 2.11319e6 0.807927
\(370\) −406515. −0.154373
\(371\) 6.58841e6 2.48511
\(372\) 2.85873e6 1.07107
\(373\) 2.90005e6 1.07928 0.539640 0.841896i \(-0.318560\pi\)
0.539640 + 0.841896i \(0.318560\pi\)
\(374\) −27742.5 −0.0102557
\(375\) −219059. −0.0804421
\(376\) −2.33946e6 −0.853388
\(377\) −3.89432e6 −1.41117
\(378\) −7.59721e6 −2.73479
\(379\) 1.01778e6 0.363960 0.181980 0.983302i \(-0.441749\pi\)
0.181980 + 0.983302i \(0.441749\pi\)
\(380\) 42615.7 0.0151395
\(381\) −1.09077e6 −0.384965
\(382\) 7.50209e6 2.63041
\(383\) −4.19874e6 −1.46259 −0.731293 0.682063i \(-0.761083\pi\)
−0.731293 + 0.682063i \(0.761083\pi\)
\(384\) 2.72305e6 0.942383
\(385\) −16027.5 −0.00551079
\(386\) −3.56495e6 −1.21783
\(387\) 3.43920e6 1.16729
\(388\) −6.07772e6 −2.04956
\(389\) 1.67191e6 0.560193 0.280097 0.959972i \(-0.409634\pi\)
0.280097 + 0.959972i \(0.409634\pi\)
\(390\) 228631. 0.0761154
\(391\) −87518.2 −0.0289505
\(392\) −6.21660e6 −2.04333
\(393\) 1.71613e6 0.560493
\(394\) −2.29738e6 −0.745576
\(395\) 222539. 0.0717651
\(396\) −151953. −0.0486935
\(397\) −1.36652e6 −0.435152 −0.217576 0.976043i \(-0.569815\pi\)
−0.217576 + 0.976043i \(0.569815\pi\)
\(398\) −4.46120e6 −1.41171
\(399\) −439030. −0.138058
\(400\) 11227.4 0.00350855
\(401\) −1.30417e6 −0.405017 −0.202509 0.979280i \(-0.564909\pi\)
−0.202509 + 0.979280i \(0.564909\pi\)
\(402\) −1.08079e6 −0.333561
\(403\) −4.32196e6 −1.32562
\(404\) −7.33003e6 −2.23436
\(405\) 21627.3 0.00655187
\(406\) 1.13427e7 3.41509
\(407\) 210745. 0.0630625
\(408\) −271574. −0.0807677
\(409\) 441139. 0.130397 0.0651984 0.997872i \(-0.479232\pi\)
0.0651984 + 0.997872i \(0.479232\pi\)
\(410\) 466019. 0.136913
\(411\) −2.91808e6 −0.852104
\(412\) 3.11317e6 0.903565
\(413\) −2.37838e6 −0.686130
\(414\) −775911. −0.222490
\(415\) −157688. −0.0449446
\(416\) −4.13136e6 −1.17047
\(417\) 1.58475e6 0.446293
\(418\) −35760.1 −0.0100106
\(419\) 1.89676e6 0.527809 0.263904 0.964549i \(-0.414990\pi\)
0.263904 + 0.964549i \(0.414990\pi\)
\(420\) −411405. −0.113801
\(421\) −1.39879e6 −0.384634 −0.192317 0.981333i \(-0.561600\pi\)
−0.192317 + 0.981333i \(0.561600\pi\)
\(422\) 2.50940e6 0.685944
\(423\) −2.07757e6 −0.564554
\(424\) 5.25325e6 1.41910
\(425\) −514534. −0.138179
\(426\) −3.76482e6 −1.00513
\(427\) 7.72026e6 2.04910
\(428\) −9.67282e6 −2.55237
\(429\) −118526. −0.0310936
\(430\) 758443. 0.197812
\(431\) −2.45338e6 −0.636168 −0.318084 0.948063i \(-0.603039\pi\)
−0.318084 + 0.948063i \(0.603039\pi\)
\(432\) 13240.1 0.00341337
\(433\) −7.33998e6 −1.88137 −0.940686 0.339277i \(-0.889818\pi\)
−0.940686 + 0.339277i \(0.889818\pi\)
\(434\) 1.25883e7 3.20806
\(435\) 192384. 0.0487468
\(436\) −3.28966e6 −0.828773
\(437\) −112811. −0.0282585
\(438\) 4.37613e6 1.08994
\(439\) 4.56023e6 1.12934 0.564671 0.825316i \(-0.309003\pi\)
0.564671 + 0.825316i \(0.309003\pi\)
\(440\) −12779.5 −0.00314689
\(441\) −5.52070e6 −1.35175
\(442\) 1.07661e6 0.262121
\(443\) 4.31520e6 1.04470 0.522350 0.852731i \(-0.325055\pi\)
0.522350 + 0.852731i \(0.325055\pi\)
\(444\) 5.40956e6 1.30228
\(445\) 211322. 0.0505877
\(446\) 1.96743e6 0.468342
\(447\) −1.31279e6 −0.310761
\(448\) 1.20070e7 2.82643
\(449\) 2.53058e6 0.592386 0.296193 0.955128i \(-0.404283\pi\)
0.296193 + 0.955128i \(0.404283\pi\)
\(450\) −4.56171e6 −1.06193
\(451\) −241593. −0.0559297
\(452\) 2.18707e6 0.503520
\(453\) −1.11167e6 −0.254524
\(454\) −7.66135e6 −1.74448
\(455\) 621981. 0.140847
\(456\) −350060. −0.0788370
\(457\) 1.61209e6 0.361077 0.180538 0.983568i \(-0.442216\pi\)
0.180538 + 0.983568i \(0.442216\pi\)
\(458\) 1.22535e7 2.72958
\(459\) −606777. −0.134430
\(460\) −105713. −0.0232934
\(461\) 360859. 0.0790833 0.0395417 0.999218i \(-0.487410\pi\)
0.0395417 + 0.999218i \(0.487410\pi\)
\(462\) 345223. 0.0752480
\(463\) −4.98842e6 −1.08146 −0.540730 0.841196i \(-0.681852\pi\)
−0.540730 + 0.841196i \(0.681852\pi\)
\(464\) −19767.7 −0.00426246
\(465\) 213510. 0.0457916
\(466\) 2.93491e6 0.626080
\(467\) −3.38295e6 −0.717800 −0.358900 0.933376i \(-0.616848\pi\)
−0.358900 + 0.933376i \(0.616848\pi\)
\(468\) 5.89687e6 1.24453
\(469\) −2.94025e6 −0.617236
\(470\) −458165. −0.0956704
\(471\) −1.04830e6 −0.217737
\(472\) −1.89640e6 −0.391809
\(473\) −393191. −0.0808073
\(474\) −4.79336e6 −0.979929
\(475\) −663235. −0.134876
\(476\) −1.93728e6 −0.391900
\(477\) 4.66518e6 0.938799
\(478\) 2.54997e6 0.510465
\(479\) −8.41871e6 −1.67651 −0.838256 0.545276i \(-0.816425\pi\)
−0.838256 + 0.545276i \(0.816425\pi\)
\(480\) 204094. 0.0404322
\(481\) −8.17841e6 −1.61178
\(482\) 6.23004e6 1.22144
\(483\) 1.08906e6 0.212415
\(484\) −8.31325e6 −1.61309
\(485\) −453926. −0.0876255
\(486\) −8.62084e6 −1.65561
\(487\) 5.35500e6 1.02314 0.511572 0.859240i \(-0.329063\pi\)
0.511572 + 0.859240i \(0.329063\pi\)
\(488\) 6.15573e6 1.17012
\(489\) 1.09683e6 0.207428
\(490\) −1.21747e6 −0.229071
\(491\) −2.09557e6 −0.392282 −0.196141 0.980576i \(-0.562841\pi\)
−0.196141 + 0.980576i \(0.562841\pi\)
\(492\) −6.20139e6 −1.15498
\(493\) 905925. 0.167871
\(494\) 1.38775e6 0.255855
\(495\) −11348.9 −0.00208181
\(496\) −21938.4 −0.00400406
\(497\) −1.02421e7 −1.85993
\(498\) 3.39651e6 0.613704
\(499\) 7.41055e6 1.33229 0.666146 0.745821i \(-0.267943\pi\)
0.666146 + 0.745821i \(0.267943\pi\)
\(500\) −1.24599e6 −0.222889
\(501\) 4.33762e6 0.772070
\(502\) −3.75441e6 −0.664940
\(503\) 8.19468e6 1.44415 0.722075 0.691815i \(-0.243189\pi\)
0.722075 + 0.691815i \(0.243189\pi\)
\(504\) −6.55005e6 −1.14860
\(505\) −547457. −0.0955260
\(506\) 88707.0 0.0154022
\(507\) 1.22308e6 0.211318
\(508\) −6.20420e6 −1.06666
\(509\) −6.72046e6 −1.14975 −0.574876 0.818240i \(-0.694950\pi\)
−0.574876 + 0.818240i \(0.694950\pi\)
\(510\) −53185.7 −0.00905460
\(511\) 1.19051e7 2.01688
\(512\) −41822.6 −0.00705075
\(513\) −782137. −0.131217
\(514\) −1.55302e7 −2.59280
\(515\) 232513. 0.0386304
\(516\) −1.00927e7 −1.66872
\(517\) 237521. 0.0390820
\(518\) 2.38207e7 3.90059
\(519\) 3.10310e6 0.505682
\(520\) 495935. 0.0804298
\(521\) −4.57081e6 −0.737732 −0.368866 0.929483i \(-0.620254\pi\)
−0.368866 + 0.929483i \(0.620254\pi\)
\(522\) 8.03166e6 1.29012
\(523\) −35425.6 −0.00566321 −0.00283161 0.999996i \(-0.500901\pi\)
−0.00283161 + 0.999996i \(0.500901\pi\)
\(524\) 9.76121e6 1.55301
\(525\) 6.40278e6 1.01384
\(526\) −1.01642e7 −1.60181
\(527\) 1.00541e6 0.157694
\(528\) −601.642 −9.39190e−5 0
\(529\) 279841. 0.0434783
\(530\) 1.02881e6 0.159091
\(531\) −1.68411e6 −0.259199
\(532\) −2.49716e6 −0.382532
\(533\) 9.37554e6 1.42948
\(534\) −4.55176e6 −0.690758
\(535\) −722433. −0.109122
\(536\) −2.34440e6 −0.352468
\(537\) 1.43640e6 0.214951
\(538\) −1.22566e6 −0.182563
\(539\) 631160. 0.0935767
\(540\) −732923. −0.108162
\(541\) 2.95315e6 0.433803 0.216902 0.976193i \(-0.430405\pi\)
0.216902 + 0.976193i \(0.430405\pi\)
\(542\) −855385. −0.125073
\(543\) −1.21312e6 −0.176564
\(544\) 961067. 0.139237
\(545\) −245695. −0.0354328
\(546\) −1.33971e7 −1.92323
\(547\) −3.04020e6 −0.434444 −0.217222 0.976122i \(-0.569700\pi\)
−0.217222 + 0.976122i \(0.569700\pi\)
\(548\) −1.65978e7 −2.36101
\(549\) 5.46664e6 0.774087
\(550\) 521523. 0.0735134
\(551\) 1.16774e6 0.163858
\(552\) 868362. 0.121298
\(553\) −1.30402e7 −1.81330
\(554\) −5.27557e6 −0.730289
\(555\) 404023. 0.0556767
\(556\) 9.01390e6 1.23659
\(557\) −3.03791e6 −0.414894 −0.207447 0.978246i \(-0.566516\pi\)
−0.207447 + 0.978246i \(0.566516\pi\)
\(558\) 8.91364e6 1.21191
\(559\) 1.52586e7 2.06531
\(560\) 3157.19 0.000425433 0
\(561\) 27572.4 0.00369886
\(562\) 3.97740e6 0.531201
\(563\) −1.05140e6 −0.139796 −0.0698982 0.997554i \(-0.522267\pi\)
−0.0698982 + 0.997554i \(0.522267\pi\)
\(564\) 6.09688e6 0.807067
\(565\) 163345. 0.0215271
\(566\) 1.39264e7 1.82726
\(567\) −1.26730e6 −0.165548
\(568\) −8.16649e6 −1.06210
\(569\) 1.67008e6 0.216250 0.108125 0.994137i \(-0.465515\pi\)
0.108125 + 0.994137i \(0.465515\pi\)
\(570\) −68556.5 −0.00883815
\(571\) 2.71320e6 0.348250 0.174125 0.984724i \(-0.444290\pi\)
0.174125 + 0.984724i \(0.444290\pi\)
\(572\) −674167. −0.0861543
\(573\) −7.45610e6 −0.948692
\(574\) −2.73075e7 −3.45941
\(575\) 1.64523e6 0.207519
\(576\) 8.50202e6 1.06774
\(577\) 8.83363e6 1.10459 0.552293 0.833650i \(-0.313753\pi\)
0.552293 + 0.833650i \(0.313753\pi\)
\(578\) 1.27416e7 1.58636
\(579\) 3.54310e6 0.439224
\(580\) 1.09426e6 0.135068
\(581\) 9.24008e6 1.13563
\(582\) 9.77731e6 1.19650
\(583\) −533353. −0.0649895
\(584\) 9.49250e6 1.15172
\(585\) 440419. 0.0532079
\(586\) −1.24756e7 −1.50078
\(587\) −1.35169e7 −1.61913 −0.809566 0.587029i \(-0.800298\pi\)
−0.809566 + 0.587029i \(0.800298\pi\)
\(588\) 1.62011e7 1.93242
\(589\) 1.29597e6 0.153924
\(590\) −371395. −0.0439244
\(591\) 2.28329e6 0.268901
\(592\) −41513.8 −0.00486843
\(593\) 848223. 0.0990543 0.0495272 0.998773i \(-0.484229\pi\)
0.0495272 + 0.998773i \(0.484229\pi\)
\(594\) 615019. 0.0715191
\(595\) −144690. −0.0167550
\(596\) −7.46703e6 −0.861057
\(597\) 4.43386e6 0.509150
\(598\) −3.44247e6 −0.393656
\(599\) −1.05945e7 −1.20646 −0.603229 0.797568i \(-0.706120\pi\)
−0.603229 + 0.797568i \(0.706120\pi\)
\(600\) 5.10524e6 0.578946
\(601\) −209181. −0.0236231 −0.0118115 0.999930i \(-0.503760\pi\)
−0.0118115 + 0.999930i \(0.503760\pi\)
\(602\) −4.44428e7 −4.99816
\(603\) −2.08196e6 −0.233173
\(604\) −6.32306e6 −0.705237
\(605\) −620891. −0.0689647
\(606\) 1.17919e7 1.30438
\(607\) 1.02964e7 1.13427 0.567133 0.823627i \(-0.308053\pi\)
0.567133 + 0.823627i \(0.308053\pi\)
\(608\) 1.23882e6 0.135909
\(609\) −1.12732e7 −1.23170
\(610\) 1.20555e6 0.131178
\(611\) −9.21753e6 −0.998876
\(612\) −1.37177e6 −0.148048
\(613\) 1.62158e7 1.74296 0.871479 0.490432i \(-0.163161\pi\)
0.871479 + 0.490432i \(0.163161\pi\)
\(614\) 2.78026e7 2.97622
\(615\) −463162. −0.0493794
\(616\) 748843. 0.0795132
\(617\) 6.25365e6 0.661334 0.330667 0.943747i \(-0.392726\pi\)
0.330667 + 0.943747i \(0.392726\pi\)
\(618\) −5.00820e6 −0.527485
\(619\) −1.06232e7 −1.11437 −0.557183 0.830390i \(-0.688118\pi\)
−0.557183 + 0.830390i \(0.688118\pi\)
\(620\) 1.21443e6 0.126880
\(621\) 1.94018e6 0.201889
\(622\) 3.04418e7 3.15496
\(623\) −1.23829e7 −1.27821
\(624\) 23348.0 0.00240043
\(625\) 9.62593e6 0.985695
\(626\) 1.23495e6 0.125955
\(627\) 35540.9 0.00361044
\(628\) −5.96261e6 −0.603305
\(629\) 1.90252e6 0.191736
\(630\) −1.28278e6 −0.128766
\(631\) 8.45928e6 0.845785 0.422893 0.906180i \(-0.361015\pi\)
0.422893 + 0.906180i \(0.361015\pi\)
\(632\) −1.03976e7 −1.03547
\(633\) −2.49402e6 −0.247395
\(634\) −2.59781e7 −2.56676
\(635\) −463373. −0.0456033
\(636\) −1.36905e7 −1.34207
\(637\) −2.44936e7 −2.39168
\(638\) −918230. −0.0893099
\(639\) −7.25231e6 −0.702626
\(640\) 1.15678e6 0.111636
\(641\) −2.03496e7 −1.95619 −0.978095 0.208157i \(-0.933253\pi\)
−0.978095 + 0.208157i \(0.933253\pi\)
\(642\) 1.55608e7 1.49003
\(643\) −6.43818e6 −0.614095 −0.307047 0.951694i \(-0.599341\pi\)
−0.307047 + 0.951694i \(0.599341\pi\)
\(644\) 6.19449e6 0.588561
\(645\) −753794. −0.0713434
\(646\) −322828. −0.0304362
\(647\) −9.20281e6 −0.864290 −0.432145 0.901804i \(-0.642243\pi\)
−0.432145 + 0.901804i \(0.642243\pi\)
\(648\) −1.01048e6 −0.0945345
\(649\) 192538. 0.0179434
\(650\) −2.02388e7 −1.87889
\(651\) −1.25111e7 −1.15703
\(652\) 6.23867e6 0.574742
\(653\) 8.14140e6 0.747164 0.373582 0.927597i \(-0.378129\pi\)
0.373582 + 0.927597i \(0.378129\pi\)
\(654\) 5.29213e6 0.483823
\(655\) 729035. 0.0663965
\(656\) 47590.5 0.00431778
\(657\) 8.42988e6 0.761917
\(658\) 2.68473e7 2.41733
\(659\) −8.11833e6 −0.728204 −0.364102 0.931359i \(-0.618624\pi\)
−0.364102 + 0.931359i \(0.618624\pi\)
\(660\) 33304.6 0.00297608
\(661\) 7.02602e6 0.625469 0.312735 0.949841i \(-0.398755\pi\)
0.312735 + 0.949841i \(0.398755\pi\)
\(662\) −2.07517e7 −1.84039
\(663\) −1.07001e6 −0.0945373
\(664\) 7.36755e6 0.648490
\(665\) −186505. −0.0163545
\(666\) 1.68672e7 1.47352
\(667\) −2.89671e6 −0.252110
\(668\) 2.46720e7 2.13926
\(669\) −1.95537e6 −0.168914
\(670\) −459132. −0.0395140
\(671\) −624981. −0.0535871
\(672\) −1.19594e7 −1.02161
\(673\) −1.13585e7 −0.966683 −0.483341 0.875432i \(-0.660577\pi\)
−0.483341 + 0.875432i \(0.660577\pi\)
\(674\) 1.06677e7 0.904530
\(675\) 1.14066e7 0.963601
\(676\) 6.95678e6 0.585520
\(677\) 7.60093e6 0.637375 0.318687 0.947860i \(-0.396758\pi\)
0.318687 + 0.947860i \(0.396758\pi\)
\(678\) −3.51837e6 −0.293946
\(679\) 2.65988e7 2.21405
\(680\) −115368. −0.00956782
\(681\) 7.61439e6 0.629169
\(682\) −1.01906e6 −0.0838957
\(683\) −7.85446e6 −0.644265 −0.322133 0.946695i \(-0.604400\pi\)
−0.322133 + 0.946695i \(0.604400\pi\)
\(684\) −1.76822e6 −0.144509
\(685\) −1.23964e6 −0.100941
\(686\) 3.65263e7 2.96344
\(687\) −1.21784e7 −0.984457
\(688\) 77453.2 0.00623833
\(689\) 2.06979e7 1.66103
\(690\) 170062. 0.0135983
\(691\) −2.73211e6 −0.217672 −0.108836 0.994060i \(-0.534712\pi\)
−0.108836 + 0.994060i \(0.534712\pi\)
\(692\) 1.76502e7 1.40115
\(693\) 665015. 0.0526015
\(694\) −2.26148e6 −0.178236
\(695\) 673220. 0.0528683
\(696\) −8.98865e6 −0.703350
\(697\) −2.18100e6 −0.170049
\(698\) −7.96016e6 −0.618419
\(699\) −2.91692e6 −0.225804
\(700\) 3.64184e7 2.80916
\(701\) −3.62295e6 −0.278463 −0.139231 0.990260i \(-0.544463\pi\)
−0.139231 + 0.990260i \(0.544463\pi\)
\(702\) −2.38671e7 −1.82792
\(703\) 2.45235e6 0.187152
\(704\) −972004. −0.0739156
\(705\) 455357. 0.0345048
\(706\) −2.90040e7 −2.19001
\(707\) 3.20795e7 2.41368
\(708\) 4.94220e6 0.370542
\(709\) −4.19896e6 −0.313708 −0.156854 0.987622i \(-0.550135\pi\)
−0.156854 + 0.987622i \(0.550135\pi\)
\(710\) −1.59934e6 −0.119068
\(711\) −9.23362e6 −0.685012
\(712\) −9.87348e6 −0.729911
\(713\) −3.21480e6 −0.236827
\(714\) 3.11654e6 0.228785
\(715\) −50351.4 −0.00368338
\(716\) 8.17009e6 0.595586
\(717\) −2.53434e6 −0.184106
\(718\) −1.35902e7 −0.983816
\(719\) 1.31323e7 0.947369 0.473684 0.880695i \(-0.342924\pi\)
0.473684 + 0.880695i \(0.342924\pi\)
\(720\) 2235.58 0.000160716 0
\(721\) −1.36246e7 −0.976083
\(722\) 2.22407e7 1.58784
\(723\) −6.19185e6 −0.440529
\(724\) −6.90009e6 −0.489224
\(725\) −1.70302e7 −1.20330
\(726\) 1.33736e7 0.941691
\(727\) −1.90858e7 −1.33929 −0.669644 0.742683i \(-0.733553\pi\)
−0.669644 + 0.742683i \(0.733553\pi\)
\(728\) −2.90605e7 −2.03224
\(729\) 7.20765e6 0.502313
\(730\) 1.85903e6 0.129116
\(731\) −3.54957e6 −0.245687
\(732\) −1.60425e7 −1.10661
\(733\) −1.58603e7 −1.09031 −0.545157 0.838334i \(-0.683530\pi\)
−0.545157 + 0.838334i \(0.683530\pi\)
\(734\) −2.07710e7 −1.42304
\(735\) 1.21001e6 0.0826172
\(736\) −3.07303e6 −0.209108
\(737\) 238022. 0.0161417
\(738\) −1.93361e7 −1.30686
\(739\) 5.61015e6 0.377888 0.188944 0.981988i \(-0.439494\pi\)
0.188944 + 0.981988i \(0.439494\pi\)
\(740\) 2.29805e6 0.154269
\(741\) −1.37924e6 −0.0922774
\(742\) −6.02854e7 −4.01978
\(743\) 4.05128e6 0.269228 0.134614 0.990898i \(-0.457021\pi\)
0.134614 + 0.990898i \(0.457021\pi\)
\(744\) −9.97571e6 −0.660711
\(745\) −557689. −0.0368130
\(746\) −2.65362e7 −1.74579
\(747\) 6.54280e6 0.429005
\(748\) 156829. 0.0102488
\(749\) 4.23326e7 2.75722
\(750\) 2.00444e6 0.130119
\(751\) −500386. −0.0323747 −0.0161873 0.999869i \(-0.505153\pi\)
−0.0161873 + 0.999869i \(0.505153\pi\)
\(752\) −46788.4 −0.00301713
\(753\) 3.73140e6 0.239819
\(754\) 3.56339e7 2.28263
\(755\) −472250. −0.0301512
\(756\) 4.29473e7 2.73295
\(757\) 2.77142e7 1.75777 0.878886 0.477032i \(-0.158288\pi\)
0.878886 + 0.477032i \(0.158288\pi\)
\(758\) −9.31287e6 −0.588722
\(759\) −88163.3 −0.00555499
\(760\) −148710. −0.00933911
\(761\) 2.10372e7 1.31682 0.658411 0.752659i \(-0.271229\pi\)
0.658411 + 0.752659i \(0.271229\pi\)
\(762\) 9.98080e6 0.622698
\(763\) 1.43971e7 0.895288
\(764\) −4.24096e7 −2.62864
\(765\) −102453. −0.00632954
\(766\) 3.84194e7 2.36580
\(767\) −7.47185e6 −0.458606
\(768\) −9.48143e6 −0.580057
\(769\) 1.52722e7 0.931295 0.465647 0.884970i \(-0.345822\pi\)
0.465647 + 0.884970i \(0.345822\pi\)
\(770\) 146655. 0.00891395
\(771\) 1.54350e7 0.935126
\(772\) 2.01528e7 1.21700
\(773\) −5.03712e6 −0.303203 −0.151602 0.988442i \(-0.548443\pi\)
−0.151602 + 0.988442i \(0.548443\pi\)
\(774\) −3.14695e7 −1.88815
\(775\) −1.89003e7 −1.13036
\(776\) 2.12085e7 1.26432
\(777\) −2.36747e7 −1.40680
\(778\) −1.52983e7 −0.906138
\(779\) −2.81132e6 −0.165984
\(780\) −1.29246e6 −0.0760641
\(781\) 829129. 0.0486401
\(782\) 800812. 0.0468288
\(783\) −2.00833e7 −1.17066
\(784\) −124330. −0.00722413
\(785\) −445329. −0.0257933
\(786\) −1.57030e7 −0.906623
\(787\) −4.02266e6 −0.231514 −0.115757 0.993278i \(-0.536929\pi\)
−0.115757 + 0.993278i \(0.536929\pi\)
\(788\) 1.29872e7 0.745073
\(789\) 1.01019e7 0.577712
\(790\) −2.03628e6 −0.116083
\(791\) −9.57161e6 −0.543931
\(792\) 530248. 0.0300377
\(793\) 2.42537e7 1.36961
\(794\) 1.25040e7 0.703879
\(795\) −1.02250e6 −0.0573781
\(796\) 2.52194e7 1.41075
\(797\) −3.18310e7 −1.77503 −0.887514 0.460781i \(-0.847569\pi\)
−0.887514 + 0.460781i \(0.847569\pi\)
\(798\) 4.01723e6 0.223316
\(799\) 2.14425e6 0.118825
\(800\) −1.80668e7 −0.998059
\(801\) −8.76821e6 −0.482869
\(802\) 1.19335e7 0.655134
\(803\) −963756. −0.0527446
\(804\) 6.10974e6 0.333336
\(805\) 462648. 0.0251629
\(806\) 3.95469e7 2.14425
\(807\) 1.21814e6 0.0658438
\(808\) 2.55785e7 1.37831
\(809\) 1.26438e7 0.679213 0.339606 0.940568i \(-0.389706\pi\)
0.339606 + 0.940568i \(0.389706\pi\)
\(810\) −197895. −0.0105979
\(811\) −1.06282e7 −0.567426 −0.283713 0.958909i \(-0.591566\pi\)
−0.283713 + 0.958909i \(0.591566\pi\)
\(812\) −6.41209e7 −3.41279
\(813\) 850142. 0.0451092
\(814\) −1.92836e6 −0.102006
\(815\) 465947. 0.0245721
\(816\) −5431.39 −0.000285552 0
\(817\) −4.57541e6 −0.239814
\(818\) −4.03652e6 −0.210923
\(819\) −2.58074e7 −1.34442
\(820\) −2.63442e6 −0.136821
\(821\) 3.63064e7 1.87986 0.939930 0.341366i \(-0.110890\pi\)
0.939930 + 0.341366i \(0.110890\pi\)
\(822\) 2.67011e7 1.37832
\(823\) −6.98474e6 −0.359460 −0.179730 0.983716i \(-0.557522\pi\)
−0.179730 + 0.983716i \(0.557522\pi\)
\(824\) −1.08636e7 −0.557384
\(825\) −518326. −0.0265136
\(826\) 2.17627e7 1.10985
\(827\) 2.85277e7 1.45045 0.725227 0.688510i \(-0.241735\pi\)
0.725227 + 0.688510i \(0.241735\pi\)
\(828\) 4.38626e6 0.222340
\(829\) −2.16227e7 −1.09276 −0.546378 0.837538i \(-0.683994\pi\)
−0.546378 + 0.837538i \(0.683994\pi\)
\(830\) 1.44288e6 0.0727000
\(831\) 5.24323e6 0.263388
\(832\) 3.77207e7 1.88917
\(833\) 5.69786e6 0.284511
\(834\) −1.45008e7 −0.721899
\(835\) 1.84267e6 0.0914602
\(836\) 202154. 0.0100038
\(837\) −2.22887e7 −1.09969
\(838\) −1.73557e7 −0.853755
\(839\) −3.42589e7 −1.68023 −0.840115 0.542408i \(-0.817513\pi\)
−0.840115 + 0.542408i \(0.817513\pi\)
\(840\) 1.43562e6 0.0702008
\(841\) 9.47345e6 0.461868
\(842\) 1.27992e7 0.622163
\(843\) −3.95302e6 −0.191584
\(844\) −1.41857e7 −0.685482
\(845\) 519581. 0.0250329
\(846\) 1.90103e7 0.913193
\(847\) 3.63825e7 1.74255
\(848\) 105063. 0.00501719
\(849\) −1.38411e7 −0.659023
\(850\) 4.70810e6 0.223511
\(851\) −6.08334e6 −0.287951
\(852\) 2.12827e7 1.00445
\(853\) −3.62802e7 −1.70725 −0.853624 0.520890i \(-0.825600\pi\)
−0.853624 + 0.520890i \(0.825600\pi\)
\(854\) −7.06421e7 −3.31451
\(855\) −132063. −0.00617824
\(856\) 3.37538e7 1.57449
\(857\) −1.47179e7 −0.684531 −0.342265 0.939603i \(-0.611194\pi\)
−0.342265 + 0.939603i \(0.611194\pi\)
\(858\) 1.08454e6 0.0502954
\(859\) −1.22343e7 −0.565713 −0.282856 0.959162i \(-0.591282\pi\)
−0.282856 + 0.959162i \(0.591282\pi\)
\(860\) −4.28751e6 −0.197678
\(861\) 2.71401e7 1.24768
\(862\) 2.24490e7 1.02903
\(863\) 2.54861e7 1.16487 0.582434 0.812878i \(-0.302100\pi\)
0.582434 + 0.812878i \(0.302100\pi\)
\(864\) −2.13057e7 −0.970984
\(865\) 1.31824e6 0.0599036
\(866\) 6.71624e7 3.04321
\(867\) −1.26635e7 −0.572143
\(868\) −7.11621e7 −3.20590
\(869\) 1.05565e6 0.0474208
\(870\) −1.76036e6 −0.0788502
\(871\) −9.23698e6 −0.412558
\(872\) 1.14795e7 0.511247
\(873\) 1.88344e7 0.836403
\(874\) 1.03225e6 0.0457094
\(875\) 5.45301e6 0.240778
\(876\) −2.47384e7 −1.08921
\(877\) 1.28458e7 0.563978 0.281989 0.959418i \(-0.409006\pi\)
0.281989 + 0.959418i \(0.409006\pi\)
\(878\) −4.17271e7 −1.82676
\(879\) 1.23991e7 0.541276
\(880\) −255.585 −1.11257e−5 0
\(881\) 2.97345e7 1.29069 0.645344 0.763892i \(-0.276714\pi\)
0.645344 + 0.763892i \(0.276714\pi\)
\(882\) 5.05156e7 2.18652
\(883\) 7.58412e6 0.327343 0.163672 0.986515i \(-0.447666\pi\)
0.163672 + 0.986515i \(0.447666\pi\)
\(884\) −6.08611e6 −0.261944
\(885\) 369118. 0.0158419
\(886\) −3.94850e7 −1.68985
\(887\) −4.35352e7 −1.85794 −0.928970 0.370155i \(-0.879305\pi\)
−0.928970 + 0.370155i \(0.879305\pi\)
\(888\) −1.88769e7 −0.803340
\(889\) 2.71524e7 1.15227
\(890\) −1.93364e6 −0.0818279
\(891\) 102592. 0.00432933
\(892\) −1.11220e7 −0.468026
\(893\) 2.76394e6 0.115985
\(894\) 1.20123e7 0.502670
\(895\) 610199. 0.0254633
\(896\) −6.77845e7 −2.82072
\(897\) 3.42137e6 0.141977
\(898\) −2.31554e7 −0.958212
\(899\) 3.32773e7 1.37325
\(900\) 2.57875e7 1.06121
\(901\) −4.81490e6 −0.197595
\(902\) 2.21063e6 0.0904689
\(903\) 4.41703e7 1.80265
\(904\) −7.63190e6 −0.310607
\(905\) −515346. −0.0209160
\(906\) 1.01720e7 0.411705
\(907\) 3.27804e7 1.32311 0.661555 0.749897i \(-0.269897\pi\)
0.661555 + 0.749897i \(0.269897\pi\)
\(908\) 4.33099e7 1.74330
\(909\) 2.27152e7 0.911815
\(910\) −5.69127e6 −0.227827
\(911\) −3.93141e7 −1.56947 −0.784733 0.619834i \(-0.787200\pi\)
−0.784733 + 0.619834i \(0.787200\pi\)
\(912\) −7001.08 −0.000278726 0
\(913\) −748014. −0.0296984
\(914\) −1.47510e7 −0.584058
\(915\) −1.19816e6 −0.0473111
\(916\) −6.92693e7 −2.72774
\(917\) −4.27195e7 −1.67766
\(918\) 5.55214e6 0.217447
\(919\) −2.97680e7 −1.16268 −0.581342 0.813660i \(-0.697472\pi\)
−0.581342 + 0.813660i \(0.697472\pi\)
\(920\) 368891. 0.0143691
\(921\) −2.76322e7 −1.07341
\(922\) −3.30194e6 −0.127921
\(923\) −3.21762e7 −1.24317
\(924\) −1.95156e6 −0.0751973
\(925\) −3.57649e7 −1.37437
\(926\) 4.56452e7 1.74931
\(927\) −9.64746e6 −0.368735
\(928\) 3.18097e7 1.21252
\(929\) 3.60951e7 1.37217 0.686086 0.727521i \(-0.259327\pi\)
0.686086 + 0.727521i \(0.259327\pi\)
\(930\) −1.95366e6 −0.0740701
\(931\) 7.34457e6 0.277710
\(932\) −1.65912e7 −0.625658
\(933\) −3.02552e7 −1.13788
\(934\) 3.09548e7 1.16108
\(935\) 11713.1 0.000438170 0
\(936\) −2.05774e7 −0.767718
\(937\) −4.14725e7 −1.54316 −0.771581 0.636131i \(-0.780534\pi\)
−0.771581 + 0.636131i \(0.780534\pi\)
\(938\) 2.69039e7 0.998408
\(939\) −1.22738e6 −0.0454272
\(940\) 2.59003e6 0.0956059
\(941\) 3.05025e6 0.112295 0.0561476 0.998422i \(-0.482118\pi\)
0.0561476 + 0.998422i \(0.482118\pi\)
\(942\) 9.59214e6 0.352199
\(943\) 6.97379e6 0.255382
\(944\) −37927.3 −0.00138523
\(945\) 3.20760e6 0.116843
\(946\) 3.59779e6 0.130710
\(947\) −2.14155e7 −0.775983 −0.387992 0.921663i \(-0.626831\pi\)
−0.387992 + 0.921663i \(0.626831\pi\)
\(948\) 2.70971e7 0.979269
\(949\) 3.74007e7 1.34807
\(950\) 6.06875e6 0.218168
\(951\) 2.58189e7 0.925734
\(952\) 6.76026e6 0.241752
\(953\) −4.74956e7 −1.69403 −0.847015 0.531569i \(-0.821602\pi\)
−0.847015 + 0.531569i \(0.821602\pi\)
\(954\) −4.26875e7 −1.51855
\(955\) −3.16744e6 −0.112383
\(956\) −1.44151e7 −0.510121
\(957\) 912601. 0.0322108
\(958\) 7.70331e7 2.71184
\(959\) 7.26394e7 2.55050
\(960\) −1.86345e6 −0.0652588
\(961\) 8.30235e6 0.289996
\(962\) 7.48343e7 2.60713
\(963\) 2.99753e7 1.04159
\(964\) −3.52187e7 −1.22062
\(965\) 1.50515e6 0.0520309
\(966\) −9.96518e6 −0.343591
\(967\) 761545. 0.0261896 0.0130948 0.999914i \(-0.495832\pi\)
0.0130948 + 0.999914i \(0.495832\pi\)
\(968\) 2.90095e7 0.995067
\(969\) 320849. 0.0109772
\(970\) 4.15352e6 0.141738
\(971\) 1.28371e7 0.436936 0.218468 0.975844i \(-0.429894\pi\)
0.218468 + 0.975844i \(0.429894\pi\)
\(972\) 4.87340e7 1.65450
\(973\) −3.94489e7 −1.33583
\(974\) −4.89994e7 −1.65498
\(975\) 2.01148e7 0.677647
\(976\) 123113. 0.00413693
\(977\) −1.89960e7 −0.636688 −0.318344 0.947975i \(-0.603127\pi\)
−0.318344 + 0.947975i \(0.603127\pi\)
\(978\) −1.00362e7 −0.335524
\(979\) 1.00244e6 0.0334272
\(980\) 6.88243e6 0.228916
\(981\) 1.01944e7 0.338213
\(982\) 1.91749e7 0.634534
\(983\) 3.27541e7 1.08114 0.540569 0.841300i \(-0.318209\pi\)
0.540569 + 0.841300i \(0.318209\pi\)
\(984\) 2.16401e7 0.712477
\(985\) 969972. 0.0318543
\(986\) −8.28941e6 −0.271539
\(987\) −2.66827e7 −0.871840
\(988\) −7.84501e6 −0.255683
\(989\) 1.13498e7 0.368976
\(990\) 103845. 0.00336742
\(991\) −7.38784e6 −0.238965 −0.119482 0.992836i \(-0.538123\pi\)
−0.119482 + 0.992836i \(0.538123\pi\)
\(992\) 3.53028e7 1.13902
\(993\) 2.06245e7 0.663760
\(994\) 9.37173e7 3.00853
\(995\) 1.88356e6 0.0603144
\(996\) −1.92006e7 −0.613291
\(997\) −3.41271e7 −1.08733 −0.543666 0.839302i \(-0.682964\pi\)
−0.543666 + 0.839302i \(0.682964\pi\)
\(998\) −6.78082e7 −2.15504
\(999\) −4.21767e7 −1.33708
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 23.6.a.a.1.1 3
3.2 odd 2 207.6.a.b.1.3 3
4.3 odd 2 368.6.a.e.1.1 3
5.4 even 2 575.6.a.b.1.3 3
23.22 odd 2 529.6.a.a.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.6.a.a.1.1 3 1.1 even 1 trivial
207.6.a.b.1.3 3 3.2 odd 2
368.6.a.e.1.1 3 4.3 odd 2
529.6.a.a.1.1 3 23.22 odd 2
575.6.a.b.1.3 3 5.4 even 2