Properties

Label 1232.4.a.s.1.3
Level $1232$
Weight $4$
Character 1232.1
Self dual yes
Analytic conductor $72.690$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.6903531271\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.522072.1
Defining polynomial: \( x^{4} - x^{3} - 12x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.148103\) of defining polynomial
Character \(\chi\) \(=\) 1232.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.77399 q^{3} +1.84418 q^{5} +7.00000 q^{7} -19.3050 q^{9} +O(q^{10})\) \(q-2.77399 q^{3} +1.84418 q^{5} +7.00000 q^{7} -19.3050 q^{9} +11.0000 q^{11} +24.6401 q^{13} -5.11574 q^{15} +17.8800 q^{17} -32.1459 q^{19} -19.4179 q^{21} -14.1248 q^{23} -121.599 q^{25} +128.450 q^{27} -41.5471 q^{29} -175.766 q^{31} -30.5139 q^{33} +12.9093 q^{35} +292.877 q^{37} -68.3513 q^{39} +154.296 q^{41} +277.144 q^{43} -35.6019 q^{45} +52.1450 q^{47} +49.0000 q^{49} -49.5989 q^{51} +82.3907 q^{53} +20.2860 q^{55} +89.1723 q^{57} -712.816 q^{59} -647.078 q^{61} -135.135 q^{63} +45.4408 q^{65} -260.867 q^{67} +39.1821 q^{69} -369.025 q^{71} +1145.77 q^{73} +337.314 q^{75} +77.0000 q^{77} -488.885 q^{79} +164.917 q^{81} -548.982 q^{83} +32.9740 q^{85} +115.251 q^{87} +105.039 q^{89} +172.481 q^{91} +487.572 q^{93} -59.2829 q^{95} -1361.91 q^{97} -212.355 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.77399 −0.533854 −0.266927 0.963717i \(-0.586008\pi\)
−0.266927 + 0.963717i \(0.586008\pi\)
\(4\) 0 0
\(5\) 1.84418 0.164949 0.0824744 0.996593i \(-0.473718\pi\)
0.0824744 + 0.996593i \(0.473718\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) −19.3050 −0.715000
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 24.6401 0.525687 0.262844 0.964838i \(-0.415340\pi\)
0.262844 + 0.964838i \(0.415340\pi\)
\(14\) 0 0
\(15\) −5.11574 −0.0880586
\(16\) 0 0
\(17\) 17.8800 0.255090 0.127545 0.991833i \(-0.459290\pi\)
0.127545 + 0.991833i \(0.459290\pi\)
\(18\) 0 0
\(19\) −32.1459 −0.388146 −0.194073 0.980987i \(-0.562170\pi\)
−0.194073 + 0.980987i \(0.562170\pi\)
\(20\) 0 0
\(21\) −19.4179 −0.201778
\(22\) 0 0
\(23\) −14.1248 −0.128054 −0.0640268 0.997948i \(-0.520394\pi\)
−0.0640268 + 0.997948i \(0.520394\pi\)
\(24\) 0 0
\(25\) −121.599 −0.972792
\(26\) 0 0
\(27\) 128.450 0.915560
\(28\) 0 0
\(29\) −41.5471 −0.266038 −0.133019 0.991113i \(-0.542467\pi\)
−0.133019 + 0.991113i \(0.542467\pi\)
\(30\) 0 0
\(31\) −175.766 −1.01834 −0.509169 0.860667i \(-0.670047\pi\)
−0.509169 + 0.860667i \(0.670047\pi\)
\(32\) 0 0
\(33\) −30.5139 −0.160963
\(34\) 0 0
\(35\) 12.9093 0.0623448
\(36\) 0 0
\(37\) 292.877 1.30131 0.650657 0.759372i \(-0.274494\pi\)
0.650657 + 0.759372i \(0.274494\pi\)
\(38\) 0 0
\(39\) −68.3513 −0.280640
\(40\) 0 0
\(41\) 154.296 0.587732 0.293866 0.955847i \(-0.405058\pi\)
0.293866 + 0.955847i \(0.405058\pi\)
\(42\) 0 0
\(43\) 277.144 0.982887 0.491443 0.870910i \(-0.336470\pi\)
0.491443 + 0.870910i \(0.336470\pi\)
\(44\) 0 0
\(45\) −35.6019 −0.117938
\(46\) 0 0
\(47\) 52.1450 0.161832 0.0809162 0.996721i \(-0.474215\pi\)
0.0809162 + 0.996721i \(0.474215\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −49.5989 −0.136181
\(52\) 0 0
\(53\) 82.3907 0.213533 0.106766 0.994284i \(-0.465950\pi\)
0.106766 + 0.994284i \(0.465950\pi\)
\(54\) 0 0
\(55\) 20.2860 0.0497339
\(56\) 0 0
\(57\) 89.1723 0.207213
\(58\) 0 0
\(59\) −712.816 −1.57289 −0.786447 0.617658i \(-0.788081\pi\)
−0.786447 + 0.617658i \(0.788081\pi\)
\(60\) 0 0
\(61\) −647.078 −1.35819 −0.679097 0.734048i \(-0.737629\pi\)
−0.679097 + 0.734048i \(0.737629\pi\)
\(62\) 0 0
\(63\) −135.135 −0.270244
\(64\) 0 0
\(65\) 45.4408 0.0867114
\(66\) 0 0
\(67\) −260.867 −0.475672 −0.237836 0.971305i \(-0.576438\pi\)
−0.237836 + 0.971305i \(0.576438\pi\)
\(68\) 0 0
\(69\) 39.1821 0.0683619
\(70\) 0 0
\(71\) −369.025 −0.616833 −0.308417 0.951251i \(-0.599799\pi\)
−0.308417 + 0.951251i \(0.599799\pi\)
\(72\) 0 0
\(73\) 1145.77 1.83702 0.918509 0.395401i \(-0.129394\pi\)
0.918509 + 0.395401i \(0.129394\pi\)
\(74\) 0 0
\(75\) 337.314 0.519329
\(76\) 0 0
\(77\) 77.0000 0.113961
\(78\) 0 0
\(79\) −488.885 −0.696251 −0.348125 0.937448i \(-0.613182\pi\)
−0.348125 + 0.937448i \(0.613182\pi\)
\(80\) 0 0
\(81\) 164.917 0.226224
\(82\) 0 0
\(83\) −548.982 −0.726008 −0.363004 0.931788i \(-0.618249\pi\)
−0.363004 + 0.931788i \(0.618249\pi\)
\(84\) 0 0
\(85\) 32.9740 0.0420769
\(86\) 0 0
\(87\) 115.251 0.142026
\(88\) 0 0
\(89\) 105.039 0.125102 0.0625510 0.998042i \(-0.480076\pi\)
0.0625510 + 0.998042i \(0.480076\pi\)
\(90\) 0 0
\(91\) 172.481 0.198691
\(92\) 0 0
\(93\) 487.572 0.543644
\(94\) 0 0
\(95\) −59.2829 −0.0640242
\(96\) 0 0
\(97\) −1361.91 −1.42557 −0.712787 0.701381i \(-0.752567\pi\)
−0.712787 + 0.701381i \(0.752567\pi\)
\(98\) 0 0
\(99\) −212.355 −0.215580
\(100\) 0 0
\(101\) −1610.32 −1.58647 −0.793234 0.608917i \(-0.791604\pi\)
−0.793234 + 0.608917i \(0.791604\pi\)
\(102\) 0 0
\(103\) 123.044 0.117708 0.0588540 0.998267i \(-0.481255\pi\)
0.0588540 + 0.998267i \(0.481255\pi\)
\(104\) 0 0
\(105\) −35.8102 −0.0332830
\(106\) 0 0
\(107\) 1740.90 1.57289 0.786446 0.617660i \(-0.211919\pi\)
0.786446 + 0.617660i \(0.211919\pi\)
\(108\) 0 0
\(109\) 248.938 0.218752 0.109376 0.994000i \(-0.465115\pi\)
0.109376 + 0.994000i \(0.465115\pi\)
\(110\) 0 0
\(111\) −812.436 −0.694712
\(112\) 0 0
\(113\) −494.465 −0.411641 −0.205820 0.978590i \(-0.565986\pi\)
−0.205820 + 0.978590i \(0.565986\pi\)
\(114\) 0 0
\(115\) −26.0488 −0.0211223
\(116\) 0 0
\(117\) −475.677 −0.375866
\(118\) 0 0
\(119\) 125.160 0.0964151
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −428.016 −0.313763
\(124\) 0 0
\(125\) −454.774 −0.325410
\(126\) 0 0
\(127\) 979.104 0.684106 0.342053 0.939681i \(-0.388878\pi\)
0.342053 + 0.939681i \(0.388878\pi\)
\(128\) 0 0
\(129\) −768.795 −0.524718
\(130\) 0 0
\(131\) 1660.85 1.10770 0.553851 0.832616i \(-0.313158\pi\)
0.553851 + 0.832616i \(0.313158\pi\)
\(132\) 0 0
\(133\) −225.021 −0.146705
\(134\) 0 0
\(135\) 236.884 0.151020
\(136\) 0 0
\(137\) −1618.17 −1.00912 −0.504559 0.863377i \(-0.668345\pi\)
−0.504559 + 0.863377i \(0.668345\pi\)
\(138\) 0 0
\(139\) −695.736 −0.424544 −0.212272 0.977211i \(-0.568086\pi\)
−0.212272 + 0.977211i \(0.568086\pi\)
\(140\) 0 0
\(141\) −144.650 −0.0863950
\(142\) 0 0
\(143\) 271.041 0.158501
\(144\) 0 0
\(145\) −76.6205 −0.0438826
\(146\) 0 0
\(147\) −135.925 −0.0762649
\(148\) 0 0
\(149\) −2081.84 −1.14464 −0.572318 0.820032i \(-0.693956\pi\)
−0.572318 + 0.820032i \(0.693956\pi\)
\(150\) 0 0
\(151\) −2679.28 −1.44395 −0.721975 0.691919i \(-0.756766\pi\)
−0.721975 + 0.691919i \(0.756766\pi\)
\(152\) 0 0
\(153\) −345.173 −0.182390
\(154\) 0 0
\(155\) −324.144 −0.167973
\(156\) 0 0
\(157\) 2410.32 1.22525 0.612627 0.790372i \(-0.290113\pi\)
0.612627 + 0.790372i \(0.290113\pi\)
\(158\) 0 0
\(159\) −228.551 −0.113995
\(160\) 0 0
\(161\) −98.8738 −0.0483997
\(162\) 0 0
\(163\) −3629.08 −1.74388 −0.871938 0.489616i \(-0.837137\pi\)
−0.871938 + 0.489616i \(0.837137\pi\)
\(164\) 0 0
\(165\) −56.2732 −0.0265507
\(166\) 0 0
\(167\) 3826.04 1.77286 0.886432 0.462859i \(-0.153176\pi\)
0.886432 + 0.462859i \(0.153176\pi\)
\(168\) 0 0
\(169\) −1589.87 −0.723653
\(170\) 0 0
\(171\) 620.576 0.277524
\(172\) 0 0
\(173\) −4310.67 −1.89442 −0.947209 0.320617i \(-0.896110\pi\)
−0.947209 + 0.320617i \(0.896110\pi\)
\(174\) 0 0
\(175\) −851.193 −0.367681
\(176\) 0 0
\(177\) 1977.34 0.839696
\(178\) 0 0
\(179\) −2491.12 −1.04019 −0.520097 0.854107i \(-0.674104\pi\)
−0.520097 + 0.854107i \(0.674104\pi\)
\(180\) 0 0
\(181\) −4315.49 −1.77220 −0.886098 0.463498i \(-0.846594\pi\)
−0.886098 + 0.463498i \(0.846594\pi\)
\(182\) 0 0
\(183\) 1794.99 0.725078
\(184\) 0 0
\(185\) 540.118 0.214650
\(186\) 0 0
\(187\) 196.680 0.0769127
\(188\) 0 0
\(189\) 899.147 0.346049
\(190\) 0 0
\(191\) −2840.41 −1.07605 −0.538024 0.842930i \(-0.680829\pi\)
−0.538024 + 0.842930i \(0.680829\pi\)
\(192\) 0 0
\(193\) −1734.68 −0.646969 −0.323485 0.946233i \(-0.604854\pi\)
−0.323485 + 0.946233i \(0.604854\pi\)
\(194\) 0 0
\(195\) −126.052 −0.0462913
\(196\) 0 0
\(197\) −3098.42 −1.12057 −0.560287 0.828298i \(-0.689309\pi\)
−0.560287 + 0.828298i \(0.689309\pi\)
\(198\) 0 0
\(199\) −4497.38 −1.60206 −0.801032 0.598622i \(-0.795715\pi\)
−0.801032 + 0.598622i \(0.795715\pi\)
\(200\) 0 0
\(201\) 723.643 0.253940
\(202\) 0 0
\(203\) −290.830 −0.100553
\(204\) 0 0
\(205\) 284.550 0.0969456
\(206\) 0 0
\(207\) 272.680 0.0915582
\(208\) 0 0
\(209\) −353.605 −0.117030
\(210\) 0 0
\(211\) −1262.32 −0.411857 −0.205929 0.978567i \(-0.566021\pi\)
−0.205929 + 0.978567i \(0.566021\pi\)
\(212\) 0 0
\(213\) 1023.67 0.329299
\(214\) 0 0
\(215\) 511.105 0.162126
\(216\) 0 0
\(217\) −1230.36 −0.384895
\(218\) 0 0
\(219\) −3178.35 −0.980700
\(220\) 0 0
\(221\) 440.565 0.134098
\(222\) 0 0
\(223\) −2931.38 −0.880268 −0.440134 0.897932i \(-0.645069\pi\)
−0.440134 + 0.897932i \(0.645069\pi\)
\(224\) 0 0
\(225\) 2347.47 0.695546
\(226\) 0 0
\(227\) −4298.37 −1.25680 −0.628399 0.777891i \(-0.716289\pi\)
−0.628399 + 0.777891i \(0.716289\pi\)
\(228\) 0 0
\(229\) 698.500 0.201564 0.100782 0.994909i \(-0.467866\pi\)
0.100782 + 0.994909i \(0.467866\pi\)
\(230\) 0 0
\(231\) −213.597 −0.0608383
\(232\) 0 0
\(233\) 1887.78 0.530783 0.265391 0.964141i \(-0.414499\pi\)
0.265391 + 0.964141i \(0.414499\pi\)
\(234\) 0 0
\(235\) 96.1649 0.0266941
\(236\) 0 0
\(237\) 1356.16 0.371697
\(238\) 0 0
\(239\) 6449.93 1.74565 0.872827 0.488030i \(-0.162284\pi\)
0.872827 + 0.488030i \(0.162284\pi\)
\(240\) 0 0
\(241\) 4636.98 1.23939 0.619697 0.784841i \(-0.287256\pi\)
0.619697 + 0.784841i \(0.287256\pi\)
\(242\) 0 0
\(243\) −3925.62 −1.03633
\(244\) 0 0
\(245\) 90.3650 0.0235641
\(246\) 0 0
\(247\) −792.078 −0.204043
\(248\) 0 0
\(249\) 1522.87 0.387582
\(250\) 0 0
\(251\) −2194.11 −0.551758 −0.275879 0.961192i \(-0.588969\pi\)
−0.275879 + 0.961192i \(0.588969\pi\)
\(252\) 0 0
\(253\) −155.373 −0.0386096
\(254\) 0 0
\(255\) −91.4695 −0.0224629
\(256\) 0 0
\(257\) −2966.97 −0.720133 −0.360067 0.932927i \(-0.617246\pi\)
−0.360067 + 0.932927i \(0.617246\pi\)
\(258\) 0 0
\(259\) 2050.14 0.491850
\(260\) 0 0
\(261\) 802.066 0.190217
\(262\) 0 0
\(263\) −915.810 −0.214720 −0.107360 0.994220i \(-0.534240\pi\)
−0.107360 + 0.994220i \(0.534240\pi\)
\(264\) 0 0
\(265\) 151.944 0.0352220
\(266\) 0 0
\(267\) −291.376 −0.0667863
\(268\) 0 0
\(269\) −164.462 −0.0372768 −0.0186384 0.999826i \(-0.505933\pi\)
−0.0186384 + 0.999826i \(0.505933\pi\)
\(270\) 0 0
\(271\) −1502.60 −0.336815 −0.168407 0.985718i \(-0.553862\pi\)
−0.168407 + 0.985718i \(0.553862\pi\)
\(272\) 0 0
\(273\) −478.459 −0.106072
\(274\) 0 0
\(275\) −1337.59 −0.293308
\(276\) 0 0
\(277\) −7500.11 −1.62685 −0.813426 0.581669i \(-0.802400\pi\)
−0.813426 + 0.581669i \(0.802400\pi\)
\(278\) 0 0
\(279\) 3393.15 0.728111
\(280\) 0 0
\(281\) −2945.12 −0.625235 −0.312618 0.949879i \(-0.601206\pi\)
−0.312618 + 0.949879i \(0.601206\pi\)
\(282\) 0 0
\(283\) −5215.29 −1.09547 −0.547733 0.836653i \(-0.684509\pi\)
−0.547733 + 0.836653i \(0.684509\pi\)
\(284\) 0 0
\(285\) 164.450 0.0341796
\(286\) 0 0
\(287\) 1080.07 0.222142
\(288\) 0 0
\(289\) −4593.31 −0.934929
\(290\) 0 0
\(291\) 3777.91 0.761049
\(292\) 0 0
\(293\) 7407.99 1.47706 0.738531 0.674219i \(-0.235520\pi\)
0.738531 + 0.674219i \(0.235520\pi\)
\(294\) 0 0
\(295\) −1314.56 −0.259447
\(296\) 0 0
\(297\) 1412.94 0.276052
\(298\) 0 0
\(299\) −348.037 −0.0673161
\(300\) 0 0
\(301\) 1940.01 0.371496
\(302\) 0 0
\(303\) 4467.02 0.846942
\(304\) 0 0
\(305\) −1193.33 −0.224033
\(306\) 0 0
\(307\) 6850.01 1.27345 0.636727 0.771089i \(-0.280288\pi\)
0.636727 + 0.771089i \(0.280288\pi\)
\(308\) 0 0
\(309\) −341.324 −0.0628390
\(310\) 0 0
\(311\) 5538.62 1.00986 0.504929 0.863161i \(-0.331519\pi\)
0.504929 + 0.863161i \(0.331519\pi\)
\(312\) 0 0
\(313\) 9361.13 1.69049 0.845243 0.534382i \(-0.179456\pi\)
0.845243 + 0.534382i \(0.179456\pi\)
\(314\) 0 0
\(315\) −249.214 −0.0445765
\(316\) 0 0
\(317\) 219.221 0.0388413 0.0194206 0.999811i \(-0.493818\pi\)
0.0194206 + 0.999811i \(0.493818\pi\)
\(318\) 0 0
\(319\) −457.018 −0.0802135
\(320\) 0 0
\(321\) −4829.24 −0.839695
\(322\) 0 0
\(323\) −574.769 −0.0990123
\(324\) 0 0
\(325\) −2996.21 −0.511384
\(326\) 0 0
\(327\) −690.551 −0.116781
\(328\) 0 0
\(329\) 365.015 0.0611669
\(330\) 0 0
\(331\) −3377.63 −0.560880 −0.280440 0.959872i \(-0.590480\pi\)
−0.280440 + 0.959872i \(0.590480\pi\)
\(332\) 0 0
\(333\) −5653.98 −0.930439
\(334\) 0 0
\(335\) −481.087 −0.0784615
\(336\) 0 0
\(337\) 7755.93 1.25369 0.626843 0.779145i \(-0.284347\pi\)
0.626843 + 0.779145i \(0.284347\pi\)
\(338\) 0 0
\(339\) 1371.64 0.219756
\(340\) 0 0
\(341\) −1933.42 −0.307040
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 72.2590 0.0112762
\(346\) 0 0
\(347\) 831.044 0.128567 0.0642835 0.997932i \(-0.479524\pi\)
0.0642835 + 0.997932i \(0.479524\pi\)
\(348\) 0 0
\(349\) 1840.94 0.282359 0.141180 0.989984i \(-0.454911\pi\)
0.141180 + 0.989984i \(0.454911\pi\)
\(350\) 0 0
\(351\) 3165.01 0.481298
\(352\) 0 0
\(353\) 3409.11 0.514019 0.257009 0.966409i \(-0.417263\pi\)
0.257009 + 0.966409i \(0.417263\pi\)
\(354\) 0 0
\(355\) −680.549 −0.101746
\(356\) 0 0
\(357\) −347.193 −0.0514716
\(358\) 0 0
\(359\) −2199.75 −0.323394 −0.161697 0.986840i \(-0.551697\pi\)
−0.161697 + 0.986840i \(0.551697\pi\)
\(360\) 0 0
\(361\) −5825.64 −0.849343
\(362\) 0 0
\(363\) −335.653 −0.0485322
\(364\) 0 0
\(365\) 2113.01 0.303014
\(366\) 0 0
\(367\) 855.008 0.121611 0.0608053 0.998150i \(-0.480633\pi\)
0.0608053 + 0.998150i \(0.480633\pi\)
\(368\) 0 0
\(369\) −2978.69 −0.420228
\(370\) 0 0
\(371\) 576.735 0.0807078
\(372\) 0 0
\(373\) 3193.40 0.443293 0.221646 0.975127i \(-0.428857\pi\)
0.221646 + 0.975127i \(0.428857\pi\)
\(374\) 0 0
\(375\) 1261.54 0.173721
\(376\) 0 0
\(377\) −1023.72 −0.139853
\(378\) 0 0
\(379\) 5614.48 0.760940 0.380470 0.924793i \(-0.375762\pi\)
0.380470 + 0.924793i \(0.375762\pi\)
\(380\) 0 0
\(381\) −2716.02 −0.365213
\(382\) 0 0
\(383\) 1736.86 0.231721 0.115861 0.993265i \(-0.463037\pi\)
0.115861 + 0.993265i \(0.463037\pi\)
\(384\) 0 0
\(385\) 142.002 0.0187977
\(386\) 0 0
\(387\) −5350.27 −0.702763
\(388\) 0 0
\(389\) 8710.78 1.13536 0.567679 0.823250i \(-0.307841\pi\)
0.567679 + 0.823250i \(0.307841\pi\)
\(390\) 0 0
\(391\) −252.552 −0.0326652
\(392\) 0 0
\(393\) −4607.17 −0.591352
\(394\) 0 0
\(395\) −901.593 −0.114846
\(396\) 0 0
\(397\) 11731.6 1.48311 0.741553 0.670894i \(-0.234089\pi\)
0.741553 + 0.670894i \(0.234089\pi\)
\(398\) 0 0
\(399\) 624.206 0.0783193
\(400\) 0 0
\(401\) −14408.8 −1.79437 −0.897183 0.441659i \(-0.854390\pi\)
−0.897183 + 0.441659i \(0.854390\pi\)
\(402\) 0 0
\(403\) −4330.88 −0.535327
\(404\) 0 0
\(405\) 304.138 0.0373154
\(406\) 0 0
\(407\) 3221.64 0.392361
\(408\) 0 0
\(409\) −2155.54 −0.260598 −0.130299 0.991475i \(-0.541594\pi\)
−0.130299 + 0.991475i \(0.541594\pi\)
\(410\) 0 0
\(411\) 4488.77 0.538722
\(412\) 0 0
\(413\) −4989.71 −0.594498
\(414\) 0 0
\(415\) −1012.42 −0.119754
\(416\) 0 0
\(417\) 1929.96 0.226644
\(418\) 0 0
\(419\) −8443.41 −0.984458 −0.492229 0.870466i \(-0.663818\pi\)
−0.492229 + 0.870466i \(0.663818\pi\)
\(420\) 0 0
\(421\) −3070.28 −0.355430 −0.177715 0.984082i \(-0.556871\pi\)
−0.177715 + 0.984082i \(0.556871\pi\)
\(422\) 0 0
\(423\) −1006.66 −0.115710
\(424\) 0 0
\(425\) −2174.19 −0.248150
\(426\) 0 0
\(427\) −4529.55 −0.513349
\(428\) 0 0
\(429\) −751.865 −0.0846163
\(430\) 0 0
\(431\) 7004.40 0.782808 0.391404 0.920219i \(-0.371990\pi\)
0.391404 + 0.920219i \(0.371990\pi\)
\(432\) 0 0
\(433\) 7486.00 0.830841 0.415421 0.909629i \(-0.363634\pi\)
0.415421 + 0.909629i \(0.363634\pi\)
\(434\) 0 0
\(435\) 212.544 0.0234269
\(436\) 0 0
\(437\) 454.055 0.0497034
\(438\) 0 0
\(439\) 13046.2 1.41836 0.709179 0.705029i \(-0.249066\pi\)
0.709179 + 0.705029i \(0.249066\pi\)
\(440\) 0 0
\(441\) −945.944 −0.102143
\(442\) 0 0
\(443\) 11781.3 1.26354 0.631768 0.775157i \(-0.282329\pi\)
0.631768 + 0.775157i \(0.282329\pi\)
\(444\) 0 0
\(445\) 193.711 0.0206354
\(446\) 0 0
\(447\) 5774.99 0.611069
\(448\) 0 0
\(449\) 7576.58 0.796349 0.398175 0.917310i \(-0.369644\pi\)
0.398175 + 0.917310i \(0.369644\pi\)
\(450\) 0 0
\(451\) 1697.26 0.177208
\(452\) 0 0
\(453\) 7432.28 0.770859
\(454\) 0 0
\(455\) 318.086 0.0327738
\(456\) 0 0
\(457\) 11793.0 1.20712 0.603560 0.797318i \(-0.293748\pi\)
0.603560 + 0.797318i \(0.293748\pi\)
\(458\) 0 0
\(459\) 2296.68 0.233551
\(460\) 0 0
\(461\) 4228.32 0.427185 0.213592 0.976923i \(-0.431484\pi\)
0.213592 + 0.976923i \(0.431484\pi\)
\(462\) 0 0
\(463\) −14448.8 −1.45031 −0.725154 0.688586i \(-0.758232\pi\)
−0.725154 + 0.688586i \(0.758232\pi\)
\(464\) 0 0
\(465\) 899.172 0.0896733
\(466\) 0 0
\(467\) −16547.5 −1.63967 −0.819836 0.572599i \(-0.805935\pi\)
−0.819836 + 0.572599i \(0.805935\pi\)
\(468\) 0 0
\(469\) −1826.07 −0.179787
\(470\) 0 0
\(471\) −6686.21 −0.654107
\(472\) 0 0
\(473\) 3048.59 0.296351
\(474\) 0 0
\(475\) 3908.91 0.377585
\(476\) 0 0
\(477\) −1590.55 −0.152676
\(478\) 0 0
\(479\) 2989.34 0.285149 0.142575 0.989784i \(-0.454462\pi\)
0.142575 + 0.989784i \(0.454462\pi\)
\(480\) 0 0
\(481\) 7216.51 0.684084
\(482\) 0 0
\(483\) 274.275 0.0258384
\(484\) 0 0
\(485\) −2511.60 −0.235147
\(486\) 0 0
\(487\) 5549.61 0.516379 0.258190 0.966094i \(-0.416874\pi\)
0.258190 + 0.966094i \(0.416874\pi\)
\(488\) 0 0
\(489\) 10067.0 0.930976
\(490\) 0 0
\(491\) 7751.20 0.712438 0.356219 0.934403i \(-0.384066\pi\)
0.356219 + 0.934403i \(0.384066\pi\)
\(492\) 0 0
\(493\) −742.862 −0.0678638
\(494\) 0 0
\(495\) −391.621 −0.0355597
\(496\) 0 0
\(497\) −2583.17 −0.233141
\(498\) 0 0
\(499\) 6841.83 0.613792 0.306896 0.951743i \(-0.400710\pi\)
0.306896 + 0.951743i \(0.400710\pi\)
\(500\) 0 0
\(501\) −10613.4 −0.946451
\(502\) 0 0
\(503\) 11518.5 1.02105 0.510523 0.859864i \(-0.329452\pi\)
0.510523 + 0.859864i \(0.329452\pi\)
\(504\) 0 0
\(505\) −2969.73 −0.261686
\(506\) 0 0
\(507\) 4410.27 0.386325
\(508\) 0 0
\(509\) 8292.40 0.722110 0.361055 0.932545i \(-0.382417\pi\)
0.361055 + 0.932545i \(0.382417\pi\)
\(510\) 0 0
\(511\) 8020.39 0.694327
\(512\) 0 0
\(513\) −4129.12 −0.355371
\(514\) 0 0
\(515\) 226.917 0.0194158
\(516\) 0 0
\(517\) 573.595 0.0487943
\(518\) 0 0
\(519\) 11957.8 1.01134
\(520\) 0 0
\(521\) 10891.6 0.915868 0.457934 0.888986i \(-0.348590\pi\)
0.457934 + 0.888986i \(0.348590\pi\)
\(522\) 0 0
\(523\) −14662.0 −1.22586 −0.612928 0.790139i \(-0.710008\pi\)
−0.612928 + 0.790139i \(0.710008\pi\)
\(524\) 0 0
\(525\) 2361.20 0.196288
\(526\) 0 0
\(527\) −3142.69 −0.259768
\(528\) 0 0
\(529\) −11967.5 −0.983602
\(530\) 0 0
\(531\) 13760.9 1.12462
\(532\) 0 0
\(533\) 3801.87 0.308963
\(534\) 0 0
\(535\) 3210.54 0.259446
\(536\) 0 0
\(537\) 6910.33 0.555313
\(538\) 0 0
\(539\) 539.000 0.0430730
\(540\) 0 0
\(541\) −19825.8 −1.57556 −0.787778 0.615960i \(-0.788768\pi\)
−0.787778 + 0.615960i \(0.788768\pi\)
\(542\) 0 0
\(543\) 11971.1 0.946094
\(544\) 0 0
\(545\) 459.087 0.0360828
\(546\) 0 0
\(547\) −12706.1 −0.993187 −0.496593 0.867983i \(-0.665416\pi\)
−0.496593 + 0.867983i \(0.665416\pi\)
\(548\) 0 0
\(549\) 12491.8 0.971109
\(550\) 0 0
\(551\) 1335.57 0.103262
\(552\) 0 0
\(553\) −3422.19 −0.263158
\(554\) 0 0
\(555\) −1498.28 −0.114592
\(556\) 0 0
\(557\) −12599.1 −0.958421 −0.479211 0.877700i \(-0.659077\pi\)
−0.479211 + 0.877700i \(0.659077\pi\)
\(558\) 0 0
\(559\) 6828.86 0.516691
\(560\) 0 0
\(561\) −545.588 −0.0410602
\(562\) 0 0
\(563\) −6004.47 −0.449482 −0.224741 0.974419i \(-0.572154\pi\)
−0.224741 + 0.974419i \(0.572154\pi\)
\(564\) 0 0
\(565\) −911.884 −0.0678996
\(566\) 0 0
\(567\) 1154.42 0.0855046
\(568\) 0 0
\(569\) −3145.89 −0.231779 −0.115890 0.993262i \(-0.536972\pi\)
−0.115890 + 0.993262i \(0.536972\pi\)
\(570\) 0 0
\(571\) −23549.1 −1.72592 −0.862960 0.505273i \(-0.831392\pi\)
−0.862960 + 0.505273i \(0.831392\pi\)
\(572\) 0 0
\(573\) 7879.27 0.574453
\(574\) 0 0
\(575\) 1717.57 0.124569
\(576\) 0 0
\(577\) 327.335 0.0236172 0.0118086 0.999930i \(-0.496241\pi\)
0.0118086 + 0.999930i \(0.496241\pi\)
\(578\) 0 0
\(579\) 4811.98 0.345387
\(580\) 0 0
\(581\) −3842.88 −0.274405
\(582\) 0 0
\(583\) 906.298 0.0643825
\(584\) 0 0
\(585\) −877.235 −0.0619986
\(586\) 0 0
\(587\) 13270.8 0.933123 0.466562 0.884489i \(-0.345493\pi\)
0.466562 + 0.884489i \(0.345493\pi\)
\(588\) 0 0
\(589\) 5650.14 0.395263
\(590\) 0 0
\(591\) 8594.98 0.598224
\(592\) 0 0
\(593\) 5098.92 0.353099 0.176549 0.984292i \(-0.443506\pi\)
0.176549 + 0.984292i \(0.443506\pi\)
\(594\) 0 0
\(595\) 230.818 0.0159036
\(596\) 0 0
\(597\) 12475.7 0.855268
\(598\) 0 0
\(599\) −19358.2 −1.32046 −0.660230 0.751064i \(-0.729541\pi\)
−0.660230 + 0.751064i \(0.729541\pi\)
\(600\) 0 0
\(601\) 1238.87 0.0840841 0.0420420 0.999116i \(-0.486614\pi\)
0.0420420 + 0.999116i \(0.486614\pi\)
\(602\) 0 0
\(603\) 5036.04 0.340105
\(604\) 0 0
\(605\) 223.146 0.0149953
\(606\) 0 0
\(607\) 14175.6 0.947888 0.473944 0.880555i \(-0.342830\pi\)
0.473944 + 0.880555i \(0.342830\pi\)
\(608\) 0 0
\(609\) 806.758 0.0536806
\(610\) 0 0
\(611\) 1284.86 0.0850732
\(612\) 0 0
\(613\) 6906.67 0.455070 0.227535 0.973770i \(-0.426933\pi\)
0.227535 + 0.973770i \(0.426933\pi\)
\(614\) 0 0
\(615\) −789.339 −0.0517549
\(616\) 0 0
\(617\) 12104.3 0.789789 0.394894 0.918727i \(-0.370781\pi\)
0.394894 + 0.918727i \(0.370781\pi\)
\(618\) 0 0
\(619\) −14945.2 −0.970437 −0.485218 0.874393i \(-0.661260\pi\)
−0.485218 + 0.874393i \(0.661260\pi\)
\(620\) 0 0
\(621\) −1814.33 −0.117241
\(622\) 0 0
\(623\) 735.271 0.0472841
\(624\) 0 0
\(625\) 14361.2 0.919116
\(626\) 0 0
\(627\) 980.895 0.0624772
\(628\) 0 0
\(629\) 5236.63 0.331953
\(630\) 0 0
\(631\) 17711.9 1.11743 0.558717 0.829358i \(-0.311294\pi\)
0.558717 + 0.829358i \(0.311294\pi\)
\(632\) 0 0
\(633\) 3501.67 0.219872
\(634\) 0 0
\(635\) 1805.65 0.112842
\(636\) 0 0
\(637\) 1207.36 0.0750982
\(638\) 0 0
\(639\) 7124.02 0.441036
\(640\) 0 0
\(641\) −17994.4 −1.10879 −0.554396 0.832253i \(-0.687051\pi\)
−0.554396 + 0.832253i \(0.687051\pi\)
\(642\) 0 0
\(643\) −27947.8 −1.71408 −0.857039 0.515251i \(-0.827699\pi\)
−0.857039 + 0.515251i \(0.827699\pi\)
\(644\) 0 0
\(645\) −1417.80 −0.0865516
\(646\) 0 0
\(647\) 14336.2 0.871122 0.435561 0.900159i \(-0.356550\pi\)
0.435561 + 0.900159i \(0.356550\pi\)
\(648\) 0 0
\(649\) −7840.97 −0.474245
\(650\) 0 0
\(651\) 3413.00 0.205478
\(652\) 0 0
\(653\) 4315.79 0.258637 0.129318 0.991603i \(-0.458721\pi\)
0.129318 + 0.991603i \(0.458721\pi\)
\(654\) 0 0
\(655\) 3062.91 0.182714
\(656\) 0 0
\(657\) −22119.1 −1.31347
\(658\) 0 0
\(659\) −4002.23 −0.236578 −0.118289 0.992979i \(-0.537741\pi\)
−0.118289 + 0.992979i \(0.537741\pi\)
\(660\) 0 0
\(661\) −12223.4 −0.719268 −0.359634 0.933094i \(-0.617098\pi\)
−0.359634 + 0.933094i \(0.617098\pi\)
\(662\) 0 0
\(663\) −1222.12 −0.0715887
\(664\) 0 0
\(665\) −414.980 −0.0241989
\(666\) 0 0
\(667\) 586.846 0.0340671
\(668\) 0 0
\(669\) 8131.61 0.469935
\(670\) 0 0
\(671\) −7117.86 −0.409511
\(672\) 0 0
\(673\) −6121.53 −0.350621 −0.175310 0.984513i \(-0.556093\pi\)
−0.175310 + 0.984513i \(0.556093\pi\)
\(674\) 0 0
\(675\) −15619.3 −0.890649
\(676\) 0 0
\(677\) 9626.46 0.546492 0.273246 0.961944i \(-0.411903\pi\)
0.273246 + 0.961944i \(0.411903\pi\)
\(678\) 0 0
\(679\) −9533.34 −0.538816
\(680\) 0 0
\(681\) 11923.6 0.670947
\(682\) 0 0
\(683\) −11554.4 −0.647315 −0.323658 0.946174i \(-0.604913\pi\)
−0.323658 + 0.946174i \(0.604913\pi\)
\(684\) 0 0
\(685\) −2984.19 −0.166453
\(686\) 0 0
\(687\) −1937.63 −0.107606
\(688\) 0 0
\(689\) 2030.12 0.112251
\(690\) 0 0
\(691\) −2991.45 −0.164689 −0.0823444 0.996604i \(-0.526241\pi\)
−0.0823444 + 0.996604i \(0.526241\pi\)
\(692\) 0 0
\(693\) −1486.48 −0.0814818
\(694\) 0 0
\(695\) −1283.06 −0.0700279
\(696\) 0 0
\(697\) 2758.82 0.149925
\(698\) 0 0
\(699\) −5236.67 −0.283361
\(700\) 0 0
\(701\) 30772.8 1.65802 0.829010 0.559234i \(-0.188905\pi\)
0.829010 + 0.559234i \(0.188905\pi\)
\(702\) 0 0
\(703\) −9414.77 −0.505099
\(704\) 0 0
\(705\) −266.760 −0.0142507
\(706\) 0 0
\(707\) −11272.3 −0.599628
\(708\) 0 0
\(709\) −21698.3 −1.14936 −0.574681 0.818377i \(-0.694874\pi\)
−0.574681 + 0.818377i \(0.694874\pi\)
\(710\) 0 0
\(711\) 9437.91 0.497819
\(712\) 0 0
\(713\) 2482.66 0.130402
\(714\) 0 0
\(715\) 499.849 0.0261445
\(716\) 0 0
\(717\) −17892.0 −0.931925
\(718\) 0 0
\(719\) 12364.2 0.641318 0.320659 0.947195i \(-0.396096\pi\)
0.320659 + 0.947195i \(0.396096\pi\)
\(720\) 0 0
\(721\) 861.311 0.0444895
\(722\) 0 0
\(723\) −12862.9 −0.661656
\(724\) 0 0
\(725\) 5052.09 0.258800
\(726\) 0 0
\(727\) 16017.2 0.817119 0.408559 0.912732i \(-0.366031\pi\)
0.408559 + 0.912732i \(0.366031\pi\)
\(728\) 0 0
\(729\) 6436.85 0.327026
\(730\) 0 0
\(731\) 4955.34 0.250725
\(732\) 0 0
\(733\) 10564.4 0.532340 0.266170 0.963926i \(-0.414242\pi\)
0.266170 + 0.963926i \(0.414242\pi\)
\(734\) 0 0
\(735\) −250.671 −0.0125798
\(736\) 0 0
\(737\) −2869.54 −0.143420
\(738\) 0 0
\(739\) −1760.32 −0.0876242 −0.0438121 0.999040i \(-0.513950\pi\)
−0.0438121 + 0.999040i \(0.513950\pi\)
\(740\) 0 0
\(741\) 2197.21 0.108929
\(742\) 0 0
\(743\) −11383.4 −0.562067 −0.281034 0.959698i \(-0.590677\pi\)
−0.281034 + 0.959698i \(0.590677\pi\)
\(744\) 0 0
\(745\) −3839.29 −0.188806
\(746\) 0 0
\(747\) 10598.1 0.519095
\(748\) 0 0
\(749\) 12186.3 0.594497
\(750\) 0 0
\(751\) 15058.3 0.731670 0.365835 0.930680i \(-0.380783\pi\)
0.365835 + 0.930680i \(0.380783\pi\)
\(752\) 0 0
\(753\) 6086.45 0.294558
\(754\) 0 0
\(755\) −4941.08 −0.238178
\(756\) 0 0
\(757\) 38073.4 1.82801 0.914004 0.405706i \(-0.132974\pi\)
0.914004 + 0.405706i \(0.132974\pi\)
\(758\) 0 0
\(759\) 431.003 0.0206119
\(760\) 0 0
\(761\) 15232.1 0.725577 0.362788 0.931872i \(-0.381825\pi\)
0.362788 + 0.931872i \(0.381825\pi\)
\(762\) 0 0
\(763\) 1742.57 0.0826803
\(764\) 0 0
\(765\) −636.563 −0.0300849
\(766\) 0 0
\(767\) −17563.8 −0.826850
\(768\) 0 0
\(769\) 12013.1 0.563332 0.281666 0.959513i \(-0.409113\pi\)
0.281666 + 0.959513i \(0.409113\pi\)
\(770\) 0 0
\(771\) 8230.33 0.384446
\(772\) 0 0
\(773\) −14258.5 −0.663443 −0.331721 0.943377i \(-0.607629\pi\)
−0.331721 + 0.943377i \(0.607629\pi\)
\(774\) 0 0
\(775\) 21372.9 0.990630
\(776\) 0 0
\(777\) −5687.05 −0.262576
\(778\) 0 0
\(779\) −4959.99 −0.228126
\(780\) 0 0
\(781\) −4059.27 −0.185982
\(782\) 0 0
\(783\) −5336.70 −0.243574
\(784\) 0 0
\(785\) 4445.08 0.202104
\(786\) 0 0
\(787\) 14377.9 0.651227 0.325613 0.945503i \(-0.394429\pi\)
0.325613 + 0.945503i \(0.394429\pi\)
\(788\) 0 0
\(789\) 2540.45 0.114629
\(790\) 0 0
\(791\) −3461.26 −0.155585
\(792\) 0 0
\(793\) −15944.1 −0.713986
\(794\) 0 0
\(795\) −421.490 −0.0188034
\(796\) 0 0
\(797\) 38515.9 1.71180 0.855899 0.517144i \(-0.173005\pi\)
0.855899 + 0.517144i \(0.173005\pi\)
\(798\) 0 0
\(799\) 932.352 0.0412819
\(800\) 0 0
\(801\) −2027.77 −0.0894479
\(802\) 0 0
\(803\) 12603.5 0.553882
\(804\) 0 0
\(805\) −182.341 −0.00798347
\(806\) 0 0
\(807\) 456.217 0.0199004
\(808\) 0 0
\(809\) 11438.6 0.497106 0.248553 0.968618i \(-0.420045\pi\)
0.248553 + 0.968618i \(0.420045\pi\)
\(810\) 0 0
\(811\) −15872.9 −0.687265 −0.343632 0.939104i \(-0.611657\pi\)
−0.343632 + 0.939104i \(0.611657\pi\)
\(812\) 0 0
\(813\) 4168.21 0.179810
\(814\) 0 0
\(815\) −6692.70 −0.287650
\(816\) 0 0
\(817\) −8909.05 −0.381503
\(818\) 0 0
\(819\) −3329.74 −0.142064
\(820\) 0 0
\(821\) −11988.8 −0.509638 −0.254819 0.966989i \(-0.582016\pi\)
−0.254819 + 0.966989i \(0.582016\pi\)
\(822\) 0 0
\(823\) 222.224 0.00941219 0.00470610 0.999989i \(-0.498502\pi\)
0.00470610 + 0.999989i \(0.498502\pi\)
\(824\) 0 0
\(825\) 3710.46 0.156584
\(826\) 0 0
\(827\) −7524.36 −0.316382 −0.158191 0.987409i \(-0.550566\pi\)
−0.158191 + 0.987409i \(0.550566\pi\)
\(828\) 0 0
\(829\) −11807.1 −0.494663 −0.247332 0.968931i \(-0.579554\pi\)
−0.247332 + 0.968931i \(0.579554\pi\)
\(830\) 0 0
\(831\) 20805.2 0.868502
\(832\) 0 0
\(833\) 876.120 0.0364415
\(834\) 0 0
\(835\) 7055.93 0.292432
\(836\) 0 0
\(837\) −22577.0 −0.932349
\(838\) 0 0
\(839\) 29112.9 1.19796 0.598980 0.800764i \(-0.295573\pi\)
0.598980 + 0.800764i \(0.295573\pi\)
\(840\) 0 0
\(841\) −22662.8 −0.929224
\(842\) 0 0
\(843\) 8169.73 0.333785
\(844\) 0 0
\(845\) −2932.00 −0.119366
\(846\) 0 0
\(847\) 847.000 0.0343604
\(848\) 0 0
\(849\) 14467.2 0.584819
\(850\) 0 0
\(851\) −4136.83 −0.166638
\(852\) 0 0
\(853\) −24892.1 −0.999169 −0.499584 0.866265i \(-0.666514\pi\)
−0.499584 + 0.866265i \(0.666514\pi\)
\(854\) 0 0
\(855\) 1144.46 0.0457773
\(856\) 0 0
\(857\) −27299.2 −1.08812 −0.544062 0.839045i \(-0.683114\pi\)
−0.544062 + 0.839045i \(0.683114\pi\)
\(858\) 0 0
\(859\) 6975.13 0.277053 0.138526 0.990359i \(-0.455763\pi\)
0.138526 + 0.990359i \(0.455763\pi\)
\(860\) 0 0
\(861\) −2996.11 −0.118591
\(862\) 0 0
\(863\) 4481.56 0.176772 0.0883858 0.996086i \(-0.471829\pi\)
0.0883858 + 0.996086i \(0.471829\pi\)
\(864\) 0 0
\(865\) −7949.67 −0.312482
\(866\) 0 0
\(867\) 12741.8 0.499116
\(868\) 0 0
\(869\) −5377.73 −0.209928
\(870\) 0 0
\(871\) −6427.80 −0.250055
\(872\) 0 0
\(873\) 26291.6 1.01928
\(874\) 0 0
\(875\) −3183.42 −0.122993
\(876\) 0 0
\(877\) −7207.96 −0.277532 −0.138766 0.990325i \(-0.544314\pi\)
−0.138766 + 0.990325i \(0.544314\pi\)
\(878\) 0 0
\(879\) −20549.7 −0.788536
\(880\) 0 0
\(881\) −38413.7 −1.46900 −0.734502 0.678607i \(-0.762584\pi\)
−0.734502 + 0.678607i \(0.762584\pi\)
\(882\) 0 0
\(883\) 8705.04 0.331764 0.165882 0.986146i \(-0.446953\pi\)
0.165882 + 0.986146i \(0.446953\pi\)
\(884\) 0 0
\(885\) 3646.58 0.138507
\(886\) 0 0
\(887\) 44100.0 1.66937 0.834687 0.550725i \(-0.185649\pi\)
0.834687 + 0.550725i \(0.185649\pi\)
\(888\) 0 0
\(889\) 6853.73 0.258568
\(890\) 0 0
\(891\) 1814.09 0.0682091
\(892\) 0 0
\(893\) −1676.25 −0.0628146
\(894\) 0 0
\(895\) −4594.08 −0.171579
\(896\) 0 0
\(897\) 965.451 0.0359370
\(898\) 0 0
\(899\) 7302.55 0.270916
\(900\) 0 0
\(901\) 1473.15 0.0544702
\(902\) 0 0
\(903\) −5381.57 −0.198325
\(904\) 0 0
\(905\) −7958.54 −0.292322
\(906\) 0 0
\(907\) −19921.2 −0.729296 −0.364648 0.931145i \(-0.618811\pi\)
−0.364648 + 0.931145i \(0.618811\pi\)
\(908\) 0 0
\(909\) 31087.3 1.13432
\(910\) 0 0
\(911\) 14446.7 0.525402 0.262701 0.964877i \(-0.415387\pi\)
0.262701 + 0.964877i \(0.415387\pi\)
\(912\) 0 0
\(913\) −6038.81 −0.218900
\(914\) 0 0
\(915\) 3310.29 0.119601
\(916\) 0 0
\(917\) 11625.9 0.418672
\(918\) 0 0
\(919\) −44381.1 −1.59303 −0.796516 0.604617i \(-0.793326\pi\)
−0.796516 + 0.604617i \(0.793326\pi\)
\(920\) 0 0
\(921\) −19001.8 −0.679839
\(922\) 0 0
\(923\) −9092.80 −0.324261
\(924\) 0 0
\(925\) −35613.5 −1.26591
\(926\) 0 0
\(927\) −2375.37 −0.0841612
\(928\) 0 0
\(929\) −3455.30 −0.122029 −0.0610145 0.998137i \(-0.519434\pi\)
−0.0610145 + 0.998137i \(0.519434\pi\)
\(930\) 0 0
\(931\) −1575.15 −0.0554494
\(932\) 0 0
\(933\) −15364.1 −0.539117
\(934\) 0 0
\(935\) 362.714 0.0126866
\(936\) 0 0
\(937\) −14962.8 −0.521678 −0.260839 0.965382i \(-0.583999\pi\)
−0.260839 + 0.965382i \(0.583999\pi\)
\(938\) 0 0
\(939\) −25967.7 −0.902474
\(940\) 0 0
\(941\) −6116.95 −0.211909 −0.105955 0.994371i \(-0.533790\pi\)
−0.105955 + 0.994371i \(0.533790\pi\)
\(942\) 0 0
\(943\) −2179.41 −0.0752611
\(944\) 0 0
\(945\) 1658.19 0.0570804
\(946\) 0 0
\(947\) −53343.7 −1.83045 −0.915225 0.402943i \(-0.867987\pi\)
−0.915225 + 0.402943i \(0.867987\pi\)
\(948\) 0 0
\(949\) 28231.9 0.965696
\(950\) 0 0
\(951\) −608.117 −0.0207356
\(952\) 0 0
\(953\) 1979.62 0.0672887 0.0336443 0.999434i \(-0.489289\pi\)
0.0336443 + 0.999434i \(0.489289\pi\)
\(954\) 0 0
\(955\) −5238.24 −0.177493
\(956\) 0 0
\(957\) 1267.76 0.0428223
\(958\) 0 0
\(959\) −11327.2 −0.381411
\(960\) 0 0
\(961\) 1102.58 0.0370104
\(962\) 0 0
\(963\) −33608.1 −1.12462
\(964\) 0 0
\(965\) −3199.07 −0.106717
\(966\) 0 0
\(967\) 38892.0 1.29336 0.646681 0.762761i \(-0.276157\pi\)
0.646681 + 0.762761i \(0.276157\pi\)
\(968\) 0 0
\(969\) 1594.40 0.0528582
\(970\) 0 0
\(971\) 47826.1 1.58065 0.790325 0.612688i \(-0.209912\pi\)
0.790325 + 0.612688i \(0.209912\pi\)
\(972\) 0 0
\(973\) −4870.15 −0.160462
\(974\) 0 0
\(975\) 8311.45 0.273005
\(976\) 0 0
\(977\) 20840.5 0.682442 0.341221 0.939983i \(-0.389160\pi\)
0.341221 + 0.939983i \(0.389160\pi\)
\(978\) 0 0
\(979\) 1155.43 0.0377197
\(980\) 0 0
\(981\) −4805.74 −0.156407
\(982\) 0 0
\(983\) 54430.5 1.76609 0.883044 0.469290i \(-0.155490\pi\)
0.883044 + 0.469290i \(0.155490\pi\)
\(984\) 0 0
\(985\) −5714.05 −0.184837
\(986\) 0 0
\(987\) −1012.55 −0.0326542
\(988\) 0 0
\(989\) −3914.62 −0.125862
\(990\) 0 0
\(991\) 35161.9 1.12710 0.563550 0.826082i \(-0.309435\pi\)
0.563550 + 0.826082i \(0.309435\pi\)
\(992\) 0 0
\(993\) 9369.50 0.299428
\(994\) 0 0
\(995\) −8293.99 −0.264258
\(996\) 0 0
\(997\) 26961.6 0.856451 0.428225 0.903672i \(-0.359139\pi\)
0.428225 + 0.903672i \(0.359139\pi\)
\(998\) 0 0
\(999\) 37619.8 1.19143
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 1232.4.a.s.1.3 4
4.3 odd 2 77.4.a.d.1.4 4
12.11 even 2 693.4.a.l.1.1 4
20.19 odd 2 1925.4.a.p.1.1 4
28.27 even 2 539.4.a.g.1.4 4
44.43 even 2 847.4.a.d.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.4.a.d.1.4 4 4.3 odd 2
539.4.a.g.1.4 4 28.27 even 2
693.4.a.l.1.1 4 12.11 even 2
847.4.a.d.1.1 4 44.43 even 2
1232.4.a.s.1.3 4 1.1 even 1 trivial
1925.4.a.p.1.1 4 20.19 odd 2