Properties

Label 1232.4.a.v.1.2
Level $1232$
Weight $4$
Character 1232.1
Self dual yes
Analytic conductor $72.690$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1232,4,Mod(1,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.6903531271\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 43x^{2} - 11x + 222 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 308)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.21425\) of defining polynomial
Character \(\chi\) \(=\) 1232.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.21425 q^{3} -20.5256 q^{5} +7.00000 q^{7} -25.5256 q^{9} +O(q^{10})\) \(q-1.21425 q^{3} -20.5256 q^{5} +7.00000 q^{7} -25.5256 q^{9} -11.0000 q^{11} +66.7086 q^{13} +24.9233 q^{15} -59.7289 q^{17} -131.071 q^{19} -8.49978 q^{21} -163.117 q^{23} +296.300 q^{25} +63.7794 q^{27} -77.2961 q^{29} -206.358 q^{31} +13.3568 q^{33} -143.679 q^{35} -215.238 q^{37} -81.0011 q^{39} -403.218 q^{41} -43.5690 q^{43} +523.928 q^{45} -69.1706 q^{47} +49.0000 q^{49} +72.5261 q^{51} +334.969 q^{53} +225.781 q^{55} +159.154 q^{57} -681.283 q^{59} -24.4899 q^{61} -178.679 q^{63} -1369.23 q^{65} +285.338 q^{67} +198.065 q^{69} +522.204 q^{71} +592.444 q^{73} -359.783 q^{75} -77.0000 q^{77} +92.7989 q^{79} +611.747 q^{81} -480.928 q^{83} +1225.97 q^{85} +93.8570 q^{87} +1197.02 q^{89} +466.960 q^{91} +250.571 q^{93} +2690.31 q^{95} +308.934 q^{97} +280.781 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + q^{5} + 28 q^{7} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} + q^{5} + 28 q^{7} - 19 q^{9} - 44 q^{11} + 98 q^{13} + 33 q^{15} + 64 q^{17} - 114 q^{19} + 21 q^{21} - 231 q^{23} + 417 q^{25} - 63 q^{27} + 268 q^{29} - 33 q^{31} - 33 q^{33} + 7 q^{35} + 357 q^{37} - 140 q^{39} + 364 q^{41} - 44 q^{43} + 912 q^{45} - 720 q^{47} + 196 q^{49} - 794 q^{51} + 740 q^{53} - 11 q^{55} + 138 q^{57} - 787 q^{59} + 1020 q^{61} - 133 q^{63} + 296 q^{65} + 995 q^{67} + 1059 q^{69} + 1011 q^{71} + 1592 q^{73} + 324 q^{75} - 308 q^{77} + 178 q^{79} - 1396 q^{81} - 324 q^{83} - 94 q^{85} + 1262 q^{87} + 19 q^{89} + 686 q^{91} + 637 q^{93} + 2418 q^{95} - 555 q^{97} + 209 q^{99}+O(q^{100}) \) 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
\(3\) −1.21425 −0.233683 −0.116842 0.993151i \(-0.537277\pi\)
−0.116842 + 0.993151i \(0.537277\pi\)
\(4\) 0 0
\(5\) −20.5256 −1.83586 −0.917932 0.396737i \(-0.870142\pi\)
−0.917932 + 0.396737i \(0.870142\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) −25.5256 −0.945392
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 66.7086 1.42320 0.711601 0.702584i \(-0.247970\pi\)
0.711601 + 0.702584i \(0.247970\pi\)
\(14\) 0 0
\(15\) 24.9233 0.429011
\(16\) 0 0
\(17\) −59.7289 −0.852141 −0.426070 0.904690i \(-0.640102\pi\)
−0.426070 + 0.904690i \(0.640102\pi\)
\(18\) 0 0
\(19\) −131.071 −1.58262 −0.791310 0.611416i \(-0.790600\pi\)
−0.791310 + 0.611416i \(0.790600\pi\)
\(20\) 0 0
\(21\) −8.49978 −0.0883240
\(22\) 0 0
\(23\) −163.117 −1.47879 −0.739395 0.673272i \(-0.764888\pi\)
−0.739395 + 0.673272i \(0.764888\pi\)
\(24\) 0 0
\(25\) 296.300 2.37040
\(26\) 0 0
\(27\) 63.7794 0.454605
\(28\) 0 0
\(29\) −77.2961 −0.494949 −0.247474 0.968894i \(-0.579601\pi\)
−0.247474 + 0.968894i \(0.579601\pi\)
\(30\) 0 0
\(31\) −206.358 −1.19558 −0.597791 0.801652i \(-0.703955\pi\)
−0.597791 + 0.801652i \(0.703955\pi\)
\(32\) 0 0
\(33\) 13.3568 0.0704581
\(34\) 0 0
\(35\) −143.679 −0.693892
\(36\) 0 0
\(37\) −215.238 −0.956348 −0.478174 0.878265i \(-0.658701\pi\)
−0.478174 + 0.878265i \(0.658701\pi\)
\(38\) 0 0
\(39\) −81.0011 −0.332578
\(40\) 0 0
\(41\) −403.218 −1.53590 −0.767952 0.640508i \(-0.778724\pi\)
−0.767952 + 0.640508i \(0.778724\pi\)
\(42\) 0 0
\(43\) −43.5690 −0.154517 −0.0772583 0.997011i \(-0.524617\pi\)
−0.0772583 + 0.997011i \(0.524617\pi\)
\(44\) 0 0
\(45\) 523.928 1.73561
\(46\) 0 0
\(47\) −69.1706 −0.214672 −0.107336 0.994223i \(-0.534232\pi\)
−0.107336 + 0.994223i \(0.534232\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 72.5261 0.199131
\(52\) 0 0
\(53\) 334.969 0.868143 0.434071 0.900878i \(-0.357077\pi\)
0.434071 + 0.900878i \(0.357077\pi\)
\(54\) 0 0
\(55\) 225.781 0.553534
\(56\) 0 0
\(57\) 159.154 0.369832
\(58\) 0 0
\(59\) −681.283 −1.50331 −0.751656 0.659555i \(-0.770745\pi\)
−0.751656 + 0.659555i \(0.770745\pi\)
\(60\) 0 0
\(61\) −24.4899 −0.0514034 −0.0257017 0.999670i \(-0.508182\pi\)
−0.0257017 + 0.999670i \(0.508182\pi\)
\(62\) 0 0
\(63\) −178.679 −0.357325
\(64\) 0 0
\(65\) −1369.23 −2.61281
\(66\) 0 0
\(67\) 285.338 0.520292 0.260146 0.965569i \(-0.416229\pi\)
0.260146 + 0.965569i \(0.416229\pi\)
\(68\) 0 0
\(69\) 198.065 0.345568
\(70\) 0 0
\(71\) 522.204 0.872876 0.436438 0.899734i \(-0.356240\pi\)
0.436438 + 0.899734i \(0.356240\pi\)
\(72\) 0 0
\(73\) 592.444 0.949867 0.474934 0.880022i \(-0.342472\pi\)
0.474934 + 0.880022i \(0.342472\pi\)
\(74\) 0 0
\(75\) −359.783 −0.553922
\(76\) 0 0
\(77\) −77.0000 −0.113961
\(78\) 0 0
\(79\) 92.7989 0.132161 0.0660803 0.997814i \(-0.478951\pi\)
0.0660803 + 0.997814i \(0.478951\pi\)
\(80\) 0 0
\(81\) 611.747 0.839158
\(82\) 0 0
\(83\) −480.928 −0.636008 −0.318004 0.948089i \(-0.603013\pi\)
−0.318004 + 0.948089i \(0.603013\pi\)
\(84\) 0 0
\(85\) 1225.97 1.56441
\(86\) 0 0
\(87\) 93.8570 0.115661
\(88\) 0 0
\(89\) 1197.02 1.42566 0.712831 0.701336i \(-0.247413\pi\)
0.712831 + 0.701336i \(0.247413\pi\)
\(90\) 0 0
\(91\) 466.960 0.537920
\(92\) 0 0
\(93\) 250.571 0.279388
\(94\) 0 0
\(95\) 2690.31 2.90547
\(96\) 0 0
\(97\) 308.934 0.323377 0.161688 0.986842i \(-0.448306\pi\)
0.161688 + 0.986842i \(0.448306\pi\)
\(98\) 0 0
\(99\) 280.781 0.285046
\(100\) 0 0
\(101\) 1742.73 1.71691 0.858454 0.512891i \(-0.171425\pi\)
0.858454 + 0.512891i \(0.171425\pi\)
\(102\) 0 0
\(103\) −1043.91 −0.998635 −0.499318 0.866419i \(-0.666416\pi\)
−0.499318 + 0.866419i \(0.666416\pi\)
\(104\) 0 0
\(105\) 174.463 0.162151
\(106\) 0 0
\(107\) −204.369 −0.184646 −0.0923230 0.995729i \(-0.529429\pi\)
−0.0923230 + 0.995729i \(0.529429\pi\)
\(108\) 0 0
\(109\) 944.275 0.829772 0.414886 0.909873i \(-0.363821\pi\)
0.414886 + 0.909873i \(0.363821\pi\)
\(110\) 0 0
\(111\) 261.353 0.223483
\(112\) 0 0
\(113\) −1905.29 −1.58615 −0.793073 0.609127i \(-0.791520\pi\)
−0.793073 + 0.609127i \(0.791520\pi\)
\(114\) 0 0
\(115\) 3348.06 2.71486
\(116\) 0 0
\(117\) −1702.78 −1.34548
\(118\) 0 0
\(119\) −418.102 −0.322079
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) 489.609 0.358915
\(124\) 0 0
\(125\) −3516.03 −2.51587
\(126\) 0 0
\(127\) −547.690 −0.382674 −0.191337 0.981524i \(-0.561282\pi\)
−0.191337 + 0.981524i \(0.561282\pi\)
\(128\) 0 0
\(129\) 52.9038 0.0361079
\(130\) 0 0
\(131\) 1926.59 1.28494 0.642468 0.766312i \(-0.277910\pi\)
0.642468 + 0.766312i \(0.277910\pi\)
\(132\) 0 0
\(133\) −917.498 −0.598174
\(134\) 0 0
\(135\) −1309.11 −0.834594
\(136\) 0 0
\(137\) −1440.77 −0.898491 −0.449245 0.893408i \(-0.648307\pi\)
−0.449245 + 0.893408i \(0.648307\pi\)
\(138\) 0 0
\(139\) 2430.80 1.48330 0.741648 0.670790i \(-0.234045\pi\)
0.741648 + 0.670790i \(0.234045\pi\)
\(140\) 0 0
\(141\) 83.9906 0.0501652
\(142\) 0 0
\(143\) −733.794 −0.429112
\(144\) 0 0
\(145\) 1586.55 0.908659
\(146\) 0 0
\(147\) −59.4984 −0.0333833
\(148\) 0 0
\(149\) 686.765 0.377597 0.188798 0.982016i \(-0.439541\pi\)
0.188798 + 0.982016i \(0.439541\pi\)
\(150\) 0 0
\(151\) −1759.45 −0.948225 −0.474113 0.880464i \(-0.657231\pi\)
−0.474113 + 0.880464i \(0.657231\pi\)
\(152\) 0 0
\(153\) 1524.62 0.805607
\(154\) 0 0
\(155\) 4235.63 2.19493
\(156\) 0 0
\(157\) 1833.85 0.932211 0.466105 0.884729i \(-0.345657\pi\)
0.466105 + 0.884729i \(0.345657\pi\)
\(158\) 0 0
\(159\) −406.738 −0.202870
\(160\) 0 0
\(161\) −1141.82 −0.558930
\(162\) 0 0
\(163\) 2306.20 1.10820 0.554098 0.832452i \(-0.313063\pi\)
0.554098 + 0.832452i \(0.313063\pi\)
\(164\) 0 0
\(165\) −274.156 −0.129352
\(166\) 0 0
\(167\) −2243.47 −1.03955 −0.519774 0.854304i \(-0.673984\pi\)
−0.519774 + 0.854304i \(0.673984\pi\)
\(168\) 0 0
\(169\) 2253.03 1.02550
\(170\) 0 0
\(171\) 3345.67 1.49620
\(172\) 0 0
\(173\) −2499.07 −1.09827 −0.549136 0.835733i \(-0.685043\pi\)
−0.549136 + 0.835733i \(0.685043\pi\)
\(174\) 0 0
\(175\) 2074.10 0.895926
\(176\) 0 0
\(177\) 827.250 0.351299
\(178\) 0 0
\(179\) −270.765 −0.113061 −0.0565304 0.998401i \(-0.518004\pi\)
−0.0565304 + 0.998401i \(0.518004\pi\)
\(180\) 0 0
\(181\) 1480.59 0.608018 0.304009 0.952669i \(-0.401675\pi\)
0.304009 + 0.952669i \(0.401675\pi\)
\(182\) 0 0
\(183\) 29.7369 0.0120121
\(184\) 0 0
\(185\) 4417.88 1.75573
\(186\) 0 0
\(187\) 657.018 0.256930
\(188\) 0 0
\(189\) 446.456 0.171825
\(190\) 0 0
\(191\) −2461.01 −0.932317 −0.466158 0.884701i \(-0.654362\pi\)
−0.466158 + 0.884701i \(0.654362\pi\)
\(192\) 0 0
\(193\) −1419.76 −0.529518 −0.264759 0.964315i \(-0.585292\pi\)
−0.264759 + 0.964315i \(0.585292\pi\)
\(194\) 0 0
\(195\) 1662.60 0.610569
\(196\) 0 0
\(197\) 2918.28 1.05542 0.527712 0.849423i \(-0.323050\pi\)
0.527712 + 0.849423i \(0.323050\pi\)
\(198\) 0 0
\(199\) −3767.60 −1.34210 −0.671050 0.741412i \(-0.734156\pi\)
−0.671050 + 0.741412i \(0.734156\pi\)
\(200\) 0 0
\(201\) −346.472 −0.121583
\(202\) 0 0
\(203\) −541.072 −0.187073
\(204\) 0 0
\(205\) 8276.28 2.81971
\(206\) 0 0
\(207\) 4163.65 1.39804
\(208\) 0 0
\(209\) 1441.78 0.477178
\(210\) 0 0
\(211\) 3565.15 1.16320 0.581600 0.813475i \(-0.302427\pi\)
0.581600 + 0.813475i \(0.302427\pi\)
\(212\) 0 0
\(213\) −634.088 −0.203976
\(214\) 0 0
\(215\) 894.280 0.283672
\(216\) 0 0
\(217\) −1444.51 −0.451888
\(218\) 0 0
\(219\) −719.377 −0.221968
\(220\) 0 0
\(221\) −3984.43 −1.21277
\(222\) 0 0
\(223\) −2531.28 −0.760122 −0.380061 0.924962i \(-0.624097\pi\)
−0.380061 + 0.924962i \(0.624097\pi\)
\(224\) 0 0
\(225\) −7563.23 −2.24096
\(226\) 0 0
\(227\) −1207.47 −0.353050 −0.176525 0.984296i \(-0.556486\pi\)
−0.176525 + 0.984296i \(0.556486\pi\)
\(228\) 0 0
\(229\) −3998.70 −1.15389 −0.576946 0.816782i \(-0.695756\pi\)
−0.576946 + 0.816782i \(0.695756\pi\)
\(230\) 0 0
\(231\) 93.4975 0.0266307
\(232\) 0 0
\(233\) 1429.63 0.401966 0.200983 0.979595i \(-0.435586\pi\)
0.200983 + 0.979595i \(0.435586\pi\)
\(234\) 0 0
\(235\) 1419.77 0.394108
\(236\) 0 0
\(237\) −112.681 −0.0308837
\(238\) 0 0
\(239\) −3223.01 −0.872297 −0.436148 0.899875i \(-0.643658\pi\)
−0.436148 + 0.899875i \(0.643658\pi\)
\(240\) 0 0
\(241\) 580.569 0.155177 0.0775887 0.996985i \(-0.475278\pi\)
0.0775887 + 0.996985i \(0.475278\pi\)
\(242\) 0 0
\(243\) −2464.86 −0.650703
\(244\) 0 0
\(245\) −1005.75 −0.262266
\(246\) 0 0
\(247\) −8743.56 −2.25239
\(248\) 0 0
\(249\) 583.968 0.148624
\(250\) 0 0
\(251\) −1277.24 −0.321190 −0.160595 0.987020i \(-0.551341\pi\)
−0.160595 + 0.987020i \(0.551341\pi\)
\(252\) 0 0
\(253\) 1794.28 0.445872
\(254\) 0 0
\(255\) −1488.64 −0.365577
\(256\) 0 0
\(257\) 2517.10 0.610943 0.305471 0.952201i \(-0.401186\pi\)
0.305471 + 0.952201i \(0.401186\pi\)
\(258\) 0 0
\(259\) −1506.67 −0.361466
\(260\) 0 0
\(261\) 1973.03 0.467921
\(262\) 0 0
\(263\) 5684.42 1.33276 0.666381 0.745611i \(-0.267842\pi\)
0.666381 + 0.745611i \(0.267842\pi\)
\(264\) 0 0
\(265\) −6875.44 −1.59379
\(266\) 0 0
\(267\) −1453.49 −0.333153
\(268\) 0 0
\(269\) 870.076 0.197210 0.0986049 0.995127i \(-0.468562\pi\)
0.0986049 + 0.995127i \(0.468562\pi\)
\(270\) 0 0
\(271\) −190.969 −0.0428065 −0.0214033 0.999771i \(-0.506813\pi\)
−0.0214033 + 0.999771i \(0.506813\pi\)
\(272\) 0 0
\(273\) −567.008 −0.125703
\(274\) 0 0
\(275\) −3259.30 −0.714702
\(276\) 0 0
\(277\) 5601.27 1.21497 0.607487 0.794330i \(-0.292178\pi\)
0.607487 + 0.794330i \(0.292178\pi\)
\(278\) 0 0
\(279\) 5267.42 1.13029
\(280\) 0 0
\(281\) 6878.31 1.46023 0.730117 0.683322i \(-0.239466\pi\)
0.730117 + 0.683322i \(0.239466\pi\)
\(282\) 0 0
\(283\) −5722.15 −1.20193 −0.600965 0.799275i \(-0.705217\pi\)
−0.600965 + 0.799275i \(0.705217\pi\)
\(284\) 0 0
\(285\) −3266.72 −0.678961
\(286\) 0 0
\(287\) −2822.53 −0.580517
\(288\) 0 0
\(289\) −1345.46 −0.273856
\(290\) 0 0
\(291\) −375.125 −0.0755677
\(292\) 0 0
\(293\) 8278.49 1.65063 0.825315 0.564673i \(-0.190998\pi\)
0.825315 + 0.564673i \(0.190998\pi\)
\(294\) 0 0
\(295\) 13983.7 2.75988
\(296\) 0 0
\(297\) −701.573 −0.137069
\(298\) 0 0
\(299\) −10881.3 −2.10462
\(300\) 0 0
\(301\) −304.983 −0.0584018
\(302\) 0 0
\(303\) −2116.11 −0.401213
\(304\) 0 0
\(305\) 502.669 0.0943696
\(306\) 0 0
\(307\) 645.184 0.119943 0.0599717 0.998200i \(-0.480899\pi\)
0.0599717 + 0.998200i \(0.480899\pi\)
\(308\) 0 0
\(309\) 1267.57 0.233364
\(310\) 0 0
\(311\) −7930.96 −1.44606 −0.723028 0.690819i \(-0.757250\pi\)
−0.723028 + 0.690819i \(0.757250\pi\)
\(312\) 0 0
\(313\) 6600.26 1.19191 0.595956 0.803017i \(-0.296773\pi\)
0.595956 + 0.803017i \(0.296773\pi\)
\(314\) 0 0
\(315\) 3667.49 0.656000
\(316\) 0 0
\(317\) −10260.0 −1.81785 −0.908923 0.416963i \(-0.863094\pi\)
−0.908923 + 0.416963i \(0.863094\pi\)
\(318\) 0 0
\(319\) 850.257 0.149233
\(320\) 0 0
\(321\) 248.156 0.0431487
\(322\) 0 0
\(323\) 7828.73 1.34861
\(324\) 0 0
\(325\) 19765.7 3.37356
\(326\) 0 0
\(327\) −1146.59 −0.193904
\(328\) 0 0
\(329\) −484.194 −0.0811383
\(330\) 0 0
\(331\) 9625.35 1.59836 0.799180 0.601092i \(-0.205268\pi\)
0.799180 + 0.601092i \(0.205268\pi\)
\(332\) 0 0
\(333\) 5494.07 0.904124
\(334\) 0 0
\(335\) −5856.72 −0.955185
\(336\) 0 0
\(337\) 5877.46 0.950046 0.475023 0.879973i \(-0.342440\pi\)
0.475023 + 0.879973i \(0.342440\pi\)
\(338\) 0 0
\(339\) 2313.50 0.370656
\(340\) 0 0
\(341\) 2269.94 0.360482
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) −4065.40 −0.634416
\(346\) 0 0
\(347\) 10051.4 1.55501 0.777506 0.628875i \(-0.216484\pi\)
0.777506 + 0.628875i \(0.216484\pi\)
\(348\) 0 0
\(349\) −12242.2 −1.87768 −0.938839 0.344356i \(-0.888097\pi\)
−0.938839 + 0.344356i \(0.888097\pi\)
\(350\) 0 0
\(351\) 4254.63 0.646996
\(352\) 0 0
\(353\) −11468.0 −1.72912 −0.864559 0.502532i \(-0.832402\pi\)
−0.864559 + 0.502532i \(0.832402\pi\)
\(354\) 0 0
\(355\) −10718.5 −1.60248
\(356\) 0 0
\(357\) 507.682 0.0752644
\(358\) 0 0
\(359\) 8566.44 1.25939 0.629693 0.776844i \(-0.283181\pi\)
0.629693 + 0.776844i \(0.283181\pi\)
\(360\) 0 0
\(361\) 10320.6 1.50468
\(362\) 0 0
\(363\) −146.925 −0.0212439
\(364\) 0 0
\(365\) −12160.3 −1.74383
\(366\) 0 0
\(367\) −6925.30 −0.985008 −0.492504 0.870310i \(-0.663918\pi\)
−0.492504 + 0.870310i \(0.663918\pi\)
\(368\) 0 0
\(369\) 10292.4 1.45203
\(370\) 0 0
\(371\) 2344.78 0.328127
\(372\) 0 0
\(373\) 5752.39 0.798518 0.399259 0.916838i \(-0.369267\pi\)
0.399259 + 0.916838i \(0.369267\pi\)
\(374\) 0 0
\(375\) 4269.35 0.587915
\(376\) 0 0
\(377\) −5156.31 −0.704412
\(378\) 0 0
\(379\) 10010.2 1.35670 0.678348 0.734740i \(-0.262696\pi\)
0.678348 + 0.734740i \(0.262696\pi\)
\(380\) 0 0
\(381\) 665.034 0.0894245
\(382\) 0 0
\(383\) −4686.10 −0.625192 −0.312596 0.949886i \(-0.601199\pi\)
−0.312596 + 0.949886i \(0.601199\pi\)
\(384\) 0 0
\(385\) 1580.47 0.209216
\(386\) 0 0
\(387\) 1112.13 0.146079
\(388\) 0 0
\(389\) −11018.9 −1.43619 −0.718095 0.695945i \(-0.754986\pi\)
−0.718095 + 0.695945i \(0.754986\pi\)
\(390\) 0 0
\(391\) 9742.77 1.26014
\(392\) 0 0
\(393\) −2339.37 −0.300268
\(394\) 0 0
\(395\) −1904.75 −0.242629
\(396\) 0 0
\(397\) −9780.07 −1.23639 −0.618196 0.786024i \(-0.712136\pi\)
−0.618196 + 0.786024i \(0.712136\pi\)
\(398\) 0 0
\(399\) 1114.07 0.139783
\(400\) 0 0
\(401\) 5288.90 0.658641 0.329320 0.944218i \(-0.393180\pi\)
0.329320 + 0.944218i \(0.393180\pi\)
\(402\) 0 0
\(403\) −13765.9 −1.70156
\(404\) 0 0
\(405\) −12556.5 −1.54058
\(406\) 0 0
\(407\) 2367.62 0.288350
\(408\) 0 0
\(409\) 7276.22 0.879672 0.439836 0.898078i \(-0.355037\pi\)
0.439836 + 0.898078i \(0.355037\pi\)
\(410\) 0 0
\(411\) 1749.46 0.209962
\(412\) 0 0
\(413\) −4768.98 −0.568199
\(414\) 0 0
\(415\) 9871.32 1.16762
\(416\) 0 0
\(417\) −2951.61 −0.346621
\(418\) 0 0
\(419\) 11032.7 1.28636 0.643178 0.765717i \(-0.277616\pi\)
0.643178 + 0.765717i \(0.277616\pi\)
\(420\) 0 0
\(421\) 16046.8 1.85765 0.928826 0.370517i \(-0.120820\pi\)
0.928826 + 0.370517i \(0.120820\pi\)
\(422\) 0 0
\(423\) 1765.62 0.202949
\(424\) 0 0
\(425\) −17697.7 −2.01991
\(426\) 0 0
\(427\) −171.429 −0.0194286
\(428\) 0 0
\(429\) 891.012 0.100276
\(430\) 0 0
\(431\) 6726.13 0.751709 0.375854 0.926679i \(-0.377349\pi\)
0.375854 + 0.926679i \(0.377349\pi\)
\(432\) 0 0
\(433\) 4622.57 0.513041 0.256520 0.966539i \(-0.417424\pi\)
0.256520 + 0.966539i \(0.417424\pi\)
\(434\) 0 0
\(435\) −1926.47 −0.212338
\(436\) 0 0
\(437\) 21379.9 2.34036
\(438\) 0 0
\(439\) −11917.6 −1.29566 −0.647830 0.761785i \(-0.724323\pi\)
−0.647830 + 0.761785i \(0.724323\pi\)
\(440\) 0 0
\(441\) −1250.75 −0.135056
\(442\) 0 0
\(443\) −5606.28 −0.601270 −0.300635 0.953739i \(-0.597199\pi\)
−0.300635 + 0.953739i \(0.597199\pi\)
\(444\) 0 0
\(445\) −24569.6 −2.61732
\(446\) 0 0
\(447\) −833.906 −0.0882381
\(448\) 0 0
\(449\) −2144.37 −0.225388 −0.112694 0.993630i \(-0.535948\pi\)
−0.112694 + 0.993630i \(0.535948\pi\)
\(450\) 0 0
\(451\) 4435.40 0.463092
\(452\) 0 0
\(453\) 2136.42 0.221584
\(454\) 0 0
\(455\) −9584.63 −0.987548
\(456\) 0 0
\(457\) 8524.38 0.872546 0.436273 0.899814i \(-0.356298\pi\)
0.436273 + 0.899814i \(0.356298\pi\)
\(458\) 0 0
\(459\) −3809.47 −0.387388
\(460\) 0 0
\(461\) −11768.7 −1.18899 −0.594494 0.804100i \(-0.702647\pi\)
−0.594494 + 0.804100i \(0.702647\pi\)
\(462\) 0 0
\(463\) −393.751 −0.0395230 −0.0197615 0.999805i \(-0.506291\pi\)
−0.0197615 + 0.999805i \(0.506291\pi\)
\(464\) 0 0
\(465\) −5143.13 −0.512918
\(466\) 0 0
\(467\) 3258.65 0.322896 0.161448 0.986881i \(-0.448384\pi\)
0.161448 + 0.986881i \(0.448384\pi\)
\(468\) 0 0
\(469\) 1997.36 0.196652
\(470\) 0 0
\(471\) −2226.76 −0.217842
\(472\) 0 0
\(473\) 479.259 0.0465885
\(474\) 0 0
\(475\) −38836.3 −3.75144
\(476\) 0 0
\(477\) −8550.29 −0.820735
\(478\) 0 0
\(479\) −7913.65 −0.754873 −0.377436 0.926036i \(-0.623194\pi\)
−0.377436 + 0.926036i \(0.623194\pi\)
\(480\) 0 0
\(481\) −14358.2 −1.36108
\(482\) 0 0
\(483\) 1386.45 0.130612
\(484\) 0 0
\(485\) −6341.06 −0.593676
\(486\) 0 0
\(487\) −5667.52 −0.527351 −0.263676 0.964611i \(-0.584935\pi\)
−0.263676 + 0.964611i \(0.584935\pi\)
\(488\) 0 0
\(489\) −2800.32 −0.258967
\(490\) 0 0
\(491\) 389.830 0.0358305 0.0179152 0.999840i \(-0.494297\pi\)
0.0179152 + 0.999840i \(0.494297\pi\)
\(492\) 0 0
\(493\) 4616.81 0.421766
\(494\) 0 0
\(495\) −5763.20 −0.523307
\(496\) 0 0
\(497\) 3655.43 0.329916
\(498\) 0 0
\(499\) −13968.3 −1.25312 −0.626558 0.779375i \(-0.715537\pi\)
−0.626558 + 0.779375i \(0.715537\pi\)
\(500\) 0 0
\(501\) 2724.14 0.242925
\(502\) 0 0
\(503\) −15152.6 −1.34318 −0.671591 0.740922i \(-0.734389\pi\)
−0.671591 + 0.740922i \(0.734389\pi\)
\(504\) 0 0
\(505\) −35770.5 −3.15201
\(506\) 0 0
\(507\) −2735.75 −0.239643
\(508\) 0 0
\(509\) −11774.4 −1.02532 −0.512662 0.858591i \(-0.671341\pi\)
−0.512662 + 0.858591i \(0.671341\pi\)
\(510\) 0 0
\(511\) 4147.11 0.359016
\(512\) 0 0
\(513\) −8359.63 −0.719467
\(514\) 0 0
\(515\) 21426.9 1.83336
\(516\) 0 0
\(517\) 760.877 0.0647259
\(518\) 0 0
\(519\) 3034.51 0.256648
\(520\) 0 0
\(521\) −7972.13 −0.670375 −0.335188 0.942151i \(-0.608800\pi\)
−0.335188 + 0.942151i \(0.608800\pi\)
\(522\) 0 0
\(523\) 20833.0 1.74181 0.870904 0.491454i \(-0.163534\pi\)
0.870904 + 0.491454i \(0.163534\pi\)
\(524\) 0 0
\(525\) −2518.48 −0.209363
\(526\) 0 0
\(527\) 12325.6 1.01880
\(528\) 0 0
\(529\) 14440.0 1.18682
\(530\) 0 0
\(531\) 17390.1 1.42122
\(532\) 0 0
\(533\) −26898.1 −2.18590
\(534\) 0 0
\(535\) 4194.80 0.338985
\(536\) 0 0
\(537\) 328.777 0.0264204
\(538\) 0 0
\(539\) −539.000 −0.0430730
\(540\) 0 0
\(541\) 9288.86 0.738187 0.369093 0.929392i \(-0.379668\pi\)
0.369093 + 0.929392i \(0.379668\pi\)
\(542\) 0 0
\(543\) −1797.81 −0.142084
\(544\) 0 0
\(545\) −19381.8 −1.52335
\(546\) 0 0
\(547\) −22942.9 −1.79336 −0.896681 0.442678i \(-0.854028\pi\)
−0.896681 + 0.442678i \(0.854028\pi\)
\(548\) 0 0
\(549\) 625.118 0.0485963
\(550\) 0 0
\(551\) 10131.3 0.783316
\(552\) 0 0
\(553\) 649.592 0.0499520
\(554\) 0 0
\(555\) −5364.43 −0.410284
\(556\) 0 0
\(557\) −4841.26 −0.368278 −0.184139 0.982900i \(-0.558950\pi\)
−0.184139 + 0.982900i \(0.558950\pi\)
\(558\) 0 0
\(559\) −2906.43 −0.219908
\(560\) 0 0
\(561\) −797.787 −0.0600402
\(562\) 0 0
\(563\) 2784.32 0.208428 0.104214 0.994555i \(-0.466767\pi\)
0.104214 + 0.994555i \(0.466767\pi\)
\(564\) 0 0
\(565\) 39107.2 2.91195
\(566\) 0 0
\(567\) 4282.23 0.317172
\(568\) 0 0
\(569\) −2229.27 −0.164246 −0.0821228 0.996622i \(-0.526170\pi\)
−0.0821228 + 0.996622i \(0.526170\pi\)
\(570\) 0 0
\(571\) 6752.10 0.494863 0.247431 0.968905i \(-0.420414\pi\)
0.247431 + 0.968905i \(0.420414\pi\)
\(572\) 0 0
\(573\) 2988.29 0.217867
\(574\) 0 0
\(575\) −48331.4 −3.50532
\(576\) 0 0
\(577\) −18454.6 −1.33150 −0.665749 0.746175i \(-0.731888\pi\)
−0.665749 + 0.746175i \(0.731888\pi\)
\(578\) 0 0
\(579\) 1723.95 0.123739
\(580\) 0 0
\(581\) −3366.49 −0.240388
\(582\) 0 0
\(583\) −3684.66 −0.261755
\(584\) 0 0
\(585\) 34950.5 2.47013
\(586\) 0 0
\(587\) −11807.7 −0.830250 −0.415125 0.909764i \(-0.636262\pi\)
−0.415125 + 0.909764i \(0.636262\pi\)
\(588\) 0 0
\(589\) 27047.6 1.89215
\(590\) 0 0
\(591\) −3543.53 −0.246635
\(592\) 0 0
\(593\) −2300.54 −0.159311 −0.0796557 0.996822i \(-0.525382\pi\)
−0.0796557 + 0.996822i \(0.525382\pi\)
\(594\) 0 0
\(595\) 8581.80 0.591293
\(596\) 0 0
\(597\) 4574.82 0.313626
\(598\) 0 0
\(599\) 6654.41 0.453909 0.226955 0.973905i \(-0.427123\pi\)
0.226955 + 0.973905i \(0.427123\pi\)
\(600\) 0 0
\(601\) −10631.7 −0.721588 −0.360794 0.932646i \(-0.617494\pi\)
−0.360794 + 0.932646i \(0.617494\pi\)
\(602\) 0 0
\(603\) −7283.41 −0.491880
\(604\) 0 0
\(605\) −2483.60 −0.166897
\(606\) 0 0
\(607\) −14834.8 −0.991971 −0.495985 0.868331i \(-0.665193\pi\)
−0.495985 + 0.868331i \(0.665193\pi\)
\(608\) 0 0
\(609\) 656.999 0.0437158
\(610\) 0 0
\(611\) −4614.27 −0.305521
\(612\) 0 0
\(613\) −21769.1 −1.43433 −0.717167 0.696902i \(-0.754561\pi\)
−0.717167 + 0.696902i \(0.754561\pi\)
\(614\) 0 0
\(615\) −10049.5 −0.658919
\(616\) 0 0
\(617\) 11615.5 0.757899 0.378949 0.925417i \(-0.376285\pi\)
0.378949 + 0.925417i \(0.376285\pi\)
\(618\) 0 0
\(619\) −25928.4 −1.68360 −0.841800 0.539789i \(-0.818504\pi\)
−0.841800 + 0.539789i \(0.818504\pi\)
\(620\) 0 0
\(621\) −10403.5 −0.672266
\(622\) 0 0
\(623\) 8379.15 0.538850
\(624\) 0 0
\(625\) 35131.1 2.24839
\(626\) 0 0
\(627\) −1750.69 −0.111508
\(628\) 0 0
\(629\) 12855.9 0.814943
\(630\) 0 0
\(631\) 1291.82 0.0815003 0.0407502 0.999169i \(-0.487025\pi\)
0.0407502 + 0.999169i \(0.487025\pi\)
\(632\) 0 0
\(633\) −4329.00 −0.271820
\(634\) 0 0
\(635\) 11241.7 0.702538
\(636\) 0 0
\(637\) 3268.72 0.203315
\(638\) 0 0
\(639\) −13329.6 −0.825210
\(640\) 0 0
\(641\) −5836.16 −0.359617 −0.179808 0.983702i \(-0.557548\pi\)
−0.179808 + 0.983702i \(0.557548\pi\)
\(642\) 0 0
\(643\) 1068.38 0.0655254 0.0327627 0.999463i \(-0.489569\pi\)
0.0327627 + 0.999463i \(0.489569\pi\)
\(644\) 0 0
\(645\) −1085.88 −0.0662893
\(646\) 0 0
\(647\) 14064.4 0.854607 0.427303 0.904108i \(-0.359464\pi\)
0.427303 + 0.904108i \(0.359464\pi\)
\(648\) 0 0
\(649\) 7494.11 0.453266
\(650\) 0 0
\(651\) 1754.00 0.105599
\(652\) 0 0
\(653\) 23428.7 1.40404 0.702018 0.712160i \(-0.252283\pi\)
0.702018 + 0.712160i \(0.252283\pi\)
\(654\) 0 0
\(655\) −39544.3 −2.35897
\(656\) 0 0
\(657\) −15122.5 −0.897997
\(658\) 0 0
\(659\) −23122.9 −1.36683 −0.683414 0.730031i \(-0.739506\pi\)
−0.683414 + 0.730031i \(0.739506\pi\)
\(660\) 0 0
\(661\) −30074.8 −1.76970 −0.884851 0.465873i \(-0.845740\pi\)
−0.884851 + 0.465873i \(0.845740\pi\)
\(662\) 0 0
\(663\) 4838.11 0.283404
\(664\) 0 0
\(665\) 18832.2 1.09817
\(666\) 0 0
\(667\) 12608.3 0.731925
\(668\) 0 0
\(669\) 3073.62 0.177628
\(670\) 0 0
\(671\) 269.388 0.0154987
\(672\) 0 0
\(673\) −27970.6 −1.60206 −0.801032 0.598621i \(-0.795715\pi\)
−0.801032 + 0.598621i \(0.795715\pi\)
\(674\) 0 0
\(675\) 18897.8 1.07760
\(676\) 0 0
\(677\) −3911.49 −0.222054 −0.111027 0.993817i \(-0.535414\pi\)
−0.111027 + 0.993817i \(0.535414\pi\)
\(678\) 0 0
\(679\) 2162.54 0.122225
\(680\) 0 0
\(681\) 1466.17 0.0825019
\(682\) 0 0
\(683\) 14125.7 0.791367 0.395683 0.918387i \(-0.370508\pi\)
0.395683 + 0.918387i \(0.370508\pi\)
\(684\) 0 0
\(685\) 29572.6 1.64951
\(686\) 0 0
\(687\) 4855.43 0.269645
\(688\) 0 0
\(689\) 22345.3 1.23554
\(690\) 0 0
\(691\) −20549.5 −1.13132 −0.565658 0.824640i \(-0.691378\pi\)
−0.565658 + 0.824640i \(0.691378\pi\)
\(692\) 0 0
\(693\) 1965.47 0.107737
\(694\) 0 0
\(695\) −49893.7 −2.72313
\(696\) 0 0
\(697\) 24083.8 1.30881
\(698\) 0 0
\(699\) −1735.93 −0.0939328
\(700\) 0 0
\(701\) 24071.4 1.29695 0.648475 0.761236i \(-0.275407\pi\)
0.648475 + 0.761236i \(0.275407\pi\)
\(702\) 0 0
\(703\) 28211.5 1.51354
\(704\) 0 0
\(705\) −1723.96 −0.0920964
\(706\) 0 0
\(707\) 12199.1 0.648930
\(708\) 0 0
\(709\) −29125.6 −1.54278 −0.771392 0.636360i \(-0.780439\pi\)
−0.771392 + 0.636360i \(0.780439\pi\)
\(710\) 0 0
\(711\) −2368.75 −0.124944
\(712\) 0 0
\(713\) 33660.5 1.76801
\(714\) 0 0
\(715\) 15061.6 0.787791
\(716\) 0 0
\(717\) 3913.55 0.203841
\(718\) 0 0
\(719\) 34567.0 1.79295 0.896475 0.443094i \(-0.146119\pi\)
0.896475 + 0.443094i \(0.146119\pi\)
\(720\) 0 0
\(721\) −7307.36 −0.377449
\(722\) 0 0
\(723\) −704.958 −0.0362623
\(724\) 0 0
\(725\) −22902.8 −1.17323
\(726\) 0 0
\(727\) 16450.2 0.839206 0.419603 0.907708i \(-0.362169\pi\)
0.419603 + 0.907708i \(0.362169\pi\)
\(728\) 0 0
\(729\) −13524.2 −0.687100
\(730\) 0 0
\(731\) 2602.33 0.131670
\(732\) 0 0
\(733\) −21032.1 −1.05981 −0.529904 0.848058i \(-0.677772\pi\)
−0.529904 + 0.848058i \(0.677772\pi\)
\(734\) 0 0
\(735\) 1221.24 0.0612872
\(736\) 0 0
\(737\) −3138.71 −0.156874
\(738\) 0 0
\(739\) 33830.2 1.68398 0.841991 0.539491i \(-0.181383\pi\)
0.841991 + 0.539491i \(0.181383\pi\)
\(740\) 0 0
\(741\) 10616.9 0.526345
\(742\) 0 0
\(743\) −30148.6 −1.48862 −0.744311 0.667833i \(-0.767222\pi\)
−0.744311 + 0.667833i \(0.767222\pi\)
\(744\) 0 0
\(745\) −14096.2 −0.693217
\(746\) 0 0
\(747\) 12276.0 0.601277
\(748\) 0 0
\(749\) −1430.58 −0.0697896
\(750\) 0 0
\(751\) 25821.0 1.25462 0.627311 0.778769i \(-0.284156\pi\)
0.627311 + 0.778769i \(0.284156\pi\)
\(752\) 0 0
\(753\) 1550.90 0.0750568
\(754\) 0 0
\(755\) 36113.8 1.74081
\(756\) 0 0
\(757\) 8504.19 0.408309 0.204155 0.978939i \(-0.434555\pi\)
0.204155 + 0.978939i \(0.434555\pi\)
\(758\) 0 0
\(759\) −2178.71 −0.104193
\(760\) 0 0
\(761\) 4597.99 0.219024 0.109512 0.993985i \(-0.465071\pi\)
0.109512 + 0.993985i \(0.465071\pi\)
\(762\) 0 0
\(763\) 6609.93 0.313624
\(764\) 0 0
\(765\) −31293.6 −1.47899
\(766\) 0 0
\(767\) −45447.4 −2.13952
\(768\) 0 0
\(769\) 38871.6 1.82282 0.911408 0.411505i \(-0.134997\pi\)
0.911408 + 0.411505i \(0.134997\pi\)
\(770\) 0 0
\(771\) −3056.39 −0.142767
\(772\) 0 0
\(773\) 21974.1 1.02245 0.511224 0.859447i \(-0.329192\pi\)
0.511224 + 0.859447i \(0.329192\pi\)
\(774\) 0 0
\(775\) −61144.0 −2.83401
\(776\) 0 0
\(777\) 1829.47 0.0844685
\(778\) 0 0
\(779\) 52850.2 2.43075
\(780\) 0 0
\(781\) −5744.24 −0.263182
\(782\) 0 0
\(783\) −4929.90 −0.225007
\(784\) 0 0
\(785\) −37640.8 −1.71141
\(786\) 0 0
\(787\) 14241.7 0.645061 0.322531 0.946559i \(-0.395466\pi\)
0.322531 + 0.946559i \(0.395466\pi\)
\(788\) 0 0
\(789\) −6902.33 −0.311444
\(790\) 0 0
\(791\) −13337.0 −0.599507
\(792\) 0 0
\(793\) −1633.68 −0.0731574
\(794\) 0 0
\(795\) 8348.53 0.372443
\(796\) 0 0
\(797\) 39470.7 1.75424 0.877118 0.480276i \(-0.159463\pi\)
0.877118 + 0.480276i \(0.159463\pi\)
\(798\) 0 0
\(799\) 4131.48 0.182930
\(800\) 0 0
\(801\) −30554.7 −1.34781
\(802\) 0 0
\(803\) −6516.88 −0.286396
\(804\) 0 0
\(805\) 23436.4 1.02612
\(806\) 0 0
\(807\) −1056.49 −0.0460846
\(808\) 0 0
\(809\) 11073.5 0.481238 0.240619 0.970620i \(-0.422650\pi\)
0.240619 + 0.970620i \(0.422650\pi\)
\(810\) 0 0
\(811\) 38855.7 1.68238 0.841189 0.540741i \(-0.181856\pi\)
0.841189 + 0.540741i \(0.181856\pi\)
\(812\) 0 0
\(813\) 231.885 0.0100032
\(814\) 0 0
\(815\) −47336.2 −2.03450
\(816\) 0 0
\(817\) 5710.64 0.244541
\(818\) 0 0
\(819\) −11919.4 −0.508545
\(820\) 0 0
\(821\) −9523.67 −0.404846 −0.202423 0.979298i \(-0.564882\pi\)
−0.202423 + 0.979298i \(0.564882\pi\)
\(822\) 0 0
\(823\) 2542.13 0.107671 0.0538355 0.998550i \(-0.482855\pi\)
0.0538355 + 0.998550i \(0.482855\pi\)
\(824\) 0 0
\(825\) 3957.61 0.167014
\(826\) 0 0
\(827\) −8794.30 −0.369779 −0.184890 0.982759i \(-0.559193\pi\)
−0.184890 + 0.982759i \(0.559193\pi\)
\(828\) 0 0
\(829\) −18268.3 −0.765362 −0.382681 0.923881i \(-0.624999\pi\)
−0.382681 + 0.923881i \(0.624999\pi\)
\(830\) 0 0
\(831\) −6801.36 −0.283919
\(832\) 0 0
\(833\) −2926.72 −0.121734
\(834\) 0 0
\(835\) 46048.5 1.90847
\(836\) 0 0
\(837\) −13161.4 −0.543519
\(838\) 0 0
\(839\) −6216.04 −0.255783 −0.127891 0.991788i \(-0.540821\pi\)
−0.127891 + 0.991788i \(0.540821\pi\)
\(840\) 0 0
\(841\) −18414.3 −0.755026
\(842\) 0 0
\(843\) −8352.02 −0.341232
\(844\) 0 0
\(845\) −46244.8 −1.88269
\(846\) 0 0
\(847\) 847.000 0.0343604
\(848\) 0 0
\(849\) 6948.14 0.280871
\(850\) 0 0
\(851\) 35108.9 1.41424
\(852\) 0 0
\(853\) 3081.20 0.123679 0.0618396 0.998086i \(-0.480303\pi\)
0.0618396 + 0.998086i \(0.480303\pi\)
\(854\) 0 0
\(855\) −68671.8 −2.74681
\(856\) 0 0
\(857\) 2992.90 0.119295 0.0596473 0.998220i \(-0.481002\pi\)
0.0596473 + 0.998220i \(0.481002\pi\)
\(858\) 0 0
\(859\) −1740.43 −0.0691300 −0.0345650 0.999402i \(-0.511005\pi\)
−0.0345650 + 0.999402i \(0.511005\pi\)
\(860\) 0 0
\(861\) 3427.26 0.135657
\(862\) 0 0
\(863\) 35296.5 1.39225 0.696123 0.717923i \(-0.254907\pi\)
0.696123 + 0.717923i \(0.254907\pi\)
\(864\) 0 0
\(865\) 51294.9 2.01628
\(866\) 0 0
\(867\) 1633.73 0.0639956
\(868\) 0 0
\(869\) −1020.79 −0.0398479
\(870\) 0 0
\(871\) 19034.5 0.740480
\(872\) 0 0
\(873\) −7885.73 −0.305718
\(874\) 0 0
\(875\) −24612.2 −0.950908
\(876\) 0 0
\(877\) 1103.09 0.0424727 0.0212364 0.999774i \(-0.493240\pi\)
0.0212364 + 0.999774i \(0.493240\pi\)
\(878\) 0 0
\(879\) −10052.2 −0.385725
\(880\) 0 0
\(881\) −37691.0 −1.44136 −0.720682 0.693266i \(-0.756171\pi\)
−0.720682 + 0.693266i \(0.756171\pi\)
\(882\) 0 0
\(883\) −23031.4 −0.877769 −0.438884 0.898544i \(-0.644626\pi\)
−0.438884 + 0.898544i \(0.644626\pi\)
\(884\) 0 0
\(885\) −16979.8 −0.644937
\(886\) 0 0
\(887\) −47839.7 −1.81094 −0.905468 0.424414i \(-0.860480\pi\)
−0.905468 + 0.424414i \(0.860480\pi\)
\(888\) 0 0
\(889\) −3833.83 −0.144637
\(890\) 0 0
\(891\) −6729.21 −0.253016
\(892\) 0 0
\(893\) 9066.26 0.339744
\(894\) 0 0
\(895\) 5557.60 0.207564
\(896\) 0 0
\(897\) 13212.6 0.491813
\(898\) 0 0
\(899\) 15950.7 0.591752
\(900\) 0 0
\(901\) −20007.4 −0.739780
\(902\) 0 0
\(903\) 370.327 0.0136475
\(904\) 0 0
\(905\) −30389.9 −1.11624
\(906\) 0 0
\(907\) −35960.6 −1.31649 −0.658243 0.752805i \(-0.728700\pi\)
−0.658243 + 0.752805i \(0.728700\pi\)
\(908\) 0 0
\(909\) −44484.1 −1.62315
\(910\) 0 0
\(911\) 25190.4 0.916132 0.458066 0.888918i \(-0.348542\pi\)
0.458066 + 0.888918i \(0.348542\pi\)
\(912\) 0 0
\(913\) 5290.20 0.191764
\(914\) 0 0
\(915\) −610.367 −0.0220526
\(916\) 0 0
\(917\) 13486.1 0.485660
\(918\) 0 0
\(919\) 42686.9 1.53222 0.766111 0.642709i \(-0.222189\pi\)
0.766111 + 0.642709i \(0.222189\pi\)
\(920\) 0 0
\(921\) −783.417 −0.0280287
\(922\) 0 0
\(923\) 34835.5 1.24228
\(924\) 0 0
\(925\) −63774.9 −2.26693
\(926\) 0 0
\(927\) 26646.4 0.944102
\(928\) 0 0
\(929\) 29206.6 1.03147 0.515736 0.856748i \(-0.327519\pi\)
0.515736 + 0.856748i \(0.327519\pi\)
\(930\) 0 0
\(931\) −6422.48 −0.226088
\(932\) 0 0
\(933\) 9630.19 0.337919
\(934\) 0 0
\(935\) −13485.7 −0.471689
\(936\) 0 0
\(937\) −27326.2 −0.952730 −0.476365 0.879248i \(-0.658046\pi\)
−0.476365 + 0.879248i \(0.658046\pi\)
\(938\) 0 0
\(939\) −8014.39 −0.278530
\(940\) 0 0
\(941\) 6802.35 0.235654 0.117827 0.993034i \(-0.462407\pi\)
0.117827 + 0.993034i \(0.462407\pi\)
\(942\) 0 0
\(943\) 65771.5 2.27128
\(944\) 0 0
\(945\) −9163.77 −0.315447
\(946\) 0 0
\(947\) −1090.60 −0.0374231 −0.0187116 0.999825i \(-0.505956\pi\)
−0.0187116 + 0.999825i \(0.505956\pi\)
\(948\) 0 0
\(949\) 39521.1 1.35185
\(950\) 0 0
\(951\) 12458.2 0.424800
\(952\) 0 0
\(953\) 48392.3 1.64489 0.822446 0.568843i \(-0.192609\pi\)
0.822446 + 0.568843i \(0.192609\pi\)
\(954\) 0 0
\(955\) 50513.7 1.71161
\(956\) 0 0
\(957\) −1032.43 −0.0348732
\(958\) 0 0
\(959\) −10085.4 −0.339598
\(960\) 0 0
\(961\) 12792.8 0.429418
\(962\) 0 0
\(963\) 5216.65 0.174563
\(964\) 0 0
\(965\) 29141.5 0.972123
\(966\) 0 0
\(967\) −27678.0 −0.920438 −0.460219 0.887805i \(-0.652229\pi\)
−0.460219 + 0.887805i \(0.652229\pi\)
\(968\) 0 0
\(969\) −9506.07 −0.315149
\(970\) 0 0
\(971\) 17642.8 0.583096 0.291548 0.956556i \(-0.405830\pi\)
0.291548 + 0.956556i \(0.405830\pi\)
\(972\) 0 0
\(973\) 17015.6 0.560633
\(974\) 0 0
\(975\) −24000.6 −0.788343
\(976\) 0 0
\(977\) −4539.48 −0.148650 −0.0743249 0.997234i \(-0.523680\pi\)
−0.0743249 + 0.997234i \(0.523680\pi\)
\(978\) 0 0
\(979\) −13167.2 −0.429853
\(980\) 0 0
\(981\) −24103.2 −0.784460
\(982\) 0 0
\(983\) −7250.49 −0.235254 −0.117627 0.993058i \(-0.537529\pi\)
−0.117627 + 0.993058i \(0.537529\pi\)
\(984\) 0 0
\(985\) −59899.4 −1.93762
\(986\) 0 0
\(987\) 587.935 0.0189607
\(988\) 0 0
\(989\) 7106.83 0.228497
\(990\) 0 0
\(991\) −851.834 −0.0273052 −0.0136526 0.999907i \(-0.504346\pi\)
−0.0136526 + 0.999907i \(0.504346\pi\)
\(992\) 0 0
\(993\) −11687.6 −0.373510
\(994\) 0 0
\(995\) 77332.1 2.46391
\(996\) 0 0
\(997\) 52462.3 1.66650 0.833249 0.552897i \(-0.186478\pi\)
0.833249 + 0.552897i \(0.186478\pi\)
\(998\) 0 0
\(999\) −13727.7 −0.434761
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.v.1.2 4
4.3 odd 2 308.4.a.d.1.3 4
28.27 even 2 2156.4.a.f.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
308.4.a.d.1.3 4 4.3 odd 2
1232.4.a.v.1.2 4 1.1 even 1 trivial
2156.4.a.f.1.2 4 28.27 even 2