Properties

Label 1840.4.a.bb.1.4
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $10$
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] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, 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("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + 1222472 x^{2} + 627136 x - 43712 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.66698\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.66698 q^{3} -5.00000 q^{5} +32.5699 q^{7} -19.8872 q^{9} +O(q^{10})\) \(q-2.66698 q^{3} -5.00000 q^{5} +32.5699 q^{7} -19.8872 q^{9} +51.2916 q^{11} +40.1984 q^{13} +13.3349 q^{15} +93.3680 q^{17} +61.9187 q^{19} -86.8634 q^{21} -23.0000 q^{23} +25.0000 q^{25} +125.047 q^{27} -160.031 q^{29} +295.881 q^{31} -136.794 q^{33} -162.849 q^{35} +401.688 q^{37} -107.208 q^{39} +310.046 q^{41} -506.024 q^{43} +99.4360 q^{45} -188.492 q^{47} +717.796 q^{49} -249.011 q^{51} +162.682 q^{53} -256.458 q^{55} -165.136 q^{57} +206.969 q^{59} -856.676 q^{61} -647.723 q^{63} -200.992 q^{65} -435.055 q^{67} +61.3407 q^{69} -132.120 q^{71} -675.451 q^{73} -66.6746 q^{75} +1670.56 q^{77} -802.451 q^{79} +203.455 q^{81} +657.368 q^{83} -466.840 q^{85} +426.799 q^{87} +1588.25 q^{89} +1309.26 q^{91} -789.109 q^{93} -309.594 q^{95} -1549.02 q^{97} -1020.05 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35} + 62 q^{37} - 283 q^{39} + 767 q^{41} - 332 q^{43} - 695 q^{45} + 323 q^{47} + 1162 q^{49} - 176 q^{51} + 558 q^{53} - 70 q^{55} + 46 q^{57} - 822 q^{59} + 318 q^{61} - 2698 q^{63} - 55 q^{65} - 1152 q^{67} - 23 q^{69} - 1247 q^{71} + 1941 q^{73} + 25 q^{75} + 528 q^{77} - 3134 q^{79} + 6210 q^{81} - 482 q^{83} - 340 q^{85} - 1797 q^{87} + 4734 q^{89} - 4992 q^{91} + 4647 q^{93} + 570 q^{95} + 2326 q^{97} - 4356 q^{99}+O(q^{100}) \) 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
\(3\) −2.66698 −0.513261 −0.256631 0.966510i \(-0.582612\pi\)
−0.256631 + 0.966510i \(0.582612\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 32.5699 1.75861 0.879304 0.476261i \(-0.158008\pi\)
0.879304 + 0.476261i \(0.158008\pi\)
\(8\) 0 0
\(9\) −19.8872 −0.736563
\(10\) 0 0
\(11\) 51.2916 1.40591 0.702955 0.711235i \(-0.251864\pi\)
0.702955 + 0.711235i \(0.251864\pi\)
\(12\) 0 0
\(13\) 40.1984 0.857617 0.428808 0.903395i \(-0.358934\pi\)
0.428808 + 0.903395i \(0.358934\pi\)
\(14\) 0 0
\(15\) 13.3349 0.229538
\(16\) 0 0
\(17\) 93.3680 1.33206 0.666032 0.745923i \(-0.267992\pi\)
0.666032 + 0.745923i \(0.267992\pi\)
\(18\) 0 0
\(19\) 61.9187 0.747638 0.373819 0.927502i \(-0.378048\pi\)
0.373819 + 0.927502i \(0.378048\pi\)
\(20\) 0 0
\(21\) −86.8634 −0.902626
\(22\) 0 0
\(23\) −23.0000 −0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 125.047 0.891311
\(28\) 0 0
\(29\) −160.031 −1.02472 −0.512361 0.858770i \(-0.671229\pi\)
−0.512361 + 0.858770i \(0.671229\pi\)
\(30\) 0 0
\(31\) 295.881 1.71425 0.857124 0.515109i \(-0.172249\pi\)
0.857124 + 0.515109i \(0.172249\pi\)
\(32\) 0 0
\(33\) −136.794 −0.721599
\(34\) 0 0
\(35\) −162.849 −0.786473
\(36\) 0 0
\(37\) 401.688 1.78479 0.892393 0.451258i \(-0.149025\pi\)
0.892393 + 0.451258i \(0.149025\pi\)
\(38\) 0 0
\(39\) −107.208 −0.440182
\(40\) 0 0
\(41\) 310.046 1.18100 0.590501 0.807037i \(-0.298930\pi\)
0.590501 + 0.807037i \(0.298930\pi\)
\(42\) 0 0
\(43\) −506.024 −1.79460 −0.897302 0.441417i \(-0.854476\pi\)
−0.897302 + 0.441417i \(0.854476\pi\)
\(44\) 0 0
\(45\) 99.4360 0.329401
\(46\) 0 0
\(47\) −188.492 −0.584987 −0.292494 0.956267i \(-0.594485\pi\)
−0.292494 + 0.956267i \(0.594485\pi\)
\(48\) 0 0
\(49\) 717.796 2.09270
\(50\) 0 0
\(51\) −249.011 −0.683697
\(52\) 0 0
\(53\) 162.682 0.421623 0.210812 0.977527i \(-0.432389\pi\)
0.210812 + 0.977527i \(0.432389\pi\)
\(54\) 0 0
\(55\) −256.458 −0.628742
\(56\) 0 0
\(57\) −165.136 −0.383734
\(58\) 0 0
\(59\) 206.969 0.456697 0.228348 0.973579i \(-0.426667\pi\)
0.228348 + 0.973579i \(0.426667\pi\)
\(60\) 0 0
\(61\) −856.676 −1.79813 −0.899067 0.437811i \(-0.855754\pi\)
−0.899067 + 0.437811i \(0.855754\pi\)
\(62\) 0 0
\(63\) −647.723 −1.29532
\(64\) 0 0
\(65\) −200.992 −0.383538
\(66\) 0 0
\(67\) −435.055 −0.793289 −0.396645 0.917972i \(-0.629825\pi\)
−0.396645 + 0.917972i \(0.629825\pi\)
\(68\) 0 0
\(69\) 61.3407 0.107022
\(70\) 0 0
\(71\) −132.120 −0.220841 −0.110421 0.993885i \(-0.535220\pi\)
−0.110421 + 0.993885i \(0.535220\pi\)
\(72\) 0 0
\(73\) −675.451 −1.08295 −0.541477 0.840716i \(-0.682135\pi\)
−0.541477 + 0.840716i \(0.682135\pi\)
\(74\) 0 0
\(75\) −66.6746 −0.102652
\(76\) 0 0
\(77\) 1670.56 2.47244
\(78\) 0 0
\(79\) −802.451 −1.14282 −0.571410 0.820665i \(-0.693603\pi\)
−0.571410 + 0.820665i \(0.693603\pi\)
\(80\) 0 0
\(81\) 203.455 0.279087
\(82\) 0 0
\(83\) 657.368 0.869344 0.434672 0.900589i \(-0.356864\pi\)
0.434672 + 0.900589i \(0.356864\pi\)
\(84\) 0 0
\(85\) −466.840 −0.595717
\(86\) 0 0
\(87\) 426.799 0.525950
\(88\) 0 0
\(89\) 1588.25 1.89163 0.945813 0.324713i \(-0.105268\pi\)
0.945813 + 0.324713i \(0.105268\pi\)
\(90\) 0 0
\(91\) 1309.26 1.50821
\(92\) 0 0
\(93\) −789.109 −0.879858
\(94\) 0 0
\(95\) −309.594 −0.334354
\(96\) 0 0
\(97\) −1549.02 −1.62144 −0.810719 0.585436i \(-0.800923\pi\)
−0.810719 + 0.585436i \(0.800923\pi\)
\(98\) 0 0
\(99\) −1020.05 −1.03554
\(100\) 0 0
\(101\) 520.718 0.513003 0.256502 0.966544i \(-0.417430\pi\)
0.256502 + 0.966544i \(0.417430\pi\)
\(102\) 0 0
\(103\) −233.521 −0.223394 −0.111697 0.993742i \(-0.535629\pi\)
−0.111697 + 0.993742i \(0.535629\pi\)
\(104\) 0 0
\(105\) 434.317 0.403666
\(106\) 0 0
\(107\) −632.552 −0.571506 −0.285753 0.958303i \(-0.592244\pi\)
−0.285753 + 0.958303i \(0.592244\pi\)
\(108\) 0 0
\(109\) 608.347 0.534579 0.267289 0.963616i \(-0.413872\pi\)
0.267289 + 0.963616i \(0.413872\pi\)
\(110\) 0 0
\(111\) −1071.30 −0.916062
\(112\) 0 0
\(113\) −1508.50 −1.25582 −0.627910 0.778286i \(-0.716090\pi\)
−0.627910 + 0.778286i \(0.716090\pi\)
\(114\) 0 0
\(115\) 115.000 0.0932505
\(116\) 0 0
\(117\) −799.432 −0.631688
\(118\) 0 0
\(119\) 3040.98 2.34258
\(120\) 0 0
\(121\) 1299.83 0.976581
\(122\) 0 0
\(123\) −826.889 −0.606163
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 399.499 0.279132 0.139566 0.990213i \(-0.455429\pi\)
0.139566 + 0.990213i \(0.455429\pi\)
\(128\) 0 0
\(129\) 1349.56 0.921101
\(130\) 0 0
\(131\) 1687.36 1.12538 0.562692 0.826666i \(-0.309766\pi\)
0.562692 + 0.826666i \(0.309766\pi\)
\(132\) 0 0
\(133\) 2016.68 1.31480
\(134\) 0 0
\(135\) −625.237 −0.398606
\(136\) 0 0
\(137\) 1788.43 1.11530 0.557649 0.830077i \(-0.311704\pi\)
0.557649 + 0.830077i \(0.311704\pi\)
\(138\) 0 0
\(139\) −3172.31 −1.93577 −0.967885 0.251392i \(-0.919112\pi\)
−0.967885 + 0.251392i \(0.919112\pi\)
\(140\) 0 0
\(141\) 502.706 0.300252
\(142\) 0 0
\(143\) 2061.84 1.20573
\(144\) 0 0
\(145\) 800.153 0.458269
\(146\) 0 0
\(147\) −1914.35 −1.07410
\(148\) 0 0
\(149\) −578.448 −0.318043 −0.159021 0.987275i \(-0.550834\pi\)
−0.159021 + 0.987275i \(0.550834\pi\)
\(150\) 0 0
\(151\) −76.8056 −0.0413930 −0.0206965 0.999786i \(-0.506588\pi\)
−0.0206965 + 0.999786i \(0.506588\pi\)
\(152\) 0 0
\(153\) −1856.83 −0.981148
\(154\) 0 0
\(155\) −1479.40 −0.766635
\(156\) 0 0
\(157\) 2206.33 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(158\) 0 0
\(159\) −433.869 −0.216403
\(160\) 0 0
\(161\) −749.107 −0.366695
\(162\) 0 0
\(163\) 1613.69 0.775425 0.387713 0.921780i \(-0.373265\pi\)
0.387713 + 0.921780i \(0.373265\pi\)
\(164\) 0 0
\(165\) 683.970 0.322709
\(166\) 0 0
\(167\) 432.118 0.200229 0.100115 0.994976i \(-0.468079\pi\)
0.100115 + 0.994976i \(0.468079\pi\)
\(168\) 0 0
\(169\) −581.093 −0.264494
\(170\) 0 0
\(171\) −1231.39 −0.550682
\(172\) 0 0
\(173\) 56.1129 0.0246600 0.0123300 0.999924i \(-0.496075\pi\)
0.0123300 + 0.999924i \(0.496075\pi\)
\(174\) 0 0
\(175\) 814.247 0.351722
\(176\) 0 0
\(177\) −551.984 −0.234405
\(178\) 0 0
\(179\) −3253.75 −1.35864 −0.679320 0.733842i \(-0.737725\pi\)
−0.679320 + 0.733842i \(0.737725\pi\)
\(180\) 0 0
\(181\) −934.727 −0.383855 −0.191927 0.981409i \(-0.561474\pi\)
−0.191927 + 0.981409i \(0.561474\pi\)
\(182\) 0 0
\(183\) 2284.74 0.922913
\(184\) 0 0
\(185\) −2008.44 −0.798181
\(186\) 0 0
\(187\) 4789.00 1.87276
\(188\) 0 0
\(189\) 4072.78 1.56747
\(190\) 0 0
\(191\) 592.378 0.224414 0.112207 0.993685i \(-0.464208\pi\)
0.112207 + 0.993685i \(0.464208\pi\)
\(192\) 0 0
\(193\) −533.680 −0.199042 −0.0995211 0.995035i \(-0.531731\pi\)
−0.0995211 + 0.995035i \(0.531731\pi\)
\(194\) 0 0
\(195\) 536.042 0.196855
\(196\) 0 0
\(197\) 1721.56 0.622621 0.311311 0.950308i \(-0.399232\pi\)
0.311311 + 0.950308i \(0.399232\pi\)
\(198\) 0 0
\(199\) 3237.34 1.15321 0.576605 0.817023i \(-0.304377\pi\)
0.576605 + 0.817023i \(0.304377\pi\)
\(200\) 0 0
\(201\) 1160.28 0.407165
\(202\) 0 0
\(203\) −5212.17 −1.80208
\(204\) 0 0
\(205\) −1550.23 −0.528160
\(206\) 0 0
\(207\) 457.405 0.153584
\(208\) 0 0
\(209\) 3175.91 1.05111
\(210\) 0 0
\(211\) −4559.29 −1.48756 −0.743779 0.668426i \(-0.766969\pi\)
−0.743779 + 0.668426i \(0.766969\pi\)
\(212\) 0 0
\(213\) 352.362 0.113349
\(214\) 0 0
\(215\) 2530.12 0.802571
\(216\) 0 0
\(217\) 9636.79 3.01469
\(218\) 0 0
\(219\) 1801.42 0.555838
\(220\) 0 0
\(221\) 3753.24 1.14240
\(222\) 0 0
\(223\) −373.549 −0.112173 −0.0560867 0.998426i \(-0.517862\pi\)
−0.0560867 + 0.998426i \(0.517862\pi\)
\(224\) 0 0
\(225\) −497.180 −0.147313
\(226\) 0 0
\(227\) 1609.88 0.470712 0.235356 0.971909i \(-0.424374\pi\)
0.235356 + 0.971909i \(0.424374\pi\)
\(228\) 0 0
\(229\) 3940.09 1.13698 0.568490 0.822690i \(-0.307528\pi\)
0.568490 + 0.822690i \(0.307528\pi\)
\(230\) 0 0
\(231\) −4455.36 −1.26901
\(232\) 0 0
\(233\) −1589.71 −0.446976 −0.223488 0.974707i \(-0.571744\pi\)
−0.223488 + 0.974707i \(0.571744\pi\)
\(234\) 0 0
\(235\) 942.461 0.261614
\(236\) 0 0
\(237\) 2140.12 0.586565
\(238\) 0 0
\(239\) 6822.07 1.84637 0.923186 0.384354i \(-0.125576\pi\)
0.923186 + 0.384354i \(0.125576\pi\)
\(240\) 0 0
\(241\) 1652.64 0.441726 0.220863 0.975305i \(-0.429113\pi\)
0.220863 + 0.975305i \(0.429113\pi\)
\(242\) 0 0
\(243\) −3918.89 −1.03456
\(244\) 0 0
\(245\) −3588.98 −0.935884
\(246\) 0 0
\(247\) 2489.03 0.641187
\(248\) 0 0
\(249\) −1753.19 −0.446201
\(250\) 0 0
\(251\) 416.739 0.104798 0.0523991 0.998626i \(-0.483313\pi\)
0.0523991 + 0.998626i \(0.483313\pi\)
\(252\) 0 0
\(253\) −1179.71 −0.293152
\(254\) 0 0
\(255\) 1245.06 0.305759
\(256\) 0 0
\(257\) 6716.52 1.63021 0.815107 0.579310i \(-0.196678\pi\)
0.815107 + 0.579310i \(0.196678\pi\)
\(258\) 0 0
\(259\) 13082.9 3.13874
\(260\) 0 0
\(261\) 3182.56 0.754772
\(262\) 0 0
\(263\) −3740.77 −0.877055 −0.438528 0.898718i \(-0.644500\pi\)
−0.438528 + 0.898718i \(0.644500\pi\)
\(264\) 0 0
\(265\) −813.408 −0.188556
\(266\) 0 0
\(267\) −4235.85 −0.970898
\(268\) 0 0
\(269\) −7097.85 −1.60879 −0.804393 0.594097i \(-0.797509\pi\)
−0.804393 + 0.594097i \(0.797509\pi\)
\(270\) 0 0
\(271\) 6263.03 1.40388 0.701941 0.712235i \(-0.252317\pi\)
0.701941 + 0.712235i \(0.252317\pi\)
\(272\) 0 0
\(273\) −3491.76 −0.774107
\(274\) 0 0
\(275\) 1282.29 0.281182
\(276\) 0 0
\(277\) −2907.85 −0.630742 −0.315371 0.948968i \(-0.602129\pi\)
−0.315371 + 0.948968i \(0.602129\pi\)
\(278\) 0 0
\(279\) −5884.23 −1.26265
\(280\) 0 0
\(281\) 8189.11 1.73851 0.869255 0.494364i \(-0.164599\pi\)
0.869255 + 0.494364i \(0.164599\pi\)
\(282\) 0 0
\(283\) −1639.18 −0.344307 −0.172154 0.985070i \(-0.555073\pi\)
−0.172154 + 0.985070i \(0.555073\pi\)
\(284\) 0 0
\(285\) 825.681 0.171611
\(286\) 0 0
\(287\) 10098.2 2.07692
\(288\) 0 0
\(289\) 3804.59 0.774393
\(290\) 0 0
\(291\) 4131.22 0.832222
\(292\) 0 0
\(293\) 5752.39 1.14696 0.573478 0.819221i \(-0.305594\pi\)
0.573478 + 0.819221i \(0.305594\pi\)
\(294\) 0 0
\(295\) −1034.85 −0.204241
\(296\) 0 0
\(297\) 6413.88 1.25310
\(298\) 0 0
\(299\) −924.562 −0.178825
\(300\) 0 0
\(301\) −16481.1 −3.15600
\(302\) 0 0
\(303\) −1388.75 −0.263305
\(304\) 0 0
\(305\) 4283.38 0.804150
\(306\) 0 0
\(307\) 158.191 0.0294086 0.0147043 0.999892i \(-0.495319\pi\)
0.0147043 + 0.999892i \(0.495319\pi\)
\(308\) 0 0
\(309\) 622.798 0.114659
\(310\) 0 0
\(311\) 4931.03 0.899078 0.449539 0.893261i \(-0.351588\pi\)
0.449539 + 0.893261i \(0.351588\pi\)
\(312\) 0 0
\(313\) 7376.79 1.33214 0.666072 0.745888i \(-0.267974\pi\)
0.666072 + 0.745888i \(0.267974\pi\)
\(314\) 0 0
\(315\) 3238.62 0.579287
\(316\) 0 0
\(317\) −10018.1 −1.77499 −0.887497 0.460813i \(-0.847558\pi\)
−0.887497 + 0.460813i \(0.847558\pi\)
\(318\) 0 0
\(319\) −8208.23 −1.44067
\(320\) 0 0
\(321\) 1687.01 0.293332
\(322\) 0 0
\(323\) 5781.23 0.995901
\(324\) 0 0
\(325\) 1004.96 0.171523
\(326\) 0 0
\(327\) −1622.45 −0.274379
\(328\) 0 0
\(329\) −6139.17 −1.02876
\(330\) 0 0
\(331\) −4863.38 −0.807599 −0.403800 0.914848i \(-0.632311\pi\)
−0.403800 + 0.914848i \(0.632311\pi\)
\(332\) 0 0
\(333\) −7988.45 −1.31461
\(334\) 0 0
\(335\) 2175.27 0.354770
\(336\) 0 0
\(337\) 3986.67 0.644415 0.322207 0.946669i \(-0.395575\pi\)
0.322207 + 0.946669i \(0.395575\pi\)
\(338\) 0 0
\(339\) 4023.15 0.644564
\(340\) 0 0
\(341\) 15176.2 2.41008
\(342\) 0 0
\(343\) 12207.1 1.92163
\(344\) 0 0
\(345\) −306.703 −0.0478619
\(346\) 0 0
\(347\) −8106.64 −1.25414 −0.627071 0.778962i \(-0.715747\pi\)
−0.627071 + 0.778962i \(0.715747\pi\)
\(348\) 0 0
\(349\) −7883.34 −1.20913 −0.604564 0.796557i \(-0.706652\pi\)
−0.604564 + 0.796557i \(0.706652\pi\)
\(350\) 0 0
\(351\) 5026.70 0.764403
\(352\) 0 0
\(353\) 8183.51 1.23389 0.616946 0.787005i \(-0.288370\pi\)
0.616946 + 0.787005i \(0.288370\pi\)
\(354\) 0 0
\(355\) 660.599 0.0987633
\(356\) 0 0
\(357\) −8110.26 −1.20235
\(358\) 0 0
\(359\) −4731.13 −0.695542 −0.347771 0.937580i \(-0.613061\pi\)
−0.347771 + 0.937580i \(0.613061\pi\)
\(360\) 0 0
\(361\) −3025.07 −0.441037
\(362\) 0 0
\(363\) −3466.63 −0.501242
\(364\) 0 0
\(365\) 3377.26 0.484312
\(366\) 0 0
\(367\) −9595.07 −1.36474 −0.682368 0.731009i \(-0.739050\pi\)
−0.682368 + 0.731009i \(0.739050\pi\)
\(368\) 0 0
\(369\) −6165.95 −0.869882
\(370\) 0 0
\(371\) 5298.52 0.741470
\(372\) 0 0
\(373\) −9329.41 −1.29506 −0.647532 0.762038i \(-0.724199\pi\)
−0.647532 + 0.762038i \(0.724199\pi\)
\(374\) 0 0
\(375\) 333.373 0.0459075
\(376\) 0 0
\(377\) −6432.96 −0.878818
\(378\) 0 0
\(379\) −1735.50 −0.235216 −0.117608 0.993060i \(-0.537523\pi\)
−0.117608 + 0.993060i \(0.537523\pi\)
\(380\) 0 0
\(381\) −1065.46 −0.143268
\(382\) 0 0
\(383\) 4477.97 0.597425 0.298712 0.954343i \(-0.403443\pi\)
0.298712 + 0.954343i \(0.403443\pi\)
\(384\) 0 0
\(385\) −8352.81 −1.10571
\(386\) 0 0
\(387\) 10063.4 1.32184
\(388\) 0 0
\(389\) 4131.24 0.538463 0.269232 0.963075i \(-0.413230\pi\)
0.269232 + 0.963075i \(0.413230\pi\)
\(390\) 0 0
\(391\) −2147.46 −0.277754
\(392\) 0 0
\(393\) −4500.17 −0.577617
\(394\) 0 0
\(395\) 4012.25 0.511084
\(396\) 0 0
\(397\) 7836.84 0.990730 0.495365 0.868685i \(-0.335034\pi\)
0.495365 + 0.868685i \(0.335034\pi\)
\(398\) 0 0
\(399\) −5378.47 −0.674837
\(400\) 0 0
\(401\) −4396.01 −0.547447 −0.273723 0.961808i \(-0.588255\pi\)
−0.273723 + 0.961808i \(0.588255\pi\)
\(402\) 0 0
\(403\) 11893.9 1.47017
\(404\) 0 0
\(405\) −1017.27 −0.124812
\(406\) 0 0
\(407\) 20603.2 2.50925
\(408\) 0 0
\(409\) 7113.24 0.859968 0.429984 0.902836i \(-0.358519\pi\)
0.429984 + 0.902836i \(0.358519\pi\)
\(410\) 0 0
\(411\) −4769.71 −0.572439
\(412\) 0 0
\(413\) 6740.96 0.803150
\(414\) 0 0
\(415\) −3286.84 −0.388782
\(416\) 0 0
\(417\) 8460.51 0.993557
\(418\) 0 0
\(419\) −6600.86 −0.769625 −0.384812 0.922995i \(-0.625734\pi\)
−0.384812 + 0.922995i \(0.625734\pi\)
\(420\) 0 0
\(421\) 5950.49 0.688858 0.344429 0.938812i \(-0.388073\pi\)
0.344429 + 0.938812i \(0.388073\pi\)
\(422\) 0 0
\(423\) 3748.58 0.430880
\(424\) 0 0
\(425\) 2334.20 0.266413
\(426\) 0 0
\(427\) −27901.8 −3.16221
\(428\) 0 0
\(429\) −5498.89 −0.618855
\(430\) 0 0
\(431\) 10387.9 1.16094 0.580470 0.814281i \(-0.302869\pi\)
0.580470 + 0.814281i \(0.302869\pi\)
\(432\) 0 0
\(433\) −953.901 −0.105870 −0.0529348 0.998598i \(-0.516858\pi\)
−0.0529348 + 0.998598i \(0.516858\pi\)
\(434\) 0 0
\(435\) −2134.00 −0.235212
\(436\) 0 0
\(437\) −1424.13 −0.155893
\(438\) 0 0
\(439\) −1178.30 −0.128103 −0.0640516 0.997947i \(-0.520402\pi\)
−0.0640516 + 0.997947i \(0.520402\pi\)
\(440\) 0 0
\(441\) −14275.0 −1.54141
\(442\) 0 0
\(443\) 7649.98 0.820455 0.410228 0.911983i \(-0.365449\pi\)
0.410228 + 0.911983i \(0.365449\pi\)
\(444\) 0 0
\(445\) −7941.27 −0.845960
\(446\) 0 0
\(447\) 1542.71 0.163239
\(448\) 0 0
\(449\) −8823.48 −0.927407 −0.463704 0.885990i \(-0.653480\pi\)
−0.463704 + 0.885990i \(0.653480\pi\)
\(450\) 0 0
\(451\) 15902.8 1.66038
\(452\) 0 0
\(453\) 204.839 0.0212454
\(454\) 0 0
\(455\) −6546.28 −0.674493
\(456\) 0 0
\(457\) 8649.10 0.885312 0.442656 0.896691i \(-0.354036\pi\)
0.442656 + 0.896691i \(0.354036\pi\)
\(458\) 0 0
\(459\) 11675.4 1.18728
\(460\) 0 0
\(461\) 6286.95 0.635168 0.317584 0.948230i \(-0.397128\pi\)
0.317584 + 0.948230i \(0.397128\pi\)
\(462\) 0 0
\(463\) −5621.96 −0.564309 −0.282154 0.959369i \(-0.591049\pi\)
−0.282154 + 0.959369i \(0.591049\pi\)
\(464\) 0 0
\(465\) 3945.54 0.393484
\(466\) 0 0
\(467\) 6971.08 0.690757 0.345378 0.938464i \(-0.387751\pi\)
0.345378 + 0.938464i \(0.387751\pi\)
\(468\) 0 0
\(469\) −14169.7 −1.39508
\(470\) 0 0
\(471\) −5884.25 −0.575651
\(472\) 0 0
\(473\) −25954.8 −2.52305
\(474\) 0 0
\(475\) 1547.97 0.149528
\(476\) 0 0
\(477\) −3235.28 −0.310552
\(478\) 0 0
\(479\) −17953.1 −1.71252 −0.856259 0.516546i \(-0.827217\pi\)
−0.856259 + 0.516546i \(0.827217\pi\)
\(480\) 0 0
\(481\) 16147.2 1.53066
\(482\) 0 0
\(483\) 1997.86 0.188210
\(484\) 0 0
\(485\) 7745.12 0.725129
\(486\) 0 0
\(487\) 7211.61 0.671025 0.335513 0.942036i \(-0.391090\pi\)
0.335513 + 0.942036i \(0.391090\pi\)
\(488\) 0 0
\(489\) −4303.70 −0.397996
\(490\) 0 0
\(491\) −17059.5 −1.56799 −0.783997 0.620764i \(-0.786822\pi\)
−0.783997 + 0.620764i \(0.786822\pi\)
\(492\) 0 0
\(493\) −14941.7 −1.36499
\(494\) 0 0
\(495\) 5100.23 0.463108
\(496\) 0 0
\(497\) −4303.13 −0.388374
\(498\) 0 0
\(499\) 7033.90 0.631023 0.315512 0.948922i \(-0.397824\pi\)
0.315512 + 0.948922i \(0.397824\pi\)
\(500\) 0 0
\(501\) −1152.45 −0.102770
\(502\) 0 0
\(503\) 6905.46 0.612125 0.306063 0.952011i \(-0.400988\pi\)
0.306063 + 0.952011i \(0.400988\pi\)
\(504\) 0 0
\(505\) −2603.59 −0.229422
\(506\) 0 0
\(507\) 1549.77 0.135754
\(508\) 0 0
\(509\) −17053.4 −1.48502 −0.742512 0.669833i \(-0.766366\pi\)
−0.742512 + 0.669833i \(0.766366\pi\)
\(510\) 0 0
\(511\) −21999.4 −1.90449
\(512\) 0 0
\(513\) 7742.78 0.666378
\(514\) 0 0
\(515\) 1167.61 0.0999047
\(516\) 0 0
\(517\) −9668.07 −0.822439
\(518\) 0 0
\(519\) −149.652 −0.0126570
\(520\) 0 0
\(521\) 8408.62 0.707080 0.353540 0.935419i \(-0.384978\pi\)
0.353540 + 0.935419i \(0.384978\pi\)
\(522\) 0 0
\(523\) 10039.2 0.839358 0.419679 0.907672i \(-0.362143\pi\)
0.419679 + 0.907672i \(0.362143\pi\)
\(524\) 0 0
\(525\) −2171.58 −0.180525
\(526\) 0 0
\(527\) 27625.8 2.28349
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −4116.04 −0.336386
\(532\) 0 0
\(533\) 12463.3 1.01285
\(534\) 0 0
\(535\) 3162.76 0.255585
\(536\) 0 0
\(537\) 8677.70 0.697338
\(538\) 0 0
\(539\) 36816.9 2.94215
\(540\) 0 0
\(541\) −20681.5 −1.64356 −0.821780 0.569805i \(-0.807019\pi\)
−0.821780 + 0.569805i \(0.807019\pi\)
\(542\) 0 0
\(543\) 2492.90 0.197018
\(544\) 0 0
\(545\) −3041.74 −0.239071
\(546\) 0 0
\(547\) −14433.2 −1.12819 −0.564095 0.825710i \(-0.690775\pi\)
−0.564095 + 0.825710i \(0.690775\pi\)
\(548\) 0 0
\(549\) 17036.9 1.32444
\(550\) 0 0
\(551\) −9908.89 −0.766121
\(552\) 0 0
\(553\) −26135.7 −2.00977
\(554\) 0 0
\(555\) 5356.48 0.409675
\(556\) 0 0
\(557\) −3260.33 −0.248016 −0.124008 0.992281i \(-0.539575\pi\)
−0.124008 + 0.992281i \(0.539575\pi\)
\(558\) 0 0
\(559\) −20341.3 −1.53908
\(560\) 0 0
\(561\) −12772.2 −0.961216
\(562\) 0 0
\(563\) 20472.8 1.53255 0.766275 0.642513i \(-0.222108\pi\)
0.766275 + 0.642513i \(0.222108\pi\)
\(564\) 0 0
\(565\) 7542.50 0.561620
\(566\) 0 0
\(567\) 6626.49 0.490805
\(568\) 0 0
\(569\) 6713.38 0.494621 0.247311 0.968936i \(-0.420453\pi\)
0.247311 + 0.968936i \(0.420453\pi\)
\(570\) 0 0
\(571\) −16917.1 −1.23986 −0.619930 0.784657i \(-0.712839\pi\)
−0.619930 + 0.784657i \(0.712839\pi\)
\(572\) 0 0
\(573\) −1579.86 −0.115183
\(574\) 0 0
\(575\) −575.000 −0.0417029
\(576\) 0 0
\(577\) 24924.6 1.79831 0.899155 0.437631i \(-0.144182\pi\)
0.899155 + 0.437631i \(0.144182\pi\)
\(578\) 0 0
\(579\) 1423.32 0.102161
\(580\) 0 0
\(581\) 21410.4 1.52883
\(582\) 0 0
\(583\) 8344.20 0.592764
\(584\) 0 0
\(585\) 3997.16 0.282500
\(586\) 0 0
\(587\) 14178.0 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(588\) 0 0
\(589\) 18320.5 1.28164
\(590\) 0 0
\(591\) −4591.39 −0.319568
\(592\) 0 0
\(593\) −202.904 −0.0140510 −0.00702552 0.999975i \(-0.502236\pi\)
−0.00702552 + 0.999975i \(0.502236\pi\)
\(594\) 0 0
\(595\) −15204.9 −1.04763
\(596\) 0 0
\(597\) −8633.92 −0.591898
\(598\) 0 0
\(599\) −11092.1 −0.756614 −0.378307 0.925680i \(-0.623494\pi\)
−0.378307 + 0.925680i \(0.623494\pi\)
\(600\) 0 0
\(601\) 8006.00 0.543380 0.271690 0.962385i \(-0.412417\pi\)
0.271690 + 0.962385i \(0.412417\pi\)
\(602\) 0 0
\(603\) 8652.02 0.584307
\(604\) 0 0
\(605\) −6499.15 −0.436740
\(606\) 0 0
\(607\) −19772.8 −1.32216 −0.661080 0.750315i \(-0.729902\pi\)
−0.661080 + 0.750315i \(0.729902\pi\)
\(608\) 0 0
\(609\) 13900.8 0.924940
\(610\) 0 0
\(611\) −7577.08 −0.501695
\(612\) 0 0
\(613\) 16760.8 1.10434 0.552171 0.833731i \(-0.313800\pi\)
0.552171 + 0.833731i \(0.313800\pi\)
\(614\) 0 0
\(615\) 4134.44 0.271084
\(616\) 0 0
\(617\) −2059.18 −0.134359 −0.0671793 0.997741i \(-0.521400\pi\)
−0.0671793 + 0.997741i \(0.521400\pi\)
\(618\) 0 0
\(619\) 9225.99 0.599070 0.299535 0.954085i \(-0.403169\pi\)
0.299535 + 0.954085i \(0.403169\pi\)
\(620\) 0 0
\(621\) −2876.09 −0.185851
\(622\) 0 0
\(623\) 51729.2 3.32663
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −8470.11 −0.539495
\(628\) 0 0
\(629\) 37504.8 2.37745
\(630\) 0 0
\(631\) −16197.8 −1.02191 −0.510954 0.859608i \(-0.670708\pi\)
−0.510954 + 0.859608i \(0.670708\pi\)
\(632\) 0 0
\(633\) 12159.6 0.763506
\(634\) 0 0
\(635\) −1997.49 −0.124832
\(636\) 0 0
\(637\) 28854.2 1.79474
\(638\) 0 0
\(639\) 2627.49 0.162664
\(640\) 0 0
\(641\) −17673.4 −1.08901 −0.544506 0.838757i \(-0.683283\pi\)
−0.544506 + 0.838757i \(0.683283\pi\)
\(642\) 0 0
\(643\) −26012.7 −1.59540 −0.797698 0.603057i \(-0.793949\pi\)
−0.797698 + 0.603057i \(0.793949\pi\)
\(644\) 0 0
\(645\) −6747.80 −0.411929
\(646\) 0 0
\(647\) −23339.4 −1.41818 −0.709092 0.705116i \(-0.750895\pi\)
−0.709092 + 0.705116i \(0.750895\pi\)
\(648\) 0 0
\(649\) 10615.8 0.642074
\(650\) 0 0
\(651\) −25701.2 −1.54732
\(652\) 0 0
\(653\) −13540.1 −0.811434 −0.405717 0.913999i \(-0.632978\pi\)
−0.405717 + 0.913999i \(0.632978\pi\)
\(654\) 0 0
\(655\) −8436.80 −0.503287
\(656\) 0 0
\(657\) 13432.8 0.797663
\(658\) 0 0
\(659\) −4110.58 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(660\) 0 0
\(661\) 6854.43 0.403338 0.201669 0.979454i \(-0.435363\pi\)
0.201669 + 0.979454i \(0.435363\pi\)
\(662\) 0 0
\(663\) −10009.8 −0.586350
\(664\) 0 0
\(665\) −10083.4 −0.587997
\(666\) 0 0
\(667\) 3680.70 0.213669
\(668\) 0 0
\(669\) 996.249 0.0575743
\(670\) 0 0
\(671\) −43940.3 −2.52801
\(672\) 0 0
\(673\) −604.990 −0.0346518 −0.0173259 0.999850i \(-0.505515\pi\)
−0.0173259 + 0.999850i \(0.505515\pi\)
\(674\) 0 0
\(675\) 3126.19 0.178262
\(676\) 0 0
\(677\) −17954.2 −1.01925 −0.509627 0.860395i \(-0.670217\pi\)
−0.509627 + 0.860395i \(0.670217\pi\)
\(678\) 0 0
\(679\) −50451.5 −2.85147
\(680\) 0 0
\(681\) −4293.53 −0.241598
\(682\) 0 0
\(683\) 17456.8 0.977987 0.488993 0.872288i \(-0.337364\pi\)
0.488993 + 0.872288i \(0.337364\pi\)
\(684\) 0 0
\(685\) −8942.14 −0.498776
\(686\) 0 0
\(687\) −10508.2 −0.583568
\(688\) 0 0
\(689\) 6539.53 0.361591
\(690\) 0 0
\(691\) 17092.8 0.941013 0.470507 0.882397i \(-0.344071\pi\)
0.470507 + 0.882397i \(0.344071\pi\)
\(692\) 0 0
\(693\) −33222.8 −1.82111
\(694\) 0 0
\(695\) 15861.6 0.865703
\(696\) 0 0
\(697\) 28948.4 1.57317
\(698\) 0 0
\(699\) 4239.73 0.229415
\(700\) 0 0
\(701\) −24311.0 −1.30986 −0.654932 0.755687i \(-0.727303\pi\)
−0.654932 + 0.755687i \(0.727303\pi\)
\(702\) 0 0
\(703\) 24872.0 1.33437
\(704\) 0 0
\(705\) −2513.53 −0.134277
\(706\) 0 0
\(707\) 16959.7 0.902172
\(708\) 0 0
\(709\) −26445.7 −1.40083 −0.700415 0.713736i \(-0.747002\pi\)
−0.700415 + 0.713736i \(0.747002\pi\)
\(710\) 0 0
\(711\) 15958.5 0.841758
\(712\) 0 0
\(713\) −6805.25 −0.357446
\(714\) 0 0
\(715\) −10309.2 −0.539219
\(716\) 0 0
\(717\) −18194.3 −0.947671
\(718\) 0 0
\(719\) 3042.17 0.157794 0.0788970 0.996883i \(-0.474860\pi\)
0.0788970 + 0.996883i \(0.474860\pi\)
\(720\) 0 0
\(721\) −7605.76 −0.392862
\(722\) 0 0
\(723\) −4407.57 −0.226721
\(724\) 0 0
\(725\) −4000.76 −0.204944
\(726\) 0 0
\(727\) 2285.89 0.116615 0.0583073 0.998299i \(-0.481430\pi\)
0.0583073 + 0.998299i \(0.481430\pi\)
\(728\) 0 0
\(729\) 4958.35 0.251910
\(730\) 0 0
\(731\) −47246.5 −2.39053
\(732\) 0 0
\(733\) 11996.9 0.604523 0.302262 0.953225i \(-0.402258\pi\)
0.302262 + 0.953225i \(0.402258\pi\)
\(734\) 0 0
\(735\) 9571.76 0.480353
\(736\) 0 0
\(737\) −22314.7 −1.11529
\(738\) 0 0
\(739\) −6071.38 −0.302218 −0.151109 0.988517i \(-0.548284\pi\)
−0.151109 + 0.988517i \(0.548284\pi\)
\(740\) 0 0
\(741\) −6638.21 −0.329097
\(742\) 0 0
\(743\) −4243.30 −0.209517 −0.104759 0.994498i \(-0.533407\pi\)
−0.104759 + 0.994498i \(0.533407\pi\)
\(744\) 0 0
\(745\) 2892.24 0.142233
\(746\) 0 0
\(747\) −13073.2 −0.640326
\(748\) 0 0
\(749\) −20602.1 −1.00505
\(750\) 0 0
\(751\) −34214.9 −1.66248 −0.831238 0.555917i \(-0.812367\pi\)
−0.831238 + 0.555917i \(0.812367\pi\)
\(752\) 0 0
\(753\) −1111.44 −0.0537889
\(754\) 0 0
\(755\) 384.028 0.0185115
\(756\) 0 0
\(757\) 13048.3 0.626485 0.313243 0.949673i \(-0.398585\pi\)
0.313243 + 0.949673i \(0.398585\pi\)
\(758\) 0 0
\(759\) 3146.26 0.150464
\(760\) 0 0
\(761\) 12372.4 0.589353 0.294676 0.955597i \(-0.404788\pi\)
0.294676 + 0.955597i \(0.404788\pi\)
\(762\) 0 0
\(763\) 19813.8 0.940115
\(764\) 0 0
\(765\) 9284.14 0.438783
\(766\) 0 0
\(767\) 8319.82 0.391671
\(768\) 0 0
\(769\) 16603.3 0.778585 0.389292 0.921114i \(-0.372720\pi\)
0.389292 + 0.921114i \(0.372720\pi\)
\(770\) 0 0
\(771\) −17912.9 −0.836726
\(772\) 0 0
\(773\) −16184.3 −0.753050 −0.376525 0.926406i \(-0.622881\pi\)
−0.376525 + 0.926406i \(0.622881\pi\)
\(774\) 0 0
\(775\) 7397.01 0.342850
\(776\) 0 0
\(777\) −34892.0 −1.61099
\(778\) 0 0
\(779\) 19197.7 0.882962
\(780\) 0 0
\(781\) −6776.64 −0.310483
\(782\) 0 0
\(783\) −20011.4 −0.913345
\(784\) 0 0
\(785\) −11031.6 −0.501575
\(786\) 0 0
\(787\) −8065.72 −0.365326 −0.182663 0.983176i \(-0.558472\pi\)
−0.182663 + 0.983176i \(0.558472\pi\)
\(788\) 0 0
\(789\) 9976.57 0.450159
\(790\) 0 0
\(791\) −49131.6 −2.20850
\(792\) 0 0
\(793\) −34437.0 −1.54211
\(794\) 0 0
\(795\) 2169.35 0.0967784
\(796\) 0 0
\(797\) −543.463 −0.0241536 −0.0120768 0.999927i \(-0.503844\pi\)
−0.0120768 + 0.999927i \(0.503844\pi\)
\(798\) 0 0
\(799\) −17599.1 −0.779240
\(800\) 0 0
\(801\) −31585.9 −1.39330
\(802\) 0 0
\(803\) −34645.0 −1.52253
\(804\) 0 0
\(805\) 3745.54 0.163991
\(806\) 0 0
\(807\) 18929.9 0.825728
\(808\) 0 0
\(809\) 7792.84 0.338667 0.169333 0.985559i \(-0.445839\pi\)
0.169333 + 0.985559i \(0.445839\pi\)
\(810\) 0 0
\(811\) −20977.8 −0.908300 −0.454150 0.890925i \(-0.650057\pi\)
−0.454150 + 0.890925i \(0.650057\pi\)
\(812\) 0 0
\(813\) −16703.4 −0.720559
\(814\) 0 0
\(815\) −8068.47 −0.346781
\(816\) 0 0
\(817\) −31332.4 −1.34171
\(818\) 0 0
\(819\) −26037.4 −1.11089
\(820\) 0 0
\(821\) −21076.3 −0.895940 −0.447970 0.894049i \(-0.647853\pi\)
−0.447970 + 0.894049i \(0.647853\pi\)
\(822\) 0 0
\(823\) −23531.1 −0.996650 −0.498325 0.866990i \(-0.666051\pi\)
−0.498325 + 0.866990i \(0.666051\pi\)
\(824\) 0 0
\(825\) −3419.85 −0.144320
\(826\) 0 0
\(827\) −16665.2 −0.700733 −0.350367 0.936613i \(-0.613943\pi\)
−0.350367 + 0.936613i \(0.613943\pi\)
\(828\) 0 0
\(829\) −7011.28 −0.293741 −0.146871 0.989156i \(-0.546920\pi\)
−0.146871 + 0.989156i \(0.546920\pi\)
\(830\) 0 0
\(831\) 7755.18 0.323735
\(832\) 0 0
\(833\) 67019.2 2.78761
\(834\) 0 0
\(835\) −2160.59 −0.0895452
\(836\) 0 0
\(837\) 36999.1 1.52793
\(838\) 0 0
\(839\) 14366.1 0.591149 0.295575 0.955320i \(-0.404489\pi\)
0.295575 + 0.955320i \(0.404489\pi\)
\(840\) 0 0
\(841\) 1220.78 0.0500544
\(842\) 0 0
\(843\) −21840.2 −0.892310
\(844\) 0 0
\(845\) 2905.46 0.118285
\(846\) 0 0
\(847\) 42335.3 1.71742
\(848\) 0 0
\(849\) 4371.66 0.176720
\(850\) 0 0
\(851\) −9238.82 −0.372154
\(852\) 0 0
\(853\) 15450.3 0.620172 0.310086 0.950709i \(-0.399642\pi\)
0.310086 + 0.950709i \(0.399642\pi\)
\(854\) 0 0
\(855\) 6156.95 0.246273
\(856\) 0 0
\(857\) 615.237 0.0245229 0.0122614 0.999925i \(-0.496097\pi\)
0.0122614 + 0.999925i \(0.496097\pi\)
\(858\) 0 0
\(859\) 40420.2 1.60550 0.802748 0.596319i \(-0.203370\pi\)
0.802748 + 0.596319i \(0.203370\pi\)
\(860\) 0 0
\(861\) −26931.7 −1.06600
\(862\) 0 0
\(863\) 41.1816 0.00162438 0.000812189 1.00000i \(-0.499741\pi\)
0.000812189 1.00000i \(0.499741\pi\)
\(864\) 0 0
\(865\) −280.564 −0.0110283
\(866\) 0 0
\(867\) −10146.8 −0.397466
\(868\) 0 0
\(869\) −41159.0 −1.60670
\(870\) 0 0
\(871\) −17488.5 −0.680338
\(872\) 0 0
\(873\) 30805.7 1.19429
\(874\) 0 0
\(875\) −4071.23 −0.157295
\(876\) 0 0
\(877\) 16812.3 0.647332 0.323666 0.946171i \(-0.395085\pi\)
0.323666 + 0.946171i \(0.395085\pi\)
\(878\) 0 0
\(879\) −15341.5 −0.588689
\(880\) 0 0
\(881\) 7044.58 0.269396 0.134698 0.990887i \(-0.456994\pi\)
0.134698 + 0.990887i \(0.456994\pi\)
\(882\) 0 0
\(883\) −43578.5 −1.66085 −0.830427 0.557127i \(-0.811904\pi\)
−0.830427 + 0.557127i \(0.811904\pi\)
\(884\) 0 0
\(885\) 2759.92 0.104829
\(886\) 0 0
\(887\) 7888.43 0.298611 0.149305 0.988791i \(-0.452296\pi\)
0.149305 + 0.988791i \(0.452296\pi\)
\(888\) 0 0
\(889\) 13011.6 0.490884
\(890\) 0 0
\(891\) 10435.5 0.392371
\(892\) 0 0
\(893\) −11671.2 −0.437359
\(894\) 0 0
\(895\) 16268.7 0.607602
\(896\) 0 0
\(897\) 2465.79 0.0917842
\(898\) 0 0
\(899\) −47349.9 −1.75663
\(900\) 0 0
\(901\) 15189.3 0.561629
\(902\) 0 0
\(903\) 43955.0 1.61986
\(904\) 0 0
\(905\) 4673.63 0.171665
\(906\) 0 0
\(907\) −25762.6 −0.943146 −0.471573 0.881827i \(-0.656314\pi\)
−0.471573 + 0.881827i \(0.656314\pi\)
\(908\) 0 0
\(909\) −10355.6 −0.377859
\(910\) 0 0
\(911\) 14667.6 0.533437 0.266718 0.963775i \(-0.414061\pi\)
0.266718 + 0.963775i \(0.414061\pi\)
\(912\) 0 0
\(913\) 33717.5 1.22222
\(914\) 0 0
\(915\) −11423.7 −0.412739
\(916\) 0 0
\(917\) 54957.1 1.97911
\(918\) 0 0
\(919\) 13259.9 0.475957 0.237979 0.971270i \(-0.423515\pi\)
0.237979 + 0.971270i \(0.423515\pi\)
\(920\) 0 0
\(921\) −421.893 −0.0150943
\(922\) 0 0
\(923\) −5311.00 −0.189397
\(924\) 0 0
\(925\) 10042.2 0.356957
\(926\) 0 0
\(927\) 4644.09 0.164543
\(928\) 0 0
\(929\) −22354.5 −0.789482 −0.394741 0.918792i \(-0.629166\pi\)
−0.394741 + 0.918792i \(0.629166\pi\)
\(930\) 0 0
\(931\) 44445.0 1.56458
\(932\) 0 0
\(933\) −13151.0 −0.461462
\(934\) 0 0
\(935\) −23945.0 −0.837524
\(936\) 0 0
\(937\) 32943.0 1.14856 0.574281 0.818659i \(-0.305282\pi\)
0.574281 + 0.818659i \(0.305282\pi\)
\(938\) 0 0
\(939\) −19673.8 −0.683738
\(940\) 0 0
\(941\) 25878.6 0.896514 0.448257 0.893905i \(-0.352045\pi\)
0.448257 + 0.893905i \(0.352045\pi\)
\(942\) 0 0
\(943\) −7131.06 −0.246256
\(944\) 0 0
\(945\) −20363.9 −0.700992
\(946\) 0 0
\(947\) −4350.14 −0.149272 −0.0746360 0.997211i \(-0.523779\pi\)
−0.0746360 + 0.997211i \(0.523779\pi\)
\(948\) 0 0
\(949\) −27152.0 −0.928759
\(950\) 0 0
\(951\) 26718.2 0.911037
\(952\) 0 0
\(953\) 31637.6 1.07539 0.537693 0.843140i \(-0.319296\pi\)
0.537693 + 0.843140i \(0.319296\pi\)
\(954\) 0 0
\(955\) −2961.89 −0.100361
\(956\) 0 0
\(957\) 21891.2 0.739438
\(958\) 0 0
\(959\) 58248.9 1.96137
\(960\) 0 0
\(961\) 57754.3 1.93865
\(962\) 0 0
\(963\) 12579.7 0.420950
\(964\) 0 0
\(965\) 2668.40 0.0890144
\(966\) 0 0
\(967\) 882.920 0.0293617 0.0146809 0.999892i \(-0.495327\pi\)
0.0146809 + 0.999892i \(0.495327\pi\)
\(968\) 0 0
\(969\) −15418.5 −0.511158
\(970\) 0 0
\(971\) −22292.9 −0.736781 −0.368391 0.929671i \(-0.620091\pi\)
−0.368391 + 0.929671i \(0.620091\pi\)
\(972\) 0 0
\(973\) −103322. −3.40426
\(974\) 0 0
\(975\) −2680.21 −0.0880363
\(976\) 0 0
\(977\) 43227.9 1.41554 0.707770 0.706443i \(-0.249702\pi\)
0.707770 + 0.706443i \(0.249702\pi\)
\(978\) 0 0
\(979\) 81464.1 2.65945
\(980\) 0 0
\(981\) −12098.3 −0.393751
\(982\) 0 0
\(983\) 5466.42 0.177367 0.0886835 0.996060i \(-0.471734\pi\)
0.0886835 + 0.996060i \(0.471734\pi\)
\(984\) 0 0
\(985\) −8607.82 −0.278445
\(986\) 0 0
\(987\) 16373.1 0.528025
\(988\) 0 0
\(989\) 11638.6 0.374201
\(990\) 0 0
\(991\) 18988.2 0.608658 0.304329 0.952567i \(-0.401568\pi\)
0.304329 + 0.952567i \(0.401568\pi\)
\(992\) 0 0
\(993\) 12970.5 0.414509
\(994\) 0 0
\(995\) −16186.7 −0.515731
\(996\) 0 0
\(997\) −38646.6 −1.22763 −0.613817 0.789449i \(-0.710367\pi\)
−0.613817 + 0.789449i \(0.710367\pi\)
\(998\) 0 0
\(999\) 50230.1 1.59080
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 1840.4.a.bb.1.4 10
4.3 odd 2 920.4.a.g.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.4.a.g.1.7 10 4.3 odd 2
1840.4.a.bb.1.4 10 1.1 even 1 trivial