Properties

Label 1840.4.a.v.1.2
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.85207\) 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)
 
\(f(q)\) \(=\) \(q-8.45088 q^{3} +5.00000 q^{5} -16.9190 q^{7} +44.4174 q^{9} +O(q^{10})\) \(q-8.45088 q^{3} +5.00000 q^{5} -16.9190 q^{7} +44.4174 q^{9} +15.9363 q^{11} -47.2304 q^{13} -42.2544 q^{15} +88.4553 q^{17} -19.0686 q^{19} +142.980 q^{21} -23.0000 q^{23} +25.0000 q^{25} -147.192 q^{27} +178.581 q^{29} -249.161 q^{31} -134.676 q^{33} -84.5949 q^{35} -206.047 q^{37} +399.138 q^{39} -83.5198 q^{41} +436.295 q^{43} +222.087 q^{45} -158.379 q^{47} -56.7479 q^{49} -747.525 q^{51} -373.971 q^{53} +79.6813 q^{55} +161.146 q^{57} -586.887 q^{59} +338.828 q^{61} -751.497 q^{63} -236.152 q^{65} +462.384 q^{67} +194.370 q^{69} +1148.86 q^{71} -887.052 q^{73} -211.272 q^{75} -269.626 q^{77} +460.823 q^{79} +44.6341 q^{81} -940.362 q^{83} +442.276 q^{85} -1509.16 q^{87} +309.068 q^{89} +799.090 q^{91} +2105.63 q^{93} -95.3428 q^{95} -1660.47 q^{97} +707.847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9} - 41 q^{11} + 28 q^{13} + 71 q^{17} - 177 q^{19} + 292 q^{21} - 184 q^{23} + 200 q^{25} + 495 q^{27} + 225 q^{29} + 36 q^{31} - 53 q^{33} - 55 q^{35} - 348 q^{37} + 1077 q^{39} + 620 q^{41} + 390 q^{43} + 790 q^{45} - 123 q^{47} + 881 q^{49} - 957 q^{51} + 1406 q^{53} - 205 q^{55} - 1142 q^{57} + 676 q^{59} + 1447 q^{61} + 58 q^{63} + 140 q^{65} + 1582 q^{67} - 1396 q^{71} + 17 q^{73} + 488 q^{77} - 708 q^{79} + 4316 q^{81} - 1486 q^{83} + 355 q^{85} + 803 q^{87} + 1360 q^{89} - 2693 q^{91} + 3833 q^{93} - 885 q^{95} - 855 q^{97} - 1319 q^{99}+O(q^{100}) \) 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
\(3\) −8.45088 −1.62637 −0.813186 0.582003i \(-0.802269\pi\)
−0.813186 + 0.582003i \(0.802269\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −16.9190 −0.913539 −0.456770 0.889585i \(-0.650994\pi\)
−0.456770 + 0.889585i \(0.650994\pi\)
\(8\) 0 0
\(9\) 44.4174 1.64509
\(10\) 0 0
\(11\) 15.9363 0.436815 0.218408 0.975858i \(-0.429914\pi\)
0.218408 + 0.975858i \(0.429914\pi\)
\(12\) 0 0
\(13\) −47.2304 −1.00764 −0.503821 0.863808i \(-0.668073\pi\)
−0.503821 + 0.863808i \(0.668073\pi\)
\(14\) 0 0
\(15\) −42.2544 −0.727336
\(16\) 0 0
\(17\) 88.4553 1.26197 0.630987 0.775794i \(-0.282650\pi\)
0.630987 + 0.775794i \(0.282650\pi\)
\(18\) 0 0
\(19\) −19.0686 −0.230244 −0.115122 0.993351i \(-0.536726\pi\)
−0.115122 + 0.993351i \(0.536726\pi\)
\(20\) 0 0
\(21\) 142.980 1.48576
\(22\) 0 0
\(23\) −23.0000 −0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −147.192 −1.04915
\(28\) 0 0
\(29\) 178.581 1.14350 0.571752 0.820426i \(-0.306264\pi\)
0.571752 + 0.820426i \(0.306264\pi\)
\(30\) 0 0
\(31\) −249.161 −1.44357 −0.721785 0.692117i \(-0.756678\pi\)
−0.721785 + 0.692117i \(0.756678\pi\)
\(32\) 0 0
\(33\) −134.676 −0.710424
\(34\) 0 0
\(35\) −84.5949 −0.408547
\(36\) 0 0
\(37\) −206.047 −0.915513 −0.457756 0.889078i \(-0.651347\pi\)
−0.457756 + 0.889078i \(0.651347\pi\)
\(38\) 0 0
\(39\) 399.138 1.63880
\(40\) 0 0
\(41\) −83.5198 −0.318137 −0.159068 0.987268i \(-0.550849\pi\)
−0.159068 + 0.987268i \(0.550849\pi\)
\(42\) 0 0
\(43\) 436.295 1.54731 0.773655 0.633607i \(-0.218426\pi\)
0.773655 + 0.633607i \(0.218426\pi\)
\(44\) 0 0
\(45\) 222.087 0.735706
\(46\) 0 0
\(47\) −158.379 −0.491531 −0.245765 0.969329i \(-0.579039\pi\)
−0.245765 + 0.969329i \(0.579039\pi\)
\(48\) 0 0
\(49\) −56.7479 −0.165446
\(50\) 0 0
\(51\) −747.525 −2.05244
\(52\) 0 0
\(53\) −373.971 −0.969224 −0.484612 0.874729i \(-0.661039\pi\)
−0.484612 + 0.874729i \(0.661039\pi\)
\(54\) 0 0
\(55\) 79.6813 0.195350
\(56\) 0 0
\(57\) 161.146 0.374462
\(58\) 0 0
\(59\) −586.887 −1.29502 −0.647510 0.762057i \(-0.724190\pi\)
−0.647510 + 0.762057i \(0.724190\pi\)
\(60\) 0 0
\(61\) 338.828 0.711188 0.355594 0.934640i \(-0.384279\pi\)
0.355594 + 0.934640i \(0.384279\pi\)
\(62\) 0 0
\(63\) −751.497 −1.50285
\(64\) 0 0
\(65\) −236.152 −0.450631
\(66\) 0 0
\(67\) 462.384 0.843123 0.421562 0.906800i \(-0.361482\pi\)
0.421562 + 0.906800i \(0.361482\pi\)
\(68\) 0 0
\(69\) 194.370 0.339122
\(70\) 0 0
\(71\) 1148.86 1.92034 0.960170 0.279415i \(-0.0901405\pi\)
0.960170 + 0.279415i \(0.0901405\pi\)
\(72\) 0 0
\(73\) −887.052 −1.42221 −0.711106 0.703084i \(-0.751806\pi\)
−0.711106 + 0.703084i \(0.751806\pi\)
\(74\) 0 0
\(75\) −211.272 −0.325275
\(76\) 0 0
\(77\) −269.626 −0.399048
\(78\) 0 0
\(79\) 460.823 0.656287 0.328144 0.944628i \(-0.393577\pi\)
0.328144 + 0.944628i \(0.393577\pi\)
\(80\) 0 0
\(81\) 44.6341 0.0612264
\(82\) 0 0
\(83\) −940.362 −1.24359 −0.621796 0.783179i \(-0.713597\pi\)
−0.621796 + 0.783179i \(0.713597\pi\)
\(84\) 0 0
\(85\) 442.276 0.564372
\(86\) 0 0
\(87\) −1509.16 −1.85976
\(88\) 0 0
\(89\) 309.068 0.368103 0.184051 0.982917i \(-0.441079\pi\)
0.184051 + 0.982917i \(0.441079\pi\)
\(90\) 0 0
\(91\) 799.090 0.920521
\(92\) 0 0
\(93\) 2105.63 2.34778
\(94\) 0 0
\(95\) −95.3428 −0.102968
\(96\) 0 0
\(97\) −1660.47 −1.73809 −0.869046 0.494731i \(-0.835267\pi\)
−0.869046 + 0.494731i \(0.835267\pi\)
\(98\) 0 0
\(99\) 707.847 0.718599
\(100\) 0 0
\(101\) −1555.86 −1.53281 −0.766404 0.642359i \(-0.777956\pi\)
−0.766404 + 0.642359i \(0.777956\pi\)
\(102\) 0 0
\(103\) 324.009 0.309957 0.154978 0.987918i \(-0.450469\pi\)
0.154978 + 0.987918i \(0.450469\pi\)
\(104\) 0 0
\(105\) 714.902 0.664450
\(106\) 0 0
\(107\) −1639.25 −1.48105 −0.740524 0.672030i \(-0.765422\pi\)
−0.740524 + 0.672030i \(0.765422\pi\)
\(108\) 0 0
\(109\) 1626.63 1.42938 0.714692 0.699439i \(-0.246567\pi\)
0.714692 + 0.699439i \(0.246567\pi\)
\(110\) 0 0
\(111\) 1741.28 1.48897
\(112\) 0 0
\(113\) −768.913 −0.640117 −0.320059 0.947398i \(-0.603703\pi\)
−0.320059 + 0.947398i \(0.603703\pi\)
\(114\) 0 0
\(115\) −115.000 −0.0932505
\(116\) 0 0
\(117\) −2097.85 −1.65766
\(118\) 0 0
\(119\) −1496.57 −1.15286
\(120\) 0 0
\(121\) −1077.04 −0.809193
\(122\) 0 0
\(123\) 705.816 0.517409
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −825.661 −0.576894 −0.288447 0.957496i \(-0.593139\pi\)
−0.288447 + 0.957496i \(0.593139\pi\)
\(128\) 0 0
\(129\) −3687.07 −2.51650
\(130\) 0 0
\(131\) −1456.43 −0.971364 −0.485682 0.874136i \(-0.661429\pi\)
−0.485682 + 0.874136i \(0.661429\pi\)
\(132\) 0 0
\(133\) 322.621 0.210337
\(134\) 0 0
\(135\) −735.961 −0.469196
\(136\) 0 0
\(137\) −1303.26 −0.812740 −0.406370 0.913709i \(-0.633206\pi\)
−0.406370 + 0.913709i \(0.633206\pi\)
\(138\) 0 0
\(139\) 1196.28 0.729981 0.364990 0.931011i \(-0.381072\pi\)
0.364990 + 0.931011i \(0.381072\pi\)
\(140\) 0 0
\(141\) 1338.44 0.799412
\(142\) 0 0
\(143\) −752.676 −0.440153
\(144\) 0 0
\(145\) 892.904 0.511391
\(146\) 0 0
\(147\) 479.570 0.269077
\(148\) 0 0
\(149\) 2620.96 1.44106 0.720529 0.693425i \(-0.243899\pi\)
0.720529 + 0.693425i \(0.243899\pi\)
\(150\) 0 0
\(151\) 696.513 0.375374 0.187687 0.982229i \(-0.439901\pi\)
0.187687 + 0.982229i \(0.439901\pi\)
\(152\) 0 0
\(153\) 3928.95 2.07606
\(154\) 0 0
\(155\) −1245.81 −0.645584
\(156\) 0 0
\(157\) 1778.37 0.904010 0.452005 0.892015i \(-0.350709\pi\)
0.452005 + 0.892015i \(0.350709\pi\)
\(158\) 0 0
\(159\) 3160.39 1.57632
\(160\) 0 0
\(161\) 389.137 0.190486
\(162\) 0 0
\(163\) 1569.03 0.753964 0.376982 0.926221i \(-0.376962\pi\)
0.376982 + 0.926221i \(0.376962\pi\)
\(164\) 0 0
\(165\) −673.378 −0.317711
\(166\) 0 0
\(167\) 1823.00 0.844720 0.422360 0.906428i \(-0.361202\pi\)
0.422360 + 0.906428i \(0.361202\pi\)
\(168\) 0 0
\(169\) 33.7080 0.0153427
\(170\) 0 0
\(171\) −846.975 −0.378771
\(172\) 0 0
\(173\) −2950.88 −1.29683 −0.648414 0.761287i \(-0.724568\pi\)
−0.648414 + 0.761287i \(0.724568\pi\)
\(174\) 0 0
\(175\) −422.975 −0.182708
\(176\) 0 0
\(177\) 4959.71 2.10619
\(178\) 0 0
\(179\) 4148.62 1.73230 0.866152 0.499781i \(-0.166586\pi\)
0.866152 + 0.499781i \(0.166586\pi\)
\(180\) 0 0
\(181\) 3585.90 1.47258 0.736292 0.676664i \(-0.236575\pi\)
0.736292 + 0.676664i \(0.236575\pi\)
\(182\) 0 0
\(183\) −2863.40 −1.15666
\(184\) 0 0
\(185\) −1030.24 −0.409430
\(186\) 0 0
\(187\) 1409.65 0.551249
\(188\) 0 0
\(189\) 2490.34 0.958443
\(190\) 0 0
\(191\) 2626.76 0.995108 0.497554 0.867433i \(-0.334232\pi\)
0.497554 + 0.867433i \(0.334232\pi\)
\(192\) 0 0
\(193\) 2034.69 0.758862 0.379431 0.925220i \(-0.376120\pi\)
0.379431 + 0.925220i \(0.376120\pi\)
\(194\) 0 0
\(195\) 1995.69 0.732894
\(196\) 0 0
\(197\) 3099.27 1.12088 0.560442 0.828194i \(-0.310631\pi\)
0.560442 + 0.828194i \(0.310631\pi\)
\(198\) 0 0
\(199\) 1118.39 0.398396 0.199198 0.979959i \(-0.436166\pi\)
0.199198 + 0.979959i \(0.436166\pi\)
\(200\) 0 0
\(201\) −3907.56 −1.37123
\(202\) 0 0
\(203\) −3021.41 −1.04464
\(204\) 0 0
\(205\) −417.599 −0.142275
\(206\) 0 0
\(207\) −1021.60 −0.343025
\(208\) 0 0
\(209\) −303.882 −0.100574
\(210\) 0 0
\(211\) 1560.32 0.509086 0.254543 0.967062i \(-0.418075\pi\)
0.254543 + 0.967062i \(0.418075\pi\)
\(212\) 0 0
\(213\) −9708.85 −3.12319
\(214\) 0 0
\(215\) 2181.47 0.691978
\(216\) 0 0
\(217\) 4215.55 1.31876
\(218\) 0 0
\(219\) 7496.37 2.31305
\(220\) 0 0
\(221\) −4177.77 −1.27162
\(222\) 0 0
\(223\) −410.575 −0.123292 −0.0616461 0.998098i \(-0.519635\pi\)
−0.0616461 + 0.998098i \(0.519635\pi\)
\(224\) 0 0
\(225\) 1110.43 0.329018
\(226\) 0 0
\(227\) 286.544 0.0837822 0.0418911 0.999122i \(-0.486662\pi\)
0.0418911 + 0.999122i \(0.486662\pi\)
\(228\) 0 0
\(229\) 1449.53 0.418286 0.209143 0.977885i \(-0.432933\pi\)
0.209143 + 0.977885i \(0.432933\pi\)
\(230\) 0 0
\(231\) 2278.57 0.649000
\(232\) 0 0
\(233\) −2733.70 −0.768629 −0.384314 0.923202i \(-0.625562\pi\)
−0.384314 + 0.923202i \(0.625562\pi\)
\(234\) 0 0
\(235\) −791.895 −0.219819
\(236\) 0 0
\(237\) −3894.36 −1.06737
\(238\) 0 0
\(239\) −1198.55 −0.324385 −0.162192 0.986759i \(-0.551857\pi\)
−0.162192 + 0.986759i \(0.551857\pi\)
\(240\) 0 0
\(241\) 6709.37 1.79331 0.896656 0.442727i \(-0.145989\pi\)
0.896656 + 0.442727i \(0.145989\pi\)
\(242\) 0 0
\(243\) 3596.99 0.949576
\(244\) 0 0
\(245\) −283.740 −0.0739896
\(246\) 0 0
\(247\) 900.615 0.232003
\(248\) 0 0
\(249\) 7946.89 2.02255
\(250\) 0 0
\(251\) 2033.02 0.511246 0.255623 0.966777i \(-0.417719\pi\)
0.255623 + 0.966777i \(0.417719\pi\)
\(252\) 0 0
\(253\) −366.534 −0.0910822
\(254\) 0 0
\(255\) −3737.62 −0.917879
\(256\) 0 0
\(257\) −281.707 −0.0683751 −0.0341875 0.999415i \(-0.510884\pi\)
−0.0341875 + 0.999415i \(0.510884\pi\)
\(258\) 0 0
\(259\) 3486.11 0.836357
\(260\) 0 0
\(261\) 7932.09 1.88116
\(262\) 0 0
\(263\) 4527.19 1.06144 0.530720 0.847547i \(-0.321922\pi\)
0.530720 + 0.847547i \(0.321922\pi\)
\(264\) 0 0
\(265\) −1869.86 −0.433450
\(266\) 0 0
\(267\) −2611.90 −0.598672
\(268\) 0 0
\(269\) 7346.53 1.66515 0.832575 0.553912i \(-0.186866\pi\)
0.832575 + 0.553912i \(0.186866\pi\)
\(270\) 0 0
\(271\) −7522.91 −1.68629 −0.843144 0.537687i \(-0.819298\pi\)
−0.843144 + 0.537687i \(0.819298\pi\)
\(272\) 0 0
\(273\) −6753.01 −1.49711
\(274\) 0 0
\(275\) 398.407 0.0873630
\(276\) 0 0
\(277\) −5255.63 −1.14000 −0.570000 0.821645i \(-0.693057\pi\)
−0.570000 + 0.821645i \(0.693057\pi\)
\(278\) 0 0
\(279\) −11067.1 −2.37480
\(280\) 0 0
\(281\) 995.675 0.211377 0.105689 0.994399i \(-0.466295\pi\)
0.105689 + 0.994399i \(0.466295\pi\)
\(282\) 0 0
\(283\) 1218.36 0.255916 0.127958 0.991780i \(-0.459158\pi\)
0.127958 + 0.991780i \(0.459158\pi\)
\(284\) 0 0
\(285\) 805.731 0.167464
\(286\) 0 0
\(287\) 1413.07 0.290630
\(288\) 0 0
\(289\) 2911.33 0.592578
\(290\) 0 0
\(291\) 14032.4 2.82679
\(292\) 0 0
\(293\) 8475.85 1.68998 0.844991 0.534781i \(-0.179606\pi\)
0.844991 + 0.534781i \(0.179606\pi\)
\(294\) 0 0
\(295\) −2934.44 −0.579151
\(296\) 0 0
\(297\) −2345.69 −0.458286
\(298\) 0 0
\(299\) 1086.30 0.210108
\(300\) 0 0
\(301\) −7381.67 −1.41353
\(302\) 0 0
\(303\) 13148.4 2.49292
\(304\) 0 0
\(305\) 1694.14 0.318053
\(306\) 0 0
\(307\) −6540.93 −1.21600 −0.607998 0.793938i \(-0.708027\pi\)
−0.607998 + 0.793938i \(0.708027\pi\)
\(308\) 0 0
\(309\) −2738.16 −0.504105
\(310\) 0 0
\(311\) −3060.38 −0.558001 −0.279000 0.960291i \(-0.590003\pi\)
−0.279000 + 0.960291i \(0.590003\pi\)
\(312\) 0 0
\(313\) 3426.33 0.618747 0.309374 0.950941i \(-0.399881\pi\)
0.309374 + 0.950941i \(0.399881\pi\)
\(314\) 0 0
\(315\) −3757.48 −0.672096
\(316\) 0 0
\(317\) −1072.78 −0.190073 −0.0950365 0.995474i \(-0.530297\pi\)
−0.0950365 + 0.995474i \(0.530297\pi\)
\(318\) 0 0
\(319\) 2845.91 0.499500
\(320\) 0 0
\(321\) 13853.1 2.40873
\(322\) 0 0
\(323\) −1686.71 −0.290561
\(324\) 0 0
\(325\) −1180.76 −0.201528
\(326\) 0 0
\(327\) −13746.4 −2.32471
\(328\) 0 0
\(329\) 2679.61 0.449033
\(330\) 0 0
\(331\) 3988.82 0.662374 0.331187 0.943565i \(-0.392551\pi\)
0.331187 + 0.943565i \(0.392551\pi\)
\(332\) 0 0
\(333\) −9152.08 −1.50610
\(334\) 0 0
\(335\) 2311.92 0.377056
\(336\) 0 0
\(337\) 7537.14 1.21832 0.609161 0.793047i \(-0.291506\pi\)
0.609161 + 0.793047i \(0.291506\pi\)
\(338\) 0 0
\(339\) 6497.99 1.04107
\(340\) 0 0
\(341\) −3970.70 −0.630573
\(342\) 0 0
\(343\) 6763.33 1.06468
\(344\) 0 0
\(345\) 971.851 0.151660
\(346\) 0 0
\(347\) 4451.78 0.688715 0.344358 0.938839i \(-0.388097\pi\)
0.344358 + 0.938839i \(0.388097\pi\)
\(348\) 0 0
\(349\) 7628.90 1.17010 0.585051 0.810996i \(-0.301074\pi\)
0.585051 + 0.810996i \(0.301074\pi\)
\(350\) 0 0
\(351\) 6951.94 1.05717
\(352\) 0 0
\(353\) −6548.34 −0.987346 −0.493673 0.869648i \(-0.664346\pi\)
−0.493673 + 0.869648i \(0.664346\pi\)
\(354\) 0 0
\(355\) 5744.28 0.858803
\(356\) 0 0
\(357\) 12647.4 1.87498
\(358\) 0 0
\(359\) −5823.20 −0.856091 −0.428045 0.903757i \(-0.640798\pi\)
−0.428045 + 0.903757i \(0.640798\pi\)
\(360\) 0 0
\(361\) −6495.39 −0.946988
\(362\) 0 0
\(363\) 9101.90 1.31605
\(364\) 0 0
\(365\) −4435.26 −0.636033
\(366\) 0 0
\(367\) 4942.79 0.703028 0.351514 0.936183i \(-0.385667\pi\)
0.351514 + 0.936183i \(0.385667\pi\)
\(368\) 0 0
\(369\) −3709.73 −0.523363
\(370\) 0 0
\(371\) 6327.21 0.885425
\(372\) 0 0
\(373\) 8053.14 1.11790 0.558949 0.829202i \(-0.311205\pi\)
0.558949 + 0.829202i \(0.311205\pi\)
\(374\) 0 0
\(375\) −1056.36 −0.145467
\(376\) 0 0
\(377\) −8434.44 −1.15224
\(378\) 0 0
\(379\) −10795.6 −1.46315 −0.731574 0.681762i \(-0.761214\pi\)
−0.731574 + 0.681762i \(0.761214\pi\)
\(380\) 0 0
\(381\) 6977.56 0.938245
\(382\) 0 0
\(383\) 5073.32 0.676853 0.338426 0.940993i \(-0.390105\pi\)
0.338426 + 0.940993i \(0.390105\pi\)
\(384\) 0 0
\(385\) −1348.13 −0.178460
\(386\) 0 0
\(387\) 19379.1 2.54546
\(388\) 0 0
\(389\) 7917.35 1.03194 0.515971 0.856606i \(-0.327431\pi\)
0.515971 + 0.856606i \(0.327431\pi\)
\(390\) 0 0
\(391\) −2034.47 −0.263140
\(392\) 0 0
\(393\) 12308.1 1.57980
\(394\) 0 0
\(395\) 2304.12 0.293501
\(396\) 0 0
\(397\) 3714.37 0.469570 0.234785 0.972047i \(-0.424561\pi\)
0.234785 + 0.972047i \(0.424561\pi\)
\(398\) 0 0
\(399\) −2726.43 −0.342086
\(400\) 0 0
\(401\) 12452.8 1.55078 0.775392 0.631480i \(-0.217552\pi\)
0.775392 + 0.631480i \(0.217552\pi\)
\(402\) 0 0
\(403\) 11768.0 1.45460
\(404\) 0 0
\(405\) 223.170 0.0273813
\(406\) 0 0
\(407\) −3283.63 −0.399910
\(408\) 0 0
\(409\) 6780.15 0.819699 0.409850 0.912153i \(-0.365581\pi\)
0.409850 + 0.912153i \(0.365581\pi\)
\(410\) 0 0
\(411\) 11013.7 1.32182
\(412\) 0 0
\(413\) 9929.53 1.18305
\(414\) 0 0
\(415\) −4701.81 −0.556152
\(416\) 0 0
\(417\) −10109.6 −1.18722
\(418\) 0 0
\(419\) −4233.55 −0.493610 −0.246805 0.969065i \(-0.579381\pi\)
−0.246805 + 0.969065i \(0.579381\pi\)
\(420\) 0 0
\(421\) −4801.63 −0.555860 −0.277930 0.960601i \(-0.589648\pi\)
−0.277930 + 0.960601i \(0.589648\pi\)
\(422\) 0 0
\(423\) −7034.78 −0.808611
\(424\) 0 0
\(425\) 2211.38 0.252395
\(426\) 0 0
\(427\) −5732.63 −0.649699
\(428\) 0 0
\(429\) 6360.77 0.715853
\(430\) 0 0
\(431\) −5221.81 −0.583586 −0.291793 0.956481i \(-0.594252\pi\)
−0.291793 + 0.956481i \(0.594252\pi\)
\(432\) 0 0
\(433\) 8441.99 0.936943 0.468471 0.883479i \(-0.344805\pi\)
0.468471 + 0.883479i \(0.344805\pi\)
\(434\) 0 0
\(435\) −7545.82 −0.831712
\(436\) 0 0
\(437\) 438.577 0.0480091
\(438\) 0 0
\(439\) 4179.48 0.454386 0.227193 0.973850i \(-0.427045\pi\)
0.227193 + 0.973850i \(0.427045\pi\)
\(440\) 0 0
\(441\) −2520.59 −0.272173
\(442\) 0 0
\(443\) −14990.7 −1.60774 −0.803871 0.594804i \(-0.797230\pi\)
−0.803871 + 0.594804i \(0.797230\pi\)
\(444\) 0 0
\(445\) 1545.34 0.164621
\(446\) 0 0
\(447\) −22149.4 −2.34370
\(448\) 0 0
\(449\) −12698.3 −1.33468 −0.667339 0.744754i \(-0.732567\pi\)
−0.667339 + 0.744754i \(0.732567\pi\)
\(450\) 0 0
\(451\) −1330.99 −0.138967
\(452\) 0 0
\(453\) −5886.15 −0.610497
\(454\) 0 0
\(455\) 3995.45 0.411669
\(456\) 0 0
\(457\) −16157.1 −1.65382 −0.826909 0.562335i \(-0.809903\pi\)
−0.826909 + 0.562335i \(0.809903\pi\)
\(458\) 0 0
\(459\) −13019.9 −1.32400
\(460\) 0 0
\(461\) 17397.1 1.75762 0.878812 0.477168i \(-0.158337\pi\)
0.878812 + 0.477168i \(0.158337\pi\)
\(462\) 0 0
\(463\) 14721.3 1.47766 0.738832 0.673889i \(-0.235378\pi\)
0.738832 + 0.673889i \(0.235378\pi\)
\(464\) 0 0
\(465\) 10528.2 1.04996
\(466\) 0 0
\(467\) 4696.10 0.465331 0.232666 0.972557i \(-0.425255\pi\)
0.232666 + 0.972557i \(0.425255\pi\)
\(468\) 0 0
\(469\) −7823.08 −0.770226
\(470\) 0 0
\(471\) −15028.8 −1.47026
\(472\) 0 0
\(473\) 6952.91 0.675888
\(474\) 0 0
\(475\) −476.714 −0.0460487
\(476\) 0 0
\(477\) −16610.8 −1.59446
\(478\) 0 0
\(479\) −12626.5 −1.20443 −0.602213 0.798336i \(-0.705714\pi\)
−0.602213 + 0.798336i \(0.705714\pi\)
\(480\) 0 0
\(481\) 9731.69 0.922509
\(482\) 0 0
\(483\) −3288.55 −0.309801
\(484\) 0 0
\(485\) −8302.34 −0.777299
\(486\) 0 0
\(487\) −597.241 −0.0555720 −0.0277860 0.999614i \(-0.508846\pi\)
−0.0277860 + 0.999614i \(0.508846\pi\)
\(488\) 0 0
\(489\) −13259.7 −1.22623
\(490\) 0 0
\(491\) −5444.13 −0.500387 −0.250194 0.968196i \(-0.580494\pi\)
−0.250194 + 0.968196i \(0.580494\pi\)
\(492\) 0 0
\(493\) 15796.4 1.44307
\(494\) 0 0
\(495\) 3539.24 0.321367
\(496\) 0 0
\(497\) −19437.5 −1.75431
\(498\) 0 0
\(499\) 9840.24 0.882785 0.441392 0.897314i \(-0.354485\pi\)
0.441392 + 0.897314i \(0.354485\pi\)
\(500\) 0 0
\(501\) −15406.0 −1.37383
\(502\) 0 0
\(503\) 2022.77 0.179306 0.0896528 0.995973i \(-0.471424\pi\)
0.0896528 + 0.995973i \(0.471424\pi\)
\(504\) 0 0
\(505\) −7779.28 −0.685492
\(506\) 0 0
\(507\) −284.862 −0.0249530
\(508\) 0 0
\(509\) 12171.6 1.05991 0.529956 0.848025i \(-0.322208\pi\)
0.529956 + 0.848025i \(0.322208\pi\)
\(510\) 0 0
\(511\) 15008.0 1.29925
\(512\) 0 0
\(513\) 2806.74 0.241561
\(514\) 0 0
\(515\) 1620.04 0.138617
\(516\) 0 0
\(517\) −2523.97 −0.214708
\(518\) 0 0
\(519\) 24937.6 2.10913
\(520\) 0 0
\(521\) −8718.74 −0.733157 −0.366579 0.930387i \(-0.619471\pi\)
−0.366579 + 0.930387i \(0.619471\pi\)
\(522\) 0 0
\(523\) 4412.10 0.368886 0.184443 0.982843i \(-0.440952\pi\)
0.184443 + 0.982843i \(0.440952\pi\)
\(524\) 0 0
\(525\) 3574.51 0.297151
\(526\) 0 0
\(527\) −22039.6 −1.82175
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −26068.0 −2.13042
\(532\) 0 0
\(533\) 3944.67 0.320568
\(534\) 0 0
\(535\) −8196.24 −0.662344
\(536\) 0 0
\(537\) −35059.5 −2.81737
\(538\) 0 0
\(539\) −904.350 −0.0722692
\(540\) 0 0
\(541\) −6786.56 −0.539329 −0.269665 0.962954i \(-0.586913\pi\)
−0.269665 + 0.962954i \(0.586913\pi\)
\(542\) 0 0
\(543\) −30304.0 −2.39497
\(544\) 0 0
\(545\) 8133.15 0.639240
\(546\) 0 0
\(547\) −2847.33 −0.222565 −0.111282 0.993789i \(-0.535496\pi\)
−0.111282 + 0.993789i \(0.535496\pi\)
\(548\) 0 0
\(549\) 15049.9 1.16997
\(550\) 0 0
\(551\) −3405.28 −0.263284
\(552\) 0 0
\(553\) −7796.67 −0.599544
\(554\) 0 0
\(555\) 8706.41 0.665885
\(556\) 0 0
\(557\) −951.795 −0.0724037 −0.0362018 0.999344i \(-0.511526\pi\)
−0.0362018 + 0.999344i \(0.511526\pi\)
\(558\) 0 0
\(559\) −20606.4 −1.55913
\(560\) 0 0
\(561\) −11912.8 −0.896537
\(562\) 0 0
\(563\) 15851.1 1.18658 0.593288 0.804990i \(-0.297829\pi\)
0.593288 + 0.804990i \(0.297829\pi\)
\(564\) 0 0
\(565\) −3844.56 −0.286269
\(566\) 0 0
\(567\) −755.163 −0.0559328
\(568\) 0 0
\(569\) −24461.5 −1.80225 −0.901125 0.433560i \(-0.857257\pi\)
−0.901125 + 0.433560i \(0.857257\pi\)
\(570\) 0 0
\(571\) 21116.1 1.54760 0.773801 0.633429i \(-0.218353\pi\)
0.773801 + 0.633429i \(0.218353\pi\)
\(572\) 0 0
\(573\) −22198.4 −1.61842
\(574\) 0 0
\(575\) −575.000 −0.0417029
\(576\) 0 0
\(577\) −14543.0 −1.04928 −0.524640 0.851324i \(-0.675800\pi\)
−0.524640 + 0.851324i \(0.675800\pi\)
\(578\) 0 0
\(579\) −17194.9 −1.23419
\(580\) 0 0
\(581\) 15910.0 1.13607
\(582\) 0 0
\(583\) −5959.70 −0.423372
\(584\) 0 0
\(585\) −10489.2 −0.741328
\(586\) 0 0
\(587\) −817.198 −0.0574606 −0.0287303 0.999587i \(-0.509146\pi\)
−0.0287303 + 0.999587i \(0.509146\pi\)
\(588\) 0 0
\(589\) 4751.15 0.332373
\(590\) 0 0
\(591\) −26191.6 −1.82297
\(592\) 0 0
\(593\) 18784.4 1.30081 0.650406 0.759586i \(-0.274599\pi\)
0.650406 + 0.759586i \(0.274599\pi\)
\(594\) 0 0
\(595\) −7482.87 −0.515576
\(596\) 0 0
\(597\) −9451.41 −0.647940
\(598\) 0 0
\(599\) −5703.48 −0.389045 −0.194522 0.980898i \(-0.562316\pi\)
−0.194522 + 0.980898i \(0.562316\pi\)
\(600\) 0 0
\(601\) −12055.2 −0.818206 −0.409103 0.912488i \(-0.634158\pi\)
−0.409103 + 0.912488i \(0.634158\pi\)
\(602\) 0 0
\(603\) 20537.9 1.38701
\(604\) 0 0
\(605\) −5385.18 −0.361882
\(606\) 0 0
\(607\) −24047.6 −1.60801 −0.804005 0.594622i \(-0.797302\pi\)
−0.804005 + 0.594622i \(0.797302\pi\)
\(608\) 0 0
\(609\) 25533.5 1.69897
\(610\) 0 0
\(611\) 7480.30 0.495287
\(612\) 0 0
\(613\) 7229.82 0.476362 0.238181 0.971221i \(-0.423449\pi\)
0.238181 + 0.971221i \(0.423449\pi\)
\(614\) 0 0
\(615\) 3529.08 0.231392
\(616\) 0 0
\(617\) 13799.4 0.900397 0.450198 0.892929i \(-0.351353\pi\)
0.450198 + 0.892929i \(0.351353\pi\)
\(618\) 0 0
\(619\) −29923.6 −1.94302 −0.971510 0.237000i \(-0.923836\pi\)
−0.971510 + 0.237000i \(0.923836\pi\)
\(620\) 0 0
\(621\) 3385.42 0.218764
\(622\) 0 0
\(623\) −5229.12 −0.336276
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 2568.07 0.163571
\(628\) 0 0
\(629\) −18226.0 −1.15535
\(630\) 0 0
\(631\) −15319.2 −0.966479 −0.483240 0.875488i \(-0.660540\pi\)
−0.483240 + 0.875488i \(0.660540\pi\)
\(632\) 0 0
\(633\) −13186.1 −0.827963
\(634\) 0 0
\(635\) −4128.31 −0.257995
\(636\) 0 0
\(637\) 2680.22 0.166710
\(638\) 0 0
\(639\) 51029.2 3.15913
\(640\) 0 0
\(641\) 4845.99 0.298604 0.149302 0.988792i \(-0.452297\pi\)
0.149302 + 0.988792i \(0.452297\pi\)
\(642\) 0 0
\(643\) −3001.58 −0.184091 −0.0920457 0.995755i \(-0.529341\pi\)
−0.0920457 + 0.995755i \(0.529341\pi\)
\(644\) 0 0
\(645\) −18435.4 −1.12541
\(646\) 0 0
\(647\) 13518.7 0.821445 0.410723 0.911760i \(-0.365276\pi\)
0.410723 + 0.911760i \(0.365276\pi\)
\(648\) 0 0
\(649\) −9352.79 −0.565684
\(650\) 0 0
\(651\) −35625.1 −2.14479
\(652\) 0 0
\(653\) 16824.5 1.00826 0.504129 0.863628i \(-0.331814\pi\)
0.504129 + 0.863628i \(0.331814\pi\)
\(654\) 0 0
\(655\) −7282.14 −0.434407
\(656\) 0 0
\(657\) −39400.5 −2.33967
\(658\) 0 0
\(659\) −990.892 −0.0585731 −0.0292865 0.999571i \(-0.509324\pi\)
−0.0292865 + 0.999571i \(0.509324\pi\)
\(660\) 0 0
\(661\) 16013.5 0.942287 0.471144 0.882056i \(-0.343841\pi\)
0.471144 + 0.882056i \(0.343841\pi\)
\(662\) 0 0
\(663\) 35305.9 2.06812
\(664\) 0 0
\(665\) 1613.10 0.0940654
\(666\) 0 0
\(667\) −4107.36 −0.238437
\(668\) 0 0
\(669\) 3469.72 0.200519
\(670\) 0 0
\(671\) 5399.66 0.310658
\(672\) 0 0
\(673\) −15026.3 −0.860653 −0.430327 0.902673i \(-0.641602\pi\)
−0.430327 + 0.902673i \(0.641602\pi\)
\(674\) 0 0
\(675\) −3679.80 −0.209831
\(676\) 0 0
\(677\) −1662.23 −0.0943645 −0.0471822 0.998886i \(-0.515024\pi\)
−0.0471822 + 0.998886i \(0.515024\pi\)
\(678\) 0 0
\(679\) 28093.4 1.58782
\(680\) 0 0
\(681\) −2421.54 −0.136261
\(682\) 0 0
\(683\) 8169.66 0.457692 0.228846 0.973463i \(-0.426505\pi\)
0.228846 + 0.973463i \(0.426505\pi\)
\(684\) 0 0
\(685\) −6516.32 −0.363468
\(686\) 0 0
\(687\) −12249.8 −0.680288
\(688\) 0 0
\(689\) 17662.8 0.976631
\(690\) 0 0
\(691\) 31434.8 1.73059 0.865294 0.501265i \(-0.167132\pi\)
0.865294 + 0.501265i \(0.167132\pi\)
\(692\) 0 0
\(693\) −11976.1 −0.656469
\(694\) 0 0
\(695\) 5981.41 0.326457
\(696\) 0 0
\(697\) −7387.76 −0.401480
\(698\) 0 0
\(699\) 23102.2 1.25008
\(700\) 0 0
\(701\) −27244.8 −1.46793 −0.733967 0.679185i \(-0.762333\pi\)
−0.733967 + 0.679185i \(0.762333\pi\)
\(702\) 0 0
\(703\) 3929.03 0.210791
\(704\) 0 0
\(705\) 6692.21 0.357508
\(706\) 0 0
\(707\) 26323.5 1.40028
\(708\) 0 0
\(709\) 8420.90 0.446056 0.223028 0.974812i \(-0.428406\pi\)
0.223028 + 0.974812i \(0.428406\pi\)
\(710\) 0 0
\(711\) 20468.6 1.07965
\(712\) 0 0
\(713\) 5730.71 0.301005
\(714\) 0 0
\(715\) −3763.38 −0.196843
\(716\) 0 0
\(717\) 10128.8 0.527570
\(718\) 0 0
\(719\) −14850.3 −0.770269 −0.385134 0.922860i \(-0.625845\pi\)
−0.385134 + 0.922860i \(0.625845\pi\)
\(720\) 0 0
\(721\) −5481.90 −0.283158
\(722\) 0 0
\(723\) −56700.1 −2.91659
\(724\) 0 0
\(725\) 4464.52 0.228701
\(726\) 0 0
\(727\) −14777.2 −0.753857 −0.376929 0.926242i \(-0.623020\pi\)
−0.376929 + 0.926242i \(0.623020\pi\)
\(728\) 0 0
\(729\) −31602.9 −1.60559
\(730\) 0 0
\(731\) 38592.6 1.95266
\(732\) 0 0
\(733\) 2071.41 0.104378 0.0521891 0.998637i \(-0.483380\pi\)
0.0521891 + 0.998637i \(0.483380\pi\)
\(734\) 0 0
\(735\) 2397.85 0.120335
\(736\) 0 0
\(737\) 7368.68 0.368289
\(738\) 0 0
\(739\) 30447.9 1.51562 0.757810 0.652475i \(-0.226269\pi\)
0.757810 + 0.652475i \(0.226269\pi\)
\(740\) 0 0
\(741\) −7610.99 −0.377324
\(742\) 0 0
\(743\) 3975.35 0.196287 0.0981436 0.995172i \(-0.468710\pi\)
0.0981436 + 0.995172i \(0.468710\pi\)
\(744\) 0 0
\(745\) 13104.8 0.644461
\(746\) 0 0
\(747\) −41768.4 −2.04582
\(748\) 0 0
\(749\) 27734.4 1.35299
\(750\) 0 0
\(751\) 22811.5 1.10839 0.554196 0.832386i \(-0.313026\pi\)
0.554196 + 0.832386i \(0.313026\pi\)
\(752\) 0 0
\(753\) −17180.8 −0.831477
\(754\) 0 0
\(755\) 3482.56 0.167872
\(756\) 0 0
\(757\) 22835.0 1.09637 0.548185 0.836357i \(-0.315319\pi\)
0.548185 + 0.836357i \(0.315319\pi\)
\(758\) 0 0
\(759\) 3097.54 0.148134
\(760\) 0 0
\(761\) −15837.9 −0.754432 −0.377216 0.926125i \(-0.623119\pi\)
−0.377216 + 0.926125i \(0.623119\pi\)
\(762\) 0 0
\(763\) −27520.9 −1.30580
\(764\) 0 0
\(765\) 19644.8 0.928441
\(766\) 0 0
\(767\) 27718.9 1.30492
\(768\) 0 0
\(769\) 7610.08 0.356861 0.178431 0.983952i \(-0.442898\pi\)
0.178431 + 0.983952i \(0.442898\pi\)
\(770\) 0 0
\(771\) 2380.67 0.111203
\(772\) 0 0
\(773\) 18824.4 0.875895 0.437948 0.899001i \(-0.355706\pi\)
0.437948 + 0.899001i \(0.355706\pi\)
\(774\) 0 0
\(775\) −6229.03 −0.288714
\(776\) 0 0
\(777\) −29460.7 −1.36023
\(778\) 0 0
\(779\) 1592.60 0.0732489
\(780\) 0 0
\(781\) 18308.5 0.838834
\(782\) 0 0
\(783\) −26285.7 −1.19971
\(784\) 0 0
\(785\) 8891.86 0.404286
\(786\) 0 0
\(787\) 5468.65 0.247696 0.123848 0.992301i \(-0.460477\pi\)
0.123848 + 0.992301i \(0.460477\pi\)
\(788\) 0 0
\(789\) −38258.8 −1.72630
\(790\) 0 0
\(791\) 13009.2 0.584772
\(792\) 0 0
\(793\) −16003.0 −0.716624
\(794\) 0 0
\(795\) 15801.9 0.704952
\(796\) 0 0
\(797\) 17054.5 0.757968 0.378984 0.925403i \(-0.376273\pi\)
0.378984 + 0.925403i \(0.376273\pi\)
\(798\) 0 0
\(799\) −14009.5 −0.620299
\(800\) 0 0
\(801\) 13728.0 0.605562
\(802\) 0 0
\(803\) −14136.3 −0.621244
\(804\) 0 0
\(805\) 1945.68 0.0851880
\(806\) 0 0
\(807\) −62084.6 −2.70816
\(808\) 0 0
\(809\) 16137.0 0.701296 0.350648 0.936507i \(-0.385961\pi\)
0.350648 + 0.936507i \(0.385961\pi\)
\(810\) 0 0
\(811\) 27672.2 1.19816 0.599078 0.800691i \(-0.295534\pi\)
0.599078 + 0.800691i \(0.295534\pi\)
\(812\) 0 0
\(813\) 63575.2 2.74253
\(814\) 0 0
\(815\) 7845.17 0.337183
\(816\) 0 0
\(817\) −8319.51 −0.356258
\(818\) 0 0
\(819\) 35493.5 1.51434
\(820\) 0 0
\(821\) 817.964 0.0347712 0.0173856 0.999849i \(-0.494466\pi\)
0.0173856 + 0.999849i \(0.494466\pi\)
\(822\) 0 0
\(823\) 29069.7 1.23123 0.615617 0.788045i \(-0.288907\pi\)
0.615617 + 0.788045i \(0.288907\pi\)
\(824\) 0 0
\(825\) −3366.89 −0.142085
\(826\) 0 0
\(827\) 3394.17 0.142717 0.0713584 0.997451i \(-0.477267\pi\)
0.0713584 + 0.997451i \(0.477267\pi\)
\(828\) 0 0
\(829\) 12569.4 0.526602 0.263301 0.964714i \(-0.415189\pi\)
0.263301 + 0.964714i \(0.415189\pi\)
\(830\) 0 0
\(831\) 44414.7 1.85406
\(832\) 0 0
\(833\) −5019.65 −0.208788
\(834\) 0 0
\(835\) 9115.02 0.377770
\(836\) 0 0
\(837\) 36674.6 1.51453
\(838\) 0 0
\(839\) 8644.40 0.355706 0.177853 0.984057i \(-0.443085\pi\)
0.177853 + 0.984057i \(0.443085\pi\)
\(840\) 0 0
\(841\) 7502.10 0.307602
\(842\) 0 0
\(843\) −8414.33 −0.343778
\(844\) 0 0
\(845\) 168.540 0.00686148
\(846\) 0 0
\(847\) 18222.3 0.739229
\(848\) 0 0
\(849\) −10296.2 −0.416214
\(850\) 0 0
\(851\) 4739.09 0.190898
\(852\) 0 0
\(853\) −24763.3 −0.993996 −0.496998 0.867752i \(-0.665564\pi\)
−0.496998 + 0.867752i \(0.665564\pi\)
\(854\) 0 0
\(855\) −4234.88 −0.169392
\(856\) 0 0
\(857\) −10811.6 −0.430942 −0.215471 0.976510i \(-0.569129\pi\)
−0.215471 + 0.976510i \(0.569129\pi\)
\(858\) 0 0
\(859\) −40250.6 −1.59876 −0.799379 0.600828i \(-0.794838\pi\)
−0.799379 + 0.600828i \(0.794838\pi\)
\(860\) 0 0
\(861\) −11941.7 −0.472673
\(862\) 0 0
\(863\) −37822.0 −1.49186 −0.745931 0.666023i \(-0.767995\pi\)
−0.745931 + 0.666023i \(0.767995\pi\)
\(864\) 0 0
\(865\) −14754.4 −0.579960
\(866\) 0 0
\(867\) −24603.3 −0.963752
\(868\) 0 0
\(869\) 7343.81 0.286676
\(870\) 0 0
\(871\) −21838.6 −0.849566
\(872\) 0 0
\(873\) −73753.6 −2.85931
\(874\) 0 0
\(875\) −2114.87 −0.0817094
\(876\) 0 0
\(877\) −4369.24 −0.168231 −0.0841157 0.996456i \(-0.526807\pi\)
−0.0841157 + 0.996456i \(0.526807\pi\)
\(878\) 0 0
\(879\) −71628.4 −2.74854
\(880\) 0 0
\(881\) 6206.68 0.237353 0.118677 0.992933i \(-0.462135\pi\)
0.118677 + 0.992933i \(0.462135\pi\)
\(882\) 0 0
\(883\) −13024.5 −0.496385 −0.248192 0.968711i \(-0.579837\pi\)
−0.248192 + 0.968711i \(0.579837\pi\)
\(884\) 0 0
\(885\) 24798.6 0.941915
\(886\) 0 0
\(887\) −44532.0 −1.68573 −0.842863 0.538128i \(-0.819132\pi\)
−0.842863 + 0.538128i \(0.819132\pi\)
\(888\) 0 0
\(889\) 13969.3 0.527016
\(890\) 0 0
\(891\) 711.301 0.0267446
\(892\) 0 0
\(893\) 3020.06 0.113172
\(894\) 0 0
\(895\) 20743.1 0.774710
\(896\) 0 0
\(897\) −9180.18 −0.341714
\(898\) 0 0
\(899\) −44495.4 −1.65073
\(900\) 0 0
\(901\) −33079.7 −1.22314
\(902\) 0 0
\(903\) 62381.6 2.29892
\(904\) 0 0
\(905\) 17929.5 0.658560
\(906\) 0 0
\(907\) 36618.2 1.34056 0.670279 0.742109i \(-0.266174\pi\)
0.670279 + 0.742109i \(0.266174\pi\)
\(908\) 0 0
\(909\) −69107.1 −2.52160
\(910\) 0 0
\(911\) 31521.0 1.14636 0.573182 0.819428i \(-0.305709\pi\)
0.573182 + 0.819428i \(0.305709\pi\)
\(912\) 0 0
\(913\) −14985.9 −0.543220
\(914\) 0 0
\(915\) −14317.0 −0.517273
\(916\) 0 0
\(917\) 24641.3 0.887379
\(918\) 0 0
\(919\) 7870.23 0.282497 0.141249 0.989974i \(-0.454888\pi\)
0.141249 + 0.989974i \(0.454888\pi\)
\(920\) 0 0
\(921\) 55276.7 1.97766
\(922\) 0 0
\(923\) −54260.9 −1.93502
\(924\) 0 0
\(925\) −5151.18 −0.183103
\(926\) 0 0
\(927\) 14391.6 0.509906
\(928\) 0 0
\(929\) −5845.33 −0.206436 −0.103218 0.994659i \(-0.532914\pi\)
−0.103218 + 0.994659i \(0.532914\pi\)
\(930\) 0 0
\(931\) 1082.10 0.0380928
\(932\) 0 0
\(933\) 25862.9 0.907517
\(934\) 0 0
\(935\) 7048.23 0.246526
\(936\) 0 0
\(937\) −2361.81 −0.0823448 −0.0411724 0.999152i \(-0.513109\pi\)
−0.0411724 + 0.999152i \(0.513109\pi\)
\(938\) 0 0
\(939\) −28955.5 −1.00631
\(940\) 0 0
\(941\) 29730.7 1.02996 0.514981 0.857202i \(-0.327799\pi\)
0.514981 + 0.857202i \(0.327799\pi\)
\(942\) 0 0
\(943\) 1920.96 0.0663361
\(944\) 0 0
\(945\) 12451.7 0.428629
\(946\) 0 0
\(947\) −8584.79 −0.294581 −0.147290 0.989093i \(-0.547055\pi\)
−0.147290 + 0.989093i \(0.547055\pi\)
\(948\) 0 0
\(949\) 41895.8 1.43308
\(950\) 0 0
\(951\) 9065.90 0.309129
\(952\) 0 0
\(953\) 51072.2 1.73598 0.867992 0.496579i \(-0.165411\pi\)
0.867992 + 0.496579i \(0.165411\pi\)
\(954\) 0 0
\(955\) 13133.8 0.445026
\(956\) 0 0
\(957\) −24050.5 −0.812373
\(958\) 0 0
\(959\) 22049.9 0.742470
\(960\) 0 0
\(961\) 32290.3 1.08389
\(962\) 0 0
\(963\) −72811.1 −2.43645
\(964\) 0 0
\(965\) 10173.5 0.339374
\(966\) 0 0
\(967\) 2670.23 0.0887992 0.0443996 0.999014i \(-0.485863\pi\)
0.0443996 + 0.999014i \(0.485863\pi\)
\(968\) 0 0
\(969\) 14254.2 0.472561
\(970\) 0 0
\(971\) −40190.1 −1.32828 −0.664141 0.747607i \(-0.731203\pi\)
−0.664141 + 0.747607i \(0.731203\pi\)
\(972\) 0 0
\(973\) −20239.9 −0.666866
\(974\) 0 0
\(975\) 9978.46 0.327760
\(976\) 0 0
\(977\) −6199.68 −0.203015 −0.101507 0.994835i \(-0.532367\pi\)
−0.101507 + 0.994835i \(0.532367\pi\)
\(978\) 0 0
\(979\) 4925.39 0.160793
\(980\) 0 0
\(981\) 72250.6 2.35146
\(982\) 0 0
\(983\) 42606.5 1.38244 0.691219 0.722645i \(-0.257074\pi\)
0.691219 + 0.722645i \(0.257074\pi\)
\(984\) 0 0
\(985\) 15496.4 0.501274
\(986\) 0 0
\(987\) −22645.1 −0.730295
\(988\) 0 0
\(989\) −10034.8 −0.322636
\(990\) 0 0
\(991\) 6898.57 0.221131 0.110565 0.993869i \(-0.464734\pi\)
0.110565 + 0.993869i \(0.464734\pi\)
\(992\) 0 0
\(993\) −33709.1 −1.07727
\(994\) 0 0
\(995\) 5591.97 0.178168
\(996\) 0 0
\(997\) −49114.1 −1.56014 −0.780070 0.625693i \(-0.784816\pi\)
−0.780070 + 0.625693i \(0.784816\pi\)
\(998\) 0 0
\(999\) 30328.6 0.960513
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.v.1.2 8
4.3 odd 2 115.4.a.f.1.2 8
12.11 even 2 1035.4.a.r.1.7 8
20.3 even 4 575.4.b.k.24.12 16
20.7 even 4 575.4.b.k.24.5 16
20.19 odd 2 575.4.a.n.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.f.1.2 8 4.3 odd 2
575.4.a.n.1.7 8 20.19 odd 2
575.4.b.k.24.5 16 20.7 even 4
575.4.b.k.24.12 16 20.3 even 4
1035.4.a.r.1.7 8 12.11 even 2
1840.4.a.v.1.2 8 1.1 even 1 trivial