Properties

Label 9680.2.a.cg.1.3
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
Dimension $3$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4840)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.14134 q^{3} +1.00000 q^{5} -3.14134 q^{7} +6.86799 q^{9} +O(q^{10})\) \(q+3.14134 q^{3} +1.00000 q^{5} -3.14134 q^{7} +6.86799 q^{9} +7.00933 q^{13} +3.14134 q^{15} -5.14134 q^{17} -0.414680 q^{19} -9.86799 q^{21} +2.72666 q^{23} +1.00000 q^{25} +12.1507 q^{27} -5.86799 q^{29} +9.86799 q^{31} -3.14134 q^{35} -3.86799 q^{37} +22.0187 q^{39} +6.28267 q^{41} +6.72666 q^{43} +6.86799 q^{45} +9.00933 q^{47} +2.86799 q^{49} -16.1507 q^{51} -3.86799 q^{53} -1.30265 q^{57} -1.45331 q^{59} -10.6974 q^{61} -21.5747 q^{63} +7.00933 q^{65} -7.29200 q^{67} +8.56534 q^{69} +6.69735 q^{71} +8.72666 q^{73} +3.14134 q^{75} +12.0000 q^{79} +17.5653 q^{81} -4.17997 q^{83} -5.14134 q^{85} -18.4333 q^{87} -16.4333 q^{89} -22.0187 q^{91} +30.9987 q^{93} -0.414680 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9} + q^{15} - 7 q^{17} + 3 q^{19} - 17 q^{21} + 4 q^{23} + 3 q^{25} + 7 q^{27} - 5 q^{29} + 17 q^{31} - q^{35} + q^{37} + 24 q^{39} + 2 q^{41} + 16 q^{43} + 8 q^{45} + 6 q^{47} - 4 q^{49} - 19 q^{51} + q^{53} - 25 q^{57} + 4 q^{59} - 11 q^{61} - 10 q^{63} + 16 q^{67} - 8 q^{69} - q^{71} + 22 q^{73} + q^{75} + 36 q^{79} + 19 q^{81} - 7 q^{85} - 9 q^{87} - 3 q^{89} - 24 q^{91} + 13 q^{93} + 3 q^{95} + 18 q^{97}+O(q^{100}) \) 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
\(3\) 3.14134 1.81365 0.906826 0.421506i \(-0.138498\pi\)
0.906826 + 0.421506i \(0.138498\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.14134 −1.18731 −0.593657 0.804718i \(-0.702316\pi\)
−0.593657 + 0.804718i \(0.702316\pi\)
\(8\) 0 0
\(9\) 6.86799 2.28933
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 7.00933 1.94404 0.972019 0.234902i \(-0.0754770\pi\)
0.972019 + 0.234902i \(0.0754770\pi\)
\(14\) 0 0
\(15\) 3.14134 0.811089
\(16\) 0 0
\(17\) −5.14134 −1.24696 −0.623479 0.781840i \(-0.714281\pi\)
−0.623479 + 0.781840i \(0.714281\pi\)
\(18\) 0 0
\(19\) −0.414680 −0.0951340 −0.0475670 0.998868i \(-0.515147\pi\)
−0.0475670 + 0.998868i \(0.515147\pi\)
\(20\) 0 0
\(21\) −9.86799 −2.15337
\(22\) 0 0
\(23\) 2.72666 0.568547 0.284274 0.958743i \(-0.408248\pi\)
0.284274 + 0.958743i \(0.408248\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 12.1507 2.33840
\(28\) 0 0
\(29\) −5.86799 −1.08966 −0.544829 0.838547i \(-0.683406\pi\)
−0.544829 + 0.838547i \(0.683406\pi\)
\(30\) 0 0
\(31\) 9.86799 1.77234 0.886172 0.463357i \(-0.153355\pi\)
0.886172 + 0.463357i \(0.153355\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.14134 −0.530983
\(36\) 0 0
\(37\) −3.86799 −0.635894 −0.317947 0.948108i \(-0.602993\pi\)
−0.317947 + 0.948108i \(0.602993\pi\)
\(38\) 0 0
\(39\) 22.0187 3.52581
\(40\) 0 0
\(41\) 6.28267 0.981189 0.490594 0.871388i \(-0.336780\pi\)
0.490594 + 0.871388i \(0.336780\pi\)
\(42\) 0 0
\(43\) 6.72666 1.02581 0.512903 0.858447i \(-0.328570\pi\)
0.512903 + 0.858447i \(0.328570\pi\)
\(44\) 0 0
\(45\) 6.86799 1.02382
\(46\) 0 0
\(47\) 9.00933 1.31415 0.657073 0.753827i \(-0.271794\pi\)
0.657073 + 0.753827i \(0.271794\pi\)
\(48\) 0 0
\(49\) 2.86799 0.409713
\(50\) 0 0
\(51\) −16.1507 −2.26155
\(52\) 0 0
\(53\) −3.86799 −0.531310 −0.265655 0.964068i \(-0.585588\pi\)
−0.265655 + 0.964068i \(0.585588\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.30265 −0.172540
\(58\) 0 0
\(59\) −1.45331 −0.189205 −0.0946026 0.995515i \(-0.530158\pi\)
−0.0946026 + 0.995515i \(0.530158\pi\)
\(60\) 0 0
\(61\) −10.6974 −1.36966 −0.684828 0.728705i \(-0.740123\pi\)
−0.684828 + 0.728705i \(0.740123\pi\)
\(62\) 0 0
\(63\) −21.5747 −2.71815
\(64\) 0 0
\(65\) 7.00933 0.869400
\(66\) 0 0
\(67\) −7.29200 −0.890860 −0.445430 0.895317i \(-0.646949\pi\)
−0.445430 + 0.895317i \(0.646949\pi\)
\(68\) 0 0
\(69\) 8.56534 1.03115
\(70\) 0 0
\(71\) 6.69735 0.794829 0.397415 0.917639i \(-0.369907\pi\)
0.397415 + 0.917639i \(0.369907\pi\)
\(72\) 0 0
\(73\) 8.72666 1.02138 0.510689 0.859766i \(-0.329390\pi\)
0.510689 + 0.859766i \(0.329390\pi\)
\(74\) 0 0
\(75\) 3.14134 0.362730
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 17.5653 1.95170
\(82\) 0 0
\(83\) −4.17997 −0.458811 −0.229406 0.973331i \(-0.573678\pi\)
−0.229406 + 0.973331i \(0.573678\pi\)
\(84\) 0 0
\(85\) −5.14134 −0.557656
\(86\) 0 0
\(87\) −18.4333 −1.97626
\(88\) 0 0
\(89\) −16.4333 −1.74193 −0.870965 0.491345i \(-0.836505\pi\)
−0.870965 + 0.491345i \(0.836505\pi\)
\(90\) 0 0
\(91\) −22.0187 −2.30818
\(92\) 0 0
\(93\) 30.9987 3.21441
\(94\) 0 0
\(95\) −0.414680 −0.0425452
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.28267 0.227134 0.113567 0.993530i \(-0.463772\pi\)
0.113567 + 0.993530i \(0.463772\pi\)
\(102\) 0 0
\(103\) 17.0093 1.67598 0.837989 0.545686i \(-0.183731\pi\)
0.837989 + 0.545686i \(0.183731\pi\)
\(104\) 0 0
\(105\) −9.86799 −0.963017
\(106\) 0 0
\(107\) −0.443984 −0.0429216 −0.0214608 0.999770i \(-0.506832\pi\)
−0.0214608 + 0.999770i \(0.506832\pi\)
\(108\) 0 0
\(109\) 17.1893 1.64644 0.823218 0.567725i \(-0.192176\pi\)
0.823218 + 0.567725i \(0.192176\pi\)
\(110\) 0 0
\(111\) −12.1507 −1.15329
\(112\) 0 0
\(113\) −2.82936 −0.266164 −0.133082 0.991105i \(-0.542487\pi\)
−0.133082 + 0.991105i \(0.542487\pi\)
\(114\) 0 0
\(115\) 2.72666 0.254262
\(116\) 0 0
\(117\) 48.1400 4.45055
\(118\) 0 0
\(119\) 16.1507 1.48053
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 19.7360 1.77953
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.72666 −0.241952 −0.120976 0.992655i \(-0.538602\pi\)
−0.120976 + 0.992655i \(0.538602\pi\)
\(128\) 0 0
\(129\) 21.1307 1.86045
\(130\) 0 0
\(131\) 6.13201 0.535756 0.267878 0.963453i \(-0.413678\pi\)
0.267878 + 0.963453i \(0.413678\pi\)
\(132\) 0 0
\(133\) 1.30265 0.112954
\(134\) 0 0
\(135\) 12.1507 1.04576
\(136\) 0 0
\(137\) −5.73599 −0.490058 −0.245029 0.969516i \(-0.578798\pi\)
−0.245029 + 0.969516i \(0.578798\pi\)
\(138\) 0 0
\(139\) 18.5840 1.57627 0.788137 0.615500i \(-0.211046\pi\)
0.788137 + 0.615500i \(0.211046\pi\)
\(140\) 0 0
\(141\) 28.3013 2.38340
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −5.86799 −0.487310
\(146\) 0 0
\(147\) 9.00933 0.743077
\(148\) 0 0
\(149\) −20.1507 −1.65081 −0.825403 0.564543i \(-0.809052\pi\)
−0.825403 + 0.564543i \(0.809052\pi\)
\(150\) 0 0
\(151\) −2.01866 −0.164276 −0.0821380 0.996621i \(-0.526175\pi\)
−0.0821380 + 0.996621i \(0.526175\pi\)
\(152\) 0 0
\(153\) −35.3107 −2.85470
\(154\) 0 0
\(155\) 9.86799 0.792616
\(156\) 0 0
\(157\) 4.69735 0.374889 0.187445 0.982275i \(-0.439979\pi\)
0.187445 + 0.982275i \(0.439979\pi\)
\(158\) 0 0
\(159\) −12.1507 −0.963610
\(160\) 0 0
\(161\) −8.56534 −0.675044
\(162\) 0 0
\(163\) −20.3306 −1.59242 −0.796209 0.605022i \(-0.793164\pi\)
−0.796209 + 0.605022i \(0.793164\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.15999 −0.399292 −0.199646 0.979868i \(-0.563979\pi\)
−0.199646 + 0.979868i \(0.563979\pi\)
\(168\) 0 0
\(169\) 36.1307 2.77928
\(170\) 0 0
\(171\) −2.84802 −0.217793
\(172\) 0 0
\(173\) 4.72666 0.359361 0.179681 0.983725i \(-0.442494\pi\)
0.179681 + 0.983725i \(0.442494\pi\)
\(174\) 0 0
\(175\) −3.14134 −0.237463
\(176\) 0 0
\(177\) −4.56534 −0.343152
\(178\) 0 0
\(179\) 10.8480 0.810819 0.405409 0.914135i \(-0.367129\pi\)
0.405409 + 0.914135i \(0.367129\pi\)
\(180\) 0 0
\(181\) −1.17064 −0.0870130 −0.0435065 0.999053i \(-0.513853\pi\)
−0.0435065 + 0.999053i \(0.513853\pi\)
\(182\) 0 0
\(183\) −33.6040 −2.48408
\(184\) 0 0
\(185\) −3.86799 −0.284380
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −38.1693 −2.77641
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −5.14134 −0.370081 −0.185041 0.982731i \(-0.559242\pi\)
−0.185041 + 0.982731i \(0.559242\pi\)
\(194\) 0 0
\(195\) 22.0187 1.57679
\(196\) 0 0
\(197\) 6.38538 0.454939 0.227470 0.973785i \(-0.426955\pi\)
0.227470 + 0.973785i \(0.426955\pi\)
\(198\) 0 0
\(199\) −0.414680 −0.0293959 −0.0146979 0.999892i \(-0.504679\pi\)
−0.0146979 + 0.999892i \(0.504679\pi\)
\(200\) 0 0
\(201\) −22.9066 −1.61571
\(202\) 0 0
\(203\) 18.4333 1.29377
\(204\) 0 0
\(205\) 6.28267 0.438801
\(206\) 0 0
\(207\) 18.7267 1.30159
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −9.86799 −0.679341 −0.339670 0.940545i \(-0.610315\pi\)
−0.339670 + 0.940545i \(0.610315\pi\)
\(212\) 0 0
\(213\) 21.0386 1.44154
\(214\) 0 0
\(215\) 6.72666 0.458754
\(216\) 0 0
\(217\) −30.9987 −2.10433
\(218\) 0 0
\(219\) 27.4134 1.85242
\(220\) 0 0
\(221\) −36.0373 −2.42413
\(222\) 0 0
\(223\) 6.46264 0.432770 0.216385 0.976308i \(-0.430573\pi\)
0.216385 + 0.976308i \(0.430573\pi\)
\(224\) 0 0
\(225\) 6.86799 0.457866
\(226\) 0 0
\(227\) −15.5560 −1.03249 −0.516245 0.856441i \(-0.672670\pi\)
−0.516245 + 0.856441i \(0.672670\pi\)
\(228\) 0 0
\(229\) 11.3947 0.752983 0.376492 0.926420i \(-0.377130\pi\)
0.376492 + 0.926420i \(0.377130\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.2534 −1.58889 −0.794445 0.607335i \(-0.792238\pi\)
−0.794445 + 0.607335i \(0.792238\pi\)
\(234\) 0 0
\(235\) 9.00933 0.587704
\(236\) 0 0
\(237\) 37.6960 2.44862
\(238\) 0 0
\(239\) 7.73599 0.500399 0.250200 0.968194i \(-0.419504\pi\)
0.250200 + 0.968194i \(0.419504\pi\)
\(240\) 0 0
\(241\) −9.71733 −0.625948 −0.312974 0.949762i \(-0.601325\pi\)
−0.312974 + 0.949762i \(0.601325\pi\)
\(242\) 0 0
\(243\) 18.7267 1.20132
\(244\) 0 0
\(245\) 2.86799 0.183229
\(246\) 0 0
\(247\) −2.90663 −0.184944
\(248\) 0 0
\(249\) −13.1307 −0.832124
\(250\) 0 0
\(251\) −16.5653 −1.04560 −0.522798 0.852457i \(-0.675112\pi\)
−0.522798 + 0.852457i \(0.675112\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −16.1507 −1.01139
\(256\) 0 0
\(257\) 26.3013 1.64063 0.820316 0.571911i \(-0.193798\pi\)
0.820316 + 0.571911i \(0.193798\pi\)
\(258\) 0 0
\(259\) 12.1507 0.755006
\(260\) 0 0
\(261\) −40.3013 −2.49459
\(262\) 0 0
\(263\) 5.68802 0.350739 0.175369 0.984503i \(-0.443888\pi\)
0.175369 + 0.984503i \(0.443888\pi\)
\(264\) 0 0
\(265\) −3.86799 −0.237609
\(266\) 0 0
\(267\) −51.6226 −3.15925
\(268\) 0 0
\(269\) −22.3013 −1.35974 −0.679868 0.733334i \(-0.737963\pi\)
−0.679868 + 0.733334i \(0.737963\pi\)
\(270\) 0 0
\(271\) 10.0187 0.608590 0.304295 0.952578i \(-0.401579\pi\)
0.304295 + 0.952578i \(0.401579\pi\)
\(272\) 0 0
\(273\) −69.1680 −4.18624
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.74531 0.164950 0.0824750 0.996593i \(-0.473718\pi\)
0.0824750 + 0.996593i \(0.473718\pi\)
\(278\) 0 0
\(279\) 67.7733 4.05748
\(280\) 0 0
\(281\) −22.2827 −1.32927 −0.664636 0.747167i \(-0.731414\pi\)
−0.664636 + 0.747167i \(0.731414\pi\)
\(282\) 0 0
\(283\) 27.2920 1.62234 0.811171 0.584810i \(-0.198831\pi\)
0.811171 + 0.584810i \(0.198831\pi\)
\(284\) 0 0
\(285\) −1.30265 −0.0771622
\(286\) 0 0
\(287\) −19.7360 −1.16498
\(288\) 0 0
\(289\) 9.43334 0.554902
\(290\) 0 0
\(291\) 18.8480 1.10489
\(292\) 0 0
\(293\) −21.8573 −1.27692 −0.638460 0.769655i \(-0.720428\pi\)
−0.638460 + 0.769655i \(0.720428\pi\)
\(294\) 0 0
\(295\) −1.45331 −0.0846152
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 19.1120 1.10528
\(300\) 0 0
\(301\) −21.1307 −1.21795
\(302\) 0 0
\(303\) 7.17064 0.411943
\(304\) 0 0
\(305\) −10.6974 −0.612529
\(306\) 0 0
\(307\) −22.6680 −1.29373 −0.646867 0.762603i \(-0.723921\pi\)
−0.646867 + 0.762603i \(0.723921\pi\)
\(308\) 0 0
\(309\) 53.4320 3.03964
\(310\) 0 0
\(311\) −14.6974 −0.833410 −0.416705 0.909042i \(-0.636815\pi\)
−0.416705 + 0.909042i \(0.636815\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 0 0
\(315\) −21.5747 −1.21560
\(316\) 0 0
\(317\) 32.4333 1.82164 0.910819 0.412806i \(-0.135451\pi\)
0.910819 + 0.412806i \(0.135451\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.39470 −0.0778448
\(322\) 0 0
\(323\) 2.13201 0.118628
\(324\) 0 0
\(325\) 7.00933 0.388808
\(326\) 0 0
\(327\) 53.9974 2.98606
\(328\) 0 0
\(329\) −28.3013 −1.56030
\(330\) 0 0
\(331\) −20.0373 −1.10135 −0.550675 0.834720i \(-0.685630\pi\)
−0.550675 + 0.834720i \(0.685630\pi\)
\(332\) 0 0
\(333\) −26.5653 −1.45577
\(334\) 0 0
\(335\) −7.29200 −0.398405
\(336\) 0 0
\(337\) 16.2534 0.885377 0.442689 0.896675i \(-0.354025\pi\)
0.442689 + 0.896675i \(0.354025\pi\)
\(338\) 0 0
\(339\) −8.88797 −0.482728
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 12.9800 0.700855
\(344\) 0 0
\(345\) 8.56534 0.461143
\(346\) 0 0
\(347\) 18.4626 0.991127 0.495563 0.868572i \(-0.334962\pi\)
0.495563 + 0.868572i \(0.334962\pi\)
\(348\) 0 0
\(349\) 11.7360 0.628213 0.314106 0.949388i \(-0.398295\pi\)
0.314106 + 0.949388i \(0.398295\pi\)
\(350\) 0 0
\(351\) 85.1680 4.54593
\(352\) 0 0
\(353\) 24.6426 1.31159 0.655797 0.754937i \(-0.272333\pi\)
0.655797 + 0.754937i \(0.272333\pi\)
\(354\) 0 0
\(355\) 6.69735 0.355459
\(356\) 0 0
\(357\) 50.7347 2.68516
\(358\) 0 0
\(359\) 29.9600 1.58123 0.790615 0.612313i \(-0.209761\pi\)
0.790615 + 0.612313i \(0.209761\pi\)
\(360\) 0 0
\(361\) −18.8280 −0.990950
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.72666 0.456774
\(366\) 0 0
\(367\) −18.1986 −0.949961 −0.474980 0.879996i \(-0.657545\pi\)
−0.474980 + 0.879996i \(0.657545\pi\)
\(368\) 0 0
\(369\) 43.1493 2.24627
\(370\) 0 0
\(371\) 12.1507 0.630831
\(372\) 0 0
\(373\) 4.72666 0.244737 0.122368 0.992485i \(-0.460951\pi\)
0.122368 + 0.992485i \(0.460951\pi\)
\(374\) 0 0
\(375\) 3.14134 0.162218
\(376\) 0 0
\(377\) −41.1307 −2.11834
\(378\) 0 0
\(379\) −19.1120 −0.981719 −0.490860 0.871239i \(-0.663317\pi\)
−0.490860 + 0.871239i \(0.663317\pi\)
\(380\) 0 0
\(381\) −8.56534 −0.438816
\(382\) 0 0
\(383\) −3.55602 −0.181704 −0.0908520 0.995864i \(-0.528959\pi\)
−0.0908520 + 0.995864i \(0.528959\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 46.1986 2.34841
\(388\) 0 0
\(389\) −2.30133 −0.116682 −0.0583410 0.998297i \(-0.518581\pi\)
−0.0583410 + 0.998297i \(0.518581\pi\)
\(390\) 0 0
\(391\) −14.0187 −0.708954
\(392\) 0 0
\(393\) 19.2627 0.971674
\(394\) 0 0
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 15.6587 0.785889 0.392944 0.919562i \(-0.371457\pi\)
0.392944 + 0.919562i \(0.371457\pi\)
\(398\) 0 0
\(399\) 4.09206 0.204859
\(400\) 0 0
\(401\) −0.697352 −0.0348241 −0.0174120 0.999848i \(-0.505543\pi\)
−0.0174120 + 0.999848i \(0.505543\pi\)
\(402\) 0 0
\(403\) 69.1680 3.44550
\(404\) 0 0
\(405\) 17.5653 0.872829
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −33.3947 −1.65126 −0.825631 0.564211i \(-0.809181\pi\)
−0.825631 + 0.564211i \(0.809181\pi\)
\(410\) 0 0
\(411\) −18.0187 −0.888795
\(412\) 0 0
\(413\) 4.56534 0.224646
\(414\) 0 0
\(415\) −4.17997 −0.205187
\(416\) 0 0
\(417\) 58.3786 2.85881
\(418\) 0 0
\(419\) −1.18930 −0.0581010 −0.0290505 0.999578i \(-0.509248\pi\)
−0.0290505 + 0.999578i \(0.509248\pi\)
\(420\) 0 0
\(421\) −11.6587 −0.568211 −0.284106 0.958793i \(-0.591697\pi\)
−0.284106 + 0.958793i \(0.591697\pi\)
\(422\) 0 0
\(423\) 61.8760 3.00851
\(424\) 0 0
\(425\) −5.14134 −0.249391
\(426\) 0 0
\(427\) 33.6040 1.62621
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 29.4906 1.42051 0.710257 0.703943i \(-0.248579\pi\)
0.710257 + 0.703943i \(0.248579\pi\)
\(432\) 0 0
\(433\) −9.47197 −0.455194 −0.227597 0.973755i \(-0.573087\pi\)
−0.227597 + 0.973755i \(0.573087\pi\)
\(434\) 0 0
\(435\) −18.4333 −0.883811
\(436\) 0 0
\(437\) −1.13069 −0.0540882
\(438\) 0 0
\(439\) 3.73599 0.178309 0.0891544 0.996018i \(-0.471584\pi\)
0.0891544 + 0.996018i \(0.471584\pi\)
\(440\) 0 0
\(441\) 19.6974 0.937969
\(442\) 0 0
\(443\) −24.9507 −1.18544 −0.592722 0.805407i \(-0.701947\pi\)
−0.592722 + 0.805407i \(0.701947\pi\)
\(444\) 0 0
\(445\) −16.4333 −0.779015
\(446\) 0 0
\(447\) −63.3000 −2.99399
\(448\) 0 0
\(449\) 2.56534 0.121066 0.0605330 0.998166i \(-0.480720\pi\)
0.0605330 + 0.998166i \(0.480720\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −6.34128 −0.297939
\(454\) 0 0
\(455\) −22.0187 −1.03225
\(456\) 0 0
\(457\) 13.1413 0.614726 0.307363 0.951592i \(-0.400553\pi\)
0.307363 + 0.951592i \(0.400553\pi\)
\(458\) 0 0
\(459\) −62.4707 −2.91588
\(460\) 0 0
\(461\) 8.45199 0.393649 0.196824 0.980439i \(-0.436937\pi\)
0.196824 + 0.980439i \(0.436937\pi\)
\(462\) 0 0
\(463\) 18.1986 0.845762 0.422881 0.906185i \(-0.361019\pi\)
0.422881 + 0.906185i \(0.361019\pi\)
\(464\) 0 0
\(465\) 30.9987 1.43753
\(466\) 0 0
\(467\) −5.98935 −0.277154 −0.138577 0.990352i \(-0.544253\pi\)
−0.138577 + 0.990352i \(0.544253\pi\)
\(468\) 0 0
\(469\) 22.9066 1.05773
\(470\) 0 0
\(471\) 14.7560 0.679919
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.414680 −0.0190268
\(476\) 0 0
\(477\) −26.5653 −1.21634
\(478\) 0 0
\(479\) −26.6426 −1.21733 −0.608666 0.793427i \(-0.708295\pi\)
−0.608666 + 0.793427i \(0.708295\pi\)
\(480\) 0 0
\(481\) −27.1120 −1.23620
\(482\) 0 0
\(483\) −26.9066 −1.22429
\(484\) 0 0
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) 3.61462 0.163794 0.0818971 0.996641i \(-0.473902\pi\)
0.0818971 + 0.996641i \(0.473902\pi\)
\(488\) 0 0
\(489\) −63.8654 −2.88809
\(490\) 0 0
\(491\) −17.9053 −0.808055 −0.404028 0.914747i \(-0.632390\pi\)
−0.404028 + 0.914747i \(0.632390\pi\)
\(492\) 0 0
\(493\) 30.1693 1.35876
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −21.0386 −0.943712
\(498\) 0 0
\(499\) −13.1893 −0.590434 −0.295217 0.955430i \(-0.595392\pi\)
−0.295217 + 0.955430i \(0.595392\pi\)
\(500\) 0 0
\(501\) −16.2093 −0.724177
\(502\) 0 0
\(503\) −14.1613 −0.631422 −0.315711 0.948855i \(-0.602243\pi\)
−0.315711 + 0.948855i \(0.602243\pi\)
\(504\) 0 0
\(505\) 2.28267 0.101578
\(506\) 0 0
\(507\) 113.499 5.04065
\(508\) 0 0
\(509\) −8.34128 −0.369721 −0.184860 0.982765i \(-0.559183\pi\)
−0.184860 + 0.982765i \(0.559183\pi\)
\(510\) 0 0
\(511\) −27.4134 −1.21270
\(512\) 0 0
\(513\) −5.03863 −0.222461
\(514\) 0 0
\(515\) 17.0093 0.749521
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 14.8480 0.651756
\(520\) 0 0
\(521\) −9.47197 −0.414975 −0.207487 0.978238i \(-0.566529\pi\)
−0.207487 + 0.978238i \(0.566529\pi\)
\(522\) 0 0
\(523\) 7.91595 0.346141 0.173070 0.984909i \(-0.444631\pi\)
0.173070 + 0.984909i \(0.444631\pi\)
\(524\) 0 0
\(525\) −9.86799 −0.430675
\(526\) 0 0
\(527\) −50.7347 −2.21004
\(528\) 0 0
\(529\) −15.5653 −0.676754
\(530\) 0 0
\(531\) −9.98134 −0.433153
\(532\) 0 0
\(533\) 44.0373 1.90747
\(534\) 0 0
\(535\) −0.443984 −0.0191951
\(536\) 0 0
\(537\) 34.0773 1.47054
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4.11335 0.176847 0.0884234 0.996083i \(-0.471817\pi\)
0.0884234 + 0.996083i \(0.471817\pi\)
\(542\) 0 0
\(543\) −3.67738 −0.157811
\(544\) 0 0
\(545\) 17.1893 0.736309
\(546\) 0 0
\(547\) −5.89730 −0.252150 −0.126075 0.992021i \(-0.540238\pi\)
−0.126075 + 0.992021i \(0.540238\pi\)
\(548\) 0 0
\(549\) −73.4693 −3.13559
\(550\) 0 0
\(551\) 2.43334 0.103664
\(552\) 0 0
\(553\) −37.6960 −1.60300
\(554\) 0 0
\(555\) −12.1507 −0.515767
\(556\) 0 0
\(557\) −18.4440 −0.781497 −0.390748 0.920498i \(-0.627784\pi\)
−0.390748 + 0.920498i \(0.627784\pi\)
\(558\) 0 0
\(559\) 47.1493 1.99420
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.1027 0.425778 0.212889 0.977076i \(-0.431713\pi\)
0.212889 + 0.977076i \(0.431713\pi\)
\(564\) 0 0
\(565\) −2.82936 −0.119032
\(566\) 0 0
\(567\) −55.1787 −2.31729
\(568\) 0 0
\(569\) 41.3947 1.73536 0.867678 0.497126i \(-0.165611\pi\)
0.867678 + 0.497126i \(0.165611\pi\)
\(570\) 0 0
\(571\) −37.6413 −1.57524 −0.787620 0.616162i \(-0.788687\pi\)
−0.787620 + 0.616162i \(0.788687\pi\)
\(572\) 0 0
\(573\) 37.6960 1.57477
\(574\) 0 0
\(575\) 2.72666 0.113709
\(576\) 0 0
\(577\) −19.9600 −0.830948 −0.415474 0.909605i \(-0.636384\pi\)
−0.415474 + 0.909605i \(0.636384\pi\)
\(578\) 0 0
\(579\) −16.1507 −0.671199
\(580\) 0 0
\(581\) 13.1307 0.544753
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 48.1400 1.99034
\(586\) 0 0
\(587\) −2.78140 −0.114801 −0.0574003 0.998351i \(-0.518281\pi\)
−0.0574003 + 0.998351i \(0.518281\pi\)
\(588\) 0 0
\(589\) −4.09206 −0.168610
\(590\) 0 0
\(591\) 20.0586 0.825101
\(592\) 0 0
\(593\) 36.7640 1.50972 0.754858 0.655889i \(-0.227706\pi\)
0.754858 + 0.655889i \(0.227706\pi\)
\(594\) 0 0
\(595\) 16.1507 0.662113
\(596\) 0 0
\(597\) −1.30265 −0.0533138
\(598\) 0 0
\(599\) −1.80938 −0.0739294 −0.0369647 0.999317i \(-0.511769\pi\)
−0.0369647 + 0.999317i \(0.511769\pi\)
\(600\) 0 0
\(601\) −9.71733 −0.396378 −0.198189 0.980164i \(-0.563506\pi\)
−0.198189 + 0.980164i \(0.563506\pi\)
\(602\) 0 0
\(603\) −50.0814 −2.03947
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −5.98935 −0.243100 −0.121550 0.992585i \(-0.538787\pi\)
−0.121550 + 0.992585i \(0.538787\pi\)
\(608\) 0 0
\(609\) 57.9053 2.34644
\(610\) 0 0
\(611\) 63.1493 2.55475
\(612\) 0 0
\(613\) −25.9160 −1.04674 −0.523368 0.852107i \(-0.675325\pi\)
−0.523368 + 0.852107i \(0.675325\pi\)
\(614\) 0 0
\(615\) 19.7360 0.795832
\(616\) 0 0
\(617\) −23.6960 −0.953966 −0.476983 0.878912i \(-0.658270\pi\)
−0.476983 + 0.878912i \(0.658270\pi\)
\(618\) 0 0
\(619\) 32.6613 1.31277 0.656384 0.754427i \(-0.272085\pi\)
0.656384 + 0.754427i \(0.272085\pi\)
\(620\) 0 0
\(621\) 33.1307 1.32949
\(622\) 0 0
\(623\) 51.6226 2.06822
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 19.8867 0.792933
\(630\) 0 0
\(631\) −6.99868 −0.278613 −0.139307 0.990249i \(-0.544487\pi\)
−0.139307 + 0.990249i \(0.544487\pi\)
\(632\) 0 0
\(633\) −30.9987 −1.23209
\(634\) 0 0
\(635\) −2.72666 −0.108204
\(636\) 0 0
\(637\) 20.1027 0.796498
\(638\) 0 0
\(639\) 45.9974 1.81963
\(640\) 0 0
\(641\) −21.2627 −0.839826 −0.419913 0.907564i \(-0.637939\pi\)
−0.419913 + 0.907564i \(0.637939\pi\)
\(642\) 0 0
\(643\) −36.8587 −1.45356 −0.726782 0.686868i \(-0.758985\pi\)
−0.726782 + 0.686868i \(0.758985\pi\)
\(644\) 0 0
\(645\) 21.1307 0.832020
\(646\) 0 0
\(647\) −28.2173 −1.10934 −0.554668 0.832072i \(-0.687155\pi\)
−0.554668 + 0.832072i \(0.687155\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −97.3773 −3.81652
\(652\) 0 0
\(653\) −40.7347 −1.59407 −0.797035 0.603933i \(-0.793600\pi\)
−0.797035 + 0.603933i \(0.793600\pi\)
\(654\) 0 0
\(655\) 6.13201 0.239597
\(656\) 0 0
\(657\) 59.9346 2.33827
\(658\) 0 0
\(659\) −9.80938 −0.382119 −0.191060 0.981578i \(-0.561192\pi\)
−0.191060 + 0.981578i \(0.561192\pi\)
\(660\) 0 0
\(661\) 40.3786 1.57055 0.785273 0.619150i \(-0.212523\pi\)
0.785273 + 0.619150i \(0.212523\pi\)
\(662\) 0 0
\(663\) −113.205 −4.39653
\(664\) 0 0
\(665\) 1.30265 0.0505145
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 0 0
\(669\) 20.3013 0.784895
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −17.1787 −0.662188 −0.331094 0.943598i \(-0.607418\pi\)
−0.331094 + 0.943598i \(0.607418\pi\)
\(674\) 0 0
\(675\) 12.1507 0.467679
\(676\) 0 0
\(677\) −5.85735 −0.225116 −0.112558 0.993645i \(-0.535904\pi\)
−0.112558 + 0.993645i \(0.535904\pi\)
\(678\) 0 0
\(679\) −18.8480 −0.723320
\(680\) 0 0
\(681\) −48.8667 −1.87258
\(682\) 0 0
\(683\) −23.1787 −0.886906 −0.443453 0.896298i \(-0.646247\pi\)
−0.443453 + 0.896298i \(0.646247\pi\)
\(684\) 0 0
\(685\) −5.73599 −0.219161
\(686\) 0 0
\(687\) 35.7946 1.36565
\(688\) 0 0
\(689\) −27.1120 −1.03289
\(690\) 0 0
\(691\) 32.8667 1.25031 0.625154 0.780502i \(-0.285036\pi\)
0.625154 + 0.780502i \(0.285036\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.5840 0.704931
\(696\) 0 0
\(697\) −32.3013 −1.22350
\(698\) 0 0
\(699\) −76.1880 −2.88169
\(700\) 0 0
\(701\) 29.9053 1.12951 0.564754 0.825259i \(-0.308971\pi\)
0.564754 + 0.825259i \(0.308971\pi\)
\(702\) 0 0
\(703\) 1.60398 0.0604952
\(704\) 0 0
\(705\) 28.3013 1.06589
\(706\) 0 0
\(707\) −7.17064 −0.269680
\(708\) 0 0
\(709\) −15.0934 −0.566844 −0.283422 0.958995i \(-0.591470\pi\)
−0.283422 + 0.958995i \(0.591470\pi\)
\(710\) 0 0
\(711\) 82.4159 3.09084
\(712\) 0 0
\(713\) 26.9066 1.00766
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.3013 0.907550
\(718\) 0 0
\(719\) 48.9800 1.82665 0.913323 0.407235i \(-0.133507\pi\)
0.913323 + 0.407235i \(0.133507\pi\)
\(720\) 0 0
\(721\) −53.4320 −1.98991
\(722\) 0 0
\(723\) −30.5254 −1.13525
\(724\) 0 0
\(725\) −5.86799 −0.217932
\(726\) 0 0
\(727\) −40.7826 −1.51254 −0.756272 0.654257i \(-0.772981\pi\)
−0.756272 + 0.654257i \(0.772981\pi\)
\(728\) 0 0
\(729\) 6.13069 0.227063
\(730\) 0 0
\(731\) −34.5840 −1.27914
\(732\) 0 0
\(733\) 8.46264 0.312575 0.156287 0.987712i \(-0.450047\pi\)
0.156287 + 0.987712i \(0.450047\pi\)
\(734\) 0 0
\(735\) 9.00933 0.332314
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −19.1120 −0.703047 −0.351524 0.936179i \(-0.614336\pi\)
−0.351524 + 0.936179i \(0.614336\pi\)
\(740\) 0 0
\(741\) −9.13069 −0.335424
\(742\) 0 0
\(743\) 45.4613 1.66781 0.833907 0.551905i \(-0.186099\pi\)
0.833907 + 0.551905i \(0.186099\pi\)
\(744\) 0 0
\(745\) −20.1507 −0.738263
\(746\) 0 0
\(747\) −28.7080 −1.05037
\(748\) 0 0
\(749\) 1.39470 0.0509614
\(750\) 0 0
\(751\) −17.0759 −0.623110 −0.311555 0.950228i \(-0.600850\pi\)
−0.311555 + 0.950228i \(0.600850\pi\)
\(752\) 0 0
\(753\) −52.0373 −1.89635
\(754\) 0 0
\(755\) −2.01866 −0.0734665
\(756\) 0 0
\(757\) −37.5093 −1.36330 −0.681649 0.731679i \(-0.738737\pi\)
−0.681649 + 0.731679i \(0.738737\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 37.3947 1.35556 0.677778 0.735266i \(-0.262943\pi\)
0.677778 + 0.735266i \(0.262943\pi\)
\(762\) 0 0
\(763\) −53.9974 −1.95484
\(764\) 0 0
\(765\) −35.3107 −1.27666
\(766\) 0 0
\(767\) −10.1867 −0.367822
\(768\) 0 0
\(769\) 4.46942 0.161171 0.0805857 0.996748i \(-0.474321\pi\)
0.0805857 + 0.996748i \(0.474321\pi\)
\(770\) 0 0
\(771\) 82.6213 2.97553
\(772\) 0 0
\(773\) 3.60398 0.129626 0.0648130 0.997897i \(-0.479355\pi\)
0.0648130 + 0.997897i \(0.479355\pi\)
\(774\) 0 0
\(775\) 9.86799 0.354469
\(776\) 0 0
\(777\) 38.1693 1.36932
\(778\) 0 0
\(779\) −2.60530 −0.0933444
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −71.3000 −2.54805
\(784\) 0 0
\(785\) 4.69735 0.167656
\(786\) 0 0
\(787\) 24.1587 0.861164 0.430582 0.902552i \(-0.358308\pi\)
0.430582 + 0.902552i \(0.358308\pi\)
\(788\) 0 0
\(789\) 17.8680 0.636117
\(790\) 0 0
\(791\) 8.88797 0.316020
\(792\) 0 0
\(793\) −74.9813 −2.66266
\(794\) 0 0
\(795\) −12.1507 −0.430940
\(796\) 0 0
\(797\) 31.6587 1.12141 0.560705 0.828016i \(-0.310530\pi\)
0.560705 + 0.828016i \(0.310530\pi\)
\(798\) 0 0
\(799\) −46.3200 −1.63868
\(800\) 0 0
\(801\) −112.864 −3.98785
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −8.56534 −0.301889
\(806\) 0 0
\(807\) −70.0560 −2.46609
\(808\) 0 0
\(809\) −34.0560 −1.19734 −0.598672 0.800994i \(-0.704305\pi\)
−0.598672 + 0.800994i \(0.704305\pi\)
\(810\) 0 0
\(811\) 20.2093 0.709644 0.354822 0.934934i \(-0.384542\pi\)
0.354822 + 0.934934i \(0.384542\pi\)
\(812\) 0 0
\(813\) 31.4720 1.10377
\(814\) 0 0
\(815\) −20.3306 −0.712151
\(816\) 0 0
\(817\) −2.78941 −0.0975890
\(818\) 0 0
\(819\) −151.224 −5.28419
\(820\) 0 0
\(821\) −52.3013 −1.82533 −0.912664 0.408710i \(-0.865979\pi\)
−0.912664 + 0.408710i \(0.865979\pi\)
\(822\) 0 0
\(823\) −41.2520 −1.43796 −0.718978 0.695033i \(-0.755390\pi\)
−0.718978 + 0.695033i \(0.755390\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41.0466 −1.42733 −0.713666 0.700486i \(-0.752966\pi\)
−0.713666 + 0.700486i \(0.752966\pi\)
\(828\) 0 0
\(829\) 2.26401 0.0786325 0.0393162 0.999227i \(-0.487482\pi\)
0.0393162 + 0.999227i \(0.487482\pi\)
\(830\) 0 0
\(831\) 8.62395 0.299162
\(832\) 0 0
\(833\) −14.7453 −0.510895
\(834\) 0 0
\(835\) −5.15999 −0.178569
\(836\) 0 0
\(837\) 119.903 4.14444
\(838\) 0 0
\(839\) −28.9253 −0.998612 −0.499306 0.866426i \(-0.666412\pi\)
−0.499306 + 0.866426i \(0.666412\pi\)
\(840\) 0 0
\(841\) 5.43334 0.187356
\(842\) 0 0
\(843\) −69.9974 −2.41084
\(844\) 0 0
\(845\) 36.1307 1.24293
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 85.7333 2.94236
\(850\) 0 0
\(851\) −10.5467 −0.361536
\(852\) 0 0
\(853\) −3.57467 −0.122394 −0.0611972 0.998126i \(-0.519492\pi\)
−0.0611972 + 0.998126i \(0.519492\pi\)
\(854\) 0 0
\(855\) −2.84802 −0.0974001
\(856\) 0 0
\(857\) −31.5199 −1.07670 −0.538350 0.842721i \(-0.680952\pi\)
−0.538350 + 0.842721i \(0.680952\pi\)
\(858\) 0 0
\(859\) 7.11203 0.242659 0.121330 0.992612i \(-0.461284\pi\)
0.121330 + 0.992612i \(0.461284\pi\)
\(860\) 0 0
\(861\) −61.9974 −2.11286
\(862\) 0 0
\(863\) 17.5374 0.596979 0.298489 0.954413i \(-0.403517\pi\)
0.298489 + 0.954413i \(0.403517\pi\)
\(864\) 0 0
\(865\) 4.72666 0.160711
\(866\) 0 0
\(867\) 29.6333 1.00640
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −51.1120 −1.73187
\(872\) 0 0
\(873\) 41.2080 1.39468
\(874\) 0 0
\(875\) −3.14134 −0.106197
\(876\) 0 0
\(877\) 8.04409 0.271630 0.135815 0.990734i \(-0.456635\pi\)
0.135815 + 0.990734i \(0.456635\pi\)
\(878\) 0 0
\(879\) −68.6613 −2.31589
\(880\) 0 0
\(881\) 23.6960 0.798340 0.399170 0.916877i \(-0.369298\pi\)
0.399170 + 0.916877i \(0.369298\pi\)
\(882\) 0 0
\(883\) −14.8187 −0.498689 −0.249345 0.968415i \(-0.580215\pi\)
−0.249345 + 0.968415i \(0.580215\pi\)
\(884\) 0 0
\(885\) −4.56534 −0.153462
\(886\) 0 0
\(887\) 32.1800 1.08050 0.540249 0.841505i \(-0.318330\pi\)
0.540249 + 0.841505i \(0.318330\pi\)
\(888\) 0 0
\(889\) 8.56534 0.287272
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.73599 −0.125020
\(894\) 0 0
\(895\) 10.8480 0.362609
\(896\) 0 0
\(897\) 60.0373 2.00459
\(898\) 0 0
\(899\) −57.9053 −1.93125
\(900\) 0 0
\(901\) 19.8867 0.662520
\(902\) 0 0
\(903\) −66.3786 −2.20894
\(904\) 0 0
\(905\) −1.17064 −0.0389134
\(906\) 0 0
\(907\) 24.8374 0.824711 0.412356 0.911023i \(-0.364706\pi\)
0.412356 + 0.911023i \(0.364706\pi\)
\(908\) 0 0
\(909\) 15.6774 0.519986
\(910\) 0 0
\(911\) −36.2466 −1.20090 −0.600452 0.799661i \(-0.705012\pi\)
−0.600452 + 0.799661i \(0.705012\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −33.6040 −1.11091
\(916\) 0 0
\(917\) −19.2627 −0.636110
\(918\) 0 0
\(919\) 12.5653 0.414492 0.207246 0.978289i \(-0.433550\pi\)
0.207246 + 0.978289i \(0.433550\pi\)
\(920\) 0 0
\(921\) −71.2080 −2.34638
\(922\) 0 0
\(923\) 46.9439 1.54518
\(924\) 0 0
\(925\) −3.86799 −0.127179
\(926\) 0 0
\(927\) 116.820 3.83687
\(928\) 0 0
\(929\) −42.9214 −1.40821 −0.704103 0.710098i \(-0.748651\pi\)
−0.704103 + 0.710098i \(0.748651\pi\)
\(930\) 0 0
\(931\) −1.18930 −0.0389777
\(932\) 0 0
\(933\) −46.1693 −1.51152
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.7453 0.743057 0.371529 0.928422i \(-0.378834\pi\)
0.371529 + 0.928422i \(0.378834\pi\)
\(938\) 0 0
\(939\) 6.28267 0.205027
\(940\) 0 0
\(941\) 47.0946 1.53524 0.767620 0.640905i \(-0.221441\pi\)
0.767620 + 0.640905i \(0.221441\pi\)
\(942\) 0 0
\(943\) 17.1307 0.557852
\(944\) 0 0
\(945\) −38.1693 −1.24165
\(946\) 0 0
\(947\) 13.7466 0.446706 0.223353 0.974738i \(-0.428300\pi\)
0.223353 + 0.974738i \(0.428300\pi\)
\(948\) 0 0
\(949\) 61.1680 1.98560
\(950\) 0 0
\(951\) 101.884 3.30382
\(952\) 0 0
\(953\) −1.72797 −0.0559746 −0.0279873 0.999608i \(-0.508910\pi\)
−0.0279873 + 0.999608i \(0.508910\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.0187 0.581853
\(960\) 0 0
\(961\) 66.3773 2.14120
\(962\) 0 0
\(963\) −3.04928 −0.0982617
\(964\) 0 0
\(965\) −5.14134 −0.165505
\(966\) 0 0
\(967\) −49.4240 −1.58937 −0.794684 0.607023i \(-0.792364\pi\)
−0.794684 + 0.607023i \(0.792364\pi\)
\(968\) 0 0
\(969\) 6.69735 0.215150
\(970\) 0 0
\(971\) −10.4881 −0.336578 −0.168289 0.985738i \(-0.553824\pi\)
−0.168289 + 0.985738i \(0.553824\pi\)
\(972\) 0 0
\(973\) −58.3786 −1.87153
\(974\) 0 0
\(975\) 22.0187 0.705161
\(976\) 0 0
\(977\) −36.3786 −1.16385 −0.581927 0.813241i \(-0.697701\pi\)
−0.581927 + 0.813241i \(0.697701\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 118.056 3.76924
\(982\) 0 0
\(983\) 8.70800 0.277742 0.138871 0.990310i \(-0.455653\pi\)
0.138871 + 0.990310i \(0.455653\pi\)
\(984\) 0 0
\(985\) 6.38538 0.203455
\(986\) 0 0
\(987\) −88.9040 −2.82985
\(988\) 0 0
\(989\) 18.3413 0.583219
\(990\) 0 0
\(991\) 27.4720 0.872676 0.436338 0.899783i \(-0.356275\pi\)
0.436338 + 0.899783i \(0.356275\pi\)
\(992\) 0 0
\(993\) −62.9439 −1.99747
\(994\) 0 0
\(995\) −0.414680 −0.0131462
\(996\) 0 0
\(997\) −47.6120 −1.50789 −0.753943 0.656939i \(-0.771851\pi\)
−0.753943 + 0.656939i \(0.771851\pi\)
\(998\) 0 0
\(999\) −46.9987 −1.48697
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 9680.2.a.cg.1.3 3
4.3 odd 2 4840.2.a.r.1.1 yes 3
11.10 odd 2 9680.2.a.ch.1.3 3
44.43 even 2 4840.2.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.q.1.1 3 44.43 even 2
4840.2.a.r.1.1 yes 3 4.3 odd 2
9680.2.a.cg.1.3 3 1.1 even 1 trivial
9680.2.a.ch.1.3 3 11.10 odd 2