Properties

Label 1323.4.a.bf.1.1
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,4,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 42x^{4} + 369x^{2} - 112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.45019\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.45019 q^{2} +21.7046 q^{4} +19.3432 q^{5} -74.6929 q^{8} +O(q^{10})\) \(q-5.45019 q^{2} +21.7046 q^{4} +19.3432 q^{5} -74.6929 q^{8} -105.424 q^{10} +11.2132 q^{11} -46.3102 q^{13} +233.454 q^{16} -97.6541 q^{17} -98.9176 q^{19} +419.836 q^{20} -61.1139 q^{22} +138.173 q^{23} +249.158 q^{25} +252.400 q^{26} +180.123 q^{29} -31.9046 q^{31} -674.825 q^{32} +532.234 q^{34} -205.452 q^{37} +539.120 q^{38} -1444.80 q^{40} +234.404 q^{41} -320.568 q^{43} +243.377 q^{44} -753.067 q^{46} -312.715 q^{47} -1357.96 q^{50} -1005.15 q^{52} +53.7942 q^{53} +216.898 q^{55} -981.706 q^{58} +400.291 q^{59} -97.3536 q^{61} +173.886 q^{62} +1810.30 q^{64} -895.786 q^{65} +257.525 q^{67} -2119.55 q^{68} +253.601 q^{71} -1161.99 q^{73} +1119.75 q^{74} -2146.97 q^{76} -1070.16 q^{79} +4515.73 q^{80} -1277.55 q^{82} -889.495 q^{83} -1888.94 q^{85} +1747.16 q^{86} -837.543 q^{88} -647.303 q^{89} +2998.98 q^{92} +1704.36 q^{94} -1913.38 q^{95} +673.379 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 36 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 36 q^{4} - 180 q^{10} - 108 q^{13} + 420 q^{16} - 198 q^{19} - 84 q^{22} + 420 q^{25} + 90 q^{31} + 648 q^{34} - 402 q^{37} - 2844 q^{40} - 660 q^{43} - 1332 q^{46} - 1224 q^{52} - 846 q^{55} - 1800 q^{58} - 1152 q^{61} + 2964 q^{64} + 924 q^{67} - 1260 q^{73} - 5868 q^{76} - 1500 q^{79} - 4500 q^{82} - 2232 q^{85} - 2460 q^{88} + 4968 q^{94} - 3312 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.45019 −1.92693 −0.963467 0.267826i \(-0.913695\pi\)
−0.963467 + 0.267826i \(0.913695\pi\)
\(3\) 0 0
\(4\) 21.7046 2.71308
\(5\) 19.3432 1.73011 0.865053 0.501681i \(-0.167285\pi\)
0.865053 + 0.501681i \(0.167285\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −74.6929 −3.30099
\(9\) 0 0
\(10\) −105.424 −3.33380
\(11\) 11.2132 0.307354 0.153677 0.988121i \(-0.450888\pi\)
0.153677 + 0.988121i \(0.450888\pi\)
\(12\) 0 0
\(13\) −46.3102 −0.988010 −0.494005 0.869459i \(-0.664468\pi\)
−0.494005 + 0.869459i \(0.664468\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 233.454 3.64771
\(17\) −97.6541 −1.39321 −0.696606 0.717454i \(-0.745307\pi\)
−0.696606 + 0.717454i \(0.745307\pi\)
\(18\) 0 0
\(19\) −98.9176 −1.19438 −0.597191 0.802099i \(-0.703716\pi\)
−0.597191 + 0.802099i \(0.703716\pi\)
\(20\) 419.836 4.69391
\(21\) 0 0
\(22\) −61.1139 −0.592251
\(23\) 138.173 1.25265 0.626325 0.779562i \(-0.284558\pi\)
0.626325 + 0.779562i \(0.284558\pi\)
\(24\) 0 0
\(25\) 249.158 1.99327
\(26\) 252.400 1.90383
\(27\) 0 0
\(28\) 0 0
\(29\) 180.123 1.15338 0.576690 0.816963i \(-0.304344\pi\)
0.576690 + 0.816963i \(0.304344\pi\)
\(30\) 0 0
\(31\) −31.9046 −0.184846 −0.0924230 0.995720i \(-0.529461\pi\)
−0.0924230 + 0.995720i \(0.529461\pi\)
\(32\) −674.825 −3.72792
\(33\) 0 0
\(34\) 532.234 2.68463
\(35\) 0 0
\(36\) 0 0
\(37\) −205.452 −0.912867 −0.456433 0.889758i \(-0.650873\pi\)
−0.456433 + 0.889758i \(0.650873\pi\)
\(38\) 539.120 2.30149
\(39\) 0 0
\(40\) −1444.80 −5.71106
\(41\) 234.404 0.892874 0.446437 0.894815i \(-0.352693\pi\)
0.446437 + 0.894815i \(0.352693\pi\)
\(42\) 0 0
\(43\) −320.568 −1.13689 −0.568443 0.822723i \(-0.692454\pi\)
−0.568443 + 0.822723i \(0.692454\pi\)
\(44\) 243.377 0.833875
\(45\) 0 0
\(46\) −753.067 −2.41378
\(47\) −312.715 −0.970516 −0.485258 0.874371i \(-0.661274\pi\)
−0.485258 + 0.874371i \(0.661274\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1357.96 −3.84089
\(51\) 0 0
\(52\) −1005.15 −2.68055
\(53\) 53.7942 0.139419 0.0697094 0.997567i \(-0.477793\pi\)
0.0697094 + 0.997567i \(0.477793\pi\)
\(54\) 0 0
\(55\) 216.898 0.531755
\(56\) 0 0
\(57\) 0 0
\(58\) −981.706 −2.22249
\(59\) 400.291 0.883278 0.441639 0.897193i \(-0.354397\pi\)
0.441639 + 0.897193i \(0.354397\pi\)
\(60\) 0 0
\(61\) −97.3536 −0.204342 −0.102171 0.994767i \(-0.532579\pi\)
−0.102171 + 0.994767i \(0.532579\pi\)
\(62\) 173.886 0.356186
\(63\) 0 0
\(64\) 1810.30 3.53574
\(65\) −895.786 −1.70936
\(66\) 0 0
\(67\) 257.525 0.469577 0.234789 0.972046i \(-0.424560\pi\)
0.234789 + 0.972046i \(0.424560\pi\)
\(68\) −2119.55 −3.77989
\(69\) 0 0
\(70\) 0 0
\(71\) 253.601 0.423900 0.211950 0.977280i \(-0.432019\pi\)
0.211950 + 0.977280i \(0.432019\pi\)
\(72\) 0 0
\(73\) −1161.99 −1.86303 −0.931514 0.363706i \(-0.881511\pi\)
−0.931514 + 0.363706i \(0.881511\pi\)
\(74\) 1119.75 1.75904
\(75\) 0 0
\(76\) −2146.97 −3.24045
\(77\) 0 0
\(78\) 0 0
\(79\) −1070.16 −1.52408 −0.762039 0.647531i \(-0.775802\pi\)
−0.762039 + 0.647531i \(0.775802\pi\)
\(80\) 4515.73 6.31093
\(81\) 0 0
\(82\) −1277.55 −1.72051
\(83\) −889.495 −1.17632 −0.588161 0.808744i \(-0.700148\pi\)
−0.588161 + 0.808744i \(0.700148\pi\)
\(84\) 0 0
\(85\) −1888.94 −2.41040
\(86\) 1747.16 2.19070
\(87\) 0 0
\(88\) −837.543 −1.01457
\(89\) −647.303 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2998.98 3.39854
\(93\) 0 0
\(94\) 1704.36 1.87012
\(95\) −1913.38 −2.06641
\(96\) 0 0
\(97\) 673.379 0.704859 0.352429 0.935838i \(-0.385356\pi\)
0.352429 + 0.935838i \(0.385356\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5407.89 5.40789
\(101\) −243.254 −0.239650 −0.119825 0.992795i \(-0.538233\pi\)
−0.119825 + 0.992795i \(0.538233\pi\)
\(102\) 0 0
\(103\) 486.156 0.465072 0.232536 0.972588i \(-0.425298\pi\)
0.232536 + 0.972588i \(0.425298\pi\)
\(104\) 3459.04 3.26141
\(105\) 0 0
\(106\) −293.189 −0.268651
\(107\) −384.679 −0.347555 −0.173777 0.984785i \(-0.555597\pi\)
−0.173777 + 0.984785i \(0.555597\pi\)
\(108\) 0 0
\(109\) 994.781 0.874153 0.437077 0.899424i \(-0.356014\pi\)
0.437077 + 0.899424i \(0.356014\pi\)
\(110\) −1182.14 −1.02466
\(111\) 0 0
\(112\) 0 0
\(113\) 189.932 0.158118 0.0790589 0.996870i \(-0.474808\pi\)
0.0790589 + 0.996870i \(0.474808\pi\)
\(114\) 0 0
\(115\) 2672.70 2.16722
\(116\) 3909.50 3.12921
\(117\) 0 0
\(118\) −2181.66 −1.70202
\(119\) 0 0
\(120\) 0 0
\(121\) −1205.27 −0.905534
\(122\) 530.596 0.393754
\(123\) 0 0
\(124\) −692.476 −0.501502
\(125\) 2401.62 1.71846
\(126\) 0 0
\(127\) 1679.82 1.17370 0.586851 0.809695i \(-0.300367\pi\)
0.586851 + 0.809695i \(0.300367\pi\)
\(128\) −4467.88 −3.08522
\(129\) 0 0
\(130\) 4882.21 3.29383
\(131\) 237.457 0.158372 0.0791859 0.996860i \(-0.474768\pi\)
0.0791859 + 0.996860i \(0.474768\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1403.56 −0.904845
\(135\) 0 0
\(136\) 7294.07 4.59898
\(137\) −526.657 −0.328433 −0.164217 0.986424i \(-0.552510\pi\)
−0.164217 + 0.986424i \(0.552510\pi\)
\(138\) 0 0
\(139\) 580.607 0.354291 0.177145 0.984185i \(-0.443314\pi\)
0.177145 + 0.984185i \(0.443314\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1382.18 −0.816828
\(143\) −519.283 −0.303669
\(144\) 0 0
\(145\) 3484.15 1.99547
\(146\) 6333.09 3.58993
\(147\) 0 0
\(148\) −4459.26 −2.47668
\(149\) −2781.48 −1.52931 −0.764657 0.644438i \(-0.777091\pi\)
−0.764657 + 0.644438i \(0.777091\pi\)
\(150\) 0 0
\(151\) −1413.35 −0.761702 −0.380851 0.924636i \(-0.624369\pi\)
−0.380851 + 0.924636i \(0.624369\pi\)
\(152\) 7388.44 3.94264
\(153\) 0 0
\(154\) 0 0
\(155\) −617.135 −0.319803
\(156\) 0 0
\(157\) −2356.59 −1.19794 −0.598970 0.800772i \(-0.704423\pi\)
−0.598970 + 0.800772i \(0.704423\pi\)
\(158\) 5832.57 2.93680
\(159\) 0 0
\(160\) −13053.3 −6.44969
\(161\) 0 0
\(162\) 0 0
\(163\) 1309.51 0.629255 0.314627 0.949215i \(-0.398120\pi\)
0.314627 + 0.949215i \(0.398120\pi\)
\(164\) 5087.66 2.42244
\(165\) 0 0
\(166\) 4847.92 2.26670
\(167\) 1623.77 0.752401 0.376201 0.926538i \(-0.377230\pi\)
0.376201 + 0.926538i \(0.377230\pi\)
\(168\) 0 0
\(169\) −52.3668 −0.0238356
\(170\) 10295.1 4.64469
\(171\) 0 0
\(172\) −6957.80 −3.08446
\(173\) −2219.78 −0.975529 −0.487765 0.872975i \(-0.662188\pi\)
−0.487765 + 0.872975i \(0.662188\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2617.75 1.12114
\(177\) 0 0
\(178\) 3527.92 1.48556
\(179\) −1124.59 −0.469584 −0.234792 0.972046i \(-0.575441\pi\)
−0.234792 + 0.972046i \(0.575441\pi\)
\(180\) 0 0
\(181\) −3951.04 −1.62253 −0.811267 0.584676i \(-0.801222\pi\)
−0.811267 + 0.584676i \(0.801222\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −10320.5 −4.13499
\(185\) −3974.09 −1.57936
\(186\) 0 0
\(187\) −1095.01 −0.428209
\(188\) −6787.37 −2.63308
\(189\) 0 0
\(190\) 10428.3 3.98183
\(191\) −1177.40 −0.446039 −0.223019 0.974814i \(-0.571591\pi\)
−0.223019 + 0.974814i \(0.571591\pi\)
\(192\) 0 0
\(193\) −1772.08 −0.660917 −0.330458 0.943821i \(-0.607203\pi\)
−0.330458 + 0.943821i \(0.607203\pi\)
\(194\) −3670.05 −1.35822
\(195\) 0 0
\(196\) 0 0
\(197\) 794.274 0.287257 0.143629 0.989632i \(-0.454123\pi\)
0.143629 + 0.989632i \(0.454123\pi\)
\(198\) 0 0
\(199\) −3662.14 −1.30453 −0.652267 0.757990i \(-0.726182\pi\)
−0.652267 + 0.757990i \(0.726182\pi\)
\(200\) −18610.3 −6.57975
\(201\) 0 0
\(202\) 1325.78 0.461791
\(203\) 0 0
\(204\) 0 0
\(205\) 4534.12 1.54477
\(206\) −2649.64 −0.896162
\(207\) 0 0
\(208\) −10811.3 −3.60398
\(209\) −1109.18 −0.367098
\(210\) 0 0
\(211\) 2063.69 0.673318 0.336659 0.941627i \(-0.390703\pi\)
0.336659 + 0.941627i \(0.390703\pi\)
\(212\) 1167.58 0.378254
\(213\) 0 0
\(214\) 2096.58 0.669715
\(215\) −6200.79 −1.96693
\(216\) 0 0
\(217\) 0 0
\(218\) −5421.75 −1.68444
\(219\) 0 0
\(220\) 4707.69 1.44269
\(221\) 4522.38 1.37651
\(222\) 0 0
\(223\) 3728.28 1.11957 0.559784 0.828638i \(-0.310884\pi\)
0.559784 + 0.828638i \(0.310884\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1035.17 −0.304683
\(227\) −4558.22 −1.33277 −0.666387 0.745606i \(-0.732160\pi\)
−0.666387 + 0.745606i \(0.732160\pi\)
\(228\) 0 0
\(229\) −2068.22 −0.596819 −0.298410 0.954438i \(-0.596456\pi\)
−0.298410 + 0.954438i \(0.596456\pi\)
\(230\) −14566.7 −4.17609
\(231\) 0 0
\(232\) −13453.9 −3.80730
\(233\) −1459.56 −0.410383 −0.205191 0.978722i \(-0.565782\pi\)
−0.205191 + 0.978722i \(0.565782\pi\)
\(234\) 0 0
\(235\) −6048.91 −1.67909
\(236\) 8688.16 2.39640
\(237\) 0 0
\(238\) 0 0
\(239\) 2220.74 0.601035 0.300518 0.953776i \(-0.402841\pi\)
0.300518 + 0.953776i \(0.402841\pi\)
\(240\) 0 0
\(241\) 5064.37 1.35363 0.676815 0.736153i \(-0.263360\pi\)
0.676815 + 0.736153i \(0.263360\pi\)
\(242\) 6568.93 1.74490
\(243\) 0 0
\(244\) −2113.02 −0.554395
\(245\) 0 0
\(246\) 0 0
\(247\) 4580.89 1.18006
\(248\) 2383.04 0.610175
\(249\) 0 0
\(250\) −13089.3 −3.31135
\(251\) −1797.86 −0.452112 −0.226056 0.974114i \(-0.572583\pi\)
−0.226056 + 0.974114i \(0.572583\pi\)
\(252\) 0 0
\(253\) 1549.35 0.385007
\(254\) −9155.37 −2.26165
\(255\) 0 0
\(256\) 9868.42 2.40928
\(257\) −2734.28 −0.663656 −0.331828 0.943340i \(-0.607665\pi\)
−0.331828 + 0.943340i \(0.607665\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −19442.7 −4.63763
\(261\) 0 0
\(262\) −1294.19 −0.305172
\(263\) −8197.71 −1.92203 −0.961013 0.276504i \(-0.910824\pi\)
−0.961013 + 0.276504i \(0.910824\pi\)
\(264\) 0 0
\(265\) 1040.55 0.241209
\(266\) 0 0
\(267\) 0 0
\(268\) 5589.48 1.27400
\(269\) 237.512 0.0538342 0.0269171 0.999638i \(-0.491431\pi\)
0.0269171 + 0.999638i \(0.491431\pi\)
\(270\) 0 0
\(271\) 4349.95 0.975057 0.487528 0.873107i \(-0.337899\pi\)
0.487528 + 0.873107i \(0.337899\pi\)
\(272\) −22797.7 −5.08204
\(273\) 0 0
\(274\) 2870.38 0.632869
\(275\) 2793.85 0.612638
\(276\) 0 0
\(277\) 6228.51 1.35103 0.675515 0.737347i \(-0.263921\pi\)
0.675515 + 0.737347i \(0.263921\pi\)
\(278\) −3164.42 −0.682695
\(279\) 0 0
\(280\) 0 0
\(281\) 3360.95 0.713515 0.356757 0.934197i \(-0.383882\pi\)
0.356757 + 0.934197i \(0.383882\pi\)
\(282\) 0 0
\(283\) −8009.37 −1.68236 −0.841180 0.540755i \(-0.818138\pi\)
−0.841180 + 0.540755i \(0.818138\pi\)
\(284\) 5504.32 1.15007
\(285\) 0 0
\(286\) 2830.19 0.585150
\(287\) 0 0
\(288\) 0 0
\(289\) 4623.33 0.941041
\(290\) −18989.3 −3.84514
\(291\) 0 0
\(292\) −25220.6 −5.05454
\(293\) 9247.12 1.84376 0.921881 0.387472i \(-0.126652\pi\)
0.921881 + 0.387472i \(0.126652\pi\)
\(294\) 0 0
\(295\) 7742.89 1.52816
\(296\) 15345.8 3.01336
\(297\) 0 0
\(298\) 15159.6 2.94689
\(299\) −6398.80 −1.23763
\(300\) 0 0
\(301\) 0 0
\(302\) 7703.04 1.46775
\(303\) 0 0
\(304\) −23092.7 −4.35676
\(305\) −1883.13 −0.353533
\(306\) 0 0
\(307\) −3367.11 −0.625965 −0.312983 0.949759i \(-0.601328\pi\)
−0.312983 + 0.949759i \(0.601328\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3363.51 0.616240
\(311\) −6555.92 −1.19534 −0.597672 0.801740i \(-0.703908\pi\)
−0.597672 + 0.801740i \(0.703908\pi\)
\(312\) 0 0
\(313\) 616.028 0.111246 0.0556229 0.998452i \(-0.482286\pi\)
0.0556229 + 0.998452i \(0.482286\pi\)
\(314\) 12843.9 2.30835
\(315\) 0 0
\(316\) −23227.4 −4.13494
\(317\) −3481.49 −0.616845 −0.308422 0.951250i \(-0.599801\pi\)
−0.308422 + 0.951250i \(0.599801\pi\)
\(318\) 0 0
\(319\) 2019.75 0.354496
\(320\) 35016.9 6.11720
\(321\) 0 0
\(322\) 0 0
\(323\) 9659.71 1.66403
\(324\) 0 0
\(325\) −11538.6 −1.96937
\(326\) −7137.07 −1.21253
\(327\) 0 0
\(328\) −17508.3 −2.94737
\(329\) 0 0
\(330\) 0 0
\(331\) −7187.44 −1.19353 −0.596763 0.802417i \(-0.703547\pi\)
−0.596763 + 0.802417i \(0.703547\pi\)
\(332\) −19306.2 −3.19145
\(333\) 0 0
\(334\) −8849.86 −1.44983
\(335\) 4981.35 0.812418
\(336\) 0 0
\(337\) 3575.89 0.578015 0.289007 0.957327i \(-0.406675\pi\)
0.289007 + 0.957327i \(0.406675\pi\)
\(338\) 285.410 0.0459297
\(339\) 0 0
\(340\) −40998.8 −6.53962
\(341\) −357.751 −0.0568131
\(342\) 0 0
\(343\) 0 0
\(344\) 23944.1 3.75285
\(345\) 0 0
\(346\) 12098.2 1.87978
\(347\) 574.458 0.0888719 0.0444359 0.999012i \(-0.485851\pi\)
0.0444359 + 0.999012i \(0.485851\pi\)
\(348\) 0 0
\(349\) 11491.0 1.76246 0.881229 0.472689i \(-0.156717\pi\)
0.881229 + 0.472689i \(0.156717\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −7566.92 −1.14579
\(353\) −1369.32 −0.206464 −0.103232 0.994657i \(-0.532918\pi\)
−0.103232 + 0.994657i \(0.532918\pi\)
\(354\) 0 0
\(355\) 4905.45 0.733392
\(356\) −14049.5 −2.09163
\(357\) 0 0
\(358\) 6129.21 0.904857
\(359\) 12915.0 1.89868 0.949339 0.314254i \(-0.101754\pi\)
0.949339 + 0.314254i \(0.101754\pi\)
\(360\) 0 0
\(361\) 2925.68 0.426546
\(362\) 21533.9 3.12652
\(363\) 0 0
\(364\) 0 0
\(365\) −22476.6 −3.22323
\(366\) 0 0
\(367\) −1186.30 −0.168731 −0.0843653 0.996435i \(-0.526886\pi\)
−0.0843653 + 0.996435i \(0.526886\pi\)
\(368\) 32256.9 4.56931
\(369\) 0 0
\(370\) 21659.6 3.04332
\(371\) 0 0
\(372\) 0 0
\(373\) −6462.68 −0.897117 −0.448559 0.893753i \(-0.648062\pi\)
−0.448559 + 0.893753i \(0.648062\pi\)
\(374\) 5968.02 0.825131
\(375\) 0 0
\(376\) 23357.6 3.20366
\(377\) −8341.54 −1.13955
\(378\) 0 0
\(379\) −8555.31 −1.15952 −0.579758 0.814789i \(-0.696853\pi\)
−0.579758 + 0.814789i \(0.696853\pi\)
\(380\) −41529.2 −5.60632
\(381\) 0 0
\(382\) 6417.04 0.859488
\(383\) 570.982 0.0761770 0.0380885 0.999274i \(-0.487873\pi\)
0.0380885 + 0.999274i \(0.487873\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 9658.17 1.27354
\(387\) 0 0
\(388\) 14615.4 1.91234
\(389\) 14437.2 1.88174 0.940869 0.338771i \(-0.110011\pi\)
0.940869 + 0.338771i \(0.110011\pi\)
\(390\) 0 0
\(391\) −13493.1 −1.74521
\(392\) 0 0
\(393\) 0 0
\(394\) −4328.95 −0.553526
\(395\) −20700.3 −2.63682
\(396\) 0 0
\(397\) −9384.81 −1.18642 −0.593212 0.805047i \(-0.702140\pi\)
−0.593212 + 0.805047i \(0.702140\pi\)
\(398\) 19959.4 2.51375
\(399\) 0 0
\(400\) 58166.9 7.27086
\(401\) −8167.50 −1.01712 −0.508560 0.861026i \(-0.669822\pi\)
−0.508560 + 0.861026i \(0.669822\pi\)
\(402\) 0 0
\(403\) 1477.51 0.182630
\(404\) −5279.74 −0.650190
\(405\) 0 0
\(406\) 0 0
\(407\) −2303.76 −0.280573
\(408\) 0 0
\(409\) 11655.7 1.40913 0.704567 0.709637i \(-0.251141\pi\)
0.704567 + 0.709637i \(0.251141\pi\)
\(410\) −24711.9 −2.97666
\(411\) 0 0
\(412\) 10551.8 1.26178
\(413\) 0 0
\(414\) 0 0
\(415\) −17205.7 −2.03516
\(416\) 31251.3 3.68322
\(417\) 0 0
\(418\) 6045.23 0.707373
\(419\) 10257.6 1.19598 0.597992 0.801502i \(-0.295965\pi\)
0.597992 + 0.801502i \(0.295965\pi\)
\(420\) 0 0
\(421\) −8160.70 −0.944723 −0.472361 0.881405i \(-0.656598\pi\)
−0.472361 + 0.881405i \(0.656598\pi\)
\(422\) −11247.5 −1.29744
\(423\) 0 0
\(424\) −4018.04 −0.460220
\(425\) −24331.3 −2.77704
\(426\) 0 0
\(427\) 0 0
\(428\) −8349.32 −0.942943
\(429\) 0 0
\(430\) 33795.5 3.79015
\(431\) 8722.62 0.974835 0.487417 0.873169i \(-0.337939\pi\)
0.487417 + 0.873169i \(0.337939\pi\)
\(432\) 0 0
\(433\) −12628.4 −1.40157 −0.700787 0.713371i \(-0.747168\pi\)
−0.700787 + 0.713371i \(0.747168\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 21591.3 2.37165
\(437\) −13667.7 −1.49614
\(438\) 0 0
\(439\) 3487.47 0.379153 0.189576 0.981866i \(-0.439289\pi\)
0.189576 + 0.981866i \(0.439289\pi\)
\(440\) −16200.7 −1.75532
\(441\) 0 0
\(442\) −24647.9 −2.65244
\(443\) 14138.0 1.51629 0.758144 0.652087i \(-0.226106\pi\)
0.758144 + 0.652087i \(0.226106\pi\)
\(444\) 0 0
\(445\) −12520.9 −1.33381
\(446\) −20319.8 −2.15734
\(447\) 0 0
\(448\) 0 0
\(449\) −14007.0 −1.47223 −0.736114 0.676858i \(-0.763341\pi\)
−0.736114 + 0.676858i \(0.763341\pi\)
\(450\) 0 0
\(451\) 2628.41 0.274428
\(452\) 4122.40 0.428986
\(453\) 0 0
\(454\) 24843.2 2.56817
\(455\) 0 0
\(456\) 0 0
\(457\) 4837.29 0.495140 0.247570 0.968870i \(-0.420368\pi\)
0.247570 + 0.968870i \(0.420368\pi\)
\(458\) 11272.2 1.15003
\(459\) 0 0
\(460\) 58009.9 5.87983
\(461\) −10034.5 −1.01378 −0.506890 0.862011i \(-0.669205\pi\)
−0.506890 + 0.862011i \(0.669205\pi\)
\(462\) 0 0
\(463\) −15501.9 −1.55602 −0.778008 0.628254i \(-0.783770\pi\)
−0.778008 + 0.628254i \(0.783770\pi\)
\(464\) 42050.4 4.20720
\(465\) 0 0
\(466\) 7954.91 0.790781
\(467\) −9616.13 −0.952851 −0.476426 0.879215i \(-0.658068\pi\)
−0.476426 + 0.879215i \(0.658068\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 32967.7 3.23551
\(471\) 0 0
\(472\) −29898.8 −2.91569
\(473\) −3594.57 −0.349426
\(474\) 0 0
\(475\) −24646.1 −2.38072
\(476\) 0 0
\(477\) 0 0
\(478\) −12103.4 −1.15816
\(479\) 14159.7 1.35067 0.675337 0.737509i \(-0.263998\pi\)
0.675337 + 0.737509i \(0.263998\pi\)
\(480\) 0 0
\(481\) 9514.51 0.901922
\(482\) −27601.8 −2.60836
\(483\) 0 0
\(484\) −26159.8 −2.45678
\(485\) 13025.3 1.21948
\(486\) 0 0
\(487\) 15584.6 1.45012 0.725058 0.688688i \(-0.241813\pi\)
0.725058 + 0.688688i \(0.241813\pi\)
\(488\) 7271.62 0.674530
\(489\) 0 0
\(490\) 0 0
\(491\) 12898.0 1.18549 0.592747 0.805389i \(-0.298043\pi\)
0.592747 + 0.805389i \(0.298043\pi\)
\(492\) 0 0
\(493\) −17589.8 −1.60690
\(494\) −24966.7 −2.27390
\(495\) 0 0
\(496\) −7448.23 −0.674265
\(497\) 0 0
\(498\) 0 0
\(499\) 6707.11 0.601707 0.300853 0.953670i \(-0.402729\pi\)
0.300853 + 0.953670i \(0.402729\pi\)
\(500\) 52126.1 4.66230
\(501\) 0 0
\(502\) 9798.70 0.871190
\(503\) 14186.3 1.25753 0.628763 0.777597i \(-0.283562\pi\)
0.628763 + 0.777597i \(0.283562\pi\)
\(504\) 0 0
\(505\) −4705.31 −0.414620
\(506\) −8444.26 −0.741884
\(507\) 0 0
\(508\) 36459.9 3.18435
\(509\) −12086.8 −1.05253 −0.526263 0.850322i \(-0.676407\pi\)
−0.526263 + 0.850322i \(0.676407\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −18041.8 −1.55731
\(513\) 0 0
\(514\) 14902.3 1.27882
\(515\) 9403.80 0.804623
\(516\) 0 0
\(517\) −3506.53 −0.298292
\(518\) 0 0
\(519\) 0 0
\(520\) 66908.8 5.64259
\(521\) 12142.6 1.02107 0.510534 0.859857i \(-0.329448\pi\)
0.510534 + 0.859857i \(0.329448\pi\)
\(522\) 0 0
\(523\) 3408.38 0.284967 0.142484 0.989797i \(-0.454491\pi\)
0.142484 + 0.989797i \(0.454491\pi\)
\(524\) 5153.91 0.429675
\(525\) 0 0
\(526\) 44679.1 3.70362
\(527\) 3115.61 0.257530
\(528\) 0 0
\(529\) 6924.66 0.569135
\(530\) −5671.20 −0.464795
\(531\) 0 0
\(532\) 0 0
\(533\) −10855.3 −0.882168
\(534\) 0 0
\(535\) −7440.92 −0.601306
\(536\) −19235.3 −1.55007
\(537\) 0 0
\(538\) −1294.49 −0.103735
\(539\) 0 0
\(540\) 0 0
\(541\) −16737.5 −1.33013 −0.665066 0.746785i \(-0.731597\pi\)
−0.665066 + 0.746785i \(0.731597\pi\)
\(542\) −23708.1 −1.87887
\(543\) 0 0
\(544\) 65899.5 5.19378
\(545\) 19242.2 1.51238
\(546\) 0 0
\(547\) −13940.6 −1.08968 −0.544840 0.838540i \(-0.683410\pi\)
−0.544840 + 0.838540i \(0.683410\pi\)
\(548\) −11430.9 −0.891065
\(549\) 0 0
\(550\) −15227.0 −1.18051
\(551\) −17817.3 −1.37758
\(552\) 0 0
\(553\) 0 0
\(554\) −33946.6 −2.60335
\(555\) 0 0
\(556\) 12601.8 0.961218
\(557\) −818.376 −0.0622544 −0.0311272 0.999515i \(-0.509910\pi\)
−0.0311272 + 0.999515i \(0.509910\pi\)
\(558\) 0 0
\(559\) 14845.5 1.12325
\(560\) 0 0
\(561\) 0 0
\(562\) −18317.9 −1.37490
\(563\) −8957.98 −0.670575 −0.335288 0.942116i \(-0.608834\pi\)
−0.335288 + 0.942116i \(0.608834\pi\)
\(564\) 0 0
\(565\) 3673.89 0.273560
\(566\) 43652.6 3.24180
\(567\) 0 0
\(568\) −18942.2 −1.39929
\(569\) −19997.8 −1.47338 −0.736688 0.676233i \(-0.763611\pi\)
−0.736688 + 0.676233i \(0.763611\pi\)
\(570\) 0 0
\(571\) −3391.26 −0.248546 −0.124273 0.992248i \(-0.539660\pi\)
−0.124273 + 0.992248i \(0.539660\pi\)
\(572\) −11270.8 −0.823877
\(573\) 0 0
\(574\) 0 0
\(575\) 34426.8 2.49687
\(576\) 0 0
\(577\) 3957.72 0.285549 0.142775 0.989755i \(-0.454398\pi\)
0.142775 + 0.989755i \(0.454398\pi\)
\(578\) −25198.1 −1.81332
\(579\) 0 0
\(580\) 75622.2 5.41387
\(581\) 0 0
\(582\) 0 0
\(583\) 603.202 0.0428509
\(584\) 86792.6 6.14983
\(585\) 0 0
\(586\) −50398.6 −3.55281
\(587\) −12647.3 −0.889283 −0.444642 0.895709i \(-0.646669\pi\)
−0.444642 + 0.895709i \(0.646669\pi\)
\(588\) 0 0
\(589\) 3155.92 0.220777
\(590\) −42200.3 −2.94467
\(591\) 0 0
\(592\) −47963.5 −3.32988
\(593\) −3910.60 −0.270808 −0.135404 0.990790i \(-0.543233\pi\)
−0.135404 + 0.990790i \(0.543233\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −60371.0 −4.14915
\(597\) 0 0
\(598\) 34874.7 2.38484
\(599\) 13933.9 0.950455 0.475228 0.879863i \(-0.342366\pi\)
0.475228 + 0.879863i \(0.342366\pi\)
\(600\) 0 0
\(601\) −16095.1 −1.09240 −0.546202 0.837654i \(-0.683927\pi\)
−0.546202 + 0.837654i \(0.683927\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −30676.3 −2.06656
\(605\) −23313.7 −1.56667
\(606\) 0 0
\(607\) 3050.91 0.204008 0.102004 0.994784i \(-0.467475\pi\)
0.102004 + 0.994784i \(0.467475\pi\)
\(608\) 66752.0 4.45255
\(609\) 0 0
\(610\) 10263.4 0.681235
\(611\) 14481.9 0.958879
\(612\) 0 0
\(613\) −1512.24 −0.0996394 −0.0498197 0.998758i \(-0.515865\pi\)
−0.0498197 + 0.998758i \(0.515865\pi\)
\(614\) 18351.4 1.20619
\(615\) 0 0
\(616\) 0 0
\(617\) −23466.1 −1.53114 −0.765569 0.643354i \(-0.777542\pi\)
−0.765569 + 0.643354i \(0.777542\pi\)
\(618\) 0 0
\(619\) −20736.1 −1.34645 −0.673226 0.739437i \(-0.735092\pi\)
−0.673226 + 0.739437i \(0.735092\pi\)
\(620\) −13394.7 −0.867651
\(621\) 0 0
\(622\) 35731.1 2.30335
\(623\) 0 0
\(624\) 0 0
\(625\) 15310.1 0.979844
\(626\) −3357.47 −0.214364
\(627\) 0 0
\(628\) −51148.9 −3.25010
\(629\) 20063.2 1.27182
\(630\) 0 0
\(631\) 15623.5 0.985673 0.492837 0.870122i \(-0.335960\pi\)
0.492837 + 0.870122i \(0.335960\pi\)
\(632\) 79933.2 5.03097
\(633\) 0 0
\(634\) 18974.8 1.18862
\(635\) 32493.1 2.03063
\(636\) 0 0
\(637\) 0 0
\(638\) −11008.0 −0.683091
\(639\) 0 0
\(640\) −86423.0 −5.33776
\(641\) −12284.6 −0.756961 −0.378481 0.925609i \(-0.623553\pi\)
−0.378481 + 0.925609i \(0.623553\pi\)
\(642\) 0 0
\(643\) 924.041 0.0566728 0.0283364 0.999598i \(-0.490979\pi\)
0.0283364 + 0.999598i \(0.490979\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −52647.3 −3.20647
\(647\) −10330.2 −0.627701 −0.313850 0.949472i \(-0.601619\pi\)
−0.313850 + 0.949472i \(0.601619\pi\)
\(648\) 0 0
\(649\) 4488.52 0.271479
\(650\) 62887.4 3.79484
\(651\) 0 0
\(652\) 28422.4 1.70722
\(653\) −3064.12 −0.183627 −0.0918134 0.995776i \(-0.529266\pi\)
−0.0918134 + 0.995776i \(0.529266\pi\)
\(654\) 0 0
\(655\) 4593.17 0.274000
\(656\) 54722.6 3.25695
\(657\) 0 0
\(658\) 0 0
\(659\) 9157.78 0.541330 0.270665 0.962674i \(-0.412756\pi\)
0.270665 + 0.962674i \(0.412756\pi\)
\(660\) 0 0
\(661\) −23367.4 −1.37502 −0.687508 0.726177i \(-0.741295\pi\)
−0.687508 + 0.726177i \(0.741295\pi\)
\(662\) 39172.9 2.29985
\(663\) 0 0
\(664\) 66438.9 3.88303
\(665\) 0 0
\(666\) 0 0
\(667\) 24888.1 1.44478
\(668\) 35243.3 2.04132
\(669\) 0 0
\(670\) −27149.3 −1.56548
\(671\) −1091.64 −0.0628053
\(672\) 0 0
\(673\) 34214.8 1.95971 0.979853 0.199719i \(-0.0640029\pi\)
0.979853 + 0.199719i \(0.0640029\pi\)
\(674\) −19489.3 −1.11380
\(675\) 0 0
\(676\) −1136.60 −0.0646679
\(677\) 19288.8 1.09502 0.547511 0.836799i \(-0.315576\pi\)
0.547511 + 0.836799i \(0.315576\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 141090. 7.95672
\(681\) 0 0
\(682\) 1949.81 0.109475
\(683\) −31119.7 −1.74343 −0.871714 0.490015i \(-0.836991\pi\)
−0.871714 + 0.490015i \(0.836991\pi\)
\(684\) 0 0
\(685\) −10187.2 −0.568224
\(686\) 0 0
\(687\) 0 0
\(688\) −74837.7 −4.14703
\(689\) −2491.22 −0.137747
\(690\) 0 0
\(691\) −12732.5 −0.700967 −0.350483 0.936569i \(-0.613983\pi\)
−0.350483 + 0.936569i \(0.613983\pi\)
\(692\) −48179.4 −2.64669
\(693\) 0 0
\(694\) −3130.91 −0.171250
\(695\) 11230.8 0.612961
\(696\) 0 0
\(697\) −22890.6 −1.24396
\(698\) −62628.1 −3.39614
\(699\) 0 0
\(700\) 0 0
\(701\) 9506.11 0.512184 0.256092 0.966652i \(-0.417565\pi\)
0.256092 + 0.966652i \(0.417565\pi\)
\(702\) 0 0
\(703\) 20322.8 1.09031
\(704\) 20299.2 1.08672
\(705\) 0 0
\(706\) 7463.08 0.397842
\(707\) 0 0
\(708\) 0 0
\(709\) 2500.48 0.132450 0.0662252 0.997805i \(-0.478904\pi\)
0.0662252 + 0.997805i \(0.478904\pi\)
\(710\) −26735.7 −1.41320
\(711\) 0 0
\(712\) 48348.9 2.54487
\(713\) −4408.33 −0.231548
\(714\) 0 0
\(715\) −10044.6 −0.525379
\(716\) −24408.7 −1.27402
\(717\) 0 0
\(718\) −70389.0 −3.65863
\(719\) −19519.3 −1.01245 −0.506223 0.862403i \(-0.668959\pi\)
−0.506223 + 0.862403i \(0.668959\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −15945.5 −0.821927
\(723\) 0 0
\(724\) −85755.9 −4.40206
\(725\) 44879.2 2.29899
\(726\) 0 0
\(727\) −35008.3 −1.78595 −0.892975 0.450105i \(-0.851386\pi\)
−0.892975 + 0.450105i \(0.851386\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 122502. 6.21096
\(731\) 31304.7 1.58392
\(732\) 0 0
\(733\) −33284.0 −1.67718 −0.838590 0.544763i \(-0.816619\pi\)
−0.838590 + 0.544763i \(0.816619\pi\)
\(734\) 6465.54 0.325133
\(735\) 0 0
\(736\) −93242.3 −4.66978
\(737\) 2887.67 0.144326
\(738\) 0 0
\(739\) −25554.2 −1.27203 −0.636013 0.771678i \(-0.719418\pi\)
−0.636013 + 0.771678i \(0.719418\pi\)
\(740\) −86256.1 −4.28492
\(741\) 0 0
\(742\) 0 0
\(743\) −3926.32 −0.193867 −0.0969333 0.995291i \(-0.530903\pi\)
−0.0969333 + 0.995291i \(0.530903\pi\)
\(744\) 0 0
\(745\) −53802.6 −2.64587
\(746\) 35222.8 1.72869
\(747\) 0 0
\(748\) −23766.8 −1.16177
\(749\) 0 0
\(750\) 0 0
\(751\) 2496.80 0.121318 0.0606589 0.998159i \(-0.480680\pi\)
0.0606589 + 0.998159i \(0.480680\pi\)
\(752\) −73004.6 −3.54016
\(753\) 0 0
\(754\) 45463.0 2.19584
\(755\) −27338.7 −1.31782
\(756\) 0 0
\(757\) 9014.06 0.432789 0.216395 0.976306i \(-0.430570\pi\)
0.216395 + 0.976306i \(0.430570\pi\)
\(758\) 46628.1 2.23431
\(759\) 0 0
\(760\) 142916. 6.82118
\(761\) 2393.59 0.114018 0.0570089 0.998374i \(-0.481844\pi\)
0.0570089 + 0.998374i \(0.481844\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −25554.9 −1.21014
\(765\) 0 0
\(766\) −3111.96 −0.146788
\(767\) −18537.5 −0.872688
\(768\) 0 0
\(769\) −9474.06 −0.444270 −0.222135 0.975016i \(-0.571303\pi\)
−0.222135 + 0.975016i \(0.571303\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −38462.3 −1.79312
\(773\) 12878.4 0.599231 0.299615 0.954060i \(-0.403142\pi\)
0.299615 + 0.954060i \(0.403142\pi\)
\(774\) 0 0
\(775\) −7949.28 −0.368447
\(776\) −50296.6 −2.32673
\(777\) 0 0
\(778\) −78685.6 −3.62599
\(779\) −23186.7 −1.06643
\(780\) 0 0
\(781\) 2843.67 0.130287
\(782\) 73540.2 3.36290
\(783\) 0 0
\(784\) 0 0
\(785\) −45584.0 −2.07256
\(786\) 0 0
\(787\) 16966.5 0.768477 0.384239 0.923234i \(-0.374464\pi\)
0.384239 + 0.923234i \(0.374464\pi\)
\(788\) 17239.4 0.779351
\(789\) 0 0
\(790\) 112820. 5.08097
\(791\) 0 0
\(792\) 0 0
\(793\) 4508.46 0.201892
\(794\) 51149.0 2.28616
\(795\) 0 0
\(796\) −79485.3 −3.53930
\(797\) 21.6935 0.000964143 0 0.000482072 1.00000i \(-0.499847\pi\)
0.000482072 1.00000i \(0.499847\pi\)
\(798\) 0 0
\(799\) 30538.0 1.35213
\(800\) −168138. −7.43073
\(801\) 0 0
\(802\) 44514.5 1.95993
\(803\) −13029.6 −0.572609
\(804\) 0 0
\(805\) 0 0
\(806\) −8052.69 −0.351916
\(807\) 0 0
\(808\) 18169.3 0.791083
\(809\) −31766.3 −1.38052 −0.690262 0.723560i \(-0.742505\pi\)
−0.690262 + 0.723560i \(0.742505\pi\)
\(810\) 0 0
\(811\) 31081.1 1.34575 0.672875 0.739756i \(-0.265059\pi\)
0.672875 + 0.739756i \(0.265059\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 12556.0 0.540646
\(815\) 25330.0 1.08868
\(816\) 0 0
\(817\) 31709.8 1.35788
\(818\) −63525.7 −2.71531
\(819\) 0 0
\(820\) 98411.5 4.19107
\(821\) −5021.74 −0.213472 −0.106736 0.994287i \(-0.534040\pi\)
−0.106736 + 0.994287i \(0.534040\pi\)
\(822\) 0 0
\(823\) 13443.5 0.569394 0.284697 0.958617i \(-0.408107\pi\)
0.284697 + 0.958617i \(0.408107\pi\)
\(824\) −36312.4 −1.53520
\(825\) 0 0
\(826\) 0 0
\(827\) 37439.2 1.57423 0.787116 0.616805i \(-0.211573\pi\)
0.787116 + 0.616805i \(0.211573\pi\)
\(828\) 0 0
\(829\) −3890.21 −0.162982 −0.0814912 0.996674i \(-0.525968\pi\)
−0.0814912 + 0.996674i \(0.525968\pi\)
\(830\) 93774.2 3.92162
\(831\) 0 0
\(832\) −83835.3 −3.49335
\(833\) 0 0
\(834\) 0 0
\(835\) 31408.9 1.30173
\(836\) −24074.3 −0.995965
\(837\) 0 0
\(838\) −55906.0 −2.30458
\(839\) 15062.0 0.619783 0.309891 0.950772i \(-0.399707\pi\)
0.309891 + 0.950772i \(0.399707\pi\)
\(840\) 0 0
\(841\) 8055.35 0.330286
\(842\) 44477.4 1.82042
\(843\) 0 0
\(844\) 44791.5 1.82676
\(845\) −1012.94 −0.0412381
\(846\) 0 0
\(847\) 0 0
\(848\) 12558.4 0.508560
\(849\) 0 0
\(850\) 132611. 5.35118
\(851\) −28387.8 −1.14350
\(852\) 0 0
\(853\) −39207.9 −1.57380 −0.786900 0.617081i \(-0.788315\pi\)
−0.786900 + 0.617081i \(0.788315\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 28732.8 1.14727
\(857\) −37029.6 −1.47597 −0.737986 0.674816i \(-0.764223\pi\)
−0.737986 + 0.674816i \(0.764223\pi\)
\(858\) 0 0
\(859\) 37646.3 1.49531 0.747657 0.664085i \(-0.231179\pi\)
0.747657 + 0.664085i \(0.231179\pi\)
\(860\) −134586. −5.33644
\(861\) 0 0
\(862\) −47540.0 −1.87844
\(863\) 26698.1 1.05309 0.526543 0.850149i \(-0.323488\pi\)
0.526543 + 0.850149i \(0.323488\pi\)
\(864\) 0 0
\(865\) −42937.5 −1.68777
\(866\) 68827.2 2.70074
\(867\) 0 0
\(868\) 0 0
\(869\) −11999.8 −0.468431
\(870\) 0 0
\(871\) −11926.0 −0.463947
\(872\) −74303.0 −2.88557
\(873\) 0 0
\(874\) 74491.6 2.88297
\(875\) 0 0
\(876\) 0 0
\(877\) −22177.5 −0.853912 −0.426956 0.904272i \(-0.640414\pi\)
−0.426956 + 0.904272i \(0.640414\pi\)
\(878\) −19007.4 −0.730602
\(879\) 0 0
\(880\) 50635.6 1.93969
\(881\) −11503.1 −0.439898 −0.219949 0.975511i \(-0.570589\pi\)
−0.219949 + 0.975511i \(0.570589\pi\)
\(882\) 0 0
\(883\) 41751.6 1.59123 0.795613 0.605806i \(-0.207149\pi\)
0.795613 + 0.605806i \(0.207149\pi\)
\(884\) 98156.6 3.73457
\(885\) 0 0
\(886\) −77054.7 −2.92179
\(887\) −4670.18 −0.176786 −0.0883930 0.996086i \(-0.528173\pi\)
−0.0883930 + 0.996086i \(0.528173\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 68241.3 2.57017
\(891\) 0 0
\(892\) 80920.8 3.03748
\(893\) 30933.0 1.15917
\(894\) 0 0
\(895\) −21753.1 −0.812430
\(896\) 0 0
\(897\) 0 0
\(898\) 76340.8 2.83689
\(899\) −5746.75 −0.213198
\(900\) 0 0
\(901\) −5253.22 −0.194240
\(902\) −14325.4 −0.528805
\(903\) 0 0
\(904\) −14186.6 −0.521945
\(905\) −76425.7 −2.80715
\(906\) 0 0
\(907\) 33783.8 1.23679 0.618397 0.785866i \(-0.287782\pi\)
0.618397 + 0.785866i \(0.287782\pi\)
\(908\) −98934.5 −3.61592
\(909\) 0 0
\(910\) 0 0
\(911\) 34880.5 1.26854 0.634271 0.773111i \(-0.281300\pi\)
0.634271 + 0.773111i \(0.281300\pi\)
\(912\) 0 0
\(913\) −9974.04 −0.361547
\(914\) −26364.2 −0.954102
\(915\) 0 0
\(916\) −44889.9 −1.61922
\(917\) 0 0
\(918\) 0 0
\(919\) −5964.51 −0.214092 −0.107046 0.994254i \(-0.534139\pi\)
−0.107046 + 0.994254i \(0.534139\pi\)
\(920\) −199631. −7.15397
\(921\) 0 0
\(922\) 54689.9 1.95349
\(923\) −11744.3 −0.418818
\(924\) 0 0
\(925\) −51190.0 −1.81959
\(926\) 84488.5 2.99834
\(927\) 0 0
\(928\) −121552. −4.29971
\(929\) 54065.1 1.90939 0.954693 0.297594i \(-0.0961841\pi\)
0.954693 + 0.297594i \(0.0961841\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −31679.3 −1.11340
\(933\) 0 0
\(934\) 52409.8 1.83608
\(935\) −21181.0 −0.740847
\(936\) 0 0
\(937\) −14678.5 −0.511766 −0.255883 0.966708i \(-0.582366\pi\)
−0.255883 + 0.966708i \(0.582366\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −131289. −4.55551
\(941\) 22198.8 0.769032 0.384516 0.923118i \(-0.374368\pi\)
0.384516 + 0.923118i \(0.374368\pi\)
\(942\) 0 0
\(943\) 32388.3 1.11846
\(944\) 93449.3 3.22194
\(945\) 0 0
\(946\) 19591.1 0.673322
\(947\) −3314.30 −0.113728 −0.0568639 0.998382i \(-0.518110\pi\)
−0.0568639 + 0.998382i \(0.518110\pi\)
\(948\) 0 0
\(949\) 53812.1 1.84069
\(950\) 134326. 4.58749
\(951\) 0 0
\(952\) 0 0
\(953\) 45137.3 1.53425 0.767125 0.641498i \(-0.221687\pi\)
0.767125 + 0.641498i \(0.221687\pi\)
\(954\) 0 0
\(955\) −22774.6 −0.771694
\(956\) 48200.2 1.63066
\(957\) 0 0
\(958\) −77173.1 −2.60266
\(959\) 0 0
\(960\) 0 0
\(961\) −28773.1 −0.965832
\(962\) −51856.0 −1.73794
\(963\) 0 0
\(964\) 109920. 3.67250
\(965\) −34277.6 −1.14346
\(966\) 0 0
\(967\) 13423.9 0.446416 0.223208 0.974771i \(-0.428347\pi\)
0.223208 + 0.974771i \(0.428347\pi\)
\(968\) 90024.7 2.98916
\(969\) 0 0
\(970\) −70990.4 −2.34986
\(971\) 50282.5 1.66183 0.830917 0.556396i \(-0.187816\pi\)
0.830917 + 0.556396i \(0.187816\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −84939.2 −2.79428
\(975\) 0 0
\(976\) −22727.6 −0.745381
\(977\) −24918.5 −0.815980 −0.407990 0.912986i \(-0.633770\pi\)
−0.407990 + 0.912986i \(0.633770\pi\)
\(978\) 0 0
\(979\) −7258.30 −0.236952
\(980\) 0 0
\(981\) 0 0
\(982\) −70296.5 −2.28437
\(983\) −5830.48 −0.189179 −0.0945897 0.995516i \(-0.530154\pi\)
−0.0945897 + 0.995516i \(0.530154\pi\)
\(984\) 0 0
\(985\) 15363.8 0.496986
\(986\) 95867.7 3.09640
\(987\) 0 0
\(988\) 99426.5 3.20160
\(989\) −44293.6 −1.42412
\(990\) 0 0
\(991\) 15669.9 0.502293 0.251146 0.967949i \(-0.419192\pi\)
0.251146 + 0.967949i \(0.419192\pi\)
\(992\) 21530.0 0.689091
\(993\) 0 0
\(994\) 0 0
\(995\) −70837.4 −2.25698
\(996\) 0 0
\(997\) 11367.5 0.361095 0.180548 0.983566i \(-0.442213\pi\)
0.180548 + 0.983566i \(0.442213\pi\)
\(998\) −36555.1 −1.15945
\(999\) 0 0
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 1323.4.a.bf.1.1 6
3.2 odd 2 inner 1323.4.a.bf.1.6 yes 6
7.6 odd 2 1323.4.a.bg.1.1 yes 6
21.20 even 2 1323.4.a.bg.1.6 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bf.1.1 6 1.1 even 1 trivial
1323.4.a.bf.1.6 yes 6 3.2 odd 2 inner
1323.4.a.bg.1.1 yes 6 7.6 odd 2
1323.4.a.bg.1.6 yes 6 21.20 even 2