Properties

Label 2842.2.a.q.1.2
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2842,2,Mod(1,2842)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2842, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2842.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.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: \(22.6934842544\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.517638\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.896575 q^{3} +1.00000 q^{4} -0.517638 q^{5} +0.896575 q^{6} -1.00000 q^{8} -2.19615 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.896575 q^{3} +1.00000 q^{4} -0.517638 q^{5} +0.896575 q^{6} -1.00000 q^{8} -2.19615 q^{9} +0.517638 q^{10} +0.464102 q^{11} -0.896575 q^{12} -5.79555 q^{13} +0.464102 q^{15} +1.00000 q^{16} +5.65685 q^{17} +2.19615 q^{18} +5.27792 q^{19} -0.517638 q^{20} -0.464102 q^{22} +2.00000 q^{23} +0.896575 q^{24} -4.73205 q^{25} +5.79555 q^{26} +4.65874 q^{27} -1.00000 q^{29} -0.464102 q^{30} +8.24504 q^{31} -1.00000 q^{32} -0.416102 q^{33} -5.65685 q^{34} -2.19615 q^{36} +4.92820 q^{37} -5.27792 q^{38} +5.19615 q^{39} +0.517638 q^{40} -7.72741 q^{41} -3.73205 q^{43} +0.464102 q^{44} +1.13681 q^{45} -2.00000 q^{46} +8.24504 q^{47} -0.896575 q^{48} +4.73205 q^{50} -5.07180 q^{51} -5.79555 q^{52} -14.4641 q^{53} -4.65874 q^{54} -0.240237 q^{55} -4.73205 q^{57} +1.00000 q^{58} +0.757875 q^{59} +0.464102 q^{60} -3.10583 q^{61} -8.24504 q^{62} +1.00000 q^{64} +3.00000 q^{65} +0.416102 q^{66} +11.4641 q^{67} +5.65685 q^{68} -1.79315 q^{69} -4.00000 q^{71} +2.19615 q^{72} -12.6264 q^{73} -4.92820 q^{74} +4.24264 q^{75} +5.27792 q^{76} -5.19615 q^{78} -15.5885 q^{79} -0.517638 q^{80} +2.41154 q^{81} +7.72741 q^{82} -15.4548 q^{83} -2.92820 q^{85} +3.73205 q^{86} +0.896575 q^{87} -0.464102 q^{88} +13.6617 q^{89} -1.13681 q^{90} +2.00000 q^{92} -7.39230 q^{93} -8.24504 q^{94} -2.73205 q^{95} +0.896575 q^{96} +12.6264 q^{97} -1.01924 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9} - 12 q^{11} - 12 q^{15} + 4 q^{16} - 12 q^{18} + 12 q^{22} + 8 q^{23} - 12 q^{25} - 4 q^{29} + 12 q^{30} - 4 q^{32} + 12 q^{36} - 8 q^{37} - 8 q^{43} - 12 q^{44} - 8 q^{46} + 12 q^{50} - 48 q^{51} - 44 q^{53} - 12 q^{57} + 4 q^{58} - 12 q^{60} + 4 q^{64} + 12 q^{65} + 32 q^{67} - 16 q^{71} - 12 q^{72} + 8 q^{74} + 72 q^{81} + 16 q^{85} + 8 q^{86} + 12 q^{88} + 8 q^{92} + 12 q^{93} - 4 q^{95} - 108 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.896575 −0.517638 −0.258819 0.965926i \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.517638 −0.231495 −0.115747 0.993279i \(-0.536926\pi\)
−0.115747 + 0.993279i \(0.536926\pi\)
\(6\) 0.896575 0.366025
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −2.19615 −0.732051
\(10\) 0.517638 0.163692
\(11\) 0.464102 0.139932 0.0699660 0.997549i \(-0.477711\pi\)
0.0699660 + 0.997549i \(0.477711\pi\)
\(12\) −0.896575 −0.258819
\(13\) −5.79555 −1.60740 −0.803699 0.595036i \(-0.797138\pi\)
−0.803699 + 0.595036i \(0.797138\pi\)
\(14\) 0 0
\(15\) 0.464102 0.119831
\(16\) 1.00000 0.250000
\(17\) 5.65685 1.37199 0.685994 0.727607i \(-0.259367\pi\)
0.685994 + 0.727607i \(0.259367\pi\)
\(18\) 2.19615 0.517638
\(19\) 5.27792 1.21084 0.605419 0.795907i \(-0.293006\pi\)
0.605419 + 0.795907i \(0.293006\pi\)
\(20\) −0.517638 −0.115747
\(21\) 0 0
\(22\) −0.464102 −0.0989468
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0.896575 0.183013
\(25\) −4.73205 −0.946410
\(26\) 5.79555 1.13660
\(27\) 4.65874 0.896575
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) −0.464102 −0.0847330
\(31\) 8.24504 1.48085 0.740427 0.672137i \(-0.234623\pi\)
0.740427 + 0.672137i \(0.234623\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.416102 −0.0724341
\(34\) −5.65685 −0.970143
\(35\) 0 0
\(36\) −2.19615 −0.366025
\(37\) 4.92820 0.810192 0.405096 0.914274i \(-0.367238\pi\)
0.405096 + 0.914274i \(0.367238\pi\)
\(38\) −5.27792 −0.856191
\(39\) 5.19615 0.832050
\(40\) 0.517638 0.0818458
\(41\) −7.72741 −1.20682 −0.603409 0.797432i \(-0.706191\pi\)
−0.603409 + 0.797432i \(0.706191\pi\)
\(42\) 0 0
\(43\) −3.73205 −0.569132 −0.284566 0.958656i \(-0.591850\pi\)
−0.284566 + 0.958656i \(0.591850\pi\)
\(44\) 0.464102 0.0699660
\(45\) 1.13681 0.169466
\(46\) −2.00000 −0.294884
\(47\) 8.24504 1.20266 0.601332 0.799000i \(-0.294637\pi\)
0.601332 + 0.799000i \(0.294637\pi\)
\(48\) −0.896575 −0.129410
\(49\) 0 0
\(50\) 4.73205 0.669213
\(51\) −5.07180 −0.710194
\(52\) −5.79555 −0.803699
\(53\) −14.4641 −1.98680 −0.993399 0.114714i \(-0.963405\pi\)
−0.993399 + 0.114714i \(0.963405\pi\)
\(54\) −4.65874 −0.633975
\(55\) −0.240237 −0.0323935
\(56\) 0 0
\(57\) −4.73205 −0.626775
\(58\) 1.00000 0.131306
\(59\) 0.757875 0.0986669 0.0493334 0.998782i \(-0.484290\pi\)
0.0493334 + 0.998782i \(0.484290\pi\)
\(60\) 0.464102 0.0599153
\(61\) −3.10583 −0.397661 −0.198830 0.980034i \(-0.563714\pi\)
−0.198830 + 0.980034i \(0.563714\pi\)
\(62\) −8.24504 −1.04712
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0.416102 0.0512186
\(67\) 11.4641 1.40056 0.700281 0.713867i \(-0.253058\pi\)
0.700281 + 0.713867i \(0.253058\pi\)
\(68\) 5.65685 0.685994
\(69\) −1.79315 −0.215870
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 2.19615 0.258819
\(73\) −12.6264 −1.47781 −0.738903 0.673811i \(-0.764656\pi\)
−0.738903 + 0.673811i \(0.764656\pi\)
\(74\) −4.92820 −0.572892
\(75\) 4.24264 0.489898
\(76\) 5.27792 0.605419
\(77\) 0 0
\(78\) −5.19615 −0.588348
\(79\) −15.5885 −1.75384 −0.876919 0.480638i \(-0.840405\pi\)
−0.876919 + 0.480638i \(0.840405\pi\)
\(80\) −0.517638 −0.0578737
\(81\) 2.41154 0.267949
\(82\) 7.72741 0.853349
\(83\) −15.4548 −1.69639 −0.848193 0.529687i \(-0.822309\pi\)
−0.848193 + 0.529687i \(0.822309\pi\)
\(84\) 0 0
\(85\) −2.92820 −0.317608
\(86\) 3.73205 0.402437
\(87\) 0.896575 0.0961230
\(88\) −0.464102 −0.0494734
\(89\) 13.6617 1.44813 0.724067 0.689730i \(-0.242271\pi\)
0.724067 + 0.689730i \(0.242271\pi\)
\(90\) −1.13681 −0.119831
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) −7.39230 −0.766546
\(94\) −8.24504 −0.850411
\(95\) −2.73205 −0.280302
\(96\) 0.896575 0.0915064
\(97\) 12.6264 1.28202 0.641008 0.767535i \(-0.278517\pi\)
0.641008 + 0.767535i \(0.278517\pi\)
\(98\) 0 0
\(99\) −1.01924 −0.102437
\(100\) −4.73205 −0.473205
\(101\) 11.3137 1.12576 0.562878 0.826540i \(-0.309694\pi\)
0.562878 + 0.826540i \(0.309694\pi\)
\(102\) 5.07180 0.502183
\(103\) −4.89898 −0.482711 −0.241355 0.970437i \(-0.577592\pi\)
−0.241355 + 0.970437i \(0.577592\pi\)
\(104\) 5.79555 0.568301
\(105\) 0 0
\(106\) 14.4641 1.40488
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 4.65874 0.448288
\(109\) 3.39230 0.324924 0.162462 0.986715i \(-0.448057\pi\)
0.162462 + 0.986715i \(0.448057\pi\)
\(110\) 0.240237 0.0229057
\(111\) −4.41851 −0.419386
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 4.73205 0.443197
\(115\) −1.03528 −0.0965400
\(116\) −1.00000 −0.0928477
\(117\) 12.7279 1.17670
\(118\) −0.757875 −0.0697680
\(119\) 0 0
\(120\) −0.464102 −0.0423665
\(121\) −10.7846 −0.980419
\(122\) 3.10583 0.281189
\(123\) 6.92820 0.624695
\(124\) 8.24504 0.740427
\(125\) 5.03768 0.450584
\(126\) 0 0
\(127\) −13.3205 −1.18200 −0.591002 0.806670i \(-0.701267\pi\)
−0.591002 + 0.806670i \(0.701267\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.34607 0.294605
\(130\) −3.00000 −0.263117
\(131\) −3.20736 −0.280229 −0.140114 0.990135i \(-0.544747\pi\)
−0.140114 + 0.990135i \(0.544747\pi\)
\(132\) −0.416102 −0.0362170
\(133\) 0 0
\(134\) −11.4641 −0.990348
\(135\) −2.41154 −0.207553
\(136\) −5.65685 −0.485071
\(137\) 7.85641 0.671218 0.335609 0.942001i \(-0.391058\pi\)
0.335609 + 0.942001i \(0.391058\pi\)
\(138\) 1.79315 0.152643
\(139\) −10.8332 −0.918863 −0.459432 0.888213i \(-0.651947\pi\)
−0.459432 + 0.888213i \(0.651947\pi\)
\(140\) 0 0
\(141\) −7.39230 −0.622544
\(142\) 4.00000 0.335673
\(143\) −2.68973 −0.224926
\(144\) −2.19615 −0.183013
\(145\) 0.517638 0.0429875
\(146\) 12.6264 1.04497
\(147\) 0 0
\(148\) 4.92820 0.405096
\(149\) 11.1962 0.917225 0.458612 0.888636i \(-0.348347\pi\)
0.458612 + 0.888636i \(0.348347\pi\)
\(150\) −4.24264 −0.346410
\(151\) 11.4641 0.932935 0.466468 0.884538i \(-0.345526\pi\)
0.466468 + 0.884538i \(0.345526\pi\)
\(152\) −5.27792 −0.428096
\(153\) −12.4233 −1.00437
\(154\) 0 0
\(155\) −4.26795 −0.342810
\(156\) 5.19615 0.416025
\(157\) −6.41473 −0.511951 −0.255976 0.966683i \(-0.582397\pi\)
−0.255976 + 0.966683i \(0.582397\pi\)
\(158\) 15.5885 1.24015
\(159\) 12.9682 1.02844
\(160\) 0.517638 0.0409229
\(161\) 0 0
\(162\) −2.41154 −0.189469
\(163\) 20.3205 1.59163 0.795813 0.605543i \(-0.207044\pi\)
0.795813 + 0.605543i \(0.207044\pi\)
\(164\) −7.72741 −0.603409
\(165\) 0.215390 0.0167681
\(166\) 15.4548 1.19953
\(167\) −12.6264 −0.977059 −0.488530 0.872547i \(-0.662467\pi\)
−0.488530 + 0.872547i \(0.662467\pi\)
\(168\) 0 0
\(169\) 20.5885 1.58373
\(170\) 2.92820 0.224583
\(171\) −11.5911 −0.886394
\(172\) −3.73205 −0.284566
\(173\) −5.00052 −0.380182 −0.190091 0.981766i \(-0.560878\pi\)
−0.190091 + 0.981766i \(0.560878\pi\)
\(174\) −0.896575 −0.0679692
\(175\) 0 0
\(176\) 0.464102 0.0349830
\(177\) −0.679492 −0.0510737
\(178\) −13.6617 −1.02398
\(179\) −25.8564 −1.93260 −0.966299 0.257421i \(-0.917127\pi\)
−0.966299 + 0.257421i \(0.917127\pi\)
\(180\) 1.13681 0.0847330
\(181\) −15.4176 −1.14598 −0.572992 0.819561i \(-0.694218\pi\)
−0.572992 + 0.819561i \(0.694218\pi\)
\(182\) 0 0
\(183\) 2.78461 0.205844
\(184\) −2.00000 −0.147442
\(185\) −2.55103 −0.187555
\(186\) 7.39230 0.542030
\(187\) 2.62536 0.191985
\(188\) 8.24504 0.601332
\(189\) 0 0
\(190\) 2.73205 0.198204
\(191\) −10.3923 −0.751961 −0.375980 0.926628i \(-0.622694\pi\)
−0.375980 + 0.926628i \(0.622694\pi\)
\(192\) −0.896575 −0.0647048
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −12.6264 −0.906522
\(195\) −2.68973 −0.192615
\(196\) 0 0
\(197\) −18.9282 −1.34858 −0.674289 0.738467i \(-0.735550\pi\)
−0.674289 + 0.738467i \(0.735550\pi\)
\(198\) 1.01924 0.0724341
\(199\) 14.6969 1.04184 0.520919 0.853606i \(-0.325589\pi\)
0.520919 + 0.853606i \(0.325589\pi\)
\(200\) 4.73205 0.334607
\(201\) −10.2784 −0.724985
\(202\) −11.3137 −0.796030
\(203\) 0 0
\(204\) −5.07180 −0.355097
\(205\) 4.00000 0.279372
\(206\) 4.89898 0.341328
\(207\) −4.39230 −0.305286
\(208\) −5.79555 −0.401849
\(209\) 2.44949 0.169435
\(210\) 0 0
\(211\) −13.3923 −0.921964 −0.460982 0.887409i \(-0.652503\pi\)
−0.460982 + 0.887409i \(0.652503\pi\)
\(212\) −14.4641 −0.993399
\(213\) 3.58630 0.245729
\(214\) 18.0000 1.23045
\(215\) 1.93185 0.131751
\(216\) −4.65874 −0.316987
\(217\) 0 0
\(218\) −3.39230 −0.229756
\(219\) 11.3205 0.764969
\(220\) −0.240237 −0.0161968
\(221\) −32.7846 −2.20533
\(222\) 4.41851 0.296551
\(223\) 19.5959 1.31224 0.656120 0.754657i \(-0.272197\pi\)
0.656120 + 0.754657i \(0.272197\pi\)
\(224\) 0 0
\(225\) 10.3923 0.692820
\(226\) 4.00000 0.266076
\(227\) −11.0363 −0.732505 −0.366253 0.930515i \(-0.619359\pi\)
−0.366253 + 0.930515i \(0.619359\pi\)
\(228\) −4.73205 −0.313388
\(229\) −10.0754 −0.665799 −0.332899 0.942962i \(-0.608027\pi\)
−0.332899 + 0.942962i \(0.608027\pi\)
\(230\) 1.03528 0.0682641
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) −24.1244 −1.58044 −0.790220 0.612824i \(-0.790034\pi\)
−0.790220 + 0.612824i \(0.790034\pi\)
\(234\) −12.7279 −0.832050
\(235\) −4.26795 −0.278410
\(236\) 0.757875 0.0493334
\(237\) 13.9762 0.907854
\(238\) 0 0
\(239\) −14.9282 −0.965625 −0.482813 0.875724i \(-0.660385\pi\)
−0.482813 + 0.875724i \(0.660385\pi\)
\(240\) 0.464102 0.0299576
\(241\) 10.6945 0.688896 0.344448 0.938805i \(-0.388066\pi\)
0.344448 + 0.938805i \(0.388066\pi\)
\(242\) 10.7846 0.693261
\(243\) −16.1384 −1.03528
\(244\) −3.10583 −0.198830
\(245\) 0 0
\(246\) −6.92820 −0.441726
\(247\) −30.5885 −1.94630
\(248\) −8.24504 −0.523561
\(249\) 13.8564 0.878114
\(250\) −5.03768 −0.318611
\(251\) 2.31079 0.145856 0.0729279 0.997337i \(-0.476766\pi\)
0.0729279 + 0.997337i \(0.476766\pi\)
\(252\) 0 0
\(253\) 0.928203 0.0583556
\(254\) 13.3205 0.835803
\(255\) 2.62536 0.164406
\(256\) 1.00000 0.0625000
\(257\) −6.83083 −0.426096 −0.213048 0.977042i \(-0.568339\pi\)
−0.213048 + 0.977042i \(0.568339\pi\)
\(258\) −3.34607 −0.208317
\(259\) 0 0
\(260\) 3.00000 0.186052
\(261\) 2.19615 0.135938
\(262\) 3.20736 0.198152
\(263\) 1.19615 0.0737579 0.0368790 0.999320i \(-0.488258\pi\)
0.0368790 + 0.999320i \(0.488258\pi\)
\(264\) 0.416102 0.0256093
\(265\) 7.48717 0.459933
\(266\) 0 0
\(267\) −12.2487 −0.749609
\(268\) 11.4641 0.700281
\(269\) −2.82843 −0.172452 −0.0862261 0.996276i \(-0.527481\pi\)
−0.0862261 + 0.996276i \(0.527481\pi\)
\(270\) 2.41154 0.146762
\(271\) 17.3867 1.05616 0.528082 0.849193i \(-0.322911\pi\)
0.528082 + 0.849193i \(0.322911\pi\)
\(272\) 5.65685 0.342997
\(273\) 0 0
\(274\) −7.85641 −0.474623
\(275\) −2.19615 −0.132433
\(276\) −1.79315 −0.107935
\(277\) 18.9282 1.13729 0.568643 0.822585i \(-0.307469\pi\)
0.568643 + 0.822585i \(0.307469\pi\)
\(278\) 10.8332 0.649734
\(279\) −18.1074 −1.08406
\(280\) 0 0
\(281\) −8.85641 −0.528329 −0.264164 0.964478i \(-0.585096\pi\)
−0.264164 + 0.964478i \(0.585096\pi\)
\(282\) 7.39230 0.440205
\(283\) −27.0459 −1.60771 −0.803857 0.594823i \(-0.797222\pi\)
−0.803857 + 0.594823i \(0.797222\pi\)
\(284\) −4.00000 −0.237356
\(285\) 2.44949 0.145095
\(286\) 2.68973 0.159047
\(287\) 0 0
\(288\) 2.19615 0.129410
\(289\) 15.0000 0.882353
\(290\) −0.517638 −0.0303968
\(291\) −11.3205 −0.663620
\(292\) −12.6264 −0.738903
\(293\) −8.20788 −0.479509 −0.239755 0.970833i \(-0.577067\pi\)
−0.239755 + 0.970833i \(0.577067\pi\)
\(294\) 0 0
\(295\) −0.392305 −0.0228409
\(296\) −4.92820 −0.286446
\(297\) 2.16213 0.125460
\(298\) −11.1962 −0.648576
\(299\) −11.5911 −0.670331
\(300\) 4.24264 0.244949
\(301\) 0 0
\(302\) −11.4641 −0.659685
\(303\) −10.1436 −0.582734
\(304\) 5.27792 0.302709
\(305\) 1.60770 0.0920564
\(306\) 12.4233 0.710194
\(307\) 6.27603 0.358192 0.179096 0.983832i \(-0.442683\pi\)
0.179096 + 0.983832i \(0.442683\pi\)
\(308\) 0 0
\(309\) 4.39230 0.249869
\(310\) 4.26795 0.242403
\(311\) 6.59059 0.373718 0.186859 0.982387i \(-0.440169\pi\)
0.186859 + 0.982387i \(0.440169\pi\)
\(312\) −5.19615 −0.294174
\(313\) −27.0088 −1.52663 −0.763313 0.646029i \(-0.776428\pi\)
−0.763313 + 0.646029i \(0.776428\pi\)
\(314\) 6.41473 0.362004
\(315\) 0 0
\(316\) −15.5885 −0.876919
\(317\) −26.3923 −1.48234 −0.741170 0.671318i \(-0.765729\pi\)
−0.741170 + 0.671318i \(0.765729\pi\)
\(318\) −12.9682 −0.727218
\(319\) −0.464102 −0.0259847
\(320\) −0.517638 −0.0289368
\(321\) 16.1384 0.900755
\(322\) 0 0
\(323\) 29.8564 1.66125
\(324\) 2.41154 0.133975
\(325\) 27.4249 1.52126
\(326\) −20.3205 −1.12545
\(327\) −3.04146 −0.168193
\(328\) 7.72741 0.426675
\(329\) 0 0
\(330\) −0.215390 −0.0118568
\(331\) 7.19615 0.395536 0.197768 0.980249i \(-0.436631\pi\)
0.197768 + 0.980249i \(0.436631\pi\)
\(332\) −15.4548 −0.848193
\(333\) −10.8231 −0.593101
\(334\) 12.6264 0.690885
\(335\) −5.93426 −0.324223
\(336\) 0 0
\(337\) 4.92820 0.268456 0.134228 0.990950i \(-0.457144\pi\)
0.134228 + 0.990950i \(0.457144\pi\)
\(338\) −20.5885 −1.11986
\(339\) 3.58630 0.194781
\(340\) −2.92820 −0.158804
\(341\) 3.82654 0.207219
\(342\) 11.5911 0.626775
\(343\) 0 0
\(344\) 3.73205 0.201219
\(345\) 0.928203 0.0499728
\(346\) 5.00052 0.268829
\(347\) −11.0718 −0.594365 −0.297183 0.954821i \(-0.596047\pi\)
−0.297183 + 0.954821i \(0.596047\pi\)
\(348\) 0.896575 0.0480615
\(349\) 26.2509 1.40518 0.702589 0.711596i \(-0.252027\pi\)
0.702589 + 0.711596i \(0.252027\pi\)
\(350\) 0 0
\(351\) −27.0000 −1.44115
\(352\) −0.464102 −0.0247367
\(353\) 2.17209 0.115609 0.0578043 0.998328i \(-0.481590\pi\)
0.0578043 + 0.998328i \(0.481590\pi\)
\(354\) 0.679492 0.0361146
\(355\) 2.07055 0.109894
\(356\) 13.6617 0.724067
\(357\) 0 0
\(358\) 25.8564 1.36655
\(359\) −3.87564 −0.204549 −0.102274 0.994756i \(-0.532612\pi\)
−0.102274 + 0.994756i \(0.532612\pi\)
\(360\) −1.13681 −0.0599153
\(361\) 8.85641 0.466127
\(362\) 15.4176 0.810334
\(363\) 9.66922 0.507502
\(364\) 0 0
\(365\) 6.53590 0.342105
\(366\) −2.78461 −0.145554
\(367\) 32.0464 1.67281 0.836405 0.548112i \(-0.184653\pi\)
0.836405 + 0.548112i \(0.184653\pi\)
\(368\) 2.00000 0.104257
\(369\) 16.9706 0.883452
\(370\) 2.55103 0.132622
\(371\) 0 0
\(372\) −7.39230 −0.383273
\(373\) 22.6603 1.17330 0.586652 0.809839i \(-0.300446\pi\)
0.586652 + 0.809839i \(0.300446\pi\)
\(374\) −2.62536 −0.135754
\(375\) −4.51666 −0.233239
\(376\) −8.24504 −0.425206
\(377\) 5.79555 0.298486
\(378\) 0 0
\(379\) −5.60770 −0.288048 −0.144024 0.989574i \(-0.546004\pi\)
−0.144024 + 0.989574i \(0.546004\pi\)
\(380\) −2.73205 −0.140151
\(381\) 11.9428 0.611850
\(382\) 10.3923 0.531717
\(383\) −8.20788 −0.419403 −0.209702 0.977765i \(-0.567249\pi\)
−0.209702 + 0.977765i \(0.567249\pi\)
\(384\) 0.896575 0.0457532
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) 8.19615 0.416634
\(388\) 12.6264 0.641008
\(389\) 11.3205 0.573973 0.286986 0.957935i \(-0.407347\pi\)
0.286986 + 0.957935i \(0.407347\pi\)
\(390\) 2.68973 0.136200
\(391\) 11.3137 0.572159
\(392\) 0 0
\(393\) 2.87564 0.145057
\(394\) 18.9282 0.953589
\(395\) 8.06918 0.406004
\(396\) −1.01924 −0.0512186
\(397\) −7.58871 −0.380866 −0.190433 0.981700i \(-0.560989\pi\)
−0.190433 + 0.981700i \(0.560989\pi\)
\(398\) −14.6969 −0.736691
\(399\) 0 0
\(400\) −4.73205 −0.236603
\(401\) −3.14359 −0.156984 −0.0784918 0.996915i \(-0.525010\pi\)
−0.0784918 + 0.996915i \(0.525010\pi\)
\(402\) 10.2784 0.512642
\(403\) −47.7846 −2.38032
\(404\) 11.3137 0.562878
\(405\) −1.24831 −0.0620288
\(406\) 0 0
\(407\) 2.28719 0.113372
\(408\) 5.07180 0.251091
\(409\) −22.6274 −1.11885 −0.559427 0.828880i \(-0.688979\pi\)
−0.559427 + 0.828880i \(0.688979\pi\)
\(410\) −4.00000 −0.197546
\(411\) −7.04386 −0.347448
\(412\) −4.89898 −0.241355
\(413\) 0 0
\(414\) 4.39230 0.215870
\(415\) 8.00000 0.392705
\(416\) 5.79555 0.284150
\(417\) 9.71281 0.475638
\(418\) −2.44949 −0.119808
\(419\) −19.7990 −0.967244 −0.483622 0.875277i \(-0.660679\pi\)
−0.483622 + 0.875277i \(0.660679\pi\)
\(420\) 0 0
\(421\) 19.4641 0.948622 0.474311 0.880357i \(-0.342697\pi\)
0.474311 + 0.880357i \(0.342697\pi\)
\(422\) 13.3923 0.651927
\(423\) −18.1074 −0.880411
\(424\) 14.4641 0.702439
\(425\) −26.7685 −1.29846
\(426\) −3.58630 −0.173757
\(427\) 0 0
\(428\) −18.0000 −0.870063
\(429\) 2.41154 0.116430
\(430\) −1.93185 −0.0931622
\(431\) −4.39230 −0.211570 −0.105785 0.994389i \(-0.533736\pi\)
−0.105785 + 0.994389i \(0.533736\pi\)
\(432\) 4.65874 0.224144
\(433\) 8.96575 0.430867 0.215433 0.976519i \(-0.430884\pi\)
0.215433 + 0.976519i \(0.430884\pi\)
\(434\) 0 0
\(435\) −0.464102 −0.0222520
\(436\) 3.39230 0.162462
\(437\) 10.5558 0.504954
\(438\) −11.3205 −0.540915
\(439\) 20.8343 0.994365 0.497183 0.867646i \(-0.334368\pi\)
0.497183 + 0.867646i \(0.334368\pi\)
\(440\) 0.240237 0.0114528
\(441\) 0 0
\(442\) 32.7846 1.55940
\(443\) −40.2487 −1.91227 −0.956137 0.292920i \(-0.905373\pi\)
−0.956137 + 0.292920i \(0.905373\pi\)
\(444\) −4.41851 −0.209693
\(445\) −7.07180 −0.335235
\(446\) −19.5959 −0.927894
\(447\) −10.0382 −0.474790
\(448\) 0 0
\(449\) 5.07180 0.239353 0.119676 0.992813i \(-0.461814\pi\)
0.119676 + 0.992813i \(0.461814\pi\)
\(450\) −10.3923 −0.489898
\(451\) −3.58630 −0.168872
\(452\) −4.00000 −0.188144
\(453\) −10.2784 −0.482923
\(454\) 11.0363 0.517960
\(455\) 0 0
\(456\) 4.73205 0.221599
\(457\) −32.9282 −1.54032 −0.770158 0.637853i \(-0.779823\pi\)
−0.770158 + 0.637853i \(0.779823\pi\)
\(458\) 10.0754 0.470791
\(459\) 26.3538 1.23009
\(460\) −1.03528 −0.0482700
\(461\) −14.6969 −0.684505 −0.342252 0.939608i \(-0.611190\pi\)
−0.342252 + 0.939608i \(0.611190\pi\)
\(462\) 0 0
\(463\) −13.8564 −0.643962 −0.321981 0.946746i \(-0.604349\pi\)
−0.321981 + 0.946746i \(0.604349\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 3.82654 0.177451
\(466\) 24.1244 1.11754
\(467\) −32.6656 −1.51158 −0.755792 0.654812i \(-0.772748\pi\)
−0.755792 + 0.654812i \(0.772748\pi\)
\(468\) 12.7279 0.588348
\(469\) 0 0
\(470\) 4.26795 0.196866
\(471\) 5.75129 0.265005
\(472\) −0.757875 −0.0348840
\(473\) −1.73205 −0.0796398
\(474\) −13.9762 −0.641949
\(475\) −24.9754 −1.14595
\(476\) 0 0
\(477\) 31.7654 1.45444
\(478\) 14.9282 0.682800
\(479\) 0.138701 0.00633740 0.00316870 0.999995i \(-0.498991\pi\)
0.00316870 + 0.999995i \(0.498991\pi\)
\(480\) −0.464102 −0.0211832
\(481\) −28.5617 −1.30230
\(482\) −10.6945 −0.487123
\(483\) 0 0
\(484\) −10.7846 −0.490210
\(485\) −6.53590 −0.296780
\(486\) 16.1384 0.732051
\(487\) −7.32051 −0.331724 −0.165862 0.986149i \(-0.553041\pi\)
−0.165862 + 0.986149i \(0.553041\pi\)
\(488\) 3.10583 0.140594
\(489\) −18.2189 −0.823886
\(490\) 0 0
\(491\) 22.3205 1.00731 0.503655 0.863905i \(-0.331988\pi\)
0.503655 + 0.863905i \(0.331988\pi\)
\(492\) 6.92820 0.312348
\(493\) −5.65685 −0.254772
\(494\) 30.5885 1.37624
\(495\) 0.527596 0.0237137
\(496\) 8.24504 0.370213
\(497\) 0 0
\(498\) −13.8564 −0.620920
\(499\) 20.9282 0.936875 0.468438 0.883497i \(-0.344817\pi\)
0.468438 + 0.883497i \(0.344817\pi\)
\(500\) 5.03768 0.225292
\(501\) 11.3205 0.505763
\(502\) −2.31079 −0.103136
\(503\) 1.37705 0.0613996 0.0306998 0.999529i \(-0.490226\pi\)
0.0306998 + 0.999529i \(0.490226\pi\)
\(504\) 0 0
\(505\) −5.85641 −0.260607
\(506\) −0.928203 −0.0412637
\(507\) −18.4591 −0.819798
\(508\) −13.3205 −0.591002
\(509\) −35.4940 −1.57325 −0.786623 0.617434i \(-0.788172\pi\)
−0.786623 + 0.617434i \(0.788172\pi\)
\(510\) −2.62536 −0.116253
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 24.5885 1.08561
\(514\) 6.83083 0.301295
\(515\) 2.53590 0.111745
\(516\) 3.34607 0.147302
\(517\) 3.82654 0.168291
\(518\) 0 0
\(519\) 4.48334 0.196797
\(520\) −3.00000 −0.131559
\(521\) 29.7356 1.30274 0.651371 0.758759i \(-0.274194\pi\)
0.651371 + 0.758759i \(0.274194\pi\)
\(522\) −2.19615 −0.0961230
\(523\) 14.4195 0.630522 0.315261 0.949005i \(-0.397908\pi\)
0.315261 + 0.949005i \(0.397908\pi\)
\(524\) −3.20736 −0.140114
\(525\) 0 0
\(526\) −1.19615 −0.0521547
\(527\) 46.6410 2.03171
\(528\) −0.416102 −0.0181085
\(529\) −19.0000 −0.826087
\(530\) −7.48717 −0.325222
\(531\) −1.66441 −0.0722292
\(532\) 0 0
\(533\) 44.7846 1.93984
\(534\) 12.2487 0.530054
\(535\) 9.31749 0.402830
\(536\) −11.4641 −0.495174
\(537\) 23.1822 1.00039
\(538\) 2.82843 0.121942
\(539\) 0 0
\(540\) −2.41154 −0.103776
\(541\) 26.9282 1.15773 0.578867 0.815422i \(-0.303495\pi\)
0.578867 + 0.815422i \(0.303495\pi\)
\(542\) −17.3867 −0.746821
\(543\) 13.8231 0.593205
\(544\) −5.65685 −0.242536
\(545\) −1.75599 −0.0752182
\(546\) 0 0
\(547\) 32.2487 1.37886 0.689428 0.724355i \(-0.257862\pi\)
0.689428 + 0.724355i \(0.257862\pi\)
\(548\) 7.85641 0.335609
\(549\) 6.82087 0.291108
\(550\) 2.19615 0.0936443
\(551\) −5.27792 −0.224847
\(552\) 1.79315 0.0763216
\(553\) 0 0
\(554\) −18.9282 −0.804182
\(555\) 2.28719 0.0970857
\(556\) −10.8332 −0.459432
\(557\) −36.9282 −1.56470 −0.782349 0.622840i \(-0.785979\pi\)
−0.782349 + 0.622840i \(0.785979\pi\)
\(558\) 18.1074 0.766546
\(559\) 21.6293 0.914822
\(560\) 0 0
\(561\) −2.35383 −0.0993787
\(562\) 8.85641 0.373585
\(563\) −16.3514 −0.689129 −0.344564 0.938763i \(-0.611973\pi\)
−0.344564 + 0.938763i \(0.611973\pi\)
\(564\) −7.39230 −0.311272
\(565\) 2.07055 0.0871088
\(566\) 27.0459 1.13682
\(567\) 0 0
\(568\) 4.00000 0.167836
\(569\) 0.535898 0.0224660 0.0112330 0.999937i \(-0.496424\pi\)
0.0112330 + 0.999937i \(0.496424\pi\)
\(570\) −2.44949 −0.102598
\(571\) 11.4641 0.479758 0.239879 0.970803i \(-0.422892\pi\)
0.239879 + 0.970803i \(0.422892\pi\)
\(572\) −2.68973 −0.112463
\(573\) 9.31749 0.389244
\(574\) 0 0
\(575\) −9.46410 −0.394680
\(576\) −2.19615 −0.0915064
\(577\) −6.69213 −0.278597 −0.139299 0.990250i \(-0.544485\pi\)
−0.139299 + 0.990250i \(0.544485\pi\)
\(578\) −15.0000 −0.623918
\(579\) 14.3452 0.596166
\(580\) 0.517638 0.0214938
\(581\) 0 0
\(582\) 11.3205 0.469250
\(583\) −6.71281 −0.278016
\(584\) 12.6264 0.522484
\(585\) −6.58846 −0.272399
\(586\) 8.20788 0.339064
\(587\) 22.7017 0.937001 0.468501 0.883463i \(-0.344794\pi\)
0.468501 + 0.883463i \(0.344794\pi\)
\(588\) 0 0
\(589\) 43.5167 1.79307
\(590\) 0.392305 0.0161509
\(591\) 16.9706 0.698076
\(592\) 4.92820 0.202548
\(593\) 26.1493 1.07383 0.536913 0.843638i \(-0.319591\pi\)
0.536913 + 0.843638i \(0.319591\pi\)
\(594\) −2.16213 −0.0887133
\(595\) 0 0
\(596\) 11.1962 0.458612
\(597\) −13.1769 −0.539295
\(598\) 11.5911 0.473996
\(599\) 22.5167 0.920006 0.460003 0.887917i \(-0.347848\pi\)
0.460003 + 0.887917i \(0.347848\pi\)
\(600\) −4.24264 −0.173205
\(601\) −2.82843 −0.115374 −0.0576870 0.998335i \(-0.518373\pi\)
−0.0576870 + 0.998335i \(0.518373\pi\)
\(602\) 0 0
\(603\) −25.1769 −1.02528
\(604\) 11.4641 0.466468
\(605\) 5.58252 0.226962
\(606\) 10.1436 0.412055
\(607\) −27.1846 −1.10339 −0.551695 0.834046i \(-0.686019\pi\)
−0.551695 + 0.834046i \(0.686019\pi\)
\(608\) −5.27792 −0.214048
\(609\) 0 0
\(610\) −1.60770 −0.0650937
\(611\) −47.7846 −1.93316
\(612\) −12.4233 −0.502183
\(613\) 22.9090 0.925284 0.462642 0.886545i \(-0.346901\pi\)
0.462642 + 0.886545i \(0.346901\pi\)
\(614\) −6.27603 −0.253280
\(615\) −3.58630 −0.144614
\(616\) 0 0
\(617\) 0.392305 0.0157936 0.00789680 0.999969i \(-0.497486\pi\)
0.00789680 + 0.999969i \(0.497486\pi\)
\(618\) −4.39230 −0.176684
\(619\) −5.41662 −0.217712 −0.108856 0.994058i \(-0.534719\pi\)
−0.108856 + 0.994058i \(0.534719\pi\)
\(620\) −4.26795 −0.171405
\(621\) 9.31749 0.373898
\(622\) −6.59059 −0.264259
\(623\) 0 0
\(624\) 5.19615 0.208013
\(625\) 21.0526 0.842102
\(626\) 27.0088 1.07949
\(627\) −2.19615 −0.0877059
\(628\) −6.41473 −0.255976
\(629\) 27.8781 1.11157
\(630\) 0 0
\(631\) 20.3923 0.811805 0.405902 0.913916i \(-0.366957\pi\)
0.405902 + 0.913916i \(0.366957\pi\)
\(632\) 15.5885 0.620076
\(633\) 12.0072 0.477244
\(634\) 26.3923 1.04817
\(635\) 6.89520 0.273628
\(636\) 12.9682 0.514221
\(637\) 0 0
\(638\) 0.464102 0.0183740
\(639\) 8.78461 0.347514
\(640\) 0.517638 0.0204614
\(641\) 28.7846 1.13692 0.568462 0.822710i \(-0.307539\pi\)
0.568462 + 0.822710i \(0.307539\pi\)
\(642\) −16.1384 −0.636930
\(643\) 6.96953 0.274852 0.137426 0.990512i \(-0.456117\pi\)
0.137426 + 0.990512i \(0.456117\pi\)
\(644\) 0 0
\(645\) −1.73205 −0.0681994
\(646\) −29.8564 −1.17468
\(647\) 6.96953 0.274001 0.137000 0.990571i \(-0.456254\pi\)
0.137000 + 0.990571i \(0.456254\pi\)
\(648\) −2.41154 −0.0947343
\(649\) 0.351731 0.0138066
\(650\) −27.4249 −1.07569
\(651\) 0 0
\(652\) 20.3205 0.795813
\(653\) 39.7128 1.55408 0.777041 0.629450i \(-0.216720\pi\)
0.777041 + 0.629450i \(0.216720\pi\)
\(654\) 3.04146 0.118930
\(655\) 1.66025 0.0648715
\(656\) −7.72741 −0.301705
\(657\) 27.7295 1.08183
\(658\) 0 0
\(659\) −41.3923 −1.61242 −0.806208 0.591633i \(-0.798484\pi\)
−0.806208 + 0.591633i \(0.798484\pi\)
\(660\) 0.215390 0.00838406
\(661\) −36.6680 −1.42622 −0.713110 0.701052i \(-0.752714\pi\)
−0.713110 + 0.701052i \(0.752714\pi\)
\(662\) −7.19615 −0.279686
\(663\) 29.3939 1.14156
\(664\) 15.4548 0.599763
\(665\) 0 0
\(666\) 10.8231 0.419386
\(667\) −2.00000 −0.0774403
\(668\) −12.6264 −0.488530
\(669\) −17.5692 −0.679265
\(670\) 5.93426 0.229260
\(671\) −1.44142 −0.0556454
\(672\) 0 0
\(673\) 37.0000 1.42625 0.713123 0.701039i \(-0.247280\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) −4.92820 −0.189827
\(675\) −22.0454 −0.848528
\(676\) 20.5885 0.791864
\(677\) 3.58630 0.137833 0.0689164 0.997622i \(-0.478046\pi\)
0.0689164 + 0.997622i \(0.478046\pi\)
\(678\) −3.58630 −0.137731
\(679\) 0 0
\(680\) 2.92820 0.112291
\(681\) 9.89488 0.379173
\(682\) −3.82654 −0.146526
\(683\) 24.3923 0.933346 0.466673 0.884430i \(-0.345453\pi\)
0.466673 + 0.884430i \(0.345453\pi\)
\(684\) −11.5911 −0.443197
\(685\) −4.06678 −0.155383
\(686\) 0 0
\(687\) 9.03332 0.344643
\(688\) −3.73205 −0.142283
\(689\) 83.8275 3.19357
\(690\) −0.928203 −0.0353361
\(691\) −29.3939 −1.11820 −0.559098 0.829102i \(-0.688852\pi\)
−0.559098 + 0.829102i \(0.688852\pi\)
\(692\) −5.00052 −0.190091
\(693\) 0 0
\(694\) 11.0718 0.420280
\(695\) 5.60770 0.212712
\(696\) −0.896575 −0.0339846
\(697\) −43.7128 −1.65574
\(698\) −26.2509 −0.993611
\(699\) 21.6293 0.818095
\(700\) 0 0
\(701\) 20.8038 0.785750 0.392875 0.919592i \(-0.371480\pi\)
0.392875 + 0.919592i \(0.371480\pi\)
\(702\) 27.0000 1.01905
\(703\) 26.0106 0.981010
\(704\) 0.464102 0.0174915
\(705\) 3.82654 0.144116
\(706\) −2.17209 −0.0817476
\(707\) 0 0
\(708\) −0.679492 −0.0255369
\(709\) 7.67949 0.288409 0.144205 0.989548i \(-0.453938\pi\)
0.144205 + 0.989548i \(0.453938\pi\)
\(710\) −2.07055 −0.0777064
\(711\) 34.2346 1.28390
\(712\) −13.6617 −0.511992
\(713\) 16.4901 0.617559
\(714\) 0 0
\(715\) 1.39230 0.0520692
\(716\) −25.8564 −0.966299
\(717\) 13.3843 0.499844
\(718\) 3.87564 0.144638
\(719\) 17.2480 0.643241 0.321620 0.946869i \(-0.395773\pi\)
0.321620 + 0.946869i \(0.395773\pi\)
\(720\) 1.13681 0.0423665
\(721\) 0 0
\(722\) −8.85641 −0.329601
\(723\) −9.58846 −0.356599
\(724\) −15.4176 −0.572992
\(725\) 4.73205 0.175744
\(726\) −9.66922 −0.358858
\(727\) −29.2180 −1.08364 −0.541818 0.840496i \(-0.682264\pi\)
−0.541818 + 0.840496i \(0.682264\pi\)
\(728\) 0 0
\(729\) 7.23463 0.267949
\(730\) −6.53590 −0.241904
\(731\) −21.1117 −0.780843
\(732\) 2.78461 0.102922
\(733\) 8.55961 0.316156 0.158078 0.987427i \(-0.449470\pi\)
0.158078 + 0.987427i \(0.449470\pi\)
\(734\) −32.0464 −1.18286
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) 5.32051 0.195983
\(738\) −16.9706 −0.624695
\(739\) 24.2679 0.892711 0.446355 0.894856i \(-0.352722\pi\)
0.446355 + 0.894856i \(0.352722\pi\)
\(740\) −2.55103 −0.0937776
\(741\) 27.4249 1.00748
\(742\) 0 0
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) 7.39230 0.271015
\(745\) −5.79555 −0.212333
\(746\) −22.6603 −0.829651
\(747\) 33.9411 1.24184
\(748\) 2.62536 0.0959925
\(749\) 0 0
\(750\) 4.51666 0.164925
\(751\) 22.6410 0.826182 0.413091 0.910690i \(-0.364449\pi\)
0.413091 + 0.910690i \(0.364449\pi\)
\(752\) 8.24504 0.300666
\(753\) −2.07180 −0.0755005
\(754\) −5.79555 −0.211062
\(755\) −5.93426 −0.215970
\(756\) 0 0
\(757\) 3.60770 0.131124 0.0655620 0.997849i \(-0.479116\pi\)
0.0655620 + 0.997849i \(0.479116\pi\)
\(758\) 5.60770 0.203681
\(759\) −0.832204 −0.0302071
\(760\) 2.73205 0.0991019
\(761\) −22.3228 −0.809201 −0.404601 0.914493i \(-0.632589\pi\)
−0.404601 + 0.914493i \(0.632589\pi\)
\(762\) −11.9428 −0.432643
\(763\) 0 0
\(764\) −10.3923 −0.375980
\(765\) 6.43078 0.232505
\(766\) 8.20788 0.296563
\(767\) −4.39230 −0.158597
\(768\) −0.896575 −0.0323524
\(769\) −39.1918 −1.41329 −0.706647 0.707566i \(-0.749793\pi\)
−0.706647 + 0.707566i \(0.749793\pi\)
\(770\) 0 0
\(771\) 6.12436 0.220563
\(772\) −16.0000 −0.575853
\(773\) 7.93048 0.285239 0.142620 0.989778i \(-0.454447\pi\)
0.142620 + 0.989778i \(0.454447\pi\)
\(774\) −8.19615 −0.294605
\(775\) −39.0160 −1.40150
\(776\) −12.6264 −0.453261
\(777\) 0 0
\(778\) −11.3205 −0.405860
\(779\) −40.7846 −1.46126
\(780\) −2.68973 −0.0963077
\(781\) −1.85641 −0.0664274
\(782\) −11.3137 −0.404577
\(783\) −4.65874 −0.166490
\(784\) 0 0
\(785\) 3.32051 0.118514
\(786\) −2.87564 −0.102571
\(787\) 20.5569 0.732773 0.366387 0.930463i \(-0.380595\pi\)
0.366387 + 0.930463i \(0.380595\pi\)
\(788\) −18.9282 −0.674289
\(789\) −1.07244 −0.0381799
\(790\) −8.06918 −0.287089
\(791\) 0 0
\(792\) 1.01924 0.0362170
\(793\) 18.0000 0.639199
\(794\) 7.58871 0.269313
\(795\) −6.71281 −0.238079
\(796\) 14.6969 0.520919
\(797\) −48.6381 −1.72285 −0.861424 0.507886i \(-0.830427\pi\)
−0.861424 + 0.507886i \(0.830427\pi\)
\(798\) 0 0
\(799\) 46.6410 1.65004
\(800\) 4.73205 0.167303
\(801\) −30.0031 −1.06011
\(802\) 3.14359 0.111004
\(803\) −5.85993 −0.206792
\(804\) −10.2784 −0.362492
\(805\) 0 0
\(806\) 47.7846 1.68314
\(807\) 2.53590 0.0892679
\(808\) −11.3137 −0.398015
\(809\) 39.0718 1.37369 0.686846 0.726803i \(-0.258995\pi\)
0.686846 + 0.726803i \(0.258995\pi\)
\(810\) 1.24831 0.0438610
\(811\) 6.69213 0.234992 0.117496 0.993073i \(-0.462513\pi\)
0.117496 + 0.993073i \(0.462513\pi\)
\(812\) 0 0
\(813\) −15.5885 −0.546711
\(814\) −2.28719 −0.0801659
\(815\) −10.5187 −0.368453
\(816\) −5.07180 −0.177548
\(817\) −19.6975 −0.689127
\(818\) 22.6274 0.791149
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 45.2487 1.57919 0.789595 0.613628i \(-0.210290\pi\)
0.789595 + 0.613628i \(0.210290\pi\)
\(822\) 7.04386 0.245683
\(823\) −13.8564 −0.483004 −0.241502 0.970400i \(-0.577640\pi\)
−0.241502 + 0.970400i \(0.577640\pi\)
\(824\) 4.89898 0.170664
\(825\) 1.96902 0.0685524
\(826\) 0 0
\(827\) −56.5167 −1.96528 −0.982638 0.185531i \(-0.940600\pi\)
−0.982638 + 0.185531i \(0.940600\pi\)
\(828\) −4.39230 −0.152643
\(829\) −26.7685 −0.929709 −0.464855 0.885387i \(-0.653893\pi\)
−0.464855 + 0.885387i \(0.653893\pi\)
\(830\) −8.00000 −0.277684
\(831\) −16.9706 −0.588702
\(832\) −5.79555 −0.200925
\(833\) 0 0
\(834\) −9.71281 −0.336327
\(835\) 6.53590 0.226184
\(836\) 2.44949 0.0847174
\(837\) 38.4115 1.32770
\(838\) 19.7990 0.683945
\(839\) −22.4887 −0.776397 −0.388198 0.921576i \(-0.626902\pi\)
−0.388198 + 0.921576i \(0.626902\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −19.4641 −0.670777
\(843\) 7.94044 0.273483
\(844\) −13.3923 −0.460982
\(845\) −10.6574 −0.366625
\(846\) 18.1074 0.622544
\(847\) 0 0
\(848\) −14.4641 −0.496699
\(849\) 24.2487 0.832214
\(850\) 26.7685 0.918153
\(851\) 9.85641 0.337873
\(852\) 3.58630 0.122865
\(853\) 6.13733 0.210138 0.105069 0.994465i \(-0.466494\pi\)
0.105069 + 0.994465i \(0.466494\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 18.0000 0.615227
\(857\) 18.4219 0.629282 0.314641 0.949211i \(-0.398116\pi\)
0.314641 + 0.949211i \(0.398116\pi\)
\(858\) −2.41154 −0.0823287
\(859\) 11.9329 0.407145 0.203572 0.979060i \(-0.434745\pi\)
0.203572 + 0.979060i \(0.434745\pi\)
\(860\) 1.93185 0.0658756
\(861\) 0 0
\(862\) 4.39230 0.149602
\(863\) 17.3205 0.589597 0.294798 0.955559i \(-0.404747\pi\)
0.294798 + 0.955559i \(0.404747\pi\)
\(864\) −4.65874 −0.158494
\(865\) 2.58846 0.0880102
\(866\) −8.96575 −0.304669
\(867\) −13.4486 −0.456739
\(868\) 0 0
\(869\) −7.23463 −0.245418
\(870\) 0.464102 0.0157345
\(871\) −66.4408 −2.25126
\(872\) −3.39230 −0.114878
\(873\) −27.7295 −0.938500
\(874\) −10.5558 −0.357056
\(875\) 0 0
\(876\) 11.3205 0.382485
\(877\) 1.33975 0.0452400 0.0226200 0.999744i \(-0.492799\pi\)
0.0226200 + 0.999744i \(0.492799\pi\)
\(878\) −20.8343 −0.703122
\(879\) 7.35898 0.248212
\(880\) −0.240237 −0.00809838
\(881\) −2.34795 −0.0791046 −0.0395523 0.999218i \(-0.512593\pi\)
−0.0395523 + 0.999218i \(0.512593\pi\)
\(882\) 0 0
\(883\) −37.7128 −1.26914 −0.634569 0.772867i \(-0.718822\pi\)
−0.634569 + 0.772867i \(0.718822\pi\)
\(884\) −32.7846 −1.10267
\(885\) 0.351731 0.0118233
\(886\) 40.2487 1.35218
\(887\) 17.4882 0.587196 0.293598 0.955929i \(-0.405147\pi\)
0.293598 + 0.955929i \(0.405147\pi\)
\(888\) 4.41851 0.148275
\(889\) 0 0
\(890\) 7.07180 0.237047
\(891\) 1.11920 0.0374946
\(892\) 19.5959 0.656120
\(893\) 43.5167 1.45623
\(894\) 10.0382 0.335727
\(895\) 13.3843 0.447386
\(896\) 0 0
\(897\) 10.3923 0.346989
\(898\) −5.07180 −0.169248
\(899\) −8.24504 −0.274988
\(900\) 10.3923 0.346410
\(901\) −81.8213 −2.72586
\(902\) 3.58630 0.119411
\(903\) 0 0
\(904\) 4.00000 0.133038
\(905\) 7.98076 0.265290
\(906\) 10.2784 0.341478
\(907\) −42.3923 −1.40761 −0.703807 0.710392i \(-0.748518\pi\)
−0.703807 + 0.710392i \(0.748518\pi\)
\(908\) −11.0363 −0.366253
\(909\) −24.8466 −0.824111
\(910\) 0 0
\(911\) 16.6603 0.551979 0.275989 0.961161i \(-0.410995\pi\)
0.275989 + 0.961161i \(0.410995\pi\)
\(912\) −4.73205 −0.156694
\(913\) −7.17260 −0.237379
\(914\) 32.9282 1.08917
\(915\) −1.44142 −0.0476519
\(916\) −10.0754 −0.332899
\(917\) 0 0
\(918\) −26.3538 −0.869806
\(919\) −49.3205 −1.62693 −0.813467 0.581611i \(-0.802422\pi\)
−0.813467 + 0.581611i \(0.802422\pi\)
\(920\) 1.03528 0.0341320
\(921\) −5.62693 −0.185414
\(922\) 14.6969 0.484018
\(923\) 23.1822 0.763052
\(924\) 0 0
\(925\) −23.3205 −0.766774
\(926\) 13.8564 0.455350
\(927\) 10.7589 0.353369
\(928\) 1.00000 0.0328266
\(929\) 27.9797 0.917983 0.458991 0.888441i \(-0.348211\pi\)
0.458991 + 0.888441i \(0.348211\pi\)
\(930\) −3.82654 −0.125477
\(931\) 0 0
\(932\) −24.1244 −0.790220
\(933\) −5.90897 −0.193451
\(934\) 32.6656 1.06885
\(935\) −1.35898 −0.0444435
\(936\) −12.7279 −0.416025
\(937\) 47.5756 1.55423 0.777113 0.629361i \(-0.216683\pi\)
0.777113 + 0.629361i \(0.216683\pi\)
\(938\) 0 0
\(939\) 24.2154 0.790239
\(940\) −4.26795 −0.139205
\(941\) 9.10446 0.296797 0.148398 0.988928i \(-0.452588\pi\)
0.148398 + 0.988928i \(0.452588\pi\)
\(942\) −5.75129 −0.187387
\(943\) −15.4548 −0.503278
\(944\) 0.757875 0.0246667
\(945\) 0 0
\(946\) 1.73205 0.0563138
\(947\) 14.6603 0.476394 0.238197 0.971217i \(-0.423444\pi\)
0.238197 + 0.971217i \(0.423444\pi\)
\(948\) 13.9762 0.453927
\(949\) 73.1769 2.37542
\(950\) 24.9754 0.810308
\(951\) 23.6627 0.767315
\(952\) 0 0
\(953\) 19.3397 0.626476 0.313238 0.949675i \(-0.398586\pi\)
0.313238 + 0.949675i \(0.398586\pi\)
\(954\) −31.7654 −1.02844
\(955\) 5.37945 0.174075
\(956\) −14.9282 −0.482813
\(957\) 0.416102 0.0134507
\(958\) −0.138701 −0.00448122
\(959\) 0 0
\(960\) 0.464102 0.0149788
\(961\) 36.9808 1.19293
\(962\) 28.5617 0.920865
\(963\) 39.5307 1.27386
\(964\) 10.6945 0.344448
\(965\) 8.28221 0.266614
\(966\) 0 0
\(967\) 6.07180 0.195256 0.0976279 0.995223i \(-0.468874\pi\)
0.0976279 + 0.995223i \(0.468874\pi\)
\(968\) 10.7846 0.346630
\(969\) −26.7685 −0.859929
\(970\) 6.53590 0.209855
\(971\) 22.6002 0.725275 0.362638 0.931930i \(-0.381876\pi\)
0.362638 + 0.931930i \(0.381876\pi\)
\(972\) −16.1384 −0.517638
\(973\) 0 0
\(974\) 7.32051 0.234564
\(975\) −24.5885 −0.787461
\(976\) −3.10583 −0.0994151
\(977\) −38.7128 −1.23853 −0.619266 0.785181i \(-0.712570\pi\)
−0.619266 + 0.785181i \(0.712570\pi\)
\(978\) 18.2189 0.582575
\(979\) 6.34040 0.202640
\(980\) 0 0
\(981\) −7.45002 −0.237861
\(982\) −22.3205 −0.712276
\(983\) 47.0579 1.50092 0.750458 0.660919i \(-0.229833\pi\)
0.750458 + 0.660919i \(0.229833\pi\)
\(984\) −6.92820 −0.220863
\(985\) 9.79796 0.312189
\(986\) 5.65685 0.180151
\(987\) 0 0
\(988\) −30.5885 −0.973148
\(989\) −7.46410 −0.237345
\(990\) −0.527596 −0.0167681
\(991\) 22.0000 0.698853 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(992\) −8.24504 −0.261780
\(993\) −6.45189 −0.204745
\(994\) 0 0
\(995\) −7.60770 −0.241180
\(996\) 13.8564 0.439057
\(997\) −11.0363 −0.349523 −0.174762 0.984611i \(-0.555915\pi\)
−0.174762 + 0.984611i \(0.555915\pi\)
\(998\) −20.9282 −0.662471
\(999\) 22.9592 0.726398
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 2842.2.a.q.1.2 4
7.6 odd 2 inner 2842.2.a.q.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2842.2.a.q.1.2 4 1.1 even 1 trivial
2842.2.a.q.1.3 yes 4 7.6 odd 2 inner