Properties

Label 9522.2.a.bq.1.1
Level $9522$
Weight $2$
Character 9522.1
Self dual yes
Analytic conductor $76.034$
Analytic rank $1$
Dimension $5$
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] = [9522,2,Mod(1,9522)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9522, 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("9522.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9522.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: \(76.0335528047\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.830830\) of defining polynomial
Character \(\chi\) \(=\) 9522.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.20362 q^{5} -1.51334 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.20362 q^{5} -1.51334 q^{7} -1.00000 q^{8} +3.20362 q^{10} -6.53843 q^{11} +2.88612 q^{13} +1.51334 q^{14} +1.00000 q^{16} -1.19647 q^{17} -0.555554 q^{19} -3.20362 q^{20} +6.53843 q^{22} +5.26315 q^{25} -2.88612 q^{26} -1.51334 q^{28} +8.66918 q^{29} -6.73559 q^{31} -1.00000 q^{32} +1.19647 q^{34} +4.84815 q^{35} +1.45380 q^{37} +0.555554 q^{38} +3.20362 q^{40} +1.84796 q^{41} -2.17379 q^{43} -6.53843 q^{44} +11.4135 q^{47} -4.70981 q^{49} -5.26315 q^{50} +2.88612 q^{52} +12.0613 q^{53} +20.9466 q^{55} +1.51334 q^{56} -8.66918 q^{58} -3.96140 q^{59} -9.03450 q^{61} +6.73559 q^{62} +1.00000 q^{64} -9.24603 q^{65} +7.66794 q^{67} -1.19647 q^{68} -4.84815 q^{70} +9.45607 q^{71} -0.627598 q^{73} -1.45380 q^{74} -0.555554 q^{76} +9.89485 q^{77} +13.7721 q^{79} -3.20362 q^{80} -1.84796 q^{82} -15.5888 q^{83} +3.83304 q^{85} +2.17379 q^{86} +6.53843 q^{88} -5.61844 q^{89} -4.36768 q^{91} -11.4135 q^{94} +1.77978 q^{95} +7.24727 q^{97} +4.70981 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8} + 7 q^{10} - 13 q^{11} - 4 q^{13} - 7 q^{14} + 5 q^{16} - 9 q^{17} + 11 q^{19} - 7 q^{20} + 13 q^{22} - 2 q^{25} + 4 q^{26} + 7 q^{28} + 7 q^{29} - 8 q^{31} - 5 q^{32} + 9 q^{34} - q^{35} + 12 q^{37} - 11 q^{38} + 7 q^{40} + 10 q^{41} + 4 q^{43} - 13 q^{44} + 24 q^{47} - 12 q^{49} + 2 q^{50} - 4 q^{52} - 9 q^{53} + 16 q^{55} - 7 q^{56} - 7 q^{58} + 14 q^{59} + 5 q^{61} + 8 q^{62} + 5 q^{64} - 12 q^{65} + 13 q^{67} - 9 q^{68} + q^{70} + 19 q^{71} + 4 q^{73} - 12 q^{74} + 11 q^{76} - 5 q^{77} + 4 q^{79} - 7 q^{80} - 10 q^{82} - 24 q^{83} + 17 q^{85} - 4 q^{86} + 13 q^{88} - 4 q^{89} - 21 q^{91} - 24 q^{94} + 11 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100}) \) 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 0
\(4\) 1.00000 0.500000
\(5\) −3.20362 −1.43270 −0.716350 0.697741i \(-0.754189\pi\)
−0.716350 + 0.697741i \(0.754189\pi\)
\(6\) 0 0
\(7\) −1.51334 −0.571988 −0.285994 0.958231i \(-0.592324\pi\)
−0.285994 + 0.958231i \(0.592324\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.20362 1.01307
\(11\) −6.53843 −1.97141 −0.985705 0.168479i \(-0.946114\pi\)
−0.985705 + 0.168479i \(0.946114\pi\)
\(12\) 0 0
\(13\) 2.88612 0.800466 0.400233 0.916413i \(-0.368929\pi\)
0.400233 + 0.916413i \(0.368929\pi\)
\(14\) 1.51334 0.404456
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.19647 −0.290188 −0.145094 0.989418i \(-0.546348\pi\)
−0.145094 + 0.989418i \(0.546348\pi\)
\(18\) 0 0
\(19\) −0.555554 −0.127453 −0.0637264 0.997967i \(-0.520299\pi\)
−0.0637264 + 0.997967i \(0.520299\pi\)
\(20\) −3.20362 −0.716350
\(21\) 0 0
\(22\) 6.53843 1.39400
\(23\) 0 0
\(24\) 0 0
\(25\) 5.26315 1.05263
\(26\) −2.88612 −0.566015
\(27\) 0 0
\(28\) −1.51334 −0.285994
\(29\) 8.66918 1.60983 0.804913 0.593393i \(-0.202212\pi\)
0.804913 + 0.593393i \(0.202212\pi\)
\(30\) 0 0
\(31\) −6.73559 −1.20975 −0.604874 0.796321i \(-0.706776\pi\)
−0.604874 + 0.796321i \(0.706776\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.19647 0.205194
\(35\) 4.84815 0.819487
\(36\) 0 0
\(37\) 1.45380 0.239003 0.119502 0.992834i \(-0.461870\pi\)
0.119502 + 0.992834i \(0.461870\pi\)
\(38\) 0.555554 0.0901228
\(39\) 0 0
\(40\) 3.20362 0.506536
\(41\) 1.84796 0.288602 0.144301 0.989534i \(-0.453907\pi\)
0.144301 + 0.989534i \(0.453907\pi\)
\(42\) 0 0
\(43\) −2.17379 −0.331500 −0.165750 0.986168i \(-0.553004\pi\)
−0.165750 + 0.986168i \(0.553004\pi\)
\(44\) −6.53843 −0.985705
\(45\) 0 0
\(46\) 0 0
\(47\) 11.4135 1.66483 0.832416 0.554152i \(-0.186957\pi\)
0.832416 + 0.554152i \(0.186957\pi\)
\(48\) 0 0
\(49\) −4.70981 −0.672830
\(50\) −5.26315 −0.744322
\(51\) 0 0
\(52\) 2.88612 0.400233
\(53\) 12.0613 1.65675 0.828375 0.560175i \(-0.189266\pi\)
0.828375 + 0.560175i \(0.189266\pi\)
\(54\) 0 0
\(55\) 20.9466 2.82444
\(56\) 1.51334 0.202228
\(57\) 0 0
\(58\) −8.66918 −1.13832
\(59\) −3.96140 −0.515730 −0.257865 0.966181i \(-0.583019\pi\)
−0.257865 + 0.966181i \(0.583019\pi\)
\(60\) 0 0
\(61\) −9.03450 −1.15675 −0.578375 0.815771i \(-0.696313\pi\)
−0.578375 + 0.815771i \(0.696313\pi\)
\(62\) 6.73559 0.855420
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −9.24603 −1.14683
\(66\) 0 0
\(67\) 7.66794 0.936788 0.468394 0.883520i \(-0.344833\pi\)
0.468394 + 0.883520i \(0.344833\pi\)
\(68\) −1.19647 −0.145094
\(69\) 0 0
\(70\) −4.84815 −0.579465
\(71\) 9.45607 1.12223 0.561114 0.827738i \(-0.310373\pi\)
0.561114 + 0.827738i \(0.310373\pi\)
\(72\) 0 0
\(73\) −0.627598 −0.0734548 −0.0367274 0.999325i \(-0.511693\pi\)
−0.0367274 + 0.999325i \(0.511693\pi\)
\(74\) −1.45380 −0.169001
\(75\) 0 0
\(76\) −0.555554 −0.0637264
\(77\) 9.89485 1.12762
\(78\) 0 0
\(79\) 13.7721 1.54949 0.774743 0.632276i \(-0.217879\pi\)
0.774743 + 0.632276i \(0.217879\pi\)
\(80\) −3.20362 −0.358175
\(81\) 0 0
\(82\) −1.84796 −0.204073
\(83\) −15.5888 −1.71110 −0.855549 0.517722i \(-0.826780\pi\)
−0.855549 + 0.517722i \(0.826780\pi\)
\(84\) 0 0
\(85\) 3.83304 0.415752
\(86\) 2.17379 0.234406
\(87\) 0 0
\(88\) 6.53843 0.696999
\(89\) −5.61844 −0.595553 −0.297777 0.954636i \(-0.596245\pi\)
−0.297777 + 0.954636i \(0.596245\pi\)
\(90\) 0 0
\(91\) −4.36768 −0.457857
\(92\) 0 0
\(93\) 0 0
\(94\) −11.4135 −1.17721
\(95\) 1.77978 0.182602
\(96\) 0 0
\(97\) 7.24727 0.735848 0.367924 0.929856i \(-0.380069\pi\)
0.367924 + 0.929856i \(0.380069\pi\)
\(98\) 4.70981 0.475763
\(99\) 0 0
\(100\) 5.26315 0.526315
\(101\) 1.02378 0.101870 0.0509348 0.998702i \(-0.483780\pi\)
0.0509348 + 0.998702i \(0.483780\pi\)
\(102\) 0 0
\(103\) −4.54925 −0.448251 −0.224125 0.974560i \(-0.571952\pi\)
−0.224125 + 0.974560i \(0.571952\pi\)
\(104\) −2.88612 −0.283008
\(105\) 0 0
\(106\) −12.0613 −1.17150
\(107\) −0.602987 −0.0582929 −0.0291465 0.999575i \(-0.509279\pi\)
−0.0291465 + 0.999575i \(0.509279\pi\)
\(108\) 0 0
\(109\) 5.87577 0.562796 0.281398 0.959591i \(-0.409202\pi\)
0.281398 + 0.959591i \(0.409202\pi\)
\(110\) −20.9466 −1.99718
\(111\) 0 0
\(112\) −1.51334 −0.142997
\(113\) 12.3076 1.15781 0.578903 0.815397i \(-0.303481\pi\)
0.578903 + 0.815397i \(0.303481\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.66918 0.804913
\(117\) 0 0
\(118\) 3.96140 0.364676
\(119\) 1.81067 0.165984
\(120\) 0 0
\(121\) 31.7511 2.88646
\(122\) 9.03450 0.817945
\(123\) 0 0
\(124\) −6.73559 −0.604874
\(125\) −0.843041 −0.0754039
\(126\) 0 0
\(127\) 8.86983 0.787070 0.393535 0.919310i \(-0.371252\pi\)
0.393535 + 0.919310i \(0.371252\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 9.24603 0.810930
\(131\) −12.5385 −1.09549 −0.547745 0.836645i \(-0.684514\pi\)
−0.547745 + 0.836645i \(0.684514\pi\)
\(132\) 0 0
\(133\) 0.840741 0.0729015
\(134\) −7.66794 −0.662409
\(135\) 0 0
\(136\) 1.19647 0.102597
\(137\) −13.6483 −1.16606 −0.583028 0.812452i \(-0.698132\pi\)
−0.583028 + 0.812452i \(0.698132\pi\)
\(138\) 0 0
\(139\) −11.6537 −0.988455 −0.494227 0.869333i \(-0.664549\pi\)
−0.494227 + 0.869333i \(0.664549\pi\)
\(140\) 4.84815 0.409743
\(141\) 0 0
\(142\) −9.45607 −0.793536
\(143\) −18.8707 −1.57805
\(144\) 0 0
\(145\) −27.7727 −2.30640
\(146\) 0.627598 0.0519404
\(147\) 0 0
\(148\) 1.45380 0.119502
\(149\) 17.3034 1.41755 0.708776 0.705434i \(-0.249248\pi\)
0.708776 + 0.705434i \(0.249248\pi\)
\(150\) 0 0
\(151\) −4.83249 −0.393262 −0.196631 0.980478i \(-0.563000\pi\)
−0.196631 + 0.980478i \(0.563000\pi\)
\(152\) 0.555554 0.0450614
\(153\) 0 0
\(154\) −9.89485 −0.797349
\(155\) 21.5782 1.73321
\(156\) 0 0
\(157\) 5.05031 0.403059 0.201529 0.979482i \(-0.435409\pi\)
0.201529 + 0.979482i \(0.435409\pi\)
\(158\) −13.7721 −1.09565
\(159\) 0 0
\(160\) 3.20362 0.253268
\(161\) 0 0
\(162\) 0 0
\(163\) 3.56743 0.279423 0.139711 0.990192i \(-0.455383\pi\)
0.139711 + 0.990192i \(0.455383\pi\)
\(164\) 1.84796 0.144301
\(165\) 0 0
\(166\) 15.5888 1.20993
\(167\) −4.56038 −0.352893 −0.176446 0.984310i \(-0.556460\pi\)
−0.176446 + 0.984310i \(0.556460\pi\)
\(168\) 0 0
\(169\) −4.67030 −0.359254
\(170\) −3.83304 −0.293981
\(171\) 0 0
\(172\) −2.17379 −0.165750
\(173\) 15.6132 1.18705 0.593525 0.804816i \(-0.297736\pi\)
0.593525 + 0.804816i \(0.297736\pi\)
\(174\) 0 0
\(175\) −7.96492 −0.602092
\(176\) −6.53843 −0.492853
\(177\) 0 0
\(178\) 5.61844 0.421120
\(179\) −6.40521 −0.478748 −0.239374 0.970927i \(-0.576942\pi\)
−0.239374 + 0.970927i \(0.576942\pi\)
\(180\) 0 0
\(181\) 22.1363 1.64538 0.822691 0.568489i \(-0.192472\pi\)
0.822691 + 0.568489i \(0.192472\pi\)
\(182\) 4.36768 0.323754
\(183\) 0 0
\(184\) 0 0
\(185\) −4.65742 −0.342420
\(186\) 0 0
\(187\) 7.82306 0.572079
\(188\) 11.4135 0.832416
\(189\) 0 0
\(190\) −1.77978 −0.129119
\(191\) −2.14915 −0.155507 −0.0777534 0.996973i \(-0.524775\pi\)
−0.0777534 + 0.996973i \(0.524775\pi\)
\(192\) 0 0
\(193\) −11.2460 −0.809506 −0.404753 0.914426i \(-0.632643\pi\)
−0.404753 + 0.914426i \(0.632643\pi\)
\(194\) −7.24727 −0.520323
\(195\) 0 0
\(196\) −4.70981 −0.336415
\(197\) −10.4125 −0.741860 −0.370930 0.928661i \(-0.620961\pi\)
−0.370930 + 0.928661i \(0.620961\pi\)
\(198\) 0 0
\(199\) −15.0998 −1.07039 −0.535197 0.844727i \(-0.679763\pi\)
−0.535197 + 0.844727i \(0.679763\pi\)
\(200\) −5.26315 −0.372161
\(201\) 0 0
\(202\) −1.02378 −0.0720327
\(203\) −13.1194 −0.920800
\(204\) 0 0
\(205\) −5.92014 −0.413480
\(206\) 4.54925 0.316961
\(207\) 0 0
\(208\) 2.88612 0.200117
\(209\) 3.63245 0.251262
\(210\) 0 0
\(211\) 11.4828 0.790512 0.395256 0.918571i \(-0.370656\pi\)
0.395256 + 0.918571i \(0.370656\pi\)
\(212\) 12.0613 0.828375
\(213\) 0 0
\(214\) 0.602987 0.0412193
\(215\) 6.96398 0.474940
\(216\) 0 0
\(217\) 10.1932 0.691961
\(218\) −5.87577 −0.397957
\(219\) 0 0
\(220\) 20.9466 1.41222
\(221\) −3.45317 −0.232285
\(222\) 0 0
\(223\) −15.8063 −1.05847 −0.529235 0.848475i \(-0.677521\pi\)
−0.529235 + 0.848475i \(0.677521\pi\)
\(224\) 1.51334 0.101114
\(225\) 0 0
\(226\) −12.3076 −0.818692
\(227\) −2.59469 −0.172216 −0.0861079 0.996286i \(-0.527443\pi\)
−0.0861079 + 0.996286i \(0.527443\pi\)
\(228\) 0 0
\(229\) −9.99374 −0.660405 −0.330202 0.943910i \(-0.607117\pi\)
−0.330202 + 0.943910i \(0.607117\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −8.66918 −0.569159
\(233\) 27.2948 1.78814 0.894070 0.447927i \(-0.147838\pi\)
0.894070 + 0.447927i \(0.147838\pi\)
\(234\) 0 0
\(235\) −36.5645 −2.38520
\(236\) −3.96140 −0.257865
\(237\) 0 0
\(238\) −1.81067 −0.117368
\(239\) −14.3187 −0.926198 −0.463099 0.886306i \(-0.653263\pi\)
−0.463099 + 0.886306i \(0.653263\pi\)
\(240\) 0 0
\(241\) −14.1957 −0.914428 −0.457214 0.889357i \(-0.651153\pi\)
−0.457214 + 0.889357i \(0.651153\pi\)
\(242\) −31.7511 −2.04103
\(243\) 0 0
\(244\) −9.03450 −0.578375
\(245\) 15.0884 0.963964
\(246\) 0 0
\(247\) −1.60340 −0.102022
\(248\) 6.73559 0.427710
\(249\) 0 0
\(250\) 0.843041 0.0533186
\(251\) 11.9645 0.755194 0.377597 0.925970i \(-0.376750\pi\)
0.377597 + 0.925970i \(0.376750\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.86983 −0.556543
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −25.9967 −1.62163 −0.810815 0.585302i \(-0.800976\pi\)
−0.810815 + 0.585302i \(0.800976\pi\)
\(258\) 0 0
\(259\) −2.20009 −0.136707
\(260\) −9.24603 −0.573414
\(261\) 0 0
\(262\) 12.5385 0.774629
\(263\) 5.74265 0.354107 0.177054 0.984201i \(-0.443343\pi\)
0.177054 + 0.984201i \(0.443343\pi\)
\(264\) 0 0
\(265\) −38.6398 −2.37363
\(266\) −0.840741 −0.0515491
\(267\) 0 0
\(268\) 7.66794 0.468394
\(269\) −8.17926 −0.498699 −0.249349 0.968414i \(-0.580217\pi\)
−0.249349 + 0.968414i \(0.580217\pi\)
\(270\) 0 0
\(271\) −13.5053 −0.820386 −0.410193 0.911999i \(-0.634539\pi\)
−0.410193 + 0.911999i \(0.634539\pi\)
\(272\) −1.19647 −0.0725469
\(273\) 0 0
\(274\) 13.6483 0.824526
\(275\) −34.4128 −2.07517
\(276\) 0 0
\(277\) 14.9697 0.899440 0.449720 0.893169i \(-0.351524\pi\)
0.449720 + 0.893169i \(0.351524\pi\)
\(278\) 11.6537 0.698943
\(279\) 0 0
\(280\) −4.84815 −0.289732
\(281\) 26.9740 1.60913 0.804566 0.593863i \(-0.202398\pi\)
0.804566 + 0.593863i \(0.202398\pi\)
\(282\) 0 0
\(283\) −13.7193 −0.815528 −0.407764 0.913087i \(-0.633691\pi\)
−0.407764 + 0.913087i \(0.633691\pi\)
\(284\) 9.45607 0.561114
\(285\) 0 0
\(286\) 18.8707 1.11585
\(287\) −2.79658 −0.165077
\(288\) 0 0
\(289\) −15.5685 −0.915791
\(290\) 27.7727 1.63087
\(291\) 0 0
\(292\) −0.627598 −0.0367274
\(293\) 9.20499 0.537761 0.268881 0.963174i \(-0.413346\pi\)
0.268881 + 0.963174i \(0.413346\pi\)
\(294\) 0 0
\(295\) 12.6908 0.738887
\(296\) −1.45380 −0.0845004
\(297\) 0 0
\(298\) −17.3034 −1.00236
\(299\) 0 0
\(300\) 0 0
\(301\) 3.28968 0.189614
\(302\) 4.83249 0.278078
\(303\) 0 0
\(304\) −0.555554 −0.0318632
\(305\) 28.9431 1.65728
\(306\) 0 0
\(307\) 1.75676 0.100264 0.0501319 0.998743i \(-0.484036\pi\)
0.0501319 + 0.998743i \(0.484036\pi\)
\(308\) 9.89485 0.563811
\(309\) 0 0
\(310\) −21.5782 −1.22556
\(311\) −30.3629 −1.72172 −0.860860 0.508841i \(-0.830074\pi\)
−0.860860 + 0.508841i \(0.830074\pi\)
\(312\) 0 0
\(313\) −11.0192 −0.622843 −0.311421 0.950272i \(-0.600805\pi\)
−0.311421 + 0.950272i \(0.600805\pi\)
\(314\) −5.05031 −0.285006
\(315\) 0 0
\(316\) 13.7721 0.774743
\(317\) 10.1471 0.569920 0.284960 0.958539i \(-0.408020\pi\)
0.284960 + 0.958539i \(0.408020\pi\)
\(318\) 0 0
\(319\) −56.6828 −3.17363
\(320\) −3.20362 −0.179088
\(321\) 0 0
\(322\) 0 0
\(323\) 0.664706 0.0369852
\(324\) 0 0
\(325\) 15.1901 0.842595
\(326\) −3.56743 −0.197582
\(327\) 0 0
\(328\) −1.84796 −0.102036
\(329\) −17.2725 −0.952263
\(330\) 0 0
\(331\) 29.7790 1.63680 0.818401 0.574648i \(-0.194861\pi\)
0.818401 + 0.574648i \(0.194861\pi\)
\(332\) −15.5888 −0.855549
\(333\) 0 0
\(334\) 4.56038 0.249533
\(335\) −24.5651 −1.34214
\(336\) 0 0
\(337\) 19.0588 1.03820 0.519100 0.854714i \(-0.326267\pi\)
0.519100 + 0.854714i \(0.326267\pi\)
\(338\) 4.67030 0.254031
\(339\) 0 0
\(340\) 3.83304 0.207876
\(341\) 44.0402 2.38491
\(342\) 0 0
\(343\) 17.7209 0.956838
\(344\) 2.17379 0.117203
\(345\) 0 0
\(346\) −15.6132 −0.839371
\(347\) 12.1929 0.654550 0.327275 0.944929i \(-0.393870\pi\)
0.327275 + 0.944929i \(0.393870\pi\)
\(348\) 0 0
\(349\) −15.4022 −0.824459 −0.412229 0.911080i \(-0.635250\pi\)
−0.412229 + 0.911080i \(0.635250\pi\)
\(350\) 7.96492 0.425743
\(351\) 0 0
\(352\) 6.53843 0.348499
\(353\) −18.5308 −0.986295 −0.493147 0.869946i \(-0.664154\pi\)
−0.493147 + 0.869946i \(0.664154\pi\)
\(354\) 0 0
\(355\) −30.2936 −1.60782
\(356\) −5.61844 −0.297777
\(357\) 0 0
\(358\) 6.40521 0.338526
\(359\) −11.7601 −0.620673 −0.310337 0.950627i \(-0.600442\pi\)
−0.310337 + 0.950627i \(0.600442\pi\)
\(360\) 0 0
\(361\) −18.6914 −0.983756
\(362\) −22.1363 −1.16346
\(363\) 0 0
\(364\) −4.36768 −0.228928
\(365\) 2.01058 0.105239
\(366\) 0 0
\(367\) 7.81907 0.408152 0.204076 0.978955i \(-0.434581\pi\)
0.204076 + 0.978955i \(0.434581\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 4.65742 0.242127
\(371\) −18.2528 −0.947640
\(372\) 0 0
\(373\) −2.13817 −0.110710 −0.0553550 0.998467i \(-0.517629\pi\)
−0.0553550 + 0.998467i \(0.517629\pi\)
\(374\) −7.82306 −0.404521
\(375\) 0 0
\(376\) −11.4135 −0.588607
\(377\) 25.0203 1.28861
\(378\) 0 0
\(379\) 21.6725 1.11324 0.556621 0.830767i \(-0.312098\pi\)
0.556621 + 0.830767i \(0.312098\pi\)
\(380\) 1.77978 0.0913009
\(381\) 0 0
\(382\) 2.14915 0.109960
\(383\) 5.00975 0.255986 0.127993 0.991775i \(-0.459146\pi\)
0.127993 + 0.991775i \(0.459146\pi\)
\(384\) 0 0
\(385\) −31.6993 −1.61555
\(386\) 11.2460 0.572407
\(387\) 0 0
\(388\) 7.24727 0.367924
\(389\) −6.02177 −0.305316 −0.152658 0.988279i \(-0.548783\pi\)
−0.152658 + 0.988279i \(0.548783\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 4.70981 0.237881
\(393\) 0 0
\(394\) 10.4125 0.524574
\(395\) −44.1206 −2.21995
\(396\) 0 0
\(397\) 24.0018 1.20461 0.602307 0.798265i \(-0.294248\pi\)
0.602307 + 0.798265i \(0.294248\pi\)
\(398\) 15.0998 0.756883
\(399\) 0 0
\(400\) 5.26315 0.263158
\(401\) −14.4766 −0.722926 −0.361463 0.932387i \(-0.617723\pi\)
−0.361463 + 0.932387i \(0.617723\pi\)
\(402\) 0 0
\(403\) −19.4397 −0.968362
\(404\) 1.02378 0.0509348
\(405\) 0 0
\(406\) 13.1194 0.651104
\(407\) −9.50557 −0.471173
\(408\) 0 0
\(409\) −32.6435 −1.61412 −0.807059 0.590471i \(-0.798942\pi\)
−0.807059 + 0.590471i \(0.798942\pi\)
\(410\) 5.92014 0.292375
\(411\) 0 0
\(412\) −4.54925 −0.224125
\(413\) 5.99493 0.294991
\(414\) 0 0
\(415\) 49.9407 2.45149
\(416\) −2.88612 −0.141504
\(417\) 0 0
\(418\) −3.63245 −0.177669
\(419\) −36.2427 −1.77057 −0.885286 0.465046i \(-0.846038\pi\)
−0.885286 + 0.465046i \(0.846038\pi\)
\(420\) 0 0
\(421\) −6.98570 −0.340462 −0.170231 0.985404i \(-0.554451\pi\)
−0.170231 + 0.985404i \(0.554451\pi\)
\(422\) −11.4828 −0.558976
\(423\) 0 0
\(424\) −12.0613 −0.585749
\(425\) −6.29722 −0.305460
\(426\) 0 0
\(427\) 13.6722 0.661646
\(428\) −0.602987 −0.0291465
\(429\) 0 0
\(430\) −6.96398 −0.335833
\(431\) −33.3054 −1.60426 −0.802132 0.597146i \(-0.796301\pi\)
−0.802132 + 0.597146i \(0.796301\pi\)
\(432\) 0 0
\(433\) 21.5646 1.03633 0.518165 0.855281i \(-0.326615\pi\)
0.518165 + 0.855281i \(0.326615\pi\)
\(434\) −10.1932 −0.489290
\(435\) 0 0
\(436\) 5.87577 0.281398
\(437\) 0 0
\(438\) 0 0
\(439\) 17.2393 0.822785 0.411393 0.911458i \(-0.365043\pi\)
0.411393 + 0.911458i \(0.365043\pi\)
\(440\) −20.9466 −0.998591
\(441\) 0 0
\(442\) 3.45317 0.164251
\(443\) −18.9166 −0.898754 −0.449377 0.893342i \(-0.648354\pi\)
−0.449377 + 0.893342i \(0.648354\pi\)
\(444\) 0 0
\(445\) 17.9993 0.853250
\(446\) 15.8063 0.748452
\(447\) 0 0
\(448\) −1.51334 −0.0714985
\(449\) −23.0511 −1.08785 −0.543924 0.839134i \(-0.683062\pi\)
−0.543924 + 0.839134i \(0.683062\pi\)
\(450\) 0 0
\(451\) −12.0827 −0.568953
\(452\) 12.3076 0.578903
\(453\) 0 0
\(454\) 2.59469 0.121775
\(455\) 13.9924 0.655972
\(456\) 0 0
\(457\) −15.1904 −0.710578 −0.355289 0.934757i \(-0.615617\pi\)
−0.355289 + 0.934757i \(0.615617\pi\)
\(458\) 9.99374 0.466977
\(459\) 0 0
\(460\) 0 0
\(461\) −19.2440 −0.896281 −0.448141 0.893963i \(-0.647914\pi\)
−0.448141 + 0.893963i \(0.647914\pi\)
\(462\) 0 0
\(463\) −3.03240 −0.140927 −0.0704637 0.997514i \(-0.522448\pi\)
−0.0704637 + 0.997514i \(0.522448\pi\)
\(464\) 8.66918 0.402456
\(465\) 0 0
\(466\) −27.2948 −1.26441
\(467\) 2.85012 0.131888 0.0659439 0.997823i \(-0.478994\pi\)
0.0659439 + 0.997823i \(0.478994\pi\)
\(468\) 0 0
\(469\) −11.6042 −0.535831
\(470\) 36.5645 1.68659
\(471\) 0 0
\(472\) 3.96140 0.182338
\(473\) 14.2132 0.653522
\(474\) 0 0
\(475\) −2.92397 −0.134161
\(476\) 1.81067 0.0829918
\(477\) 0 0
\(478\) 14.3187 0.654921
\(479\) 37.2607 1.70249 0.851243 0.524772i \(-0.175849\pi\)
0.851243 + 0.524772i \(0.175849\pi\)
\(480\) 0 0
\(481\) 4.19584 0.191314
\(482\) 14.1957 0.646598
\(483\) 0 0
\(484\) 31.7511 1.44323
\(485\) −23.2175 −1.05425
\(486\) 0 0
\(487\) −30.7455 −1.39321 −0.696606 0.717454i \(-0.745307\pi\)
−0.696606 + 0.717454i \(0.745307\pi\)
\(488\) 9.03450 0.408973
\(489\) 0 0
\(490\) −15.0884 −0.681625
\(491\) −7.81494 −0.352683 −0.176342 0.984329i \(-0.556426\pi\)
−0.176342 + 0.984329i \(0.556426\pi\)
\(492\) 0 0
\(493\) −10.3724 −0.467151
\(494\) 1.60340 0.0721403
\(495\) 0 0
\(496\) −6.73559 −0.302437
\(497\) −14.3102 −0.641901
\(498\) 0 0
\(499\) 12.9597 0.580154 0.290077 0.957003i \(-0.406319\pi\)
0.290077 + 0.957003i \(0.406319\pi\)
\(500\) −0.843041 −0.0377020
\(501\) 0 0
\(502\) −11.9645 −0.534003
\(503\) 19.6304 0.875276 0.437638 0.899151i \(-0.355815\pi\)
0.437638 + 0.899151i \(0.355815\pi\)
\(504\) 0 0
\(505\) −3.27979 −0.145949
\(506\) 0 0
\(507\) 0 0
\(508\) 8.86983 0.393535
\(509\) −6.25244 −0.277134 −0.138567 0.990353i \(-0.544250\pi\)
−0.138567 + 0.990353i \(0.544250\pi\)
\(510\) 0 0
\(511\) 0.949767 0.0420152
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 25.9967 1.14667
\(515\) 14.5740 0.642209
\(516\) 0 0
\(517\) −74.6264 −3.28207
\(518\) 2.20009 0.0966664
\(519\) 0 0
\(520\) 9.24603 0.405465
\(521\) 23.9904 1.05104 0.525518 0.850782i \(-0.323871\pi\)
0.525518 + 0.850782i \(0.323871\pi\)
\(522\) 0 0
\(523\) 7.42756 0.324784 0.162392 0.986726i \(-0.448079\pi\)
0.162392 + 0.986726i \(0.448079\pi\)
\(524\) −12.5385 −0.547745
\(525\) 0 0
\(526\) −5.74265 −0.250392
\(527\) 8.05895 0.351054
\(528\) 0 0
\(529\) 0 0
\(530\) 38.6398 1.67841
\(531\) 0 0
\(532\) 0.840741 0.0364507
\(533\) 5.33343 0.231016
\(534\) 0 0
\(535\) 1.93174 0.0835163
\(536\) −7.66794 −0.331204
\(537\) 0 0
\(538\) 8.17926 0.352633
\(539\) 30.7948 1.32642
\(540\) 0 0
\(541\) 20.4393 0.878752 0.439376 0.898303i \(-0.355200\pi\)
0.439376 + 0.898303i \(0.355200\pi\)
\(542\) 13.5053 0.580101
\(543\) 0 0
\(544\) 1.19647 0.0512984
\(545\) −18.8237 −0.806318
\(546\) 0 0
\(547\) −9.31027 −0.398078 −0.199039 0.979992i \(-0.563782\pi\)
−0.199039 + 0.979992i \(0.563782\pi\)
\(548\) −13.6483 −0.583028
\(549\) 0 0
\(550\) 34.4128 1.46736
\(551\) −4.81620 −0.205177
\(552\) 0 0
\(553\) −20.8419 −0.886287
\(554\) −14.9697 −0.636000
\(555\) 0 0
\(556\) −11.6537 −0.494227
\(557\) −31.9209 −1.35253 −0.676265 0.736658i \(-0.736403\pi\)
−0.676265 + 0.736658i \(0.736403\pi\)
\(558\) 0 0
\(559\) −6.27382 −0.265354
\(560\) 4.84815 0.204872
\(561\) 0 0
\(562\) −26.9740 −1.13783
\(563\) −16.8512 −0.710195 −0.355097 0.934829i \(-0.615552\pi\)
−0.355097 + 0.934829i \(0.615552\pi\)
\(564\) 0 0
\(565\) −39.4289 −1.65879
\(566\) 13.7193 0.576665
\(567\) 0 0
\(568\) −9.45607 −0.396768
\(569\) −2.18185 −0.0914680 −0.0457340 0.998954i \(-0.514563\pi\)
−0.0457340 + 0.998954i \(0.514563\pi\)
\(570\) 0 0
\(571\) 8.97740 0.375693 0.187846 0.982198i \(-0.439849\pi\)
0.187846 + 0.982198i \(0.439849\pi\)
\(572\) −18.8707 −0.789024
\(573\) 0 0
\(574\) 2.79658 0.116727
\(575\) 0 0
\(576\) 0 0
\(577\) −41.1842 −1.71452 −0.857261 0.514882i \(-0.827836\pi\)
−0.857261 + 0.514882i \(0.827836\pi\)
\(578\) 15.5685 0.647562
\(579\) 0 0
\(580\) −27.7727 −1.15320
\(581\) 23.5912 0.978727
\(582\) 0 0
\(583\) −78.8620 −3.26613
\(584\) 0.627598 0.0259702
\(585\) 0 0
\(586\) −9.20499 −0.380254
\(587\) −9.61116 −0.396695 −0.198348 0.980132i \(-0.563557\pi\)
−0.198348 + 0.980132i \(0.563557\pi\)
\(588\) 0 0
\(589\) 3.74198 0.154186
\(590\) −12.6908 −0.522472
\(591\) 0 0
\(592\) 1.45380 0.0597508
\(593\) 36.4575 1.49713 0.748564 0.663062i \(-0.230744\pi\)
0.748564 + 0.663062i \(0.230744\pi\)
\(594\) 0 0
\(595\) −5.80068 −0.237805
\(596\) 17.3034 0.708776
\(597\) 0 0
\(598\) 0 0
\(599\) 6.93439 0.283331 0.141666 0.989915i \(-0.454754\pi\)
0.141666 + 0.989915i \(0.454754\pi\)
\(600\) 0 0
\(601\) 16.7707 0.684089 0.342045 0.939684i \(-0.388881\pi\)
0.342045 + 0.939684i \(0.388881\pi\)
\(602\) −3.28968 −0.134077
\(603\) 0 0
\(604\) −4.83249 −0.196631
\(605\) −101.718 −4.13543
\(606\) 0 0
\(607\) −10.8715 −0.441259 −0.220629 0.975358i \(-0.570811\pi\)
−0.220629 + 0.975358i \(0.570811\pi\)
\(608\) 0.555554 0.0225307
\(609\) 0 0
\(610\) −28.9431 −1.17187
\(611\) 32.9408 1.33264
\(612\) 0 0
\(613\) −20.7458 −0.837914 −0.418957 0.908006i \(-0.637604\pi\)
−0.418957 + 0.908006i \(0.637604\pi\)
\(614\) −1.75676 −0.0708972
\(615\) 0 0
\(616\) −9.89485 −0.398675
\(617\) 3.09354 0.124541 0.0622706 0.998059i \(-0.480166\pi\)
0.0622706 + 0.998059i \(0.480166\pi\)
\(618\) 0 0
\(619\) 13.0130 0.523038 0.261519 0.965198i \(-0.415777\pi\)
0.261519 + 0.965198i \(0.415777\pi\)
\(620\) 21.5782 0.866603
\(621\) 0 0
\(622\) 30.3629 1.21744
\(623\) 8.50259 0.340649
\(624\) 0 0
\(625\) −23.6150 −0.944599
\(626\) 11.0192 0.440416
\(627\) 0 0
\(628\) 5.05031 0.201529
\(629\) −1.73943 −0.0693557
\(630\) 0 0
\(631\) −9.82295 −0.391045 −0.195523 0.980699i \(-0.562640\pi\)
−0.195523 + 0.980699i \(0.562640\pi\)
\(632\) −13.7721 −0.547826
\(633\) 0 0
\(634\) −10.1471 −0.402994
\(635\) −28.4155 −1.12764
\(636\) 0 0
\(637\) −13.5931 −0.538578
\(638\) 56.6828 2.24409
\(639\) 0 0
\(640\) 3.20362 0.126634
\(641\) 13.6360 0.538590 0.269295 0.963058i \(-0.413209\pi\)
0.269295 + 0.963058i \(0.413209\pi\)
\(642\) 0 0
\(643\) −3.57857 −0.141125 −0.0705626 0.997507i \(-0.522479\pi\)
−0.0705626 + 0.997507i \(0.522479\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.664706 −0.0261525
\(647\) 26.2492 1.03196 0.515982 0.856599i \(-0.327427\pi\)
0.515982 + 0.856599i \(0.327427\pi\)
\(648\) 0 0
\(649\) 25.9013 1.01672
\(650\) −15.1901 −0.595805
\(651\) 0 0
\(652\) 3.56743 0.139711
\(653\) −25.7390 −1.00725 −0.503623 0.863923i \(-0.668000\pi\)
−0.503623 + 0.863923i \(0.668000\pi\)
\(654\) 0 0
\(655\) 40.1684 1.56951
\(656\) 1.84796 0.0721505
\(657\) 0 0
\(658\) 17.2725 0.673352
\(659\) −10.1996 −0.397318 −0.198659 0.980069i \(-0.563659\pi\)
−0.198659 + 0.980069i \(0.563659\pi\)
\(660\) 0 0
\(661\) 29.9545 1.16509 0.582547 0.812797i \(-0.302056\pi\)
0.582547 + 0.812797i \(0.302056\pi\)
\(662\) −29.7790 −1.15739
\(663\) 0 0
\(664\) 15.5888 0.604965
\(665\) −2.69341 −0.104446
\(666\) 0 0
\(667\) 0 0
\(668\) −4.56038 −0.176446
\(669\) 0 0
\(670\) 24.5651 0.949034
\(671\) 59.0714 2.28043
\(672\) 0 0
\(673\) −2.02716 −0.0781413 −0.0390707 0.999236i \(-0.512440\pi\)
−0.0390707 + 0.999236i \(0.512440\pi\)
\(674\) −19.0588 −0.734118
\(675\) 0 0
\(676\) −4.67030 −0.179627
\(677\) 2.46315 0.0946664 0.0473332 0.998879i \(-0.484928\pi\)
0.0473332 + 0.998879i \(0.484928\pi\)
\(678\) 0 0
\(679\) −10.9676 −0.420896
\(680\) −3.83304 −0.146990
\(681\) 0 0
\(682\) −44.0402 −1.68638
\(683\) 30.3002 1.15941 0.579703 0.814828i \(-0.303169\pi\)
0.579703 + 0.814828i \(0.303169\pi\)
\(684\) 0 0
\(685\) 43.7240 1.67061
\(686\) −17.7209 −0.676587
\(687\) 0 0
\(688\) −2.17379 −0.0828749
\(689\) 34.8104 1.32617
\(690\) 0 0
\(691\) −1.71993 −0.0654294 −0.0327147 0.999465i \(-0.510415\pi\)
−0.0327147 + 0.999465i \(0.510415\pi\)
\(692\) 15.6132 0.593525
\(693\) 0 0
\(694\) −12.1929 −0.462837
\(695\) 37.3340 1.41616
\(696\) 0 0
\(697\) −2.21103 −0.0837487
\(698\) 15.4022 0.582980
\(699\) 0 0
\(700\) −7.96492 −0.301046
\(701\) −0.739062 −0.0279140 −0.0139570 0.999903i \(-0.504443\pi\)
−0.0139570 + 0.999903i \(0.504443\pi\)
\(702\) 0 0
\(703\) −0.807665 −0.0304616
\(704\) −6.53843 −0.246426
\(705\) 0 0
\(706\) 18.5308 0.697416
\(707\) −1.54932 −0.0582682
\(708\) 0 0
\(709\) −22.5819 −0.848081 −0.424040 0.905643i \(-0.639389\pi\)
−0.424040 + 0.905643i \(0.639389\pi\)
\(710\) 30.2936 1.13690
\(711\) 0 0
\(712\) 5.61844 0.210560
\(713\) 0 0
\(714\) 0 0
\(715\) 60.4545 2.26087
\(716\) −6.40521 −0.239374
\(717\) 0 0
\(718\) 11.7601 0.438882
\(719\) −10.9509 −0.408400 −0.204200 0.978929i \(-0.565459\pi\)
−0.204200 + 0.978929i \(0.565459\pi\)
\(720\) 0 0
\(721\) 6.88454 0.256394
\(722\) 18.6914 0.695620
\(723\) 0 0
\(724\) 22.1363 0.822691
\(725\) 45.6272 1.69455
\(726\) 0 0
\(727\) −37.2793 −1.38261 −0.691307 0.722562i \(-0.742965\pi\)
−0.691307 + 0.722562i \(0.742965\pi\)
\(728\) 4.36768 0.161877
\(729\) 0 0
\(730\) −2.01058 −0.0744150
\(731\) 2.60088 0.0961971
\(732\) 0 0
\(733\) −32.1008 −1.18567 −0.592835 0.805324i \(-0.701991\pi\)
−0.592835 + 0.805324i \(0.701991\pi\)
\(734\) −7.81907 −0.288607
\(735\) 0 0
\(736\) 0 0
\(737\) −50.1363 −1.84679
\(738\) 0 0
\(739\) 37.5076 1.37974 0.689870 0.723934i \(-0.257668\pi\)
0.689870 + 0.723934i \(0.257668\pi\)
\(740\) −4.65742 −0.171210
\(741\) 0 0
\(742\) 18.2528 0.670083
\(743\) 31.8791 1.16953 0.584766 0.811202i \(-0.301186\pi\)
0.584766 + 0.811202i \(0.301186\pi\)
\(744\) 0 0
\(745\) −55.4335 −2.03093
\(746\) 2.13817 0.0782838
\(747\) 0 0
\(748\) 7.82306 0.286039
\(749\) 0.912522 0.0333428
\(750\) 0 0
\(751\) 17.9494 0.654983 0.327491 0.944854i \(-0.393797\pi\)
0.327491 + 0.944854i \(0.393797\pi\)
\(752\) 11.4135 0.416208
\(753\) 0 0
\(754\) −25.0203 −0.911186
\(755\) 15.4814 0.563427
\(756\) 0 0
\(757\) 10.9617 0.398410 0.199205 0.979958i \(-0.436164\pi\)
0.199205 + 0.979958i \(0.436164\pi\)
\(758\) −21.6725 −0.787180
\(759\) 0 0
\(760\) −1.77978 −0.0645595
\(761\) 0.365855 0.0132622 0.00663111 0.999978i \(-0.497889\pi\)
0.00663111 + 0.999978i \(0.497889\pi\)
\(762\) 0 0
\(763\) −8.89201 −0.321912
\(764\) −2.14915 −0.0777534
\(765\) 0 0
\(766\) −5.00975 −0.181010
\(767\) −11.4331 −0.412825
\(768\) 0 0
\(769\) −0.305975 −0.0110337 −0.00551686 0.999985i \(-0.501756\pi\)
−0.00551686 + 0.999985i \(0.501756\pi\)
\(770\) 31.6993 1.14236
\(771\) 0 0
\(772\) −11.2460 −0.404753
\(773\) −3.53034 −0.126977 −0.0634887 0.997983i \(-0.520223\pi\)
−0.0634887 + 0.997983i \(0.520223\pi\)
\(774\) 0 0
\(775\) −35.4504 −1.27342
\(776\) −7.24727 −0.260162
\(777\) 0 0
\(778\) 6.02177 0.215891
\(779\) −1.02664 −0.0367832
\(780\) 0 0
\(781\) −61.8278 −2.21237
\(782\) 0 0
\(783\) 0 0
\(784\) −4.70981 −0.168208
\(785\) −16.1793 −0.577462
\(786\) 0 0
\(787\) 2.24943 0.0801837 0.0400918 0.999196i \(-0.487235\pi\)
0.0400918 + 0.999196i \(0.487235\pi\)
\(788\) −10.4125 −0.370930
\(789\) 0 0
\(790\) 44.1206 1.56974
\(791\) −18.6256 −0.662250
\(792\) 0 0
\(793\) −26.0747 −0.925939
\(794\) −24.0018 −0.851790
\(795\) 0 0
\(796\) −15.0998 −0.535197
\(797\) −39.6694 −1.40516 −0.702581 0.711604i \(-0.747969\pi\)
−0.702581 + 0.711604i \(0.747969\pi\)
\(798\) 0 0
\(799\) −13.6560 −0.483113
\(800\) −5.26315 −0.186081
\(801\) 0 0
\(802\) 14.4766 0.511186
\(803\) 4.10350 0.144810
\(804\) 0 0
\(805\) 0 0
\(806\) 19.4397 0.684735
\(807\) 0 0
\(808\) −1.02378 −0.0360164
\(809\) 38.4898 1.35323 0.676614 0.736338i \(-0.263447\pi\)
0.676614 + 0.736338i \(0.263447\pi\)
\(810\) 0 0
\(811\) −14.4390 −0.507021 −0.253511 0.967333i \(-0.581585\pi\)
−0.253511 + 0.967333i \(0.581585\pi\)
\(812\) −13.1194 −0.460400
\(813\) 0 0
\(814\) 9.50557 0.333170
\(815\) −11.4287 −0.400329
\(816\) 0 0
\(817\) 1.20766 0.0422506
\(818\) 32.6435 1.14135
\(819\) 0 0
\(820\) −5.92014 −0.206740
\(821\) −29.2724 −1.02161 −0.510806 0.859696i \(-0.670653\pi\)
−0.510806 + 0.859696i \(0.670653\pi\)
\(822\) 0 0
\(823\) −16.6990 −0.582091 −0.291045 0.956709i \(-0.594003\pi\)
−0.291045 + 0.956709i \(0.594003\pi\)
\(824\) 4.54925 0.158481
\(825\) 0 0
\(826\) −5.99493 −0.208590
\(827\) −20.3196 −0.706582 −0.353291 0.935514i \(-0.614937\pi\)
−0.353291 + 0.935514i \(0.614937\pi\)
\(828\) 0 0
\(829\) 0.285271 0.00990788 0.00495394 0.999988i \(-0.498423\pi\)
0.00495394 + 0.999988i \(0.498423\pi\)
\(830\) −49.9407 −1.73347
\(831\) 0 0
\(832\) 2.88612 0.100058
\(833\) 5.63517 0.195247
\(834\) 0 0
\(835\) 14.6097 0.505589
\(836\) 3.63245 0.125631
\(837\) 0 0
\(838\) 36.2427 1.25198
\(839\) −24.4524 −0.844192 −0.422096 0.906551i \(-0.638705\pi\)
−0.422096 + 0.906551i \(0.638705\pi\)
\(840\) 0 0
\(841\) 46.1546 1.59154
\(842\) 6.98570 0.240743
\(843\) 0 0
\(844\) 11.4828 0.395256
\(845\) 14.9618 0.514703
\(846\) 0 0
\(847\) −48.0500 −1.65102
\(848\) 12.0613 0.414187
\(849\) 0 0
\(850\) 6.29722 0.215993
\(851\) 0 0
\(852\) 0 0
\(853\) 28.9592 0.991545 0.495773 0.868452i \(-0.334885\pi\)
0.495773 + 0.868452i \(0.334885\pi\)
\(854\) −13.6722 −0.467855
\(855\) 0 0
\(856\) 0.602987 0.0206097
\(857\) 30.4523 1.04023 0.520116 0.854096i \(-0.325889\pi\)
0.520116 + 0.854096i \(0.325889\pi\)
\(858\) 0 0
\(859\) 42.3725 1.44573 0.722866 0.690988i \(-0.242824\pi\)
0.722866 + 0.690988i \(0.242824\pi\)
\(860\) 6.96398 0.237470
\(861\) 0 0
\(862\) 33.3054 1.13439
\(863\) 49.6854 1.69131 0.845655 0.533729i \(-0.179210\pi\)
0.845655 + 0.533729i \(0.179210\pi\)
\(864\) 0 0
\(865\) −50.0187 −1.70069
\(866\) −21.5646 −0.732796
\(867\) 0 0
\(868\) 10.1932 0.345980
\(869\) −90.0482 −3.05467
\(870\) 0 0
\(871\) 22.1306 0.749867
\(872\) −5.87577 −0.198979
\(873\) 0 0
\(874\) 0 0
\(875\) 1.27581 0.0431301
\(876\) 0 0
\(877\) 4.98560 0.168352 0.0841759 0.996451i \(-0.473174\pi\)
0.0841759 + 0.996451i \(0.473174\pi\)
\(878\) −17.2393 −0.581797
\(879\) 0 0
\(880\) 20.9466 0.706110
\(881\) 49.2503 1.65928 0.829642 0.558296i \(-0.188545\pi\)
0.829642 + 0.558296i \(0.188545\pi\)
\(882\) 0 0
\(883\) −29.1640 −0.981448 −0.490724 0.871315i \(-0.663268\pi\)
−0.490724 + 0.871315i \(0.663268\pi\)
\(884\) −3.45317 −0.116143
\(885\) 0 0
\(886\) 18.9166 0.635515
\(887\) −6.09320 −0.204590 −0.102295 0.994754i \(-0.532619\pi\)
−0.102295 + 0.994754i \(0.532619\pi\)
\(888\) 0 0
\(889\) −13.4230 −0.450194
\(890\) −17.9993 −0.603339
\(891\) 0 0
\(892\) −15.8063 −0.529235
\(893\) −6.34082 −0.212188
\(894\) 0 0
\(895\) 20.5198 0.685903
\(896\) 1.51334 0.0505570
\(897\) 0 0
\(898\) 23.0511 0.769225
\(899\) −58.3920 −1.94748
\(900\) 0 0
\(901\) −14.4310 −0.480768
\(902\) 12.0827 0.402311
\(903\) 0 0
\(904\) −12.3076 −0.409346
\(905\) −70.9163 −2.35734
\(906\) 0 0
\(907\) 2.46052 0.0817004 0.0408502 0.999165i \(-0.486993\pi\)
0.0408502 + 0.999165i \(0.486993\pi\)
\(908\) −2.59469 −0.0861079
\(909\) 0 0
\(910\) −13.9924 −0.463842
\(911\) −52.4143 −1.73656 −0.868281 0.496073i \(-0.834775\pi\)
−0.868281 + 0.496073i \(0.834775\pi\)
\(912\) 0 0
\(913\) 101.927 3.37328
\(914\) 15.1904 0.502454
\(915\) 0 0
\(916\) −9.99374 −0.330202
\(917\) 18.9749 0.626607
\(918\) 0 0
\(919\) 12.0132 0.396280 0.198140 0.980174i \(-0.436510\pi\)
0.198140 + 0.980174i \(0.436510\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 19.2440 0.633767
\(923\) 27.2914 0.898306
\(924\) 0 0
\(925\) 7.65157 0.251582
\(926\) 3.03240 0.0996507
\(927\) 0 0
\(928\) −8.66918 −0.284580
\(929\) 17.2437 0.565747 0.282874 0.959157i \(-0.408712\pi\)
0.282874 + 0.959157i \(0.408712\pi\)
\(930\) 0 0
\(931\) 2.61656 0.0857541
\(932\) 27.2948 0.894070
\(933\) 0 0
\(934\) −2.85012 −0.0932588
\(935\) −25.0621 −0.819617
\(936\) 0 0
\(937\) −19.8875 −0.649696 −0.324848 0.945766i \(-0.605313\pi\)
−0.324848 + 0.945766i \(0.605313\pi\)
\(938\) 11.6042 0.378890
\(939\) 0 0
\(940\) −36.5645 −1.19260
\(941\) −14.7942 −0.482278 −0.241139 0.970491i \(-0.577521\pi\)
−0.241139 + 0.970491i \(0.577521\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −3.96140 −0.128933
\(945\) 0 0
\(946\) −14.2132 −0.462110
\(947\) 18.1501 0.589799 0.294900 0.955528i \(-0.404714\pi\)
0.294900 + 0.955528i \(0.404714\pi\)
\(948\) 0 0
\(949\) −1.81132 −0.0587981
\(950\) 2.92397 0.0948660
\(951\) 0 0
\(952\) −1.81067 −0.0586841
\(953\) −26.1661 −0.847605 −0.423802 0.905755i \(-0.639305\pi\)
−0.423802 + 0.905755i \(0.639305\pi\)
\(954\) 0 0
\(955\) 6.88504 0.222795
\(956\) −14.3187 −0.463099
\(957\) 0 0
\(958\) −37.2607 −1.20384
\(959\) 20.6545 0.666969
\(960\) 0 0
\(961\) 14.3681 0.463488
\(962\) −4.19584 −0.135279
\(963\) 0 0
\(964\) −14.1957 −0.457214
\(965\) 36.0279 1.15978
\(966\) 0 0
\(967\) −22.8164 −0.733726 −0.366863 0.930275i \(-0.619568\pi\)
−0.366863 + 0.930275i \(0.619568\pi\)
\(968\) −31.7511 −1.02052
\(969\) 0 0
\(970\) 23.2175 0.745468
\(971\) −2.27238 −0.0729240 −0.0364620 0.999335i \(-0.511609\pi\)
−0.0364620 + 0.999335i \(0.511609\pi\)
\(972\) 0 0
\(973\) 17.6360 0.565384
\(974\) 30.7455 0.985150
\(975\) 0 0
\(976\) −9.03450 −0.289187
\(977\) −37.7855 −1.20887 −0.604433 0.796656i \(-0.706600\pi\)
−0.604433 + 0.796656i \(0.706600\pi\)
\(978\) 0 0
\(979\) 36.7358 1.17408
\(980\) 15.0884 0.481982
\(981\) 0 0
\(982\) 7.81494 0.249385
\(983\) −53.3063 −1.70021 −0.850104 0.526615i \(-0.823461\pi\)
−0.850104 + 0.526615i \(0.823461\pi\)
\(984\) 0 0
\(985\) 33.3577 1.06286
\(986\) 10.3724 0.330326
\(987\) 0 0
\(988\) −1.60340 −0.0510109
\(989\) 0 0
\(990\) 0 0
\(991\) 17.1670 0.545328 0.272664 0.962109i \(-0.412095\pi\)
0.272664 + 0.962109i \(0.412095\pi\)
\(992\) 6.73559 0.213855
\(993\) 0 0
\(994\) 14.3102 0.453893
\(995\) 48.3739 1.53356
\(996\) 0 0
\(997\) −28.8767 −0.914535 −0.457268 0.889329i \(-0.651172\pi\)
−0.457268 + 0.889329i \(0.651172\pi\)
\(998\) −12.9597 −0.410231
\(999\) 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 9522.2.a.bq.1.1 5
3.2 odd 2 3174.2.a.bd.1.5 5
23.15 odd 22 414.2.i.d.271.1 10
23.20 odd 22 414.2.i.d.55.1 10
23.22 odd 2 9522.2.a.bt.1.5 5
69.20 even 22 138.2.e.a.55.1 10
69.38 even 22 138.2.e.a.133.1 yes 10
69.68 even 2 3174.2.a.bc.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.a.55.1 10 69.20 even 22
138.2.e.a.133.1 yes 10 69.38 even 22
414.2.i.d.55.1 10 23.20 odd 22
414.2.i.d.271.1 10 23.15 odd 22
3174.2.a.bc.1.1 5 69.68 even 2
3174.2.a.bd.1.5 5 3.2 odd 2
9522.2.a.bq.1.1 5 1.1 even 1 trivial
9522.2.a.bt.1.5 5 23.22 odd 2