Properties

Label 4410.2.d.d.4409.7
Level $4410$
Weight $2$
Character 4410.4409
Analytic conductor $35.214$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4410,2,Mod(4409,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4410.4409");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4409.7
Character \(\chi\) \(=\) 4410.4409
Dual form 4410.2.d.d.4409.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +(-1.12606 - 1.93184i) q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +(-1.12606 - 1.93184i) q^{5} +1.00000 q^{8} +(-1.12606 - 1.93184i) q^{10} -0.602126i q^{11} -0.0571454 q^{13} +1.00000 q^{16} -0.347217i q^{17} -4.92239i q^{19} +(-1.12606 - 1.93184i) q^{20} -0.602126i q^{22} -1.45795 q^{23} +(-2.46398 + 4.35073i) q^{25} -0.0571454 q^{26} +6.49327i q^{29} -5.05969i q^{31} +1.00000 q^{32} -0.347217i q^{34} -1.16480i q^{37} -4.92239i q^{38} +(-1.12606 - 1.93184i) q^{40} -2.64008 q^{41} -12.0070i q^{43} -0.602126i q^{44} -1.45795 q^{46} -7.74571i q^{47} +(-2.46398 + 4.35073i) q^{50} -0.0571454 q^{52} -5.55299 q^{53} +(-1.16321 + 0.678030i) q^{55} +6.49327i q^{58} +0.890569 q^{59} -5.72460i q^{61} -5.05969i q^{62} +1.00000 q^{64} +(0.0643492 + 0.110395i) q^{65} -0.110634i q^{67} -0.347217i q^{68} +12.7816i q^{71} -0.914240 q^{73} -1.16480i q^{74} -4.92239i q^{76} +1.38275 q^{79} +(-1.12606 - 1.93184i) q^{80} -2.64008 q^{82} +0.429722i q^{83} +(-0.670765 + 0.390987i) q^{85} -12.0070i q^{86} -0.602126i q^{88} -15.9541 q^{89} -1.45795 q^{92} -7.74571i q^{94} +(-9.50925 + 5.54291i) q^{95} -13.1539 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{8} + 24 q^{16} - 32 q^{23} - 16 q^{25} + 24 q^{32} - 32 q^{46} - 16 q^{50} + 32 q^{53} + 24 q^{64} - 32 q^{85} - 32 q^{92} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(2647\) \(3431\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) −1.12606 1.93184i −0.503590 0.863943i
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.12606 1.93184i −0.356092 0.610900i
\(11\) 0.602126i 0.181548i −0.995872 0.0907739i \(-0.971066\pi\)
0.995872 0.0907739i \(-0.0289341\pi\)
\(12\) 0 0
\(13\) −0.0571454 −0.0158493 −0.00792464 0.999969i \(-0.502523\pi\)
−0.00792464 + 0.999969i \(0.502523\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.347217i 0.0842124i −0.999113 0.0421062i \(-0.986593\pi\)
0.999113 0.0421062i \(-0.0134068\pi\)
\(18\) 0 0
\(19\) 4.92239i 1.12927i −0.825339 0.564637i \(-0.809016\pi\)
0.825339 0.564637i \(-0.190984\pi\)
\(20\) −1.12606 1.93184i −0.251795 0.431972i
\(21\) 0 0
\(22\) 0.602126i 0.128374i
\(23\) −1.45795 −0.304004 −0.152002 0.988380i \(-0.548572\pi\)
−0.152002 + 0.988380i \(0.548572\pi\)
\(24\) 0 0
\(25\) −2.46398 + 4.35073i −0.492795 + 0.870145i
\(26\) −0.0571454 −0.0112071
\(27\) 0 0
\(28\) 0 0
\(29\) 6.49327i 1.20577i 0.797828 + 0.602885i \(0.205982\pi\)
−0.797828 + 0.602885i \(0.794018\pi\)
\(30\) 0 0
\(31\) 5.05969i 0.908748i −0.890811 0.454374i \(-0.849863\pi\)
0.890811 0.454374i \(-0.150137\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.347217i 0.0595472i
\(35\) 0 0
\(36\) 0 0
\(37\) 1.16480i 0.191493i −0.995406 0.0957464i \(-0.969476\pi\)
0.995406 0.0957464i \(-0.0305238\pi\)
\(38\) 4.92239i 0.798517i
\(39\) 0 0
\(40\) −1.12606 1.93184i −0.178046 0.305450i
\(41\) −2.64008 −0.412311 −0.206156 0.978519i \(-0.566095\pi\)
−0.206156 + 0.978519i \(0.566095\pi\)
\(42\) 0 0
\(43\) 12.0070i 1.83105i −0.402260 0.915526i \(-0.631775\pi\)
0.402260 0.915526i \(-0.368225\pi\)
\(44\) 0.602126i 0.0907739i
\(45\) 0 0
\(46\) −1.45795 −0.214963
\(47\) 7.74571i 1.12983i −0.825150 0.564914i \(-0.808909\pi\)
0.825150 0.564914i \(-0.191091\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.46398 + 4.35073i −0.348459 + 0.615286i
\(51\) 0 0
\(52\) −0.0571454 −0.00792464
\(53\) −5.55299 −0.762762 −0.381381 0.924418i \(-0.624551\pi\)
−0.381381 + 0.924418i \(0.624551\pi\)
\(54\) 0 0
\(55\) −1.16321 + 0.678030i −0.156847 + 0.0914256i
\(56\) 0 0
\(57\) 0 0
\(58\) 6.49327i 0.852608i
\(59\) 0.890569 0.115942 0.0579711 0.998318i \(-0.481537\pi\)
0.0579711 + 0.998318i \(0.481537\pi\)
\(60\) 0 0
\(61\) 5.72460i 0.732959i −0.930426 0.366480i \(-0.880563\pi\)
0.930426 0.366480i \(-0.119437\pi\)
\(62\) 5.05969i 0.642582i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.0643492 + 0.110395i 0.00798153 + 0.0136929i
\(66\) 0 0
\(67\) 0.110634i 0.0135160i −0.999977 0.00675802i \(-0.997849\pi\)
0.999977 0.00675802i \(-0.00215116\pi\)
\(68\) 0.347217i 0.0421062i
\(69\) 0 0
\(70\) 0 0
\(71\) 12.7816i 1.51689i 0.651735 + 0.758447i \(0.274041\pi\)
−0.651735 + 0.758447i \(0.725959\pi\)
\(72\) 0 0
\(73\) −0.914240 −0.107004 −0.0535018 0.998568i \(-0.517038\pi\)
−0.0535018 + 0.998568i \(0.517038\pi\)
\(74\) 1.16480i 0.135406i
\(75\) 0 0
\(76\) 4.92239i 0.564637i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.38275 0.155572 0.0777860 0.996970i \(-0.475215\pi\)
0.0777860 + 0.996970i \(0.475215\pi\)
\(80\) −1.12606 1.93184i −0.125897 0.215986i
\(81\) 0 0
\(82\) −2.64008 −0.291548
\(83\) 0.429722i 0.0471681i 0.999722 + 0.0235841i \(0.00750774\pi\)
−0.999722 + 0.0235841i \(0.992492\pi\)
\(84\) 0 0
\(85\) −0.670765 + 0.390987i −0.0727547 + 0.0424085i
\(86\) 12.0070i 1.29475i
\(87\) 0 0
\(88\) 0.602126i 0.0641868i
\(89\) −15.9541 −1.69113 −0.845563 0.533875i \(-0.820735\pi\)
−0.845563 + 0.533875i \(0.820735\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.45795 −0.152002
\(93\) 0 0
\(94\) 7.74571i 0.798909i
\(95\) −9.50925 + 5.54291i −0.975629 + 0.568691i
\(96\) 0 0
\(97\) −13.1539 −1.33558 −0.667788 0.744351i \(-0.732759\pi\)
−0.667788 + 0.744351i \(0.732759\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.46398 + 4.35073i −0.246398 + 0.435073i
\(101\) −13.3448 −1.32786 −0.663928 0.747796i \(-0.731112\pi\)
−0.663928 + 0.747796i \(0.731112\pi\)
\(102\) 0 0
\(103\) 6.58914 0.649248 0.324624 0.945843i \(-0.394762\pi\)
0.324624 + 0.945843i \(0.394762\pi\)
\(104\) −0.0571454 −0.00560357
\(105\) 0 0
\(106\) −5.55299 −0.539354
\(107\) 3.66504 0.354313 0.177156 0.984183i \(-0.443310\pi\)
0.177156 + 0.984183i \(0.443310\pi\)
\(108\) 0 0
\(109\) −12.6104 −1.20785 −0.603926 0.797040i \(-0.706398\pi\)
−0.603926 + 0.797040i \(0.706398\pi\)
\(110\) −1.16321 + 0.678030i −0.110908 + 0.0646476i
\(111\) 0 0
\(112\) 0 0
\(113\) 2.87823 0.270761 0.135381 0.990794i \(-0.456774\pi\)
0.135381 + 0.990794i \(0.456774\pi\)
\(114\) 0 0
\(115\) 1.64174 + 2.81652i 0.153093 + 0.262642i
\(116\) 6.49327i 0.602885i
\(117\) 0 0
\(118\) 0.890569 0.0819836
\(119\) 0 0
\(120\) 0 0
\(121\) 10.6374 0.967040
\(122\) 5.72460i 0.518281i
\(123\) 0 0
\(124\) 5.05969i 0.454374i
\(125\) 11.1795 0.139188i 0.999923 0.0124493i
\(126\) 0 0
\(127\) 14.2456i 1.26409i −0.774931 0.632046i \(-0.782215\pi\)
0.774931 0.632046i \(-0.217785\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0.0643492 + 0.110395i 0.00564380 + 0.00968232i
\(131\) −8.52449 −0.744788 −0.372394 0.928075i \(-0.621463\pi\)
−0.372394 + 0.928075i \(0.621463\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0.110634i 0.00955729i
\(135\) 0 0
\(136\) 0.347217i 0.0297736i
\(137\) −8.54900 −0.730391 −0.365195 0.930931i \(-0.618998\pi\)
−0.365195 + 0.930931i \(0.618998\pi\)
\(138\) 0 0
\(139\) 6.50047i 0.551362i −0.961249 0.275681i \(-0.911097\pi\)
0.961249 0.275681i \(-0.0889034\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.7816i 1.07261i
\(143\) 0.0344087i 0.00287740i
\(144\) 0 0
\(145\) 12.5439 7.31181i 1.04172 0.607213i
\(146\) −0.914240 −0.0756630
\(147\) 0 0
\(148\) 1.16480i 0.0957464i
\(149\) 9.98512i 0.818013i −0.912532 0.409006i \(-0.865875\pi\)
0.912532 0.409006i \(-0.134125\pi\)
\(150\) 0 0
\(151\) −9.36685 −0.762263 −0.381132 0.924521i \(-0.624466\pi\)
−0.381132 + 0.924521i \(0.624466\pi\)
\(152\) 4.92239i 0.399259i
\(153\) 0 0
\(154\) 0 0
\(155\) −9.77449 + 5.69752i −0.785106 + 0.457636i
\(156\) 0 0
\(157\) 23.4850 1.87430 0.937152 0.348921i \(-0.113452\pi\)
0.937152 + 0.348921i \(0.113452\pi\)
\(158\) 1.38275 0.110006
\(159\) 0 0
\(160\) −1.12606 1.93184i −0.0890229 0.152725i
\(161\) 0 0
\(162\) 0 0
\(163\) 0.209681i 0.0164235i 0.999966 + 0.00821173i \(0.00261391\pi\)
−0.999966 + 0.00821173i \(0.997386\pi\)
\(164\) −2.64008 −0.206156
\(165\) 0 0
\(166\) 0.429722i 0.0333529i
\(167\) 11.7957i 0.912779i −0.889780 0.456390i \(-0.849142\pi\)
0.889780 0.456390i \(-0.150858\pi\)
\(168\) 0 0
\(169\) −12.9967 −0.999749
\(170\) −0.670765 + 0.390987i −0.0514454 + 0.0299873i
\(171\) 0 0
\(172\) 12.0070i 0.915526i
\(173\) 19.8493i 1.50911i −0.656236 0.754556i \(-0.727852\pi\)
0.656236 0.754556i \(-0.272148\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.602126i 0.0453869i
\(177\) 0 0
\(178\) −15.9541 −1.19581
\(179\) 6.96012i 0.520224i 0.965578 + 0.260112i \(0.0837595\pi\)
−0.965578 + 0.260112i \(0.916241\pi\)
\(180\) 0 0
\(181\) 13.3113i 0.989422i −0.869057 0.494711i \(-0.835274\pi\)
0.869057 0.494711i \(-0.164726\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.45795 −0.107482
\(185\) −2.25021 + 1.31164i −0.165439 + 0.0964337i
\(186\) 0 0
\(187\) −0.209068 −0.0152886
\(188\) 7.74571i 0.564914i
\(189\) 0 0
\(190\) −9.50925 + 5.54291i −0.689874 + 0.402125i
\(191\) 25.0941i 1.81575i −0.419243 0.907874i \(-0.637704\pi\)
0.419243 0.907874i \(-0.362296\pi\)
\(192\) 0 0
\(193\) 16.5745i 1.19306i 0.802592 + 0.596529i \(0.203454\pi\)
−0.802592 + 0.596529i \(0.796546\pi\)
\(194\) −13.1539 −0.944395
\(195\) 0 0
\(196\) 0 0
\(197\) −0.440156 −0.0313598 −0.0156799 0.999877i \(-0.504991\pi\)
−0.0156799 + 0.999877i \(0.504991\pi\)
\(198\) 0 0
\(199\) 20.7109i 1.46815i 0.679066 + 0.734077i \(0.262385\pi\)
−0.679066 + 0.734077i \(0.737615\pi\)
\(200\) −2.46398 + 4.35073i −0.174229 + 0.307643i
\(201\) 0 0
\(202\) −13.3448 −0.938936
\(203\) 0 0
\(204\) 0 0
\(205\) 2.97289 + 5.10020i 0.207636 + 0.356213i
\(206\) 6.58914 0.459087
\(207\) 0 0
\(208\) −0.0571454 −0.00396232
\(209\) −2.96390 −0.205017
\(210\) 0 0
\(211\) −10.4319 −0.718164 −0.359082 0.933306i \(-0.616910\pi\)
−0.359082 + 0.933306i \(0.616910\pi\)
\(212\) −5.55299 −0.381381
\(213\) 0 0
\(214\) 3.66504 0.250537
\(215\) −23.1956 + 13.5206i −1.58192 + 0.922098i
\(216\) 0 0
\(217\) 0 0
\(218\) −12.6104 −0.854081
\(219\) 0 0
\(220\) −1.16321 + 0.678030i −0.0784235 + 0.0457128i
\(221\) 0.0198418i 0.00133471i
\(222\) 0 0
\(223\) 11.3950 0.763067 0.381533 0.924355i \(-0.375396\pi\)
0.381533 + 0.924355i \(0.375396\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.87823 0.191457
\(227\) 27.4366i 1.82103i 0.413475 + 0.910516i \(0.364315\pi\)
−0.413475 + 0.910516i \(0.635685\pi\)
\(228\) 0 0
\(229\) 14.2770i 0.943452i −0.881745 0.471726i \(-0.843631\pi\)
0.881745 0.471726i \(-0.156369\pi\)
\(230\) 1.64174 + 2.81652i 0.108253 + 0.185716i
\(231\) 0 0
\(232\) 6.49327i 0.426304i
\(233\) 25.2801 1.65616 0.828078 0.560613i \(-0.189435\pi\)
0.828078 + 0.560613i \(0.189435\pi\)
\(234\) 0 0
\(235\) −14.9634 + 8.72214i −0.976107 + 0.568969i
\(236\) 0.890569 0.0579711
\(237\) 0 0
\(238\) 0 0
\(239\) 19.6427i 1.27058i 0.772274 + 0.635289i \(0.219119\pi\)
−0.772274 + 0.635289i \(0.780881\pi\)
\(240\) 0 0
\(241\) 0.467621i 0.0301221i 0.999887 + 0.0150611i \(0.00479427\pi\)
−0.999887 + 0.0150611i \(0.995206\pi\)
\(242\) 10.6374 0.683801
\(243\) 0 0
\(244\) 5.72460i 0.366480i
\(245\) 0 0
\(246\) 0 0
\(247\) 0.281292i 0.0178982i
\(248\) 5.05969i 0.321291i
\(249\) 0 0
\(250\) 11.1795 0.139188i 0.707052 0.00880300i
\(251\) −12.1113 −0.764460 −0.382230 0.924067i \(-0.624844\pi\)
−0.382230 + 0.924067i \(0.624844\pi\)
\(252\) 0 0
\(253\) 0.877870i 0.0551912i
\(254\) 14.2456i 0.893848i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.4198i 1.02424i −0.858914 0.512120i \(-0.828860\pi\)
0.858914 0.512120i \(-0.171140\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.0643492 + 0.110395i 0.00399077 + 0.00684644i
\(261\) 0 0
\(262\) −8.52449 −0.526645
\(263\) −19.0087 −1.17212 −0.586062 0.810266i \(-0.699323\pi\)
−0.586062 + 0.810266i \(0.699323\pi\)
\(264\) 0 0
\(265\) 6.25300 + 10.7275i 0.384119 + 0.658983i
\(266\) 0 0
\(267\) 0 0
\(268\) 0.110634i 0.00675802i
\(269\) 1.59449 0.0972175 0.0486088 0.998818i \(-0.484521\pi\)
0.0486088 + 0.998818i \(0.484521\pi\)
\(270\) 0 0
\(271\) 7.99015i 0.485367i 0.970106 + 0.242683i \(0.0780276\pi\)
−0.970106 + 0.242683i \(0.921972\pi\)
\(272\) 0.347217i 0.0210531i
\(273\) 0 0
\(274\) −8.54900 −0.516464
\(275\) 2.61969 + 1.48362i 0.157973 + 0.0894658i
\(276\) 0 0
\(277\) 19.8876i 1.19493i −0.801894 0.597466i \(-0.796174\pi\)
0.801894 0.597466i \(-0.203826\pi\)
\(278\) 6.50047i 0.389872i
\(279\) 0 0
\(280\) 0 0
\(281\) 8.28665i 0.494340i −0.968972 0.247170i \(-0.920499\pi\)
0.968972 0.247170i \(-0.0795007\pi\)
\(282\) 0 0
\(283\) 0.939284 0.0558347 0.0279173 0.999610i \(-0.491112\pi\)
0.0279173 + 0.999610i \(0.491112\pi\)
\(284\) 12.7816i 0.758447i
\(285\) 0 0
\(286\) 0.0344087i 0.00203463i
\(287\) 0 0
\(288\) 0 0
\(289\) 16.8794 0.992908
\(290\) 12.5439 7.31181i 0.736605 0.429365i
\(291\) 0 0
\(292\) −0.914240 −0.0535018
\(293\) 13.4622i 0.786469i 0.919438 + 0.393235i \(0.128644\pi\)
−0.919438 + 0.393235i \(0.871356\pi\)
\(294\) 0 0
\(295\) −1.00283 1.72043i −0.0583873 0.100167i
\(296\) 1.16480i 0.0677029i
\(297\) 0 0
\(298\) 9.98512i 0.578422i
\(299\) 0.0833152 0.00481824
\(300\) 0 0
\(301\) 0 0
\(302\) −9.36685 −0.539002
\(303\) 0 0
\(304\) 4.92239i 0.282319i
\(305\) −11.0590 + 6.44624i −0.633235 + 0.369111i
\(306\) 0 0
\(307\) 8.59205 0.490374 0.245187 0.969476i \(-0.421151\pi\)
0.245187 + 0.969476i \(0.421151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −9.77449 + 5.69752i −0.555154 + 0.323597i
\(311\) −21.5290 −1.22080 −0.610398 0.792095i \(-0.708990\pi\)
−0.610398 + 0.792095i \(0.708990\pi\)
\(312\) 0 0
\(313\) −2.77206 −0.156686 −0.0783429 0.996926i \(-0.524963\pi\)
−0.0783429 + 0.996926i \(0.524963\pi\)
\(314\) 23.4850 1.32533
\(315\) 0 0
\(316\) 1.38275 0.0777860
\(317\) 27.6421 1.55254 0.776268 0.630403i \(-0.217110\pi\)
0.776268 + 0.630403i \(0.217110\pi\)
\(318\) 0 0
\(319\) 3.90977 0.218905
\(320\) −1.12606 1.93184i −0.0629487 0.107993i
\(321\) 0 0
\(322\) 0 0
\(323\) −1.70914 −0.0950989
\(324\) 0 0
\(325\) 0.140805 0.248624i 0.00781045 0.0137912i
\(326\) 0.209681i 0.0116131i
\(327\) 0 0
\(328\) −2.64008 −0.145774
\(329\) 0 0
\(330\) 0 0
\(331\) 33.8909 1.86281 0.931406 0.363981i \(-0.118583\pi\)
0.931406 + 0.363981i \(0.118583\pi\)
\(332\) 0.429722i 0.0235841i
\(333\) 0 0
\(334\) 11.7957i 0.645433i
\(335\) −0.213726 + 0.124580i −0.0116771 + 0.00680654i
\(336\) 0 0
\(337\) 3.42365i 0.186498i −0.995643 0.0932492i \(-0.970275\pi\)
0.995643 0.0932492i \(-0.0297253\pi\)
\(338\) −12.9967 −0.706929
\(339\) 0 0
\(340\) −0.670765 + 0.390987i −0.0363774 + 0.0212042i
\(341\) −3.04657 −0.164981
\(342\) 0 0
\(343\) 0 0
\(344\) 12.0070i 0.647374i
\(345\) 0 0
\(346\) 19.8493i 1.06710i
\(347\) 25.9747 1.39439 0.697196 0.716880i \(-0.254431\pi\)
0.697196 + 0.716880i \(0.254431\pi\)
\(348\) 0 0
\(349\) 16.0984i 0.861726i 0.902417 + 0.430863i \(0.141791\pi\)
−0.902417 + 0.430863i \(0.858209\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.602126i 0.0320934i
\(353\) 21.3823i 1.13807i −0.822315 0.569033i \(-0.807318\pi\)
0.822315 0.569033i \(-0.192682\pi\)
\(354\) 0 0
\(355\) 24.6919 14.3928i 1.31051 0.763892i
\(356\) −15.9541 −0.845563
\(357\) 0 0
\(358\) 6.96012i 0.367854i
\(359\) 19.0185i 1.00376i 0.864938 + 0.501878i \(0.167357\pi\)
−0.864938 + 0.501878i \(0.832643\pi\)
\(360\) 0 0
\(361\) −5.22995 −0.275260
\(362\) 13.3113i 0.699627i
\(363\) 0 0
\(364\) 0 0
\(365\) 1.02949 + 1.76616i 0.0538859 + 0.0924451i
\(366\) 0 0
\(367\) 36.1510 1.88707 0.943533 0.331280i \(-0.107480\pi\)
0.943533 + 0.331280i \(0.107480\pi\)
\(368\) −1.45795 −0.0760009
\(369\) 0 0
\(370\) −2.25021 + 1.31164i −0.116983 + 0.0681890i
\(371\) 0 0
\(372\) 0 0
\(373\) 21.5031i 1.11339i −0.830718 0.556693i \(-0.812070\pi\)
0.830718 0.556693i \(-0.187930\pi\)
\(374\) −0.209068 −0.0108107
\(375\) 0 0
\(376\) 7.74571i 0.399454i
\(377\) 0.371060i 0.0191106i
\(378\) 0 0
\(379\) 5.31543 0.273035 0.136518 0.990638i \(-0.456409\pi\)
0.136518 + 0.990638i \(0.456409\pi\)
\(380\) −9.50925 + 5.54291i −0.487814 + 0.284345i
\(381\) 0 0
\(382\) 25.0941i 1.28393i
\(383\) 26.3353i 1.34567i 0.739793 + 0.672835i \(0.234923\pi\)
−0.739793 + 0.672835i \(0.765077\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16.5745i 0.843619i
\(387\) 0 0
\(388\) −13.1539 −0.667788
\(389\) 3.43842i 0.174335i 0.996194 + 0.0871674i \(0.0277815\pi\)
−0.996194 + 0.0871674i \(0.972219\pi\)
\(390\) 0 0
\(391\) 0.506225i 0.0256009i
\(392\) 0 0
\(393\) 0 0
\(394\) −0.440156 −0.0221747
\(395\) −1.55707 2.67125i −0.0783445 0.134405i
\(396\) 0 0
\(397\) −27.9177 −1.40115 −0.700576 0.713578i \(-0.747073\pi\)
−0.700576 + 0.713578i \(0.747073\pi\)
\(398\) 20.7109i 1.03814i
\(399\) 0 0
\(400\) −2.46398 + 4.35073i −0.123199 + 0.217536i
\(401\) 24.2952i 1.21325i −0.794990 0.606623i \(-0.792524\pi\)
0.794990 0.606623i \(-0.207476\pi\)
\(402\) 0 0
\(403\) 0.289138i 0.0144030i
\(404\) −13.3448 −0.663928
\(405\) 0 0
\(406\) 0 0
\(407\) −0.701359 −0.0347651
\(408\) 0 0
\(409\) 14.1853i 0.701420i −0.936484 0.350710i \(-0.885940\pi\)
0.936484 0.350710i \(-0.114060\pi\)
\(410\) 2.97289 + 5.10020i 0.146821 + 0.251881i
\(411\) 0 0
\(412\) 6.58914 0.324624
\(413\) 0 0
\(414\) 0 0
\(415\) 0.830152 0.483893i 0.0407506 0.0237534i
\(416\) −0.0571454 −0.00280178
\(417\) 0 0
\(418\) −2.96390 −0.144969
\(419\) −17.9730 −0.878036 −0.439018 0.898478i \(-0.644674\pi\)
−0.439018 + 0.898478i \(0.644674\pi\)
\(420\) 0 0
\(421\) 12.9077 0.629084 0.314542 0.949244i \(-0.398149\pi\)
0.314542 + 0.949244i \(0.398149\pi\)
\(422\) −10.4319 −0.507819
\(423\) 0 0
\(424\) −5.55299 −0.269677
\(425\) 1.51064 + 0.855533i 0.0732770 + 0.0414995i
\(426\) 0 0
\(427\) 0 0
\(428\) 3.66504 0.177156
\(429\) 0 0
\(430\) −23.1956 + 13.5206i −1.11859 + 0.652022i
\(431\) 7.13005i 0.343442i 0.985146 + 0.171721i \(0.0549328\pi\)
−0.985146 + 0.171721i \(0.945067\pi\)
\(432\) 0 0
\(433\) 28.1054 1.35066 0.675330 0.737516i \(-0.264001\pi\)
0.675330 + 0.737516i \(0.264001\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −12.6104 −0.603926
\(437\) 7.17661i 0.343304i
\(438\) 0 0
\(439\) 21.2538i 1.01439i 0.861831 + 0.507195i \(0.169318\pi\)
−0.861831 + 0.507195i \(0.830682\pi\)
\(440\) −1.16321 + 0.678030i −0.0554538 + 0.0323238i
\(441\) 0 0
\(442\) 0.0198418i 0.000943780i
\(443\) 8.94032 0.424768 0.212384 0.977186i \(-0.431877\pi\)
0.212384 + 0.977186i \(0.431877\pi\)
\(444\) 0 0
\(445\) 17.9652 + 30.8206i 0.851634 + 1.46104i
\(446\) 11.3950 0.539570
\(447\) 0 0
\(448\) 0 0
\(449\) 20.6014i 0.972241i 0.873892 + 0.486121i \(0.161588\pi\)
−0.873892 + 0.486121i \(0.838412\pi\)
\(450\) 0 0
\(451\) 1.58966i 0.0748542i
\(452\) 2.87823 0.135381
\(453\) 0 0
\(454\) 27.4366i 1.28766i
\(455\) 0 0
\(456\) 0 0
\(457\) 26.8747i 1.25715i −0.777750 0.628573i \(-0.783639\pi\)
0.777750 0.628573i \(-0.216361\pi\)
\(458\) 14.2770i 0.667122i
\(459\) 0 0
\(460\) 1.64174 + 2.81652i 0.0765466 + 0.131321i
\(461\) −25.0610 −1.16721 −0.583603 0.812039i \(-0.698357\pi\)
−0.583603 + 0.812039i \(0.698357\pi\)
\(462\) 0 0
\(463\) 25.7470i 1.19656i −0.801286 0.598282i \(-0.795850\pi\)
0.801286 0.598282i \(-0.204150\pi\)
\(464\) 6.49327i 0.301442i
\(465\) 0 0
\(466\) 25.2801 1.17108
\(467\) 0.198229i 0.00917293i 0.999989 + 0.00458647i \(0.00145992\pi\)
−0.999989 + 0.00458647i \(0.998540\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −14.9634 + 8.72214i −0.690212 + 0.402322i
\(471\) 0 0
\(472\) 0.890569 0.0409918
\(473\) −7.22973 −0.332423
\(474\) 0 0
\(475\) 21.4160 + 12.1287i 0.982633 + 0.556501i
\(476\) 0 0
\(477\) 0 0
\(478\) 19.6427i 0.898435i
\(479\) 8.78904 0.401581 0.200791 0.979634i \(-0.435649\pi\)
0.200791 + 0.979634i \(0.435649\pi\)
\(480\) 0 0
\(481\) 0.0665632i 0.00303502i
\(482\) 0.467621i 0.0212996i
\(483\) 0 0
\(484\) 10.6374 0.483520
\(485\) 14.8121 + 25.4112i 0.672582 + 1.15386i
\(486\) 0 0
\(487\) 42.7286i 1.93622i −0.250527 0.968110i \(-0.580604\pi\)
0.250527 0.968110i \(-0.419396\pi\)
\(488\) 5.72460i 0.259140i
\(489\) 0 0
\(490\) 0 0
\(491\) 3.45933i 0.156117i 0.996949 + 0.0780587i \(0.0248722\pi\)
−0.996949 + 0.0780587i \(0.975128\pi\)
\(492\) 0 0
\(493\) 2.25457 0.101541
\(494\) 0.281292i 0.0126559i
\(495\) 0 0
\(496\) 5.05969i 0.227187i
\(497\) 0 0
\(498\) 0 0
\(499\) −25.9942 −1.16366 −0.581830 0.813311i \(-0.697663\pi\)
−0.581830 + 0.813311i \(0.697663\pi\)
\(500\) 11.1795 0.139188i 0.499961 0.00622466i
\(501\) 0 0
\(502\) −12.1113 −0.540555
\(503\) 9.37901i 0.418189i 0.977895 + 0.209095i \(0.0670517\pi\)
−0.977895 + 0.209095i \(0.932948\pi\)
\(504\) 0 0
\(505\) 15.0270 + 25.7799i 0.668695 + 1.14719i
\(506\) 0.877870i 0.0390261i
\(507\) 0 0
\(508\) 14.2456i 0.632046i
\(509\) 7.20399 0.319311 0.159656 0.987173i \(-0.448962\pi\)
0.159656 + 0.987173i \(0.448962\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 16.4198i 0.724248i
\(515\) −7.41978 12.7291i −0.326954 0.560913i
\(516\) 0 0
\(517\) −4.66389 −0.205118
\(518\) 0 0
\(519\) 0 0
\(520\) 0.0643492 + 0.110395i 0.00282190 + 0.00484116i
\(521\) −5.21234 −0.228357 −0.114178 0.993460i \(-0.536424\pi\)
−0.114178 + 0.993460i \(0.536424\pi\)
\(522\) 0 0
\(523\) 29.7249 1.29978 0.649890 0.760028i \(-0.274815\pi\)
0.649890 + 0.760028i \(0.274815\pi\)
\(524\) −8.52449 −0.372394
\(525\) 0 0
\(526\) −19.0087 −0.828817
\(527\) −1.75681 −0.0765278
\(528\) 0 0
\(529\) −20.8744 −0.907582
\(530\) 6.25300 + 10.7275i 0.271613 + 0.465971i
\(531\) 0 0
\(532\) 0 0
\(533\) 0.150868 0.00653484
\(534\) 0 0
\(535\) −4.12705 7.08025i −0.178428 0.306106i
\(536\) 0.110634i 0.00477864i
\(537\) 0 0
\(538\) 1.59449 0.0687432
\(539\) 0 0
\(540\) 0 0
\(541\) 28.2785 1.21579 0.607895 0.794018i \(-0.292014\pi\)
0.607895 + 0.794018i \(0.292014\pi\)
\(542\) 7.99015i 0.343206i
\(543\) 0 0
\(544\) 0.347217i 0.0148868i
\(545\) 14.2000 + 24.3611i 0.608262 + 1.04352i
\(546\) 0 0
\(547\) 18.7288i 0.800786i −0.916343 0.400393i \(-0.868874\pi\)
0.916343 0.400393i \(-0.131126\pi\)
\(548\) −8.54900 −0.365195
\(549\) 0 0
\(550\) 2.61969 + 1.48362i 0.111704 + 0.0632619i
\(551\) 31.9624 1.36164
\(552\) 0 0
\(553\) 0 0
\(554\) 19.8876i 0.844945i
\(555\) 0 0
\(556\) 6.50047i 0.275681i
\(557\) 34.0649 1.44338 0.721688 0.692219i \(-0.243367\pi\)
0.721688 + 0.692219i \(0.243367\pi\)
\(558\) 0 0
\(559\) 0.686145i 0.0290208i
\(560\) 0 0
\(561\) 0 0
\(562\) 8.28665i 0.349551i
\(563\) 24.6170i 1.03748i 0.854931 + 0.518741i \(0.173599\pi\)
−0.854931 + 0.518741i \(0.826401\pi\)
\(564\) 0 0
\(565\) −3.24106 5.56027i −0.136353 0.233922i
\(566\) 0.939284 0.0394811
\(567\) 0 0
\(568\) 12.7816i 0.536303i
\(569\) 35.1245i 1.47249i 0.676713 + 0.736247i \(0.263404\pi\)
−0.676713 + 0.736247i \(0.736596\pi\)
\(570\) 0 0
\(571\) 27.0787 1.13321 0.566605 0.823990i \(-0.308257\pi\)
0.566605 + 0.823990i \(0.308257\pi\)
\(572\) 0.0344087i 0.00143870i
\(573\) 0 0
\(574\) 0 0
\(575\) 3.59235 6.34315i 0.149812 0.264527i
\(576\) 0 0
\(577\) 19.0688 0.793845 0.396923 0.917852i \(-0.370078\pi\)
0.396923 + 0.917852i \(0.370078\pi\)
\(578\) 16.8794 0.702092
\(579\) 0 0
\(580\) 12.5439 7.31181i 0.520858 0.303607i
\(581\) 0 0
\(582\) 0 0
\(583\) 3.34360i 0.138478i
\(584\) −0.914240 −0.0378315
\(585\) 0 0
\(586\) 13.4622i 0.556118i
\(587\) 9.47721i 0.391166i −0.980687 0.195583i \(-0.937340\pi\)
0.980687 0.195583i \(-0.0626599\pi\)
\(588\) 0 0
\(589\) −24.9058 −1.02623
\(590\) −1.00283 1.72043i −0.0412861 0.0708291i
\(591\) 0 0
\(592\) 1.16480i 0.0478732i
\(593\) 23.4238i 0.961902i 0.876748 + 0.480951i \(0.159708\pi\)
−0.876748 + 0.480951i \(0.840292\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.98512i 0.409006i
\(597\) 0 0
\(598\) 0.0833152 0.00340701
\(599\) 0.818609i 0.0334474i −0.999860 0.0167237i \(-0.994676\pi\)
0.999860 0.0167237i \(-0.00532357\pi\)
\(600\) 0 0
\(601\) 7.66116i 0.312505i 0.987717 + 0.156253i \(0.0499414\pi\)
−0.987717 + 0.156253i \(0.950059\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −9.36685 −0.381132
\(605\) −11.9784 20.5498i −0.486991 0.835468i
\(606\) 0 0
\(607\) −22.6746 −0.920334 −0.460167 0.887832i \(-0.652211\pi\)
−0.460167 + 0.887832i \(0.652211\pi\)
\(608\) 4.92239i 0.199629i
\(609\) 0 0
\(610\) −11.0590 + 6.44624i −0.447765 + 0.261001i
\(611\) 0.442631i 0.0179070i
\(612\) 0 0
\(613\) 19.2088i 0.775837i 0.921694 + 0.387919i \(0.126806\pi\)
−0.921694 + 0.387919i \(0.873194\pi\)
\(614\) 8.59205 0.346747
\(615\) 0 0
\(616\) 0 0
\(617\) 19.1200 0.769744 0.384872 0.922970i \(-0.374246\pi\)
0.384872 + 0.922970i \(0.374246\pi\)
\(618\) 0 0
\(619\) 27.2018i 1.09333i 0.837351 + 0.546666i \(0.184103\pi\)
−0.837351 + 0.546666i \(0.815897\pi\)
\(620\) −9.77449 + 5.69752i −0.392553 + 0.228818i
\(621\) 0 0
\(622\) −21.5290 −0.863232
\(623\) 0 0
\(624\) 0 0
\(625\) −12.8577 21.4402i −0.514306 0.857607i
\(626\) −2.77206 −0.110794
\(627\) 0 0
\(628\) 23.4850 0.937152
\(629\) −0.404440 −0.0161261
\(630\) 0 0
\(631\) 32.7018 1.30184 0.650918 0.759148i \(-0.274384\pi\)
0.650918 + 0.759148i \(0.274384\pi\)
\(632\) 1.38275 0.0550030
\(633\) 0 0
\(634\) 27.6421 1.09781
\(635\) −27.5201 + 16.0414i −1.09210 + 0.636583i
\(636\) 0 0
\(637\) 0 0
\(638\) 3.90977 0.154789
\(639\) 0 0
\(640\) −1.12606 1.93184i −0.0445115 0.0763625i
\(641\) 42.5112i 1.67909i −0.543290 0.839545i \(-0.682822\pi\)
0.543290 0.839545i \(-0.317178\pi\)
\(642\) 0 0
\(643\) 29.8265 1.17624 0.588121 0.808773i \(-0.299868\pi\)
0.588121 + 0.808773i \(0.299868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.70914 −0.0672451
\(647\) 23.0895i 0.907741i 0.891068 + 0.453871i \(0.149957\pi\)
−0.891068 + 0.453871i \(0.850043\pi\)
\(648\) 0 0
\(649\) 0.536235i 0.0210491i
\(650\) 0.140805 0.248624i 0.00552282 0.00975184i
\(651\) 0 0
\(652\) 0.209681i 0.00821173i
\(653\) −21.1582 −0.827986 −0.413993 0.910280i \(-0.635866\pi\)
−0.413993 + 0.910280i \(0.635866\pi\)
\(654\) 0 0
\(655\) 9.59910 + 16.4679i 0.375068 + 0.643455i
\(656\) −2.64008 −0.103078
\(657\) 0 0
\(658\) 0 0
\(659\) 0.985920i 0.0384060i 0.999816 + 0.0192030i \(0.00611288\pi\)
−0.999816 + 0.0192030i \(0.993887\pi\)
\(660\) 0 0
\(661\) 7.37369i 0.286803i −0.989665 0.143402i \(-0.954196\pi\)
0.989665 0.143402i \(-0.0458041\pi\)
\(662\) 33.8909 1.31721
\(663\) 0 0
\(664\) 0.429722i 0.0166765i
\(665\) 0 0
\(666\) 0 0
\(667\) 9.46687i 0.366559i
\(668\) 11.7957i 0.456390i
\(669\) 0 0
\(670\) −0.213726 + 0.124580i −0.00825695 + 0.00481295i
\(671\) −3.44693 −0.133067
\(672\) 0 0
\(673\) 28.3692i 1.09355i 0.837279 + 0.546776i \(0.184145\pi\)
−0.837279 + 0.546776i \(0.815855\pi\)
\(674\) 3.42365i 0.131874i
\(675\) 0 0
\(676\) −12.9967 −0.499874
\(677\) 38.2412i 1.46973i −0.678213 0.734865i \(-0.737245\pi\)
0.678213 0.734865i \(-0.262755\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.670765 + 0.390987i −0.0257227 + 0.0149937i
\(681\) 0 0
\(682\) −3.04657 −0.116659
\(683\) 28.0427 1.07302 0.536512 0.843893i \(-0.319742\pi\)
0.536512 + 0.843893i \(0.319742\pi\)
\(684\) 0 0
\(685\) 9.62670 + 16.5153i 0.367817 + 0.631016i
\(686\) 0 0
\(687\) 0 0
\(688\) 12.0070i 0.457763i
\(689\) 0.317328 0.0120892
\(690\) 0 0
\(691\) 34.3370i 1.30624i 0.757254 + 0.653120i \(0.226540\pi\)
−0.757254 + 0.653120i \(0.773460\pi\)
\(692\) 19.8493i 0.754556i
\(693\) 0 0
\(694\) 25.9747 0.985984
\(695\) −12.5578 + 7.31992i −0.476346 + 0.277660i
\(696\) 0 0
\(697\) 0.916680i 0.0347217i
\(698\) 16.0984i 0.609332i
\(699\) 0 0
\(700\) 0 0
\(701\) 49.6503i 1.87527i −0.347628 0.937633i \(-0.613013\pi\)
0.347628 0.937633i \(-0.386987\pi\)
\(702\) 0 0
\(703\) −5.73363 −0.216248
\(704\) 0.602126i 0.0226935i
\(705\) 0 0
\(706\) 21.3823i 0.804734i
\(707\) 0 0
\(708\) 0 0
\(709\) −0.141183 −0.00530223 −0.00265111 0.999996i \(-0.500844\pi\)
−0.00265111 + 0.999996i \(0.500844\pi\)
\(710\) 24.6919 14.3928i 0.926670 0.540153i
\(711\) 0 0
\(712\) −15.9541 −0.597904
\(713\) 7.37678i 0.276263i
\(714\) 0 0
\(715\) 0.0664720 0.0387463i 0.00248591 0.00144903i
\(716\) 6.96012i 0.260112i
\(717\) 0 0
\(718\) 19.0185i 0.709763i
\(719\) 18.8949 0.704660 0.352330 0.935876i \(-0.385389\pi\)
0.352330 + 0.935876i \(0.385389\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −5.22995 −0.194638
\(723\) 0 0
\(724\) 13.3113i 0.494711i
\(725\) −28.2504 15.9993i −1.04920 0.594197i
\(726\) 0 0
\(727\) −42.1018 −1.56147 −0.780735 0.624862i \(-0.785155\pi\)
−0.780735 + 0.624862i \(0.785155\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.02949 + 1.76616i 0.0381031 + 0.0653685i
\(731\) −4.16903 −0.154197
\(732\) 0 0
\(733\) 10.2332 0.377974 0.188987 0.981980i \(-0.439480\pi\)
0.188987 + 0.981980i \(0.439480\pi\)
\(734\) 36.1510 1.33436
\(735\) 0 0
\(736\) −1.45795 −0.0537408
\(737\) −0.0666154 −0.00245381
\(738\) 0 0
\(739\) 21.0324 0.773688 0.386844 0.922145i \(-0.373565\pi\)
0.386844 + 0.922145i \(0.373565\pi\)
\(740\) −2.25021 + 1.31164i −0.0827194 + 0.0482169i
\(741\) 0 0
\(742\) 0 0
\(743\) 14.0317 0.514774 0.257387 0.966308i \(-0.417139\pi\)
0.257387 + 0.966308i \(0.417139\pi\)
\(744\) 0 0
\(745\) −19.2896 + 11.2438i −0.706716 + 0.411943i
\(746\) 21.5031i 0.787283i
\(747\) 0 0
\(748\) −0.209068 −0.00764429
\(749\) 0 0
\(750\) 0 0
\(751\) −38.7407 −1.41367 −0.706835 0.707379i \(-0.749878\pi\)
−0.706835 + 0.707379i \(0.749878\pi\)
\(752\) 7.74571i 0.282457i
\(753\) 0 0
\(754\) 0.371060i 0.0135132i
\(755\) 10.5476 + 18.0952i 0.383868 + 0.658552i
\(756\) 0 0
\(757\) 18.1568i 0.659919i 0.943995 + 0.329960i \(0.107035\pi\)
−0.943995 + 0.329960i \(0.892965\pi\)
\(758\) 5.31543 0.193065
\(759\) 0 0
\(760\) −9.50925 + 5.54291i −0.344937 + 0.201063i
\(761\) 25.0314 0.907386 0.453693 0.891158i \(-0.350106\pi\)
0.453693 + 0.891158i \(0.350106\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 25.0941i 0.907874i
\(765\) 0 0
\(766\) 26.3353i 0.951532i
\(767\) −0.0508919 −0.00183760
\(768\) 0 0
\(769\) 28.6902i 1.03460i −0.855805 0.517298i \(-0.826938\pi\)
0.855805 0.517298i \(-0.173062\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16.5745i 0.596529i
\(773\) 19.3034i 0.694294i 0.937811 + 0.347147i \(0.112850\pi\)
−0.937811 + 0.347147i \(0.887150\pi\)
\(774\) 0 0
\(775\) 22.0133 + 12.4670i 0.790743 + 0.447826i
\(776\) −13.1539 −0.472197
\(777\) 0 0
\(778\) 3.43842i 0.123273i
\(779\) 12.9955i 0.465612i
\(780\) 0 0
\(781\) 7.69611 0.275389
\(782\) 0.506225i 0.0181026i
\(783\) 0 0
\(784\) 0 0
\(785\) −26.4455 45.3691i −0.943880 1.61929i
\(786\) 0 0
\(787\) 41.4238 1.47660 0.738300 0.674473i \(-0.235629\pi\)
0.738300 + 0.674473i \(0.235629\pi\)
\(788\) −0.440156 −0.0156799
\(789\) 0 0
\(790\) −1.55707 2.67125i −0.0553979 0.0950390i
\(791\) 0 0
\(792\) 0 0
\(793\) 0.327134i 0.0116169i
\(794\) −27.9177 −0.990764
\(795\) 0 0
\(796\) 20.7109i 0.734077i
\(797\) 24.9918i 0.885253i −0.896706 0.442627i \(-0.854047\pi\)
0.896706 0.442627i \(-0.145953\pi\)
\(798\) 0 0
\(799\) −2.68944 −0.0951455
\(800\) −2.46398 + 4.35073i −0.0871147 + 0.153821i
\(801\) 0 0
\(802\) 24.2952i 0.857895i
\(803\) 0.550487i 0.0194263i
\(804\) 0 0
\(805\) 0 0
\(806\) 0.289138i 0.0101845i
\(807\) 0 0
\(808\) −13.3448 −0.469468
\(809\) 6.63517i 0.233280i 0.993174 + 0.116640i \(0.0372124\pi\)
−0.993174 + 0.116640i \(0.962788\pi\)
\(810\) 0 0
\(811\) 2.77141i 0.0973173i 0.998815 + 0.0486587i \(0.0154946\pi\)
−0.998815 + 0.0486587i \(0.984505\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −0.701359 −0.0245826
\(815\) 0.405069 0.236113i 0.0141889 0.00827069i
\(816\) 0 0
\(817\) −59.1032 −2.06776
\(818\) 14.1853i 0.495979i
\(819\) 0 0
\(820\) 2.97289 + 5.10020i 0.103818 + 0.178107i
\(821\) 20.9012i 0.729457i 0.931114 + 0.364728i \(0.118838\pi\)
−0.931114 + 0.364728i \(0.881162\pi\)
\(822\) 0 0
\(823\) 34.9867i 1.21956i 0.792571 + 0.609780i \(0.208742\pi\)
−0.792571 + 0.609780i \(0.791258\pi\)
\(824\) 6.58914 0.229544
\(825\) 0 0
\(826\) 0 0
\(827\) 31.8919 1.10899 0.554495 0.832187i \(-0.312911\pi\)
0.554495 + 0.832187i \(0.312911\pi\)
\(828\) 0 0
\(829\) 50.1802i 1.74283i 0.490547 + 0.871415i \(0.336797\pi\)
−0.490547 + 0.871415i \(0.663203\pi\)
\(830\) 0.830152 0.483893i 0.0288150 0.0167962i
\(831\) 0 0
\(832\) −0.0571454 −0.00198116
\(833\) 0 0
\(834\) 0 0
\(835\) −22.7874 + 13.2827i −0.788589 + 0.459666i
\(836\) −2.96390 −0.102509
\(837\) 0 0
\(838\) −17.9730 −0.620865
\(839\) 21.7782 0.751866 0.375933 0.926647i \(-0.377322\pi\)
0.375933 + 0.926647i \(0.377322\pi\)
\(840\) 0 0
\(841\) −13.1625 −0.453881
\(842\) 12.9077 0.444829
\(843\) 0 0
\(844\) −10.4319 −0.359082
\(845\) 14.6351 + 25.1076i 0.503463 + 0.863726i
\(846\) 0 0
\(847\) 0 0
\(848\) −5.55299 −0.190690
\(849\) 0 0
\(850\) 1.51064 + 0.855533i 0.0518147 + 0.0293445i
\(851\) 1.69823i 0.0582145i
\(852\) 0 0
\(853\) −41.3403 −1.41547 −0.707733 0.706480i \(-0.750282\pi\)
−0.707733 + 0.706480i \(0.750282\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.66504 0.125268
\(857\) 40.9455i 1.39867i 0.714793 + 0.699336i \(0.246521\pi\)
−0.714793 + 0.699336i \(0.753479\pi\)
\(858\) 0 0
\(859\) 39.8993i 1.36135i 0.732587 + 0.680673i \(0.238313\pi\)
−0.732587 + 0.680673i \(0.761687\pi\)
\(860\) −23.1956 + 13.5206i −0.790962 + 0.461049i
\(861\) 0 0
\(862\) 7.13005i 0.242850i
\(863\) −31.9325 −1.08700 −0.543498 0.839410i \(-0.682900\pi\)
−0.543498 + 0.839410i \(0.682900\pi\)
\(864\) 0 0
\(865\) −38.3455 + 22.3515i −1.30379 + 0.759973i
\(866\) 28.1054 0.955060
\(867\) 0 0
\(868\) 0 0
\(869\) 0.832592i 0.0282438i
\(870\) 0 0
\(871\) 0.00632220i 0.000214220i
\(872\) −12.6104 −0.427040
\(873\) 0 0
\(874\) 7.17661i 0.242752i
\(875\) 0 0
\(876\) 0 0
\(877\) 44.3366i 1.49714i −0.663055 0.748571i \(-0.730741\pi\)
0.663055 0.748571i \(-0.269259\pi\)
\(878\) 21.2538i 0.717282i
\(879\) 0 0
\(880\) −1.16321 + 0.678030i −0.0392117 + 0.0228564i
\(881\) 18.8931 0.636526 0.318263 0.948003i \(-0.396901\pi\)
0.318263 + 0.948003i \(0.396901\pi\)
\(882\) 0 0
\(883\) 33.2866i 1.12018i 0.828430 + 0.560092i \(0.189234\pi\)
−0.828430 + 0.560092i \(0.810766\pi\)
\(884\) 0.0198418i 0.000667353i
\(885\) 0 0
\(886\) 8.94032 0.300356
\(887\) 32.5007i 1.09127i 0.838024 + 0.545633i \(0.183711\pi\)
−0.838024 + 0.545633i \(0.816289\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 17.9652 + 30.8206i 0.602196 + 1.03311i
\(891\) 0 0
\(892\) 11.3950 0.381533
\(893\) −38.1274 −1.27589
\(894\) 0 0
\(895\) 13.4458 7.83752i 0.449444 0.261979i
\(896\) 0 0
\(897\) 0 0
\(898\) 20.6014i 0.687478i
\(899\) 32.8540 1.09574
\(900\) 0 0
\(901\) 1.92809i 0.0642340i
\(902\) 1.58966i 0.0529299i
\(903\) 0 0
\(904\) 2.87823 0.0957286
\(905\) −25.7153 + 14.9893i −0.854804 + 0.498263i
\(906\) 0 0
\(907\) 57.4699i 1.90826i −0.299396 0.954129i \(-0.596785\pi\)
0.299396 0.954129i \(-0.403215\pi\)
\(908\) 27.4366i 0.910516i
\(909\) 0 0
\(910\) 0 0
\(911\) 15.7785i 0.522766i −0.965235 0.261383i \(-0.915822\pi\)
0.965235 0.261383i \(-0.0841785\pi\)
\(912\) 0 0
\(913\) 0.258747 0.00856327
\(914\) 26.8747i 0.888937i
\(915\) 0 0
\(916\) 14.2770i 0.471726i
\(917\) 0 0
\(918\) 0 0
\(919\) −12.9710 −0.427873 −0.213937 0.976848i \(-0.568629\pi\)
−0.213937 + 0.976848i \(0.568629\pi\)
\(920\) 1.64174 + 2.81652i 0.0541266 + 0.0928579i
\(921\) 0 0
\(922\) −25.0610 −0.825339
\(923\) 0.730408i 0.0240417i
\(924\) 0 0
\(925\) 5.06775 + 2.87005i 0.166627 + 0.0943667i
\(926\) 25.7470i 0.846098i
\(927\) 0 0
\(928\) 6.49327i 0.213152i
\(929\) 51.1394 1.67783 0.838914 0.544264i \(-0.183191\pi\)
0.838914 + 0.544264i \(0.183191\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 25.2801 0.828078
\(933\) 0 0
\(934\) 0.198229i 0.00648624i
\(935\) 0.235423 + 0.403885i 0.00769917 + 0.0132085i
\(936\) 0 0
\(937\) −37.0919 −1.21174 −0.605870 0.795564i \(-0.707175\pi\)
−0.605870 + 0.795564i \(0.707175\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −14.9634 + 8.72214i −0.488053 + 0.284485i
\(941\) 15.2864 0.498322 0.249161 0.968462i \(-0.419845\pi\)
0.249161 + 0.968462i \(0.419845\pi\)
\(942\) 0 0
\(943\) 3.84911 0.125344
\(944\) 0.890569 0.0289856
\(945\) 0 0
\(946\) −7.22973 −0.235059
\(947\) −24.2177 −0.786970 −0.393485 0.919331i \(-0.628731\pi\)
−0.393485 + 0.919331i \(0.628731\pi\)
\(948\) 0 0
\(949\) 0.0522446 0.00169593
\(950\) 21.4160 + 12.1287i 0.694826 + 0.393505i
\(951\) 0 0
\(952\) 0 0
\(953\) −19.3359 −0.626352 −0.313176 0.949695i \(-0.601393\pi\)
−0.313176 + 0.949695i \(0.601393\pi\)
\(954\) 0 0
\(955\) −48.4777 + 28.2575i −1.56870 + 0.914392i
\(956\) 19.6427i 0.635289i
\(957\) 0 0
\(958\) 8.78904 0.283961
\(959\) 0 0
\(960\) 0 0
\(961\) 5.39950 0.174178
\(962\) 0.0665632i 0.00214608i
\(963\) 0 0
\(964\) 0.467621i 0.0150611i
\(965\) 32.0192 18.6639i 1.03073 0.600811i
\(966\) 0 0
\(967\) 46.7895i 1.50465i −0.658792 0.752325i \(-0.728932\pi\)
0.658792 0.752325i \(-0.271068\pi\)
\(968\) 10.6374 0.341900
\(969\) 0 0
\(970\) 14.8121 + 25.4112i 0.475587 + 0.815903i
\(971\) 5.69948 0.182905 0.0914525 0.995809i \(-0.470849\pi\)
0.0914525 + 0.995809i \(0.470849\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 42.7286i 1.36911i
\(975\) 0 0
\(976\) 5.72460i 0.183240i
\(977\) 35.5489 1.13731 0.568656 0.822576i \(-0.307464\pi\)
0.568656 + 0.822576i \(0.307464\pi\)
\(978\) 0 0
\(979\) 9.60635i 0.307020i
\(980\) 0 0
\(981\) 0 0
\(982\) 3.45933i 0.110392i
\(983\) 34.6900i 1.10644i −0.833035 0.553220i \(-0.813399\pi\)
0.833035 0.553220i \(-0.186601\pi\)
\(984\) 0 0
\(985\) 0.495642 + 0.850309i 0.0157925 + 0.0270931i
\(986\) 2.25457 0.0718002
\(987\) 0 0
\(988\) 0.281292i 0.00894909i
\(989\) 17.5056i 0.556646i
\(990\) 0 0
\(991\) −31.8763 −1.01258 −0.506292 0.862362i \(-0.668984\pi\)
−0.506292 + 0.862362i \(0.668984\pi\)
\(992\) 5.05969i 0.160645i
\(993\) 0 0
\(994\) 0 0
\(995\) 40.0100 23.3217i 1.26840 0.739347i
\(996\) 0 0
\(997\) −11.2004 −0.354720 −0.177360 0.984146i \(-0.556756\pi\)
−0.177360 + 0.984146i \(0.556756\pi\)
\(998\) −25.9942 −0.822831
\(999\) 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 4410.2.d.d.4409.7 yes 24
3.2 odd 2 4410.2.d.c.4409.18 yes 24
5.4 even 2 4410.2.d.c.4409.8 yes 24
7.6 odd 2 inner 4410.2.d.d.4409.18 yes 24
15.14 odd 2 inner 4410.2.d.d.4409.17 yes 24
21.20 even 2 4410.2.d.c.4409.7 24
35.34 odd 2 4410.2.d.c.4409.17 yes 24
105.104 even 2 inner 4410.2.d.d.4409.8 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4410.2.d.c.4409.7 24 21.20 even 2
4410.2.d.c.4409.8 yes 24 5.4 even 2
4410.2.d.c.4409.17 yes 24 35.34 odd 2
4410.2.d.c.4409.18 yes 24 3.2 odd 2
4410.2.d.d.4409.7 yes 24 1.1 even 1 trivial
4410.2.d.d.4409.8 yes 24 105.104 even 2 inner
4410.2.d.d.4409.17 yes 24 15.14 odd 2 inner
4410.2.d.d.4409.18 yes 24 7.6 odd 2 inner