Properties

Label 600.2.b.d.251.4
Level $600$
Weight $2$
Character 600.251
Analytic conductor $4.791$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,2,Mod(251,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.b (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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 251.4
Root \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 600.251
Dual form 600.2.b.d.251.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} +(1.72474 - 0.158919i) q^{3} -2.00000 q^{4} +(0.224745 + 2.43916i) q^{6} -2.82843i q^{8} +(2.94949 - 0.548188i) q^{9} +O(q^{10})\) \(q+1.41421i q^{2} +(1.72474 - 0.158919i) q^{3} -2.00000 q^{4} +(0.224745 + 2.43916i) q^{6} -2.82843i q^{8} +(2.94949 - 0.548188i) q^{9} +3.78194i q^{11} +(-3.44949 + 0.317837i) q^{12} +4.00000 q^{16} +8.02458i q^{17} +(0.775255 + 4.17121i) q^{18} +6.34847 q^{19} -5.34847 q^{22} +(-0.449490 - 4.87832i) q^{24} +(5.00000 - 1.41421i) q^{27} +5.65685i q^{32} +(0.601021 + 6.52288i) q^{33} -11.3485 q^{34} +(-5.89898 + 1.09638i) q^{36} +8.97809i q^{38} -10.8530i q^{41} -10.0000 q^{43} -7.56388i q^{44} +(6.89898 - 0.635674i) q^{48} +7.00000 q^{49} +(1.27526 + 13.8404i) q^{51} +(2.00000 + 7.07107i) q^{54} +(10.9495 - 1.00889i) q^{57} -14.1421i q^{59} -8.00000 q^{64} +(-9.22474 + 0.849971i) q^{66} +0.348469 q^{67} -16.0492i q^{68} +(-1.55051 - 8.34242i) q^{72} -15.6969 q^{73} -12.6969 q^{76} +(8.39898 - 3.23375i) q^{81} +15.3485 q^{82} -17.0027i q^{83} -14.1421i q^{86} +10.6969 q^{88} +18.4169i q^{89} +(0.898979 + 9.75663i) q^{96} -10.0000 q^{97} +9.89949i q^{98} +(2.07321 + 11.1548i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 8 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 8 q^{4} - 4 q^{6} + 2 q^{9} - 4 q^{12} + 16 q^{16} + 8 q^{18} - 4 q^{19} + 8 q^{22} + 8 q^{24} + 20 q^{27} + 22 q^{33} - 16 q^{34} - 4 q^{36} - 40 q^{43} + 8 q^{48} + 28 q^{49} + 10 q^{51} + 8 q^{54} + 34 q^{57} - 32 q^{64} - 32 q^{66} - 28 q^{67} - 16 q^{72} - 4 q^{73} + 8 q^{76} + 14 q^{81} + 32 q^{82} - 16 q^{88} - 16 q^{96} - 40 q^{97} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 1.72474 0.158919i 0.995782 0.0917517i
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 0.224745 + 2.43916i 0.0917517 + 0.995782i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 2.82843i 1.00000i
\(9\) 2.94949 0.548188i 0.983163 0.182729i
\(10\) 0 0
\(11\) 3.78194i 1.14030i 0.821541 + 0.570149i \(0.193114\pi\)
−0.821541 + 0.570149i \(0.806886\pi\)
\(12\) −3.44949 + 0.317837i −0.995782 + 0.0917517i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 8.02458i 1.94625i 0.230285 + 0.973123i \(0.426034\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0.775255 + 4.17121i 0.182729 + 0.983163i
\(19\) 6.34847 1.45644 0.728219 0.685344i \(-0.240348\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.34847 −1.14030
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −0.449490 4.87832i −0.0917517 0.995782i
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 1.41421i 0.962250 0.272166i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0.601021 + 6.52288i 0.104624 + 1.13549i
\(34\) −11.3485 −1.94625
\(35\) 0 0
\(36\) −5.89898 + 1.09638i −0.983163 + 0.182729i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 8.97809i 1.45644i
\(39\) 0 0
\(40\) 0 0
\(41\) 10.8530i 1.69495i −0.530831 0.847477i \(-0.678120\pi\)
0.530831 0.847477i \(-0.321880\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 7.56388i 1.14030i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 6.89898 0.635674i 0.995782 0.0917517i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 1.27526 + 13.8404i 0.178571 + 1.93804i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 2.00000 + 7.07107i 0.272166 + 0.962250i
\(55\) 0 0
\(56\) 0 0
\(57\) 10.9495 1.00889i 1.45030 0.133631i
\(58\) 0 0
\(59\) 14.1421i 1.84115i −0.390567 0.920575i \(-0.627721\pi\)
0.390567 0.920575i \(-0.372279\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) −9.22474 + 0.849971i −1.13549 + 0.104624i
\(67\) 0.348469 0.0425723 0.0212861 0.999773i \(-0.493224\pi\)
0.0212861 + 0.999773i \(0.493224\pi\)
\(68\) 16.0492i 1.94625i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.55051 8.34242i −0.182729 0.983163i
\(73\) −15.6969 −1.83719 −0.918594 0.395203i \(-0.870674\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −12.6969 −1.45644
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 8.39898 3.23375i 0.933220 0.359306i
\(82\) 15.3485 1.69495
\(83\) 17.0027i 1.86629i −0.359506 0.933143i \(-0.617055\pi\)
0.359506 0.933143i \(-0.382945\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 14.1421i 1.52499i
\(87\) 0 0
\(88\) 10.6969 1.14030
\(89\) 18.4169i 1.95219i 0.217354 + 0.976093i \(0.430258\pi\)
−0.217354 + 0.976093i \(0.569742\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.898979 + 9.75663i 0.0917517 + 0.995782i
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 9.89949i 1.00000i
\(99\) 2.07321 + 11.1548i 0.208366 + 1.12110i
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −19.5732 + 1.80348i −1.93804 + 0.178571i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.0956i 1.45935i −0.683793 0.729676i \(-0.739671\pi\)
0.683793 0.729676i \(-0.260329\pi\)
\(108\) −10.0000 + 2.82843i −0.962250 + 0.272166i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.93160i 0.934287i 0.884182 + 0.467143i \(0.154717\pi\)
−0.884182 + 0.467143i \(0.845283\pi\)
\(114\) 1.42679 + 15.4849i 0.133631 + 1.45030i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 20.0000 1.84115
\(119\) 0 0
\(120\) 0 0
\(121\) −3.30306 −0.300278
\(122\) 0 0
\(123\) −1.72474 18.7187i −0.155515 1.68781i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 11.3137i 1.00000i
\(129\) −17.2474 + 1.58919i −1.51855 + 0.139920i
\(130\) 0 0
\(131\) 14.1421i 1.23560i −0.786334 0.617802i \(-0.788023\pi\)
0.786334 0.617802i \(-0.211977\pi\)
\(132\) −1.20204 13.0458i −0.104624 1.13549i
\(133\) 0 0
\(134\) 0.492810i 0.0425723i
\(135\) 0 0
\(136\) 22.6969 1.94625
\(137\) 6.11756i 0.522658i 0.965250 + 0.261329i \(0.0841608\pi\)
−0.965250 + 0.261329i \(0.915839\pi\)
\(138\) 0 0
\(139\) 3.65153 0.309719 0.154859 0.987937i \(-0.450508\pi\)
0.154859 + 0.987937i \(0.450508\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 11.7980 2.19275i 0.983163 0.182729i
\(145\) 0 0
\(146\) 22.1988i 1.83719i
\(147\) 12.0732 1.11243i 0.995782 0.0917517i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 17.9562i 1.45644i
\(153\) 4.39898 + 23.6684i 0.355636 + 1.91348i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 4.57321 + 11.8780i 0.359306 + 0.933220i
\(163\) 21.0454 1.64840 0.824202 0.566296i \(-0.191624\pi\)
0.824202 + 0.566296i \(0.191624\pi\)
\(164\) 21.7060i 1.69495i
\(165\) 0 0
\(166\) 24.0454 1.86629
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 18.7247 3.48016i 1.43192 0.266134i
\(172\) 20.0000 1.52499
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 15.1278i 1.14030i
\(177\) −2.24745 24.3916i −0.168929 1.83338i
\(178\) −26.0454 −1.95219
\(179\) 25.4880i 1.90506i −0.304446 0.952529i \(-0.598471\pi\)
0.304446 0.952529i \(-0.401529\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −30.3485 −2.21930
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −13.7980 + 1.27135i −0.995782 + 0.0917517i
\(193\) 25.6969 1.84971 0.924853 0.380325i \(-0.124188\pi\)
0.924853 + 0.380325i \(0.124188\pi\)
\(194\) 14.1421i 1.01535i
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −15.7753 + 2.93197i −1.12110 + 0.208366i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0.601021 0.0553782i 0.0423927 0.00390608i
\(202\) 0 0
\(203\) 0 0
\(204\) −2.55051 27.6807i −0.178571 1.93804i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 24.0095i 1.66077i
\(210\) 0 0
\(211\) −29.0454 −1.99957 −0.999784 0.0207756i \(-0.993386\pi\)
−0.999784 + 0.0207756i \(0.993386\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 21.3485 1.45935
\(215\) 0 0
\(216\) −4.00000 14.1421i −0.272166 0.962250i
\(217\) 0 0
\(218\) 0 0
\(219\) −27.0732 + 2.49454i −1.82944 + 0.168565i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −14.0454 −0.934287
\(227\) 2.82843i 0.187729i 0.995585 + 0.0938647i \(0.0299221\pi\)
−0.995585 + 0.0938647i \(0.970078\pi\)
\(228\) −21.8990 + 2.01778i −1.45030 + 0.133631i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.65685i 0.370593i −0.982683 0.185296i \(-0.940675\pi\)
0.982683 0.185296i \(-0.0593245\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 28.2843i 1.84115i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −27.6969 −1.78412 −0.892058 0.451920i \(-0.850739\pi\)
−0.892058 + 0.451920i \(0.850739\pi\)
\(242\) 4.67123i 0.300278i
\(243\) 13.9722 6.91215i 0.896317 0.443415i
\(244\) 0 0
\(245\) 0 0
\(246\) 26.4722 2.43916i 1.68781 0.155515i
\(247\) 0 0
\(248\) 0 0
\(249\) −2.70204 29.3253i −0.171235 1.85841i
\(250\) 0 0
\(251\) 10.3602i 0.653930i 0.945036 + 0.326965i \(0.106026\pi\)
−0.945036 + 0.326965i \(0.893974\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 11.3137i 0.705730i 0.935674 + 0.352865i \(0.114792\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) −2.24745 24.3916i −0.139920 1.51855i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 20.0000 1.23560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 18.4495 1.69994i 1.13549 0.104624i
\(265\) 0 0
\(266\) 0 0
\(267\) 2.92679 + 31.7644i 0.179116 + 1.94395i
\(268\) −0.696938 −0.0425723
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 32.0983i 1.94625i
\(273\) 0 0
\(274\) −8.65153 −0.522658
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 5.16404i 0.309719i
\(279\) 0 0
\(280\) 0 0
\(281\) 28.2843i 1.68730i 0.536895 + 0.843649i \(0.319597\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −11.0454 −0.656581 −0.328291 0.944577i \(-0.606473\pi\)
−0.328291 + 0.944577i \(0.606473\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 3.10102 + 16.6848i 0.182729 + 0.983163i
\(289\) −47.3939 −2.78788
\(290\) 0 0
\(291\) −17.2474 + 1.58919i −1.01106 + 0.0931597i
\(292\) 31.3939 1.83719
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 1.57321 + 17.0741i 0.0917517 + 0.995782i
\(295\) 0 0
\(296\) 0 0
\(297\) 5.34847 + 18.9097i 0.310350 + 1.09725i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 25.3939 1.45644
\(305\) 0 0
\(306\) −33.4722 + 6.22110i −1.91348 + 0.355636i
\(307\) 9.65153 0.550842 0.275421 0.961324i \(-0.411183\pi\)
0.275421 + 0.961324i \(0.411183\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −2.39898 26.0361i −0.133898 1.45320i
\(322\) 0 0
\(323\) 50.9438i 2.83459i
\(324\) −16.7980 + 6.46750i −0.933220 + 0.359306i
\(325\) 0 0
\(326\) 29.7627i 1.64840i
\(327\) 0 0
\(328\) −30.6969 −1.69495
\(329\) 0 0
\(330\) 0 0
\(331\) 9.04541 0.497181 0.248590 0.968609i \(-0.420033\pi\)
0.248590 + 0.968609i \(0.420033\pi\)
\(332\) 34.0053i 1.86629i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −36.3939 −1.98250 −0.991250 0.131995i \(-0.957862\pi\)
−0.991250 + 0.131995i \(0.957862\pi\)
\(338\) 18.3848i 1.00000i
\(339\) 1.57832 + 17.1295i 0.0857224 + 0.930346i
\(340\) 0 0
\(341\) 0 0
\(342\) 4.92168 + 26.4808i 0.266134 + 1.43192i
\(343\) 0 0
\(344\) 28.2843i 1.52499i
\(345\) 0 0
\(346\) 0 0
\(347\) 13.1886i 0.708002i −0.935245 0.354001i \(-0.884821\pi\)
0.935245 0.354001i \(-0.115179\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −21.3939 −1.14030
\(353\) 22.6274i 1.20434i −0.798369 0.602168i \(-0.794304\pi\)
0.798369 0.602168i \(-0.205696\pi\)
\(354\) 34.4949 3.17837i 1.83338 0.168929i
\(355\) 0 0
\(356\) 36.8338i 1.95219i
\(357\) 0 0
\(358\) 36.0454 1.90506
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 21.3031 1.12121
\(362\) 0 0
\(363\) −5.69694 + 0.524918i −0.299012 + 0.0275510i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −5.94949 32.0108i −0.309718 1.66642i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 42.9192i 2.21930i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −26.3485 −1.35343 −0.676715 0.736245i \(-0.736597\pi\)
−0.676715 + 0.736245i \(0.736597\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.79796 19.5133i −0.0917517 0.995782i
\(385\) 0 0
\(386\) 36.3410i 1.84971i
\(387\) −29.4949 + 5.48188i −1.49931 + 0.278660i
\(388\) 20.0000 1.01535
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 19.7990i 1.00000i
\(393\) −2.24745 24.3916i −0.113369 1.23039i
\(394\) 0 0
\(395\) 0 0
\(396\) −4.14643 22.3096i −0.208366 1.12110i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.9951i 1.24820i 0.781345 + 0.624099i \(0.214534\pi\)
−0.781345 + 0.624099i \(0.785466\pi\)
\(402\) 0.0783167 + 0.849971i 0.00390608 + 0.0423927i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 39.1464 3.60697i 1.93804 0.178571i
\(409\) 40.3939 1.99735 0.998674 0.0514740i \(-0.0163919\pi\)
0.998674 + 0.0514740i \(0.0163919\pi\)
\(410\) 0 0
\(411\) 0.972194 + 10.5512i 0.0479548 + 0.520453i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.29796 0.580296i 0.308412 0.0284172i
\(418\) −33.9546 −1.66077
\(419\) 2.79632i 0.136609i −0.997665 0.0683046i \(-0.978241\pi\)
0.997665 0.0683046i \(-0.0217590\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 41.0764i 1.99957i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 30.1913i 1.45935i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 20.0000 5.65685i 0.962250 0.272166i
\(433\) −4.30306 −0.206792 −0.103396 0.994640i \(-0.532971\pi\)
−0.103396 + 0.994640i \(0.532971\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −3.52781 38.2873i −0.168565 1.82944i
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 20.6464 3.83732i 0.983163 0.182729i
\(442\) 0 0
\(443\) 34.9589i 1.66095i 0.557059 + 0.830473i \(0.311930\pi\)
−0.557059 + 0.830473i \(0.688070\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 33.5446i 1.58307i −0.611124 0.791535i \(-0.709282\pi\)
0.611124 0.791535i \(-0.290718\pi\)
\(450\) 0 0
\(451\) 41.0454 1.93275
\(452\) 19.8632i 0.934287i
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) −2.85357 30.9698i −0.133631 1.45030i
\(457\) 16.3939 0.766873 0.383437 0.923567i \(-0.374740\pi\)
0.383437 + 0.923567i \(0.374740\pi\)
\(458\) 0 0
\(459\) 11.3485 + 40.1229i 0.529701 + 1.87278i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) 31.1127i 1.43972i −0.694117 0.719862i \(-0.744205\pi\)
0.694117 0.719862i \(-0.255795\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −40.0000 −1.84115
\(473\) 37.8194i 1.73894i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 39.1694i 1.78412i
\(483\) 0 0
\(484\) 6.60612 0.300278
\(485\) 0 0
\(486\) 9.77526 + 19.7597i 0.443415 + 0.896317i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 36.2980 3.34451i 1.64145 0.151244i
\(490\) 0 0
\(491\) 14.1421i 0.638226i −0.947717 0.319113i \(-0.896615\pi\)
0.947717 0.319113i \(-0.103385\pi\)
\(492\) 3.44949 + 37.4373i 0.155515 + 1.68781i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 41.4722 3.82126i 1.85841 0.171235i
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −14.6515 −0.653930
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 22.4217 2.06594i 0.995782 0.0917517i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 31.7423 8.97809i 1.40146 0.396392i
\(514\) −16.0000 −0.705730
\(515\) 0 0
\(516\) 34.4949 3.17837i 1.51855 0.139920i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.4313i 0.763678i −0.924229 0.381839i \(-0.875291\pi\)
0.924229 0.381839i \(-0.124709\pi\)
\(522\) 0 0
\(523\) −41.0454 −1.79479 −0.897395 0.441228i \(-0.854543\pi\)
−0.897395 + 0.441228i \(0.854543\pi\)
\(524\) 28.2843i 1.23560i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 2.40408 + 26.0915i 0.104624 + 1.13549i
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −7.75255 41.7121i −0.336432 1.81015i
\(532\) 0 0
\(533\) 0 0
\(534\) −44.9217 + 4.13910i −1.94395 + 0.179116i
\(535\) 0 0
\(536\) 0.985620i 0.0425723i
\(537\) −4.05051 43.9602i −0.174792 1.89702i
\(538\) 0 0
\(539\) 26.4736i 1.14030i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −45.3939 −1.94625
\(545\) 0 0
\(546\) 0 0
\(547\) 30.3485 1.29761 0.648803 0.760956i \(-0.275270\pi\)
0.648803 + 0.760956i \(0.275270\pi\)
\(548\) 12.2351i 0.522658i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −7.30306 −0.309719
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −52.3434 + 4.82294i −2.20994 + 0.203625i
\(562\) −40.0000 −1.68730
\(563\) 36.7696i 1.54965i 0.632175 + 0.774826i \(0.282163\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 15.6206i 0.656581i
\(567\) 0 0
\(568\) 0 0
\(569\) 47.6868i 1.99913i 0.0294311 + 0.999567i \(0.490630\pi\)
−0.0294311 + 0.999567i \(0.509370\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −23.5959 + 4.38551i −0.983163 + 0.182729i
\(577\) 46.3939 1.93140 0.965701 0.259656i \(-0.0836092\pi\)
0.965701 + 0.259656i \(0.0836092\pi\)
\(578\) 67.0251i 2.78788i
\(579\) 44.3207 4.08372i 1.84190 0.169714i
\(580\) 0 0
\(581\) 0 0
\(582\) −2.24745 24.3916i −0.0931597 1.01106i
\(583\) 0 0
\(584\) 44.3976i 1.83719i
\(585\) 0 0
\(586\) 0 0
\(587\) 29.2378i 1.20677i 0.797449 + 0.603386i \(0.206182\pi\)
−0.797449 + 0.603386i \(0.793818\pi\)
\(588\) −24.1464 + 2.22486i −0.995782 + 0.0917517i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 38.2159i 1.56934i −0.619915 0.784669i \(-0.712833\pi\)
0.619915 0.784669i \(-0.287167\pi\)
\(594\) −26.7423 + 7.56388i −1.09725 + 0.310350i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 37.6969 1.53769 0.768845 0.639435i \(-0.220832\pi\)
0.768845 + 0.639435i \(0.220832\pi\)
\(602\) 0 0
\(603\) 1.02781 0.191027i 0.0418555 0.00777921i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 35.9124i 1.45644i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −8.79796 47.3368i −0.355636 1.91348i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 13.6493i 0.550842i
\(615\) 0 0
\(616\) 0 0
\(617\) 39.5980i 1.59415i −0.603877 0.797077i \(-0.706378\pi\)
0.603877 0.797077i \(-0.293622\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 14.1421i 0.565233i
\(627\) 3.81556 + 41.4103i 0.152379 + 1.65377i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −50.0959 + 4.61586i −1.99113 + 0.183464i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 28.2843i 1.11716i 0.829450 + 0.558581i \(0.188654\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 36.8207 3.39267i 1.45320 0.133898i
\(643\) 50.0000 1.97181 0.985904 0.167313i \(-0.0535092\pi\)
0.985904 + 0.167313i \(0.0535092\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −72.0454 −2.83459
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −9.14643 23.7559i −0.359306 0.933220i
\(649\) 53.4847 2.09946
\(650\) 0 0
\(651\) 0 0
\(652\) −42.0908 −1.64840
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 43.4120i 1.69495i
\(657\) −46.2980 + 8.60488i −1.80626 + 0.335708i
\(658\) 0 0
\(659\) 39.6301i 1.54377i 0.635763 + 0.771885i \(0.280686\pi\)
−0.635763 + 0.771885i \(0.719314\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 12.7921i 0.497181i
\(663\) 0 0
\(664\) −48.0908 −1.86629
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 51.4687i 1.98250i
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −24.2247 + 2.23208i −0.930346 + 0.0857224i
\(679\) 0 0
\(680\) 0 0
\(681\) 0.449490 + 4.87832i 0.0172245 + 0.186937i
\(682\) 0 0
\(683\) 20.8167i 0.796530i −0.917270 0.398265i \(-0.869613\pi\)
0.917270 0.398265i \(-0.130387\pi\)
\(684\) −37.4495 + 6.96031i −1.43192 + 0.266134i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −40.0000 −1.52499
\(689\) 0 0
\(690\) 0 0
\(691\) 0.954592 0.0363144 0.0181572 0.999835i \(-0.494220\pi\)
0.0181572 + 0.999835i \(0.494220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 18.6515 0.708002
\(695\) 0 0
\(696\) 0 0
\(697\) 87.0908 3.29880
\(698\) 0 0
\(699\) −0.898979 9.75663i −0.0340025 0.369030i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 30.2555i 1.14030i
\(705\) 0 0
\(706\) 32.0000 1.20434
\(707\) 0 0
\(708\) 4.49490 + 48.7832i 0.168929 + 1.83338i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 52.0908 1.95219
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 50.9759i 1.90506i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 30.1271i 1.12121i
\(723\) −47.7702 + 4.40156i −1.77659 + 0.163696i
\(724\) 0 0
\(725\) 0 0
\(726\) −0.742346 8.05669i −0.0275510 0.299012i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) 80.2458i 2.96800i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.31789i 0.0485451i
\(738\) 45.2702 8.41385i 1.66642 0.309718i
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −9.32066 50.1492i −0.341025 1.83486i
\(748\) 60.6969 2.21930
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 1.64643 + 17.8687i 0.0599992 + 0.651171i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 37.2624i 1.35343i
\(759\) 0 0
\(760\) 0 0
\(761\) 41.1085i 1.49018i 0.666962 + 0.745091i \(0.267594\pi\)
−0.666962 + 0.745091i \(0.732406\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 27.5959 2.54270i 0.995782 0.0917517i
\(769\) −33.0908 −1.19329 −0.596643 0.802507i \(-0.703499\pi\)
−0.596643 + 0.802507i \(0.703499\pi\)
\(770\) 0 0
\(771\) 1.79796 + 19.5133i 0.0647519 + 0.702753i
\(772\) −51.3939 −1.84971
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −7.75255 41.7121i −0.278660 1.49931i
\(775\) 0 0
\(776\) 28.2843i 1.01535i
\(777\) 0 0
\(778\) 0 0
\(779\) 68.9000i 2.46860i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) 34.4949 3.17837i 1.23039 0.113369i
\(787\) 50.0000 1.78231 0.891154 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 31.5505 5.86393i 1.12110 0.208366i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 10.0959 + 54.3204i 0.356722 + 1.91932i
\(802\) −35.3485 −1.24820
\(803\) 59.3649i 2.09494i
\(804\) −1.20204 + 0.110756i −0.0423927 + 0.00390608i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 56.5685i 1.98884i −0.105474 0.994422i \(-0.533636\pi\)
0.105474 0.994422i \(-0.466364\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 5.10102 + 55.3614i 0.178571 + 1.93804i
\(817\) −63.4847 −2.22105
\(818\) 57.1256i 1.99735i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −14.9217 + 1.37489i −0.520453 + 0.0479548i
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.8659i 1.28195i 0.767561 + 0.640976i \(0.221470\pi\)
−0.767561 + 0.640976i \(0.778530\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 56.1721i 1.94625i
\(834\) 0.820663 + 8.90666i 0.0284172 + 0.308412i
\(835\) 0 0
\(836\) 48.0190i 1.66077i
\(837\) 0 0
\(838\) 3.95459 0.136609
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 4.49490 + 48.7832i 0.154812 + 1.68018i
\(844\) 58.0908 1.99957
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −19.0505 + 1.75532i −0.653812 + 0.0602425i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −42.6969 −1.45935
\(857\) 58.0791i 1.98394i 0.126459 + 0.991972i \(0.459639\pi\)
−0.126459 + 0.991972i \(0.540361\pi\)
\(858\) 0 0
\(859\) 36.3485 1.24019 0.620097 0.784525i \(-0.287093\pi\)
0.620097 + 0.784525i \(0.287093\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 8.00000 + 28.2843i 0.272166 + 0.962250i
\(865\) 0 0
\(866\) 6.08545i 0.206792i
\(867\) −81.7423 + 7.53177i −2.77612 + 0.255792i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −29.4949 + 5.48188i −0.998251 + 0.185534i
\(874\) 0 0
\(875\) 0 0
\(876\) 54.1464 4.98907i 1.82944 0.168565i
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 56.5685i 1.90584i −0.303218 0.952921i \(-0.598061\pi\)
0.303218 0.952921i \(-0.401939\pi\)
\(882\) 5.42679 + 29.1985i 0.182729 + 0.983163i
\(883\) −52.4393 −1.76472 −0.882361 0.470573i \(-0.844047\pi\)
−0.882361 + 0.470573i \(0.844047\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −49.4393 −1.66095
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 12.2298 + 31.7644i 0.409715 + 1.06415i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 47.4393 1.58307
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 58.0470i 1.93275i
\(903\) 0 0
\(904\) 28.0908 0.934287
\(905\) 0 0
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 5.65685i 0.187729i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 43.7980 4.03556i 1.45030 0.133631i
\(913\) 64.3031 2.12812
\(914\) 23.1844i 0.766873i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −56.7423 + 16.0492i −1.87278 + 0.529701i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 16.6464 1.53381i 0.548518 0.0505407i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 28.2843i 0.927977i 0.885841 + 0.463988i \(0.153582\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) 44.4393 1.45644
\(932\) 11.3137i 0.370593i
\(933\) 0 0
\(934\) 44.0000 1.43972
\(935\) 0 0
\(936\) 0 0
\(937\) −27.0908 −0.885018 −0.442509 0.896764i \(-0.645912\pi\)
−0.442509 + 0.896764i \(0.645912\pi\)
\(938\) 0 0
\(939\) −17.2474 + 1.58919i −0.562849 + 0.0518611i
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 56.5685i 1.84115i
\(945\) 0 0
\(946\) 53.4847 1.73894
\(947\) 53.7401i 1.74632i 0.487435 + 0.873160i \(0.337933\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 13.7456i 0.445265i 0.974902 + 0.222633i \(0.0714650\pi\)
−0.974902 + 0.222633i \(0.928535\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) −8.27526 44.5245i −0.266666 1.43478i
\(964\) 55.3939 1.78412
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 9.34247i 0.300278i
\(969\) 8.09592 + 87.8651i 0.260078 + 2.82263i
\(970\) 0 0
\(971\) 62.3217i 2.00000i 0.000892350 1.00000i \(0.499716\pi\)
−0.000892350 1.00000i \(0.500284\pi\)
\(972\) −27.9444 + 13.8243i −0.896317 + 0.443415i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36.3089i 1.16162i −0.814038 0.580812i \(-0.802735\pi\)
0.814038 0.580812i \(-0.197265\pi\)
\(978\) 4.72985 + 51.3331i 0.151244 + 1.64145i
\(979\) −69.6515 −2.22607
\(980\) 0 0
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −52.9444 + 4.87832i −1.68781 + 0.155515i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 15.6010 1.43748i 0.495083 0.0456172i
\(994\) 0 0
\(995\) 0 0
\(996\) 5.40408 + 58.6505i 0.171235 + 1.85841i
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 19.7990i 0.626726i
\(999\) 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 600.2.b.d.251.4 yes 4
3.2 odd 2 inner 600.2.b.d.251.2 yes 4
4.3 odd 2 2400.2.b.b.2351.2 4
5.2 odd 4 600.2.m.b.299.1 8
5.3 odd 4 600.2.m.b.299.8 8
5.4 even 2 600.2.b.b.251.1 4
8.3 odd 2 CM 600.2.b.d.251.4 yes 4
8.5 even 2 2400.2.b.b.2351.2 4
12.11 even 2 2400.2.b.b.2351.1 4
15.2 even 4 600.2.m.b.299.7 8
15.8 even 4 600.2.m.b.299.2 8
15.14 odd 2 600.2.b.b.251.3 yes 4
20.3 even 4 2400.2.m.b.1199.3 8
20.7 even 4 2400.2.m.b.1199.6 8
20.19 odd 2 2400.2.b.d.2351.3 4
24.5 odd 2 2400.2.b.b.2351.1 4
24.11 even 2 inner 600.2.b.d.251.2 yes 4
40.3 even 4 600.2.m.b.299.8 8
40.13 odd 4 2400.2.m.b.1199.3 8
40.19 odd 2 600.2.b.b.251.1 4
40.27 even 4 600.2.m.b.299.1 8
40.29 even 2 2400.2.b.d.2351.3 4
40.37 odd 4 2400.2.m.b.1199.6 8
60.23 odd 4 2400.2.m.b.1199.5 8
60.47 odd 4 2400.2.m.b.1199.4 8
60.59 even 2 2400.2.b.d.2351.4 4
120.29 odd 2 2400.2.b.d.2351.4 4
120.53 even 4 2400.2.m.b.1199.5 8
120.59 even 2 600.2.b.b.251.3 yes 4
120.77 even 4 2400.2.m.b.1199.4 8
120.83 odd 4 600.2.m.b.299.2 8
120.107 odd 4 600.2.m.b.299.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.2.b.b.251.1 4 5.4 even 2
600.2.b.b.251.1 4 40.19 odd 2
600.2.b.b.251.3 yes 4 15.14 odd 2
600.2.b.b.251.3 yes 4 120.59 even 2
600.2.b.d.251.2 yes 4 3.2 odd 2 inner
600.2.b.d.251.2 yes 4 24.11 even 2 inner
600.2.b.d.251.4 yes 4 1.1 even 1 trivial
600.2.b.d.251.4 yes 4 8.3 odd 2 CM
600.2.m.b.299.1 8 5.2 odd 4
600.2.m.b.299.1 8 40.27 even 4
600.2.m.b.299.2 8 15.8 even 4
600.2.m.b.299.2 8 120.83 odd 4
600.2.m.b.299.7 8 15.2 even 4
600.2.m.b.299.7 8 120.107 odd 4
600.2.m.b.299.8 8 5.3 odd 4
600.2.m.b.299.8 8 40.3 even 4
2400.2.b.b.2351.1 4 12.11 even 2
2400.2.b.b.2351.1 4 24.5 odd 2
2400.2.b.b.2351.2 4 4.3 odd 2
2400.2.b.b.2351.2 4 8.5 even 2
2400.2.b.d.2351.3 4 20.19 odd 2
2400.2.b.d.2351.3 4 40.29 even 2
2400.2.b.d.2351.4 4 60.59 even 2
2400.2.b.d.2351.4 4 120.29 odd 2
2400.2.m.b.1199.3 8 20.3 even 4
2400.2.m.b.1199.3 8 40.13 odd 4
2400.2.m.b.1199.4 8 60.47 odd 4
2400.2.m.b.1199.4 8 120.77 even 4
2400.2.m.b.1199.5 8 60.23 odd 4
2400.2.m.b.1199.5 8 120.53 even 4
2400.2.m.b.1199.6 8 20.7 even 4
2400.2.m.b.1199.6 8 40.37 odd 4