Properties

Label 605.2.b.b.364.4
Level $605$
Weight $2$
Character 605.364
Analytic conductor $4.831$
Analytic rank $0$
Dimension $4$
CM discriminant -55
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(364,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("605.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.4400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 364.4
Root \(-1.54336i\) of defining polynomial
Character \(\chi\) \(=\) 605.364
Dual form 605.2.b.b.364.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.49721i q^{2} -4.23607 q^{4} +2.23607 q^{5} +3.08672i q^{7} -5.58394i q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+2.49721i q^{2} -4.23607 q^{4} +2.23607 q^{5} +3.08672i q^{7} -5.58394i q^{8} +3.00000 q^{9} +5.58394i q^{10} -1.90770i q^{13} -7.70820 q^{14} +5.47214 q^{16} +8.08115i q^{17} +7.49164i q^{18} -9.47214 q^{20} +5.00000 q^{25} +4.76393 q^{26} -13.0756i q^{28} -8.94427 q^{31} +2.49721i q^{32} -20.1803 q^{34} +6.90212i q^{35} -12.7082 q^{36} -12.4861i q^{40} -13.0756i q^{43} +6.70820 q^{45} -2.52786 q^{49} +12.4861i q^{50} +8.08115i q^{52} +17.2361 q^{56} -4.00000 q^{59} -22.3357i q^{62} +9.26017i q^{63} +4.70820 q^{64} -4.26575i q^{65} -34.2323i q^{68} -17.2361 q^{70} +8.00000 q^{71} -16.7518i q^{72} +11.8965i q^{73} +12.2361 q^{80} +9.00000 q^{81} -0.728677i q^{83} +18.0700i q^{85} +32.6525 q^{86} +13.4164 q^{89} +16.7518i q^{90} +5.88854 q^{91} -6.31261i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 12 q^{9} - 4 q^{14} + 4 q^{16} - 20 q^{20} + 20 q^{25} + 28 q^{26} - 36 q^{34} - 24 q^{36} - 28 q^{49} + 60 q^{56} - 16 q^{59} - 8 q^{64} - 60 q^{70} + 32 q^{71} + 40 q^{80} + 36 q^{81} + 68 q^{86} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.49721i 1.76580i 0.469565 + 0.882898i \(0.344411\pi\)
−0.469565 + 0.882898i \(0.655589\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −4.23607 −2.11803
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) 3.08672i 1.16667i 0.812231 + 0.583336i \(0.198253\pi\)
−0.812231 + 0.583336i \(0.801747\pi\)
\(8\) − 5.58394i − 1.97422i
\(9\) 3.00000 1.00000
\(10\) 5.58394i 1.76580i
\(11\) 0 0
\(12\) 0 0
\(13\) − 1.90770i − 0.529101i −0.964372 0.264550i \(-0.914776\pi\)
0.964372 0.264550i \(-0.0852236\pi\)
\(14\) −7.70820 −2.06010
\(15\) 0 0
\(16\) 5.47214 1.36803
\(17\) 8.08115i 1.95997i 0.199081 + 0.979983i \(0.436204\pi\)
−0.199081 + 0.979983i \(0.563796\pi\)
\(18\) 7.49164i 1.76580i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −9.47214 −2.11803
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 4.76393 0.934284
\(27\) 0 0
\(28\) − 13.0756i − 2.47105i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −8.94427 −1.60644 −0.803219 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 2.49721i 0.441449i
\(33\) 0 0
\(34\) −20.1803 −3.46090
\(35\) 6.90212i 1.16667i
\(36\) −12.7082 −2.11803
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) − 12.4861i − 1.97422i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 13.0756i − 1.99401i −0.0773627 0.997003i \(-0.524650\pi\)
0.0773627 0.997003i \(-0.475350\pi\)
\(44\) 0 0
\(45\) 6.70820 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −2.52786 −0.361123
\(50\) 12.4861i 1.76580i
\(51\) 0 0
\(52\) 8.08115i 1.12065i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 17.2361 2.30327
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) − 22.3357i − 2.83664i
\(63\) 9.26017i 1.16667i
\(64\) 4.70820 0.588525
\(65\) − 4.26575i − 0.529101i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 34.2323i − 4.15128i
\(69\) 0 0
\(70\) −17.2361 −2.06010
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) − 16.7518i − 1.97422i
\(73\) 11.8965i 1.39239i 0.717855 + 0.696193i \(0.245124\pi\)
−0.717855 + 0.696193i \(0.754876\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 12.2361 1.36803
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 0.728677i − 0.0799827i −0.999200 0.0399913i \(-0.987267\pi\)
0.999200 0.0399913i \(-0.0127330\pi\)
\(84\) 0 0
\(85\) 18.0700i 1.95997i
\(86\) 32.6525 3.52101
\(87\) 0 0
\(88\) 0 0
\(89\) 13.4164 1.42214 0.711068 0.703123i \(-0.248212\pi\)
0.711068 + 0.703123i \(0.248212\pi\)
\(90\) 16.7518i 1.76580i
\(91\) 5.88854 0.617287
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) − 6.31261i − 0.637670i
\(99\) 0 0
\(100\) −21.1803 −2.11803
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −10.6525 −1.04456
\(105\) 0 0
\(106\) 0 0
\(107\) − 19.2490i − 1.86087i −0.366453 0.930436i \(-0.619428\pi\)
0.366453 0.930436i \(-0.380572\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 16.8910i 1.59605i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 5.72310i − 0.529101i
\(118\) − 9.98885i − 0.919548i
\(119\) −24.9443 −2.28664
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 37.8885 3.40249
\(125\) 11.1803 1.00000
\(126\) −23.1246 −2.06010
\(127\) 16.8910i 1.49883i 0.662100 + 0.749416i \(0.269666\pi\)
−0.662100 + 0.749416i \(0.730334\pi\)
\(128\) 16.7518i 1.48066i
\(129\) 0 0
\(130\) 10.6525 0.934284
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 45.1246 3.86940
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) − 29.2379i − 2.47105i
\(141\) 0 0
\(142\) 19.9777i 1.67649i
\(143\) 0 0
\(144\) 16.4164 1.36803
\(145\) 0 0
\(146\) −29.7082 −2.45867
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 24.2434i 1.95997i
\(154\) 0 0
\(155\) −20.0000 −1.60644
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 5.58394i 0.441449i
\(161\) 0 0
\(162\) 22.4749i 1.76580i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 1.81966 0.141233
\(167\) − 10.7175i − 0.829347i −0.909970 0.414673i \(-0.863896\pi\)
0.909970 0.414673i \(-0.136104\pi\)
\(168\) 0 0
\(169\) 9.36068 0.720052
\(170\) −45.1246 −3.46090
\(171\) 0 0
\(172\) 55.3890i 4.22337i
\(173\) − 10.4392i − 0.793677i −0.917888 0.396839i \(-0.870107\pi\)
0.917888 0.396839i \(-0.129893\pi\)
\(174\) 0 0
\(175\) 15.4336i 1.16667i
\(176\) 0 0
\(177\) 0 0
\(178\) 33.5036i 2.51120i
\(179\) 17.8885 1.33705 0.668526 0.743689i \(-0.266925\pi\)
0.668526 + 0.743689i \(0.266925\pi\)
\(180\) −28.4164 −2.11803
\(181\) 4.47214 0.332411 0.166206 0.986091i \(-0.446848\pi\)
0.166206 + 0.986091i \(0.446848\pi\)
\(182\) 14.7049i 1.09000i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −26.8328 −1.94155 −0.970777 0.239983i \(-0.922858\pi\)
−0.970777 + 0.239983i \(0.922858\pi\)
\(192\) 0 0
\(193\) − 20.4280i − 1.47044i −0.677827 0.735221i \(-0.737078\pi\)
0.677827 0.735221i \(-0.262922\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 10.7082 0.764872
\(197\) − 21.8854i − 1.55927i −0.626234 0.779635i \(-0.715405\pi\)
0.626234 0.779635i \(-0.284595\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) − 27.9197i − 1.97422i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) − 10.4392i − 0.723828i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 48.0689 3.28592
\(215\) − 29.2379i − 1.99401i
\(216\) 0 0
\(217\) − 27.6085i − 1.87419i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.4164 1.03702
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) −7.70820 −0.515026
\(225\) 15.0000 1.00000
\(226\) 0 0
\(227\) − 25.4225i − 1.68735i −0.536855 0.843674i \(-0.680388\pi\)
0.536855 0.843674i \(-0.319612\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.7119i 1.02932i 0.857393 + 0.514662i \(0.172083\pi\)
−0.857393 + 0.514662i \(0.827917\pi\)
\(234\) 14.2918 0.934284
\(235\) 0 0
\(236\) 16.9443 1.10298
\(237\) 0 0
\(238\) − 62.2911i − 4.03773i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.65248 −0.361123
\(246\) 0 0
\(247\) 0 0
\(248\) 49.9442i 3.17146i
\(249\) 0 0
\(250\) 27.9197i 1.76580i
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) − 39.2267i − 2.47105i
\(253\) 0 0
\(254\) −42.1803 −2.64663
\(255\) 0 0
\(256\) −32.4164 −2.02603
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 18.0700i 1.12065i
\(261\) 0 0
\(262\) 0 0
\(263\) 4.54408i 0.280200i 0.990137 + 0.140100i \(0.0447424\pi\)
−0.990137 + 0.140100i \(0.955258\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 44.2211i 2.68130i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 30.4169i − 1.82757i −0.406194 0.913787i \(-0.633144\pi\)
0.406194 0.913787i \(-0.366856\pi\)
\(278\) 0 0
\(279\) −26.8328 −1.60644
\(280\) 38.5410 2.30327
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 26.8798i 1.59784i 0.601438 + 0.798920i \(0.294595\pi\)
−0.601438 + 0.798920i \(0.705405\pi\)
\(284\) −33.8885 −2.01092
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 7.49164i 0.441449i
\(289\) −48.3050 −2.84147
\(290\) 0 0
\(291\) 0 0
\(292\) − 50.3946i − 2.94912i
\(293\) − 34.2323i − 1.99987i −0.0113203 0.999936i \(-0.503603\pi\)
0.0113203 0.999936i \(-0.496397\pi\)
\(294\) 0 0
\(295\) −8.94427 −0.520756
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 40.3607 2.32635
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −60.5410 −3.46090
\(307\) 33.0533i 1.88645i 0.332155 + 0.943225i \(0.392224\pi\)
−0.332155 + 0.943225i \(0.607776\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) − 49.9442i − 2.83664i
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 20.7064i 1.16667i
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 10.5279 0.588525
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −38.1246 −2.11803
\(325\) − 9.53850i − 0.529101i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −35.7771 −1.96649 −0.983243 0.182298i \(-0.941646\pi\)
−0.983243 + 0.182298i \(0.941646\pi\)
\(332\) 3.08672i 0.169406i
\(333\) 0 0
\(334\) 26.7639 1.46446
\(335\) 0 0
\(336\) 0 0
\(337\) − 16.6126i − 0.904948i −0.891778 0.452474i \(-0.850541\pi\)
0.891778 0.452474i \(-0.149459\pi\)
\(338\) 23.3756i 1.27147i
\(339\) 0 0
\(340\) − 76.5457i − 4.15128i
\(341\) 0 0
\(342\) 0 0
\(343\) 13.8042i 0.745359i
\(344\) −73.0132 −3.93661
\(345\) 0 0
\(346\) 26.0689 1.40147
\(347\) − 11.6182i − 0.623699i −0.950132 0.311849i \(-0.899052\pi\)
0.950132 0.311849i \(-0.100948\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) −38.5410 −2.06010
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 17.8885 0.949425
\(356\) −56.8328 −3.01213
\(357\) 0 0
\(358\) 44.6715i 2.36096i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) − 37.4582i − 1.97422i
\(361\) −19.0000 −1.00000
\(362\) 11.1679i 0.586970i
\(363\) 0 0
\(364\) −24.9443 −1.30744
\(365\) 26.6015i 1.39239i
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.62379i 0.342967i 0.985187 + 0.171484i \(0.0548560\pi\)
−0.985187 + 0.171484i \(0.945144\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 67.0072i − 3.42839i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 51.0132 2.59650
\(387\) − 39.2267i − 1.99401i
\(388\) 0 0
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 14.1154i 0.712937i
\(393\) 0 0
\(394\) 54.6525 2.75335
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 59.9331i 3.00417i
\(399\) 0 0
\(400\) 27.3607 1.36803
\(401\) 4.47214 0.223328 0.111664 0.993746i \(-0.464382\pi\)
0.111664 + 0.993746i \(0.464382\pi\)
\(402\) 0 0
\(403\) 17.0630i 0.849968i
\(404\) 0 0
\(405\) 20.1246 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 12.3469i − 0.607551i
\(414\) 0 0
\(415\) − 1.62937i − 0.0799827i
\(416\) 4.76393 0.233571
\(417\) 0 0
\(418\) 0 0
\(419\) −35.7771 −1.74783 −0.873913 0.486083i \(-0.838425\pi\)
−0.873913 + 0.486083i \(0.838425\pi\)
\(420\) 0 0
\(421\) −31.3050 −1.52571 −0.762855 0.646570i \(-0.776203\pi\)
−0.762855 + 0.646570i \(0.776203\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 40.4057i 1.95997i
\(426\) 0 0
\(427\) 0 0
\(428\) 81.5402i 3.94139i
\(429\) 0 0
\(430\) 73.0132 3.52101
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 68.9443 3.30943
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −7.58359 −0.361123
\(442\) 38.4980i 1.83116i
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 30.0000 1.42214
\(446\) 0 0
\(447\) 0 0
\(448\) 14.5329i 0.686616i
\(449\) 40.2492 1.89948 0.949739 0.313042i \(-0.101348\pi\)
0.949739 + 0.313042i \(0.101348\pi\)
\(450\) 37.4582i 1.76580i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 63.4853 2.97951
\(455\) 13.1672 0.617287
\(456\) 0 0
\(457\) 36.5903i 1.71162i 0.517287 + 0.855812i \(0.326942\pi\)
−0.517287 + 0.855812i \(0.673058\pi\)
\(458\) 14.9833i 0.700122i
\(459\) 0 0
\(460\) 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
\(466\) −39.2361 −1.81758
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 24.2434i 1.12065i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 22.3357i 1.02809i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 105.666 4.84318
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) − 14.1154i − 0.637670i
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −48.9443 −2.19766
\(497\) 24.6938i 1.10767i
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −47.3607 −2.11803
\(501\) 0 0
\(502\) 69.9219i 3.12077i
\(503\) − 36.8687i − 1.64389i −0.569565 0.821946i \(-0.692888\pi\)
0.569565 0.821946i \(-0.307112\pi\)
\(504\) 51.7082 2.30327
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) − 71.5513i − 3.17458i
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) −36.7214 −1.62446
\(512\) − 47.4470i − 2.09688i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −23.8197 −1.04456
\(521\) 22.3607 0.979639 0.489820 0.871824i \(-0.337063\pi\)
0.489820 + 0.871824i \(0.337063\pi\)
\(522\) 0 0
\(523\) 45.4002i 1.98521i 0.121387 + 0.992605i \(0.461266\pi\)
−0.121387 + 0.992605i \(0.538734\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −11.3475 −0.494776
\(527\) − 72.2800i − 3.14857i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 43.0421i − 1.86087i
\(536\) 0 0
\(537\) 0 0
\(538\) − 34.9610i − 1.50727i
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −20.1803 −0.865225
\(545\) 0 0
\(546\) 0 0
\(547\) − 8.35948i − 0.357425i −0.983901 0.178713i \(-0.942807\pi\)
0.983901 0.178713i \(-0.0571933\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 75.9574 3.22712
\(555\) 0 0
\(556\) 0 0
\(557\) 22.7861i 0.965478i 0.875764 + 0.482739i \(0.160358\pi\)
−0.875764 + 0.482739i \(0.839642\pi\)
\(558\) − 67.0072i − 2.83664i
\(559\) −24.9443 −1.05503
\(560\) 37.7694i 1.59605i
\(561\) 0 0
\(562\) 0 0
\(563\) 17.7917i 0.749829i 0.927059 + 0.374915i \(0.122328\pi\)
−0.927059 + 0.374915i \(0.877672\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −67.1246 −2.82146
\(567\) 27.7805i 1.16667i
\(568\) − 44.6715i − 1.87437i
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 14.1246 0.588525
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) − 120.628i − 5.01745i
\(579\) 0 0
\(580\) 0 0
\(581\) 2.24922 0.0933135
\(582\) 0 0
\(583\) 0 0
\(584\) 66.4296 2.74887
\(585\) − 12.7972i − 0.529101i
\(586\) 85.4853 3.53136
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) − 22.3357i − 0.919548i
\(591\) 0 0
\(592\) 0 0
\(593\) − 32.7749i − 1.34591i −0.739686 0.672953i \(-0.765026\pi\)
0.739686 0.672953i \(-0.234974\pi\)
\(594\) 0 0
\(595\) −55.7771 −2.28664
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −44.7214 −1.82727 −0.913633 0.406541i \(-0.866735\pi\)
−0.913633 + 0.406541i \(0.866735\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 100.789i 4.10786i
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 24.5218i 0.995308i 0.867376 + 0.497654i \(0.165805\pi\)
−0.867376 + 0.497654i \(0.834195\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) − 102.697i − 4.15128i
\(613\) 46.5792i 1.88132i 0.339357 + 0.940658i \(0.389791\pi\)
−0.339357 + 0.940658i \(0.610209\pi\)
\(614\) −82.5410 −3.33108
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 17.8885 0.719001 0.359501 0.933145i \(-0.382947\pi\)
0.359501 + 0.933145i \(0.382947\pi\)
\(620\) 84.7214 3.40249
\(621\) 0 0
\(622\) 79.9108i 3.20413i
\(623\) 41.4127i 1.65917i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −51.7082 −2.06010
\(631\) −48.0000 −1.91085 −0.955425 0.295234i \(-0.904602\pi\)
−0.955425 + 0.295234i \(0.904602\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 37.7694i 1.49883i
\(636\) 0 0
\(637\) 4.82241i 0.191071i
\(638\) 0 0
\(639\) 24.0000 0.949425
\(640\) 37.4582i 1.48066i
\(641\) −31.3050 −1.23647 −0.618236 0.785993i \(-0.712152\pi\)
−0.618236 + 0.785993i \(0.712152\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) − 50.2554i − 1.97422i
\(649\) 0 0
\(650\) 23.8197 0.934284
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 35.6896i 1.39239i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) − 89.3430i − 3.47241i
\(663\) 0 0
\(664\) −4.06888 −0.157903
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 45.4002i 1.75659i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 48.0365i − 1.85167i −0.377925 0.925836i \(-0.623362\pi\)
0.377925 0.925836i \(-0.376638\pi\)
\(674\) 41.4853 1.59795
\(675\) 0 0
\(676\) −39.6525 −1.52510
\(677\) − 2.80839i − 0.107935i −0.998543 0.0539677i \(-0.982813\pi\)
0.998543 0.0539677i \(-0.0171868\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 100.902 3.86940
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −34.4721 −1.31615
\(687\) 0 0
\(688\) − 71.5513i − 2.72787i
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 44.2211i 1.68104i
\(693\) 0 0
\(694\) 29.0132 1.10132
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) − 65.3779i − 2.47105i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.47214 0.167955 0.0839773 0.996468i \(-0.473238\pi\)
0.0839773 + 0.996468i \(0.473238\pi\)
\(710\) 44.6715i 1.67649i
\(711\) 0 0
\(712\) − 74.9164i − 2.80761i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −75.7771 −2.83192
\(717\) 0 0
\(718\) 0 0
\(719\) −26.8328 −1.00070 −0.500348 0.865825i \(-0.666794\pi\)
−0.500348 + 0.865825i \(0.666794\pi\)
\(720\) 36.7082 1.36803
\(721\) 0 0
\(722\) − 47.4470i − 1.76580i
\(723\) 0 0
\(724\) −18.9443 −0.704058
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) − 32.8813i − 1.21866i
\(729\) 27.0000 1.00000
\(730\) −66.4296 −2.45867
\(731\) 105.666 3.90818
\(732\) 0 0
\(733\) 29.5162i 1.09021i 0.838369 + 0.545103i \(0.183509\pi\)
−0.838369 + 0.545103i \(0.816491\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 18.3483i − 0.673135i −0.941659 0.336567i \(-0.890734\pi\)
0.941659 0.336567i \(-0.109266\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −16.5410 −0.605610
\(747\) − 2.18603i − 0.0799827i
\(748\) 0 0
\(749\) 59.4164 2.17103
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) − 89.8996i − 3.26530i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 113.666 4.11228
\(765\) 54.2100i 1.95997i
\(766\) 0 0
\(767\) 7.63080i 0.275532i
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 86.5346i 3.11445i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 97.9574 3.52101
\(775\) −44.7214 −1.60644
\(776\) 0 0
\(777\) 0 0
\(778\) − 64.9275i − 2.32776i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −13.8328 −0.494029
\(785\) 0 0
\(786\) 0 0
\(787\) − 46.8575i − 1.67029i −0.550030 0.835145i \(-0.685384\pi\)
0.550030 0.835145i \(-0.314616\pi\)
\(788\) 92.7080i 3.30259i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −101.666 −3.60344
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 12.4861i 0.441449i
\(801\) 40.2492 1.42214
\(802\) 11.1679i 0.394351i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −42.6099 −1.50087
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 50.2554i 1.76580i
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 17.6656 0.617287
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 30.8328 1.07281
\(827\) 53.0310i 1.84407i 0.387110 + 0.922034i \(0.373473\pi\)
−0.387110 + 0.922034i \(0.626527\pi\)
\(828\) 0 0
\(829\) 22.3607 0.776619 0.388309 0.921529i \(-0.373059\pi\)
0.388309 + 0.921529i \(0.373059\pi\)
\(830\) 4.06888 0.141233
\(831\) 0 0
\(832\) − 8.98184i − 0.311389i
\(833\) − 20.4280i − 0.707790i
\(834\) 0 0
\(835\) − 23.9651i − 0.829347i
\(836\) 0 0
\(837\) 0 0
\(838\) − 89.3430i − 3.08630i
\(839\) −56.0000 −1.93333 −0.966667 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) − 78.1751i − 2.69409i
\(843\) 0 0
\(844\) 0 0
\(845\) 20.9311 0.720052
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −100.902 −3.46090
\(851\) 0 0
\(852\) 0 0
\(853\) 18.9707i 0.649544i 0.945792 + 0.324772i \(0.105288\pi\)
−0.945792 + 0.324772i \(0.894712\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −107.485 −3.67377
\(857\) 52.7526i 1.80200i 0.433824 + 0.900998i \(0.357164\pi\)
−0.433824 + 0.900998i \(0.642836\pi\)
\(858\) 0 0
\(859\) −35.7771 −1.22070 −0.610349 0.792132i \(-0.708971\pi\)
−0.610349 + 0.792132i \(0.708971\pi\)
\(860\) 123.854i 4.22337i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) − 23.3427i − 0.793677i
\(866\) 0 0
\(867\) 0 0
\(868\) 116.951i 3.96959i
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 34.5106i 1.16667i
\(876\) 0 0
\(877\) − 38.9484i − 1.31519i −0.753370 0.657597i \(-0.771573\pi\)
0.753370 0.657597i \(-0.228427\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) − 18.9378i − 0.637670i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −65.3050 −2.19644
\(885\) 0 0
\(886\) 0 0
\(887\) 13.9763i 0.469277i 0.972083 + 0.234639i \(0.0753906\pi\)
−0.972083 + 0.234639i \(0.924609\pi\)
\(888\) 0 0
\(889\) −52.1378 −1.74864
\(890\) 74.9164i 2.51120i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 40.0000 1.33705
\(896\) −51.7082 −1.72745
\(897\) 0 0
\(898\) 100.511i 3.35409i
\(899\) 0 0
\(900\) −63.5410 −2.11803
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 107.691i 3.57386i
\(909\) 0 0
\(910\) 32.8813i 1.09000i
\(911\) −8.94427 −0.296337 −0.148168 0.988962i \(-0.547338\pi\)
−0.148168 + 0.988962i \(0.547338\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −91.3738 −3.02238
\(915\) 0 0
\(916\) −25.4164 −0.839782
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 15.2616i − 0.502342i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 66.5569i − 2.18014i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −31.9574 −1.04456
\(937\) 8.98184i 0.293424i 0.989179 + 0.146712i \(0.0468691\pi\)
−0.989179 + 0.146712i \(0.953131\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −21.8885 −0.712411
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 22.6950 0.736712
\(950\) 0 0
\(951\) 0 0
\(952\) 139.287i 4.51432i
\(953\) − 51.8519i − 1.67965i −0.542858 0.839825i \(-0.682658\pi\)
0.542858 0.839825i \(-0.317342\pi\)
\(954\) 0 0
\(955\) −60.0000 −1.94155
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 49.0000 1.58065
\(962\) 0 0
\(963\) − 57.7471i − 1.86087i
\(964\) 0 0
\(965\) − 45.6785i − 1.47044i
\(966\) 0 0
\(967\) 61.5625i 1.97972i 0.142063 + 0.989858i \(0.454626\pi\)
−0.142063 + 0.989858i \(0.545374\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 17.8885 0.574071 0.287035 0.957920i \(-0.407330\pi\)
0.287035 + 0.957920i \(0.407330\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 23.9443 0.764872
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) − 48.9372i − 1.55927i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 62.6099 1.98887 0.994435 0.105356i \(-0.0335982\pi\)
0.994435 + 0.105356i \(0.0335982\pi\)
\(992\) − 22.3357i − 0.709161i
\(993\) 0 0
\(994\) −61.6656 −1.95592
\(995\) 53.6656 1.70131
\(996\) 0 0
\(997\) 58.0254i 1.83768i 0.394627 + 0.918841i \(0.370874\pi\)
−0.394627 + 0.918841i \(0.629126\pi\)
\(998\) − 9.98885i − 0.316191i
\(999\) 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 605.2.b.b.364.4 yes 4
5.2 odd 4 3025.2.a.bb.1.1 4
5.3 odd 4 3025.2.a.bb.1.4 4
5.4 even 2 inner 605.2.b.b.364.1 4
11.2 odd 10 605.2.j.a.444.1 8
11.3 even 5 605.2.j.c.9.2 8
11.4 even 5 605.2.j.c.269.1 8
11.5 even 5 605.2.j.a.124.1 8
11.6 odd 10 605.2.j.a.124.2 8
11.7 odd 10 605.2.j.c.269.2 8
11.8 odd 10 605.2.j.c.9.1 8
11.9 even 5 605.2.j.a.444.2 8
11.10 odd 2 inner 605.2.b.b.364.1 4
55.4 even 10 605.2.j.c.269.2 8
55.9 even 10 605.2.j.a.444.1 8
55.14 even 10 605.2.j.c.9.1 8
55.19 odd 10 605.2.j.c.9.2 8
55.24 odd 10 605.2.j.a.444.2 8
55.29 odd 10 605.2.j.c.269.1 8
55.32 even 4 3025.2.a.bb.1.4 4
55.39 odd 10 605.2.j.a.124.1 8
55.43 even 4 3025.2.a.bb.1.1 4
55.49 even 10 605.2.j.a.124.2 8
55.54 odd 2 CM 605.2.b.b.364.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.b.b.364.1 4 5.4 even 2 inner
605.2.b.b.364.1 4 11.10 odd 2 inner
605.2.b.b.364.4 yes 4 1.1 even 1 trivial
605.2.b.b.364.4 yes 4 55.54 odd 2 CM
605.2.j.a.124.1 8 11.5 even 5
605.2.j.a.124.1 8 55.39 odd 10
605.2.j.a.124.2 8 11.6 odd 10
605.2.j.a.124.2 8 55.49 even 10
605.2.j.a.444.1 8 11.2 odd 10
605.2.j.a.444.1 8 55.9 even 10
605.2.j.a.444.2 8 11.9 even 5
605.2.j.a.444.2 8 55.24 odd 10
605.2.j.c.9.1 8 11.8 odd 10
605.2.j.c.9.1 8 55.14 even 10
605.2.j.c.9.2 8 11.3 even 5
605.2.j.c.9.2 8 55.19 odd 10
605.2.j.c.269.1 8 11.4 even 5
605.2.j.c.269.1 8 55.29 odd 10
605.2.j.c.269.2 8 11.7 odd 10
605.2.j.c.269.2 8 55.4 even 10
3025.2.a.bb.1.1 4 5.2 odd 4
3025.2.a.bb.1.1 4 55.43 even 4
3025.2.a.bb.1.4 4 5.3 odd 4
3025.2.a.bb.1.4 4 55.32 even 4