Properties

Label 1900.2.l.c.1557.2
Level $1900$
Weight $2$
Character 1900.1557
Analytic conductor $15.172$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,2,Mod(493,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.493");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.l (of order \(4\), degree \(2\), 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: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 184 x^{13} + 588 x^{12} - 1440 x^{11} + 3064 x^{10} - 5344 x^{9} + 8028 x^{8} - 9824 x^{7} + 9952 x^{6} - 5472 x^{5} + 1248 x^{4} + 6112 x^{3} + \cdots + 5800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1557.2
Root \(0.129018 - 1.66674i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1557
Dual form 1900.2.l.c.493.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.79576 - 1.79576i) q^{3} +(3.30136 + 3.30136i) q^{7} +3.44949i q^{9} +O(q^{10})\) \(q+(-1.79576 - 1.79576i) q^{3} +(3.30136 + 3.30136i) q^{7} +3.44949i q^{9} +2.44949 q^{11} +(-2.60293 - 2.60293i) q^{13} +(1.48393 + 1.48393i) q^{17} +(2.66477 - 3.44949i) q^{19} -11.8569i q^{21} +(-4.78529 + 4.78529i) q^{23} +(0.807175 - 0.807175i) q^{27} +5.32955 q^{29} +6.52734i q^{31} +(-4.39869 - 4.39869i) q^{33} +(-7.00162 + 7.00162i) q^{37} +9.34847i q^{39} +11.8569i q^{41} +(1.81743 - 1.81743i) q^{43} +(-3.30136 - 3.30136i) q^{47} +14.7980i q^{49} -5.32955i q^{51} +(3.41011 + 3.41011i) q^{53} +(-10.9797 + 1.40916i) q^{57} -6.52734 q^{59} +5.34847 q^{61} +(-11.3880 + 11.3880i) q^{63} +(4.58010 - 4.58010i) q^{67} +17.1864 q^{69} -11.8569i q^{71} +(2.96786 - 2.96786i) q^{73} +(8.08665 + 8.08665i) q^{77} +11.8569 q^{79} +7.44949 q^{81} +(-6.93623 + 6.93623i) q^{83} +(-9.57058 - 9.57058i) q^{87} +5.32955 q^{89} -17.1864i q^{91} +(11.7215 - 11.7215i) q^{93} +(-3.77292 + 3.77292i) q^{97} +8.44949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{61} + 80 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.79576 1.79576i −1.03678 1.03678i −0.999297 0.0374838i \(-0.988066\pi\)
−0.0374838 0.999297i \(-0.511934\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.30136 + 3.30136i 1.24780 + 1.24780i 0.956691 + 0.291106i \(0.0940233\pi\)
0.291106 + 0.956691i \(0.405977\pi\)
\(8\) 0 0
\(9\) 3.44949i 1.14983i
\(10\) 0 0
\(11\) 2.44949 0.738549 0.369274 0.929320i \(-0.379606\pi\)
0.369274 + 0.929320i \(0.379606\pi\)
\(12\) 0 0
\(13\) −2.60293 2.60293i −0.721923 0.721923i 0.247073 0.968997i \(-0.420531\pi\)
−0.968997 + 0.247073i \(0.920531\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.48393 + 1.48393i 0.359905 + 0.359905i 0.863778 0.503873i \(-0.168092\pi\)
−0.503873 + 0.863778i \(0.668092\pi\)
\(18\) 0 0
\(19\) 2.66477 3.44949i 0.611341 0.791367i
\(20\) 0 0
\(21\) 11.8569i 2.58738i
\(22\) 0 0
\(23\) −4.78529 + 4.78529i −0.997801 + 0.997801i −0.999998 0.00219610i \(-0.999301\pi\)
0.00219610 + 0.999998i \(0.499301\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.807175 0.807175i 0.155341 0.155341i
\(28\) 0 0
\(29\) 5.32955 0.989672 0.494836 0.868986i \(-0.335228\pi\)
0.494836 + 0.868986i \(0.335228\pi\)
\(30\) 0 0
\(31\) 6.52734i 1.17234i 0.810187 + 0.586172i \(0.199366\pi\)
−0.810187 + 0.586172i \(0.800634\pi\)
\(32\) 0 0
\(33\) −4.39869 4.39869i −0.765714 0.765714i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00162 + 7.00162i −1.15106 + 1.15106i −0.164719 + 0.986341i \(0.552672\pi\)
−0.986341 + 0.164719i \(0.947328\pi\)
\(38\) 0 0
\(39\) 9.34847i 1.49695i
\(40\) 0 0
\(41\) 11.8569i 1.85173i 0.377849 + 0.925867i \(0.376664\pi\)
−0.377849 + 0.925867i \(0.623336\pi\)
\(42\) 0 0
\(43\) 1.81743 1.81743i 0.277156 0.277156i −0.554817 0.831973i \(-0.687212\pi\)
0.831973 + 0.554817i \(0.187212\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.30136 3.30136i −0.481553 0.481553i 0.424074 0.905627i \(-0.360599\pi\)
−0.905627 + 0.424074i \(0.860599\pi\)
\(48\) 0 0
\(49\) 14.7980i 2.11399i
\(50\) 0 0
\(51\) 5.32955i 0.746286i
\(52\) 0 0
\(53\) 3.41011 + 3.41011i 0.468414 + 0.468414i 0.901400 0.432986i \(-0.142540\pi\)
−0.432986 + 0.901400i \(0.642540\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −10.9797 + 1.40916i −1.45430 + 0.186648i
\(58\) 0 0
\(59\) −6.52734 −0.849787 −0.424893 0.905243i \(-0.639688\pi\)
−0.424893 + 0.905243i \(0.639688\pi\)
\(60\) 0 0
\(61\) 5.34847 0.684801 0.342401 0.939554i \(-0.388760\pi\)
0.342401 + 0.939554i \(0.388760\pi\)
\(62\) 0 0
\(63\) −11.3880 + 11.3880i −1.43475 + 1.43475i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.58010 4.58010i 0.559548 0.559548i −0.369631 0.929179i \(-0.620516\pi\)
0.929179 + 0.369631i \(0.120516\pi\)
\(68\) 0 0
\(69\) 17.1864 2.06900
\(70\) 0 0
\(71\) 11.8569i 1.40715i −0.710619 0.703577i \(-0.751585\pi\)
0.710619 0.703577i \(-0.248415\pi\)
\(72\) 0 0
\(73\) 2.96786 2.96786i 0.347361 0.347361i −0.511765 0.859126i \(-0.671008\pi\)
0.859126 + 0.511765i \(0.171008\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.08665 + 8.08665i 0.921559 + 0.921559i
\(78\) 0 0
\(79\) 11.8569 1.33400 0.667002 0.745056i \(-0.267577\pi\)
0.667002 + 0.745056i \(0.267577\pi\)
\(80\) 0 0
\(81\) 7.44949 0.827721
\(82\) 0 0
\(83\) −6.93623 + 6.93623i −0.761350 + 0.761350i −0.976566 0.215217i \(-0.930954\pi\)
0.215217 + 0.976566i \(0.430954\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.57058 9.57058i −1.02607 1.02607i
\(88\) 0 0
\(89\) 5.32955 0.564931 0.282465 0.959277i \(-0.408848\pi\)
0.282465 + 0.959277i \(0.408848\pi\)
\(90\) 0 0
\(91\) 17.1864i 1.80163i
\(92\) 0 0
\(93\) 11.7215 11.7215i 1.21546 1.21546i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.77292 + 3.77292i −0.383082 + 0.383082i −0.872211 0.489129i \(-0.837315\pi\)
0.489129 + 0.872211i \(0.337315\pi\)
\(98\) 0 0
\(99\) 8.44949i 0.849206i
\(100\) 0 0
\(101\) −1.34847 −0.134178 −0.0670889 0.997747i \(-0.521371\pi\)
−0.0670889 + 0.997747i \(0.521371\pi\)
\(102\) 0 0
\(103\) 4.58010 + 4.58010i 0.451290 + 0.451290i 0.895783 0.444492i \(-0.146616\pi\)
−0.444492 + 0.895783i \(0.646616\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.80880 7.80880i 0.754905 0.754905i −0.220485 0.975390i \(-0.570764\pi\)
0.975390 + 0.220485i \(0.0707642\pi\)
\(108\) 0 0
\(109\) −6.52734 −0.625205 −0.312603 0.949884i \(-0.601201\pi\)
−0.312603 + 0.949884i \(0.601201\pi\)
\(110\) 0 0
\(111\) 25.1464 2.38679
\(112\) 0 0
\(113\) 9.78596 + 9.78596i 0.920586 + 0.920586i 0.997071 0.0764849i \(-0.0243697\pi\)
−0.0764849 + 0.997071i \(0.524370\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.97879 8.97879i 0.830089 0.830089i
\(118\) 0 0
\(119\) 9.79796i 0.898177i
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 21.2921 21.2921i 1.91984 1.91984i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.79576 + 1.79576i −0.159348 + 0.159348i −0.782278 0.622930i \(-0.785942\pi\)
0.622930 + 0.782278i \(0.285942\pi\)
\(128\) 0 0
\(129\) −6.52734 −0.574700
\(130\) 0 0
\(131\) 9.79796 0.856052 0.428026 0.903767i \(-0.359209\pi\)
0.428026 + 0.903767i \(0.359209\pi\)
\(132\) 0 0
\(133\) 20.1854 2.59063i 1.75030 0.224636i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.08665 + 8.08665i 0.690889 + 0.690889i 0.962428 0.271539i \(-0.0875325\pi\)
−0.271539 + 0.962428i \(0.587532\pi\)
\(138\) 0 0
\(139\) 5.55051i 0.470788i −0.971900 0.235394i \(-0.924362\pi\)
0.971900 0.235394i \(-0.0756381\pi\)
\(140\) 0 0
\(141\) 11.8569i 0.998530i
\(142\) 0 0
\(143\) −6.37586 6.37586i −0.533176 0.533176i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 26.5735 26.5735i 2.19175 2.19175i
\(148\) 0 0
\(149\) 23.1464i 1.89623i 0.317929 + 0.948115i \(0.397013\pi\)
−0.317929 + 0.948115i \(0.602987\pi\)
\(150\) 0 0
\(151\) 5.32955i 0.433712i −0.976204 0.216856i \(-0.930420\pi\)
0.976204 0.216856i \(-0.0695803\pi\)
\(152\) 0 0
\(153\) −5.11879 + 5.11879i −0.413830 + 0.413830i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.7215 11.7215i −0.935479 0.935479i 0.0625624 0.998041i \(-0.480073\pi\)
−0.998041 + 0.0625624i \(0.980073\pi\)
\(158\) 0 0
\(159\) 12.2474i 0.971286i
\(160\) 0 0
\(161\) −31.5959 −2.49011
\(162\) 0 0
\(163\) 4.78529 4.78529i 0.374813 0.374813i −0.494414 0.869227i \(-0.664617\pi\)
0.869227 + 0.494414i \(0.164617\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.43294 + 1.43294i −0.110884 + 0.110884i −0.760372 0.649488i \(-0.774983\pi\)
0.649488 + 0.760372i \(0.274983\pi\)
\(168\) 0 0
\(169\) 0.550510i 0.0423469i
\(170\) 0 0
\(171\) 11.8990 + 9.19211i 0.909938 + 0.702938i
\(172\) 0 0
\(173\) 11.7631 + 11.7631i 0.894334 + 0.894334i 0.994928 0.100594i \(-0.0320742\pi\)
−0.100594 + 0.994928i \(0.532074\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 11.7215 + 11.7215i 0.881043 + 0.881043i
\(178\) 0 0
\(179\) 17.1864 1.28457 0.642287 0.766464i \(-0.277986\pi\)
0.642287 + 0.766464i \(0.277986\pi\)
\(180\) 0 0
\(181\) 5.32955i 0.396142i −0.980188 0.198071i \(-0.936532\pi\)
0.980188 0.198071i \(-0.0634677\pi\)
\(182\) 0 0
\(183\) −9.60455 9.60455i −0.709989 0.709989i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.63487 + 3.63487i 0.265808 + 0.265808i
\(188\) 0 0
\(189\) 5.32955 0.387668
\(190\) 0 0
\(191\) 1.10102 0.0796670 0.0398335 0.999206i \(-0.487317\pi\)
0.0398335 + 0.999206i \(0.487317\pi\)
\(192\) 0 0
\(193\) −0.181408 0.181408i −0.0130581 0.0130581i 0.700548 0.713606i \(-0.252939\pi\)
−0.713606 + 0.700548i \(0.752939\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.7215 + 11.7215i 0.835123 + 0.835123i 0.988212 0.153089i \(-0.0489222\pi\)
−0.153089 + 0.988212i \(0.548922\pi\)
\(198\) 0 0
\(199\) 16.6969i 1.18361i −0.806079 0.591807i \(-0.798415\pi\)
0.806079 0.591807i \(-0.201585\pi\)
\(200\) 0 0
\(201\) −16.4495 −1.16026
\(202\) 0 0
\(203\) 17.5948 + 17.5948i 1.23491 + 1.23491i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −16.5068 16.5068i −1.14730 1.14730i
\(208\) 0 0
\(209\) 6.52734 8.44949i 0.451505 0.584463i
\(210\) 0 0
\(211\) 18.3842i 1.26562i 0.774306 + 0.632811i \(0.218099\pi\)
−0.774306 + 0.632811i \(0.781901\pi\)
\(212\) 0 0
\(213\) −21.2921 + 21.2921i −1.45891 + 1.45891i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −21.5491 + 21.5491i −1.46285 + 1.46285i
\(218\) 0 0
\(219\) −10.6591 −0.720275
\(220\) 0 0
\(221\) 7.72513i 0.519648i
\(222\) 0 0
\(223\) −12.5703 12.5703i −0.841770 0.841770i 0.147319 0.989089i \(-0.452936\pi\)
−0.989089 + 0.147319i \(0.952936\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.7631 + 11.7631i −0.780746 + 0.780746i −0.979957 0.199211i \(-0.936162\pi\)
0.199211 + 0.979957i \(0.436162\pi\)
\(228\) 0 0
\(229\) 10.2474i 0.677170i 0.940936 + 0.338585i \(0.109948\pi\)
−0.940936 + 0.338585i \(0.890052\pi\)
\(230\) 0 0
\(231\) 29.0433i 1.91091i
\(232\) 0 0
\(233\) 14.6894 14.6894i 0.962333 0.962333i −0.0369834 0.999316i \(-0.511775\pi\)
0.999316 + 0.0369834i \(0.0117749\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −21.2921 21.2921i −1.38307 1.38307i
\(238\) 0 0
\(239\) 12.0000i 0.776215i 0.921614 + 0.388108i \(0.126871\pi\)
−0.921614 + 0.388108i \(0.873129\pi\)
\(240\) 0 0
\(241\) 4.13176i 0.266150i 0.991106 + 0.133075i \(0.0424851\pi\)
−0.991106 + 0.133075i \(0.957515\pi\)
\(242\) 0 0
\(243\) −15.7990 15.7990i −1.01351 1.01351i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −15.9150 + 2.04256i −1.01265 + 0.129965i
\(248\) 0 0
\(249\) 24.9116 1.57871
\(250\) 0 0
\(251\) −13.1010 −0.826929 −0.413465 0.910520i \(-0.635681\pi\)
−0.413465 + 0.910520i \(0.635681\pi\)
\(252\) 0 0
\(253\) −11.7215 + 11.7215i −0.736925 + 0.736925i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.1390 18.1390i 1.13148 1.13148i 0.141547 0.989932i \(-0.454792\pi\)
0.989932 0.141547i \(-0.0452077\pi\)
\(258\) 0 0
\(259\) −46.2297 −2.87258
\(260\) 0 0
\(261\) 18.3842i 1.13795i
\(262\) 0 0
\(263\) −7.75314 + 7.75314i −0.478079 + 0.478079i −0.904517 0.426438i \(-0.859768\pi\)
0.426438 + 0.904517i \(0.359768\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −9.57058 9.57058i −0.585710 0.585710i
\(268\) 0 0
\(269\) −24.9116 −1.51888 −0.759442 0.650575i \(-0.774528\pi\)
−0.759442 + 0.650575i \(0.774528\pi\)
\(270\) 0 0
\(271\) 27.3485 1.66130 0.830651 0.556794i \(-0.187969\pi\)
0.830651 + 0.556794i \(0.187969\pi\)
\(272\) 0 0
\(273\) −30.8627 + 30.8627i −1.86789 + 1.86789i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.6251 + 20.6251i 1.23924 + 1.23924i 0.960312 + 0.278929i \(0.0899795\pi\)
0.278929 + 0.960312i \(0.410021\pi\)
\(278\) 0 0
\(279\) −22.5160 −1.34800
\(280\) 0 0
\(281\) 27.8455i 1.66112i 0.556925 + 0.830562i \(0.311981\pi\)
−0.556925 + 0.830562i \(0.688019\pi\)
\(282\) 0 0
\(283\) 9.90408 9.90408i 0.588736 0.588736i −0.348553 0.937289i \(-0.613327\pi\)
0.937289 + 0.348553i \(0.113327\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −39.1438 + 39.1438i −2.31059 + 2.31059i
\(288\) 0 0
\(289\) 12.5959i 0.740936i
\(290\) 0 0
\(291\) 13.5505 0.794345
\(292\) 0 0
\(293\) −3.41011 3.41011i −0.199221 0.199221i 0.600445 0.799666i \(-0.294990\pi\)
−0.799666 + 0.600445i \(0.794990\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.97717 1.97717i 0.114727 0.114727i
\(298\) 0 0
\(299\) 24.9116 1.44067
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) 2.42152 + 2.42152i 0.139113 + 0.139113i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 6.55726 6.55726i 0.374243 0.374243i −0.494777 0.869020i \(-0.664750\pi\)
0.869020 + 0.494777i \(0.164750\pi\)
\(308\) 0 0
\(309\) 16.4495i 0.935779i
\(310\) 0 0
\(311\) −29.1464 −1.65274 −0.826371 0.563126i \(-0.809599\pi\)
−0.826371 + 0.563126i \(0.809599\pi\)
\(312\) 0 0
\(313\) 2.15094 2.15094i 0.121578 0.121578i −0.643700 0.765278i \(-0.722601\pi\)
0.765278 + 0.643700i \(0.222601\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.7631 + 11.7631i −0.660683 + 0.660683i −0.955541 0.294858i \(-0.904728\pi\)
0.294858 + 0.955541i \(0.404728\pi\)
\(318\) 0 0
\(319\) 13.0547 0.730921
\(320\) 0 0
\(321\) −28.0454 −1.56534
\(322\) 0 0
\(323\) 9.07312 1.16446i 0.504842 0.0647924i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.7215 + 11.7215i 0.648201 + 0.648201i
\(328\) 0 0
\(329\) 21.7980i 1.20176i
\(330\) 0 0
\(331\) 34.3729i 1.88930i −0.328075 0.944652i \(-0.606400\pi\)
0.328075 0.944652i \(-0.393600\pi\)
\(332\) 0 0
\(333\) −24.1520 24.1520i −1.32352 1.32352i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.9462 18.9462i 1.03206 1.03206i 0.0325943 0.999469i \(-0.489623\pi\)
0.999469 0.0325943i \(-0.0103769\pi\)
\(338\) 0 0
\(339\) 35.1464i 1.90889i
\(340\) 0 0
\(341\) 15.9886i 0.865834i
\(342\) 0 0
\(343\) −25.7439 + 25.7439i −1.39004 + 1.39004i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.0229 + 15.0229i 0.806470 + 0.806470i 0.984098 0.177628i \(-0.0568423\pi\)
−0.177628 + 0.984098i \(0.556842\pi\)
\(348\) 0 0
\(349\) 13.7980i 0.738588i −0.929313 0.369294i \(-0.879600\pi\)
0.929313 0.369294i \(-0.120400\pi\)
\(350\) 0 0
\(351\) −4.20204 −0.224288
\(352\) 0 0
\(353\) 11.7215 11.7215i 0.623873 0.623873i −0.322646 0.946520i \(-0.604572\pi\)
0.946520 + 0.322646i \(0.104572\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 17.5948 17.5948i 0.931213 0.931213i
\(358\) 0 0
\(359\) 24.2474i 1.27973i −0.768487 0.639866i \(-0.778990\pi\)
0.768487 0.639866i \(-0.221010\pi\)
\(360\) 0 0
\(361\) −4.79796 18.3842i −0.252524 0.967591i
\(362\) 0 0
\(363\) 8.97879 + 8.97879i 0.471264 + 0.471264i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 23.9264 + 23.9264i 1.24895 + 1.24895i 0.956184 + 0.292766i \(0.0945756\pi\)
0.292766 + 0.956184i \(0.405424\pi\)
\(368\) 0 0
\(369\) −40.9002 −2.12918
\(370\) 0 0
\(371\) 22.5160i 1.16897i
\(372\) 0 0
\(373\) 7.00162 + 7.00162i 0.362530 + 0.362530i 0.864744 0.502213i \(-0.167481\pi\)
−0.502213 + 0.864744i \(0.667481\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −13.8725 13.8725i −0.714468 0.714468i
\(378\) 0 0
\(379\) −5.32955 −0.273760 −0.136880 0.990588i \(-0.543708\pi\)
−0.136880 + 0.990588i \(0.543708\pi\)
\(380\) 0 0
\(381\) 6.44949 0.330417
\(382\) 0 0
\(383\) −27.3807 27.3807i −1.39909 1.39909i −0.802681 0.596408i \(-0.796594\pi\)
−0.596408 0.802681i \(-0.703406\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.26922 + 6.26922i 0.318682 + 0.318682i
\(388\) 0 0
\(389\) 27.7980i 1.40941i 0.709499 + 0.704706i \(0.248921\pi\)
−0.709499 + 0.704706i \(0.751079\pi\)
\(390\) 0 0
\(391\) −14.2020 −0.718228
\(392\) 0 0
\(393\) −17.5948 17.5948i −0.887538 0.887538i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8.90357 8.90357i −0.446857 0.446857i 0.447451 0.894308i \(-0.352332\pi\)
−0.894308 + 0.447451i \(0.852332\pi\)
\(398\) 0 0
\(399\) −40.9002 31.5959i −2.04757 1.58177i
\(400\) 0 0
\(401\) 34.3729i 1.71650i 0.513233 + 0.858249i \(0.328448\pi\)
−0.513233 + 0.858249i \(0.671552\pi\)
\(402\) 0 0
\(403\) 16.9902 16.9902i 0.846343 0.846343i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −17.1504 + 17.1504i −0.850114 + 0.850114i
\(408\) 0 0
\(409\) 1.19779 0.0592268 0.0296134 0.999561i \(-0.490572\pi\)
0.0296134 + 0.999561i \(0.490572\pi\)
\(410\) 0 0
\(411\) 29.0433i 1.43260i
\(412\) 0 0
\(413\) −21.5491 21.5491i −1.06036 1.06036i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −9.96737 + 9.96737i −0.488104 + 0.488104i
\(418\) 0 0
\(419\) 9.79796i 0.478662i −0.970938 0.239331i \(-0.923072\pi\)
0.970938 0.239331i \(-0.0769280\pi\)
\(420\) 0 0
\(421\) 1.19779i 0.0583766i −0.999574 0.0291883i \(-0.990708\pi\)
0.999574 0.0291883i \(-0.00929225\pi\)
\(422\) 0 0
\(423\) 11.3880 11.3880i 0.553704 0.553704i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 17.6572 + 17.6572i 0.854493 + 0.854493i
\(428\) 0 0
\(429\) 22.8990i 1.10557i
\(430\) 0 0
\(431\) 18.3842i 0.885537i 0.896636 + 0.442768i \(0.146004\pi\)
−0.896636 + 0.442768i \(0.853996\pi\)
\(432\) 0 0
\(433\) −4.58010 4.58010i −0.220105 0.220105i 0.588437 0.808543i \(-0.299743\pi\)
−0.808543 + 0.588437i \(0.799743\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.75509 + 29.2585i 0.179630 + 1.39962i
\(438\) 0 0
\(439\) −22.5160 −1.07463 −0.537315 0.843382i \(-0.680561\pi\)
−0.537315 + 0.843382i \(0.680561\pi\)
\(440\) 0 0
\(441\) −51.0454 −2.43073
\(442\) 0 0
\(443\) −12.0550 + 12.0550i −0.572751 + 0.572751i −0.932896 0.360145i \(-0.882727\pi\)
0.360145 + 0.932896i \(0.382727\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 41.5654 41.5654i 1.96597 1.96597i
\(448\) 0 0
\(449\) 6.52734 0.308044 0.154022 0.988067i \(-0.450777\pi\)
0.154022 + 0.988067i \(0.450777\pi\)
\(450\) 0 0
\(451\) 29.0433i 1.36760i
\(452\) 0 0
\(453\) −9.57058 + 9.57058i −0.449665 + 0.449665i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.63487 + 3.63487i 0.170032 + 0.170032i 0.786993 0.616962i \(-0.211637\pi\)
−0.616962 + 0.786993i \(0.711637\pi\)
\(458\) 0 0
\(459\) 2.39558 0.111816
\(460\) 0 0
\(461\) −23.3939 −1.08956 −0.544781 0.838579i \(-0.683387\pi\)
−0.544781 + 0.838579i \(0.683387\pi\)
\(462\) 0 0
\(463\) −7.75314 + 7.75314i −0.360319 + 0.360319i −0.863930 0.503611i \(-0.832004\pi\)
0.503611 + 0.863930i \(0.332004\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.9907 17.9907i −0.832512 0.832512i 0.155348 0.987860i \(-0.450350\pi\)
−0.987860 + 0.155348i \(0.950350\pi\)
\(468\) 0 0
\(469\) 30.2411 1.39640
\(470\) 0 0
\(471\) 42.0980i 1.93977i
\(472\) 0 0
\(473\) 4.45178 4.45178i 0.204693 0.204693i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −11.7631 + 11.7631i −0.538597 + 0.538597i
\(478\) 0 0
\(479\) 29.1464i 1.33173i 0.746070 + 0.665867i \(0.231938\pi\)
−0.746070 + 0.665867i \(0.768062\pi\)
\(480\) 0 0
\(481\) 36.4495 1.66195
\(482\) 0 0
\(483\) 56.7386 + 56.7386i 2.58170 + 2.58170i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −20.1977 + 20.1977i −0.915245 + 0.915245i −0.996679 0.0814340i \(-0.974050\pi\)
0.0814340 + 0.996679i \(0.474050\pi\)
\(488\) 0 0
\(489\) −17.1864 −0.777197
\(490\) 0 0
\(491\) −28.2929 −1.27684 −0.638419 0.769689i \(-0.720411\pi\)
−0.638419 + 0.769689i \(0.720411\pi\)
\(492\) 0 0
\(493\) 7.90866 + 7.90866i 0.356188 + 0.356188i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 39.1438 39.1438i 1.75584 1.75584i
\(498\) 0 0
\(499\) 42.9444i 1.92245i −0.275757 0.961227i \(-0.588928\pi\)
0.275757 0.961227i \(-0.411072\pi\)
\(500\) 0 0
\(501\) 5.14643 0.229925
\(502\) 0 0
\(503\) 14.3559 14.3559i 0.640096 0.640096i −0.310483 0.950579i \(-0.600491\pi\)
0.950579 + 0.310483i \(0.100491\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.988583 0.988583i 0.0439045 0.0439045i
\(508\) 0 0
\(509\) 23.7138 1.05109 0.525547 0.850764i \(-0.323861\pi\)
0.525547 + 0.850764i \(0.323861\pi\)
\(510\) 0 0
\(511\) 19.5959 0.866872
\(512\) 0 0
\(513\) −0.633403 4.93528i −0.0279654 0.217898i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −8.08665 8.08665i −0.355650 0.355650i
\(518\) 0 0
\(519\) 42.2474i 1.85446i
\(520\) 0 0
\(521\) 2.93397i 0.128540i 0.997933 + 0.0642698i \(0.0204718\pi\)
−0.997933 + 0.0642698i \(0.979528\pi\)
\(522\) 0 0
\(523\) −2.15857 2.15857i −0.0943879 0.0943879i 0.658336 0.752724i \(-0.271261\pi\)
−0.752724 + 0.658336i \(0.771261\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.68609 + 9.68609i −0.421933 + 0.421933i
\(528\) 0 0
\(529\) 22.7980i 0.991216i
\(530\) 0 0
\(531\) 22.5160i 0.977110i
\(532\) 0 0
\(533\) 30.8627 30.8627i 1.33681 1.33681i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −30.8627 30.8627i −1.33182 1.33182i
\(538\) 0 0
\(539\) 36.2474i 1.56129i
\(540\) 0 0
\(541\) −14.6515 −0.629919 −0.314959 0.949105i \(-0.601991\pi\)
−0.314959 + 0.949105i \(0.601991\pi\)
\(542\) 0 0
\(543\) −9.57058 + 9.57058i −0.410713 + 0.410713i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 30.1651 30.1651i 1.28976 1.28976i 0.354836 0.934928i \(-0.384537\pi\)
0.934928 0.354836i \(-0.115463\pi\)
\(548\) 0 0
\(549\) 18.4495i 0.787405i
\(550\) 0 0
\(551\) 14.2020 18.3842i 0.605027 0.783194i
\(552\) 0 0
\(553\) 39.1438 + 39.1438i 1.66457 + 1.66457i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −27.2278 27.2278i −1.15368 1.15368i −0.985809 0.167870i \(-0.946311\pi\)
−0.167870 0.985809i \(-0.553689\pi\)
\(558\) 0 0
\(559\) −9.46131 −0.400171
\(560\) 0 0
\(561\) 13.0547i 0.551169i
\(562\) 0 0
\(563\) 8.25315 + 8.25315i 0.347829 + 0.347829i 0.859300 0.511471i \(-0.170899\pi\)
−0.511471 + 0.859300i \(0.670899\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 24.5934 + 24.5934i 1.03283 + 1.03283i
\(568\) 0 0
\(569\) −27.8455 −1.16735 −0.583673 0.811989i \(-0.698385\pi\)
−0.583673 + 0.811989i \(0.698385\pi\)
\(570\) 0 0
\(571\) 20.2474 0.847329 0.423665 0.905819i \(-0.360743\pi\)
0.423665 + 0.905819i \(0.360743\pi\)
\(572\) 0 0
\(573\) −1.97717 1.97717i −0.0825973 0.0825973i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −17.6572 17.6572i −0.735080 0.735080i 0.236542 0.971621i \(-0.423986\pi\)
−0.971621 + 0.236542i \(0.923986\pi\)
\(578\) 0 0
\(579\) 0.651531i 0.0270767i
\(580\) 0 0
\(581\) −45.7980 −1.90002
\(582\) 0 0
\(583\) 8.35302 + 8.35302i 0.345947 + 0.345947i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.1417 20.1417i −0.831336 0.831336i 0.156364 0.987700i \(-0.450023\pi\)
−0.987700 + 0.156364i \(0.950023\pi\)
\(588\) 0 0
\(589\) 22.5160 + 17.3939i 0.927755 + 0.716702i
\(590\) 0 0
\(591\) 42.0980i 1.73168i
\(592\) 0 0
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −29.9837 + 29.9837i −1.22715 + 1.22715i
\(598\) 0 0
\(599\) −14.2525 −0.582340 −0.291170 0.956671i \(-0.594044\pi\)
−0.291170 + 0.956671i \(0.594044\pi\)
\(600\) 0 0
\(601\) 19.5820i 0.798767i 0.916784 + 0.399383i \(0.130776\pi\)
−0.916784 + 0.399383i \(0.869224\pi\)
\(602\) 0 0
\(603\) 15.7990 + 15.7990i 0.643385 + 0.643385i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −8.53443 + 8.53443i −0.346402 + 0.346402i −0.858767 0.512366i \(-0.828769\pi\)
0.512366 + 0.858767i \(0.328769\pi\)
\(608\) 0 0
\(609\) 63.1918i 2.56066i
\(610\) 0 0
\(611\) 17.1864i 0.695289i
\(612\) 0 0
\(613\) −20.4752 + 20.4752i −0.826984 + 0.826984i −0.987099 0.160114i \(-0.948814\pi\)
0.160114 + 0.987099i \(0.448814\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.7760 22.7760i −0.916928 0.916928i 0.0798768 0.996805i \(-0.474547\pi\)
−0.996805 + 0.0798768i \(0.974547\pi\)
\(618\) 0 0
\(619\) 14.0454i 0.564533i −0.959336 0.282266i \(-0.908914\pi\)
0.959336 0.282266i \(-0.0910862\pi\)
\(620\) 0 0
\(621\) 7.72513i 0.309999i
\(622\) 0 0
\(623\) 17.5948 + 17.5948i 0.704919 + 0.704919i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −26.8947 + 3.45172i −1.07407 + 0.137848i
\(628\) 0 0
\(629\) −20.7798 −0.828545
\(630\) 0 0
\(631\) −17.5505 −0.698675 −0.349337 0.936997i \(-0.613593\pi\)
−0.349337 + 0.936997i \(0.613593\pi\)
\(632\) 0 0
\(633\) 33.0136 33.0136i 1.31217 1.31217i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 38.5181 38.5181i 1.52614 1.52614i
\(638\) 0 0
\(639\) 40.9002 1.61799
\(640\) 0 0
\(641\) 8.92291i 0.352434i −0.984351 0.176217i \(-0.943614\pi\)
0.984351 0.176217i \(-0.0563860\pi\)
\(642\) 0 0
\(643\) 21.6256 21.6256i 0.852830 0.852830i −0.137651 0.990481i \(-0.543955\pi\)
0.990481 + 0.137651i \(0.0439551\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.26922 + 6.26922i 0.246468 + 0.246468i 0.819520 0.573051i \(-0.194240\pi\)
−0.573051 + 0.819520i \(0.694240\pi\)
\(648\) 0 0
\(649\) −15.9886 −0.627609
\(650\) 0 0
\(651\) 77.3939 3.03331
\(652\) 0 0
\(653\) 26.4109 26.4109i 1.03354 1.03354i 0.0341199 0.999418i \(-0.489137\pi\)
0.999418 0.0341199i \(-0.0108628\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 10.2376 + 10.2376i 0.399406 + 0.399406i
\(658\) 0 0
\(659\) −1.19779 −0.0466592 −0.0233296 0.999728i \(-0.507427\pi\)
−0.0233296 + 0.999728i \(0.507427\pi\)
\(660\) 0 0
\(661\) 33.1751i 1.29036i −0.764030 0.645180i \(-0.776782\pi\)
0.764030 0.645180i \(-0.223218\pi\)
\(662\) 0 0
\(663\) −13.8725 + 13.8725i −0.538761 + 0.538761i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −25.5034 + 25.5034i −0.987496 + 0.987496i
\(668\) 0 0
\(669\) 45.1464i 1.74546i
\(670\) 0 0
\(671\) 13.1010 0.505759
\(672\) 0 0
\(673\) −4.58010 4.58010i −0.176550 0.176550i 0.613300 0.789850i \(-0.289842\pi\)
−0.789850 + 0.613300i \(0.789842\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.1847 14.1847i 0.545160 0.545160i −0.379877 0.925037i \(-0.624034\pi\)
0.925037 + 0.379877i \(0.124034\pi\)
\(678\) 0 0
\(679\) −24.9116 −0.956018
\(680\) 0 0
\(681\) 42.2474 1.61893
\(682\) 0 0
\(683\) 5.38727 + 5.38727i 0.206138 + 0.206138i 0.802624 0.596486i \(-0.203437\pi\)
−0.596486 + 0.802624i \(0.703437\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 18.4019 18.4019i 0.702078 0.702078i
\(688\) 0 0
\(689\) 17.7526i 0.676318i
\(690\) 0 0
\(691\) −10.4495 −0.397517 −0.198759 0.980048i \(-0.563691\pi\)
−0.198759 + 0.980048i \(0.563691\pi\)
\(692\) 0 0
\(693\) −27.8948 + 27.8948i −1.05964 + 1.05964i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −17.5948 + 17.5948i −0.666449 + 0.666449i
\(698\) 0 0
\(699\) −52.7571 −1.99546
\(700\) 0 0
\(701\) −32.9444 −1.24429 −0.622146 0.782901i \(-0.713739\pi\)
−0.622146 + 0.782901i \(0.713739\pi\)
\(702\) 0 0
\(703\) 5.49428 + 42.8098i 0.207221 + 1.61460i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.45178 4.45178i −0.167427 0.167427i
\(708\) 0 0
\(709\) 4.00000i 0.150223i −0.997175 0.0751116i \(-0.976069\pi\)
0.997175 0.0751116i \(-0.0239313\pi\)
\(710\) 0 0
\(711\) 40.9002i 1.53388i
\(712\) 0 0
\(713\) −31.2352 31.2352i −1.16977 1.16977i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 21.5491 21.5491i 0.804765 0.804765i
\(718\) 0 0
\(719\) 0.247449i 0.00922828i 0.999989 + 0.00461414i \(0.00146873\pi\)
−0.999989 + 0.00461414i \(0.998531\pi\)
\(720\) 0 0
\(721\) 30.2411i 1.12624i
\(722\) 0 0
\(723\) 7.41964 7.41964i 0.275939 0.275939i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.9907 + 17.9907i 0.667239 + 0.667239i 0.957076 0.289837i \(-0.0936011\pi\)
−0.289837 + 0.957076i \(0.593601\pi\)
\(728\) 0 0
\(729\) 34.3939i 1.27385i
\(730\) 0 0
\(731\) 5.39388 0.199500
\(732\) 0 0
\(733\) −11.7215 + 11.7215i −0.432944 + 0.432944i −0.889629 0.456685i \(-0.849037\pi\)
0.456685 + 0.889629i \(0.349037\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.2189 11.2189i 0.413254 0.413254i
\(738\) 0 0
\(739\) 32.2929i 1.18791i 0.804498 + 0.593956i \(0.202435\pi\)
−0.804498 + 0.593956i \(0.797565\pi\)
\(740\) 0 0
\(741\) 32.2474 + 24.9116i 1.18464 + 0.915149i
\(742\) 0 0
\(743\) −22.9820 22.9820i −0.843129 0.843129i 0.146136 0.989265i \(-0.453316\pi\)
−0.989265 + 0.146136i \(0.953316\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −23.9264 23.9264i −0.875423 0.875423i
\(748\) 0 0
\(749\) 51.5593 1.88394
\(750\) 0 0
\(751\) 50.3615i 1.83772i −0.394587 0.918859i \(-0.629112\pi\)
0.394587 0.918859i \(-0.370888\pi\)
\(752\) 0 0
\(753\) 23.5263 + 23.5263i 0.857344 + 0.857344i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −11.0545 11.0545i −0.401783 0.401783i 0.477078 0.878861i \(-0.341696\pi\)
−0.878861 + 0.477078i \(0.841696\pi\)
\(758\) 0 0
\(759\) 42.0980 1.52806
\(760\) 0 0
\(761\) −13.3485 −0.483882 −0.241941 0.970291i \(-0.577784\pi\)
−0.241941 + 0.970291i \(0.577784\pi\)
\(762\) 0 0
\(763\) −21.5491 21.5491i −0.780129 0.780129i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.9902 + 16.9902i 0.613481 + 0.613481i
\(768\) 0 0
\(769\) 18.6515i 0.672591i 0.941756 + 0.336296i \(0.109174\pi\)
−0.941756 + 0.336296i \(0.890826\pi\)
\(770\) 0 0
\(771\) −65.1464 −2.34619
\(772\) 0 0
\(773\) 5.83163 + 5.83163i 0.209749 + 0.209749i 0.804161 0.594412i \(-0.202615\pi\)
−0.594412 + 0.804161i \(0.702615\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 83.0174 + 83.0174i 2.97823 + 2.97823i
\(778\) 0 0
\(779\) 40.9002 + 31.5959i 1.46540 + 1.13204i
\(780\) 0 0
\(781\) 29.0433i 1.03925i
\(782\) 0 0
\(783\) 4.30188 4.30188i 0.153736 0.153736i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −19.3905 + 19.3905i −0.691197 + 0.691197i −0.962495 0.271298i \(-0.912547\pi\)
0.271298 + 0.962495i \(0.412547\pi\)
\(788\) 0 0
\(789\) 27.8455 0.991327
\(790\) 0 0
\(791\) 64.6140i 2.29741i
\(792\) 0 0
\(793\) −13.9217 13.9217i −0.494374 0.494374i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.1847 14.1847i 0.502446 0.502446i −0.409751 0.912197i \(-0.634384\pi\)
0.912197 + 0.409751i \(0.134384\pi\)
\(798\) 0 0
\(799\) 9.79796i 0.346627i
\(800\) 0 0
\(801\) 18.3842i 0.649575i
\(802\) 0 0
\(803\) 7.26973 7.26973i 0.256543 0.256543i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 44.7351 + 44.7351i 1.57475 + 1.57475i
\(808\) 0 0
\(809\) 5.39388i 0.189639i −0.995494 0.0948193i \(-0.969773\pi\)
0.995494 0.0948193i \(-0.0302273\pi\)
\(810\) 0 0
\(811\) 35.5707i 1.24905i −0.781003 0.624527i \(-0.785292\pi\)
0.781003 0.624527i \(-0.214708\pi\)
\(812\) 0 0
\(813\) −49.1112 49.1112i −1.72241 1.72241i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.42617 11.1123i −0.0498953 0.388769i
\(818\) 0 0
\(819\) 59.2844 2.07157
\(820\) 0 0
\(821\) −51.7980 −1.80776 −0.903881 0.427785i \(-0.859294\pi\)
−0.903881 + 0.427785i \(0.859294\pi\)
\(822\) 0 0
\(823\) 15.6899 15.6899i 0.546915 0.546915i −0.378632 0.925547i \(-0.623605\pi\)
0.925547 + 0.378632i \(0.123605\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.34160 9.34160i 0.324839 0.324839i −0.525781 0.850620i \(-0.676227\pi\)
0.850620 + 0.525781i \(0.176227\pi\)
\(828\) 0 0
\(829\) 4.13176 0.143502 0.0717510 0.997423i \(-0.477141\pi\)
0.0717510 + 0.997423i \(0.477141\pi\)
\(830\) 0 0
\(831\) 74.0753i 2.56964i
\(832\) 0 0
\(833\) −21.9591 + 21.9591i −0.760838 + 0.760838i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.26870 + 5.26870i 0.182113 + 0.182113i
\(838\) 0 0
\(839\) 14.7909 0.510637 0.255319 0.966857i \(-0.417820\pi\)
0.255319 + 0.966857i \(0.417820\pi\)
\(840\) 0 0
\(841\) −0.595918 −0.0205489
\(842\) 0 0
\(843\) 50.0038 50.0038i 1.72222 1.72222i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −16.5068 16.5068i −0.567180 0.567180i
\(848\) 0 0
\(849\) −35.5707 −1.22078
\(850\) 0 0
\(851\) 67.0095i 2.29706i
\(852\) 0 0
\(853\) 13.8725 13.8725i 0.474984 0.474984i −0.428539 0.903523i \(-0.640972\pi\)
0.903523 + 0.428539i \(0.140972\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −21.0049 + 21.0049i −0.717513 + 0.717513i −0.968095 0.250583i \(-0.919378\pi\)
0.250583 + 0.968095i \(0.419378\pi\)
\(858\) 0 0
\(859\) 32.0000i 1.09183i 0.837842 + 0.545913i \(0.183817\pi\)
−0.837842 + 0.545913i \(0.816183\pi\)
\(860\) 0 0
\(861\) 140.586 4.79115
\(862\) 0 0
\(863\) −8.25315 8.25315i −0.280941 0.280941i 0.552543 0.833484i \(-0.313657\pi\)
−0.833484 + 0.552543i \(0.813657\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −22.6192 + 22.6192i −0.768189 + 0.768189i
\(868\) 0 0
\(869\) 29.0433 0.985227
\(870\) 0 0
\(871\) −23.8434 −0.807902
\(872\) 0 0
\(873\) −13.0147 13.0147i −0.440480 0.440480i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.181408 0.181408i 0.00612572 0.00612572i −0.704037 0.710163i \(-0.748621\pi\)
0.710163 + 0.704037i \(0.248621\pi\)
\(878\) 0 0
\(879\) 12.2474i 0.413096i
\(880\) 0 0
\(881\) 11.1464 0.375533 0.187766 0.982214i \(-0.439875\pi\)
0.187766 + 0.982214i \(0.439875\pi\)
\(882\) 0 0
\(883\) −29.0452 + 29.0452i −0.977450 + 0.977450i −0.999751 0.0223014i \(-0.992901\pi\)
0.0223014 + 0.999751i \(0.492901\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.4036 25.4036i 0.852968 0.852968i −0.137530 0.990498i \(-0.543916\pi\)
0.990498 + 0.137530i \(0.0439163\pi\)
\(888\) 0 0
\(889\) −11.8569 −0.397667
\(890\) 0 0
\(891\) 18.2474 0.611313
\(892\) 0 0
\(893\) −20.1854 + 2.59063i −0.675478 + 0.0866921i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −44.7351 44.7351i −1.49366 1.49366i
\(898\) 0 0
\(899\) 34.7878i 1.16024i
\(900\) 0 0
\(901\) 10.1207i 0.337169i
\(902\) 0 0
\(903\) −21.5491 21.5491i −0.717109 0.717109i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20.1977 20.1977i 0.670653 0.670653i −0.287213 0.957867i \(-0.592729\pi\)
0.957867 + 0.287213i \(0.0927289\pi\)
\(908\) 0 0
\(909\) 4.65153i 0.154282i
\(910\) 0 0
\(911\) 15.9886i 0.529727i 0.964286 + 0.264864i \(0.0853270\pi\)
−0.964286 + 0.264864i \(0.914673\pi\)
\(912\) 0 0
\(913\) −16.9902 + 16.9902i −0.562294 + 0.562294i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 32.3466 + 32.3466i 1.06818 + 1.06818i
\(918\) 0 0
\(919\) 47.5959i 1.57004i 0.619468 + 0.785022i \(0.287349\pi\)
−0.619468 + 0.785022i \(0.712651\pi\)
\(920\) 0 0
\(921\) −23.5505 −0.776016
\(922\) 0 0
\(923\) −30.8627 + 30.8627i −1.01586 + 1.01586i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −15.7990 + 15.7990i −0.518907 + 0.518907i
\(928\) 0 0
\(929\) 21.7980i 0.715168i 0.933881 + 0.357584i \(0.116399\pi\)
−0.933881 + 0.357584i \(0.883601\pi\)
\(930\) 0 0
\(931\) 51.0454 + 39.4332i 1.67295 + 1.29237i
\(932\) 0 0
\(933\) 52.3399 + 52.3399i 1.71353 + 1.71353i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 15.5063 + 15.5063i 0.506568 + 0.506568i 0.913471 0.406903i \(-0.133391\pi\)
−0.406903 + 0.913471i \(0.633391\pi\)
\(938\) 0 0
\(939\) −7.72513 −0.252100
\(940\) 0 0
\(941\) 34.3729i 1.12052i 0.828316 + 0.560262i \(0.189299\pi\)
−0.828316 + 0.560262i \(0.810701\pi\)
\(942\) 0 0
\(943\) −56.7386 56.7386i −1.84766 1.84766i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.78529 4.78529i −0.155501 0.155501i 0.625069 0.780570i \(-0.285071\pi\)
−0.780570 + 0.625069i \(0.785071\pi\)
\(948\) 0 0
\(949\) −15.4503 −0.501536
\(950\) 0 0
\(951\) 42.2474 1.36997
\(952\) 0 0
\(953\) 21.0049 + 21.0049i 0.680414 + 0.680414i 0.960094 0.279679i \(-0.0902281\pi\)
−0.279679 + 0.960094i \(0.590228\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −23.4430 23.4430i −0.757805 0.757805i
\(958\) 0 0
\(959\) 53.3939i 1.72418i
\(960\) 0 0
\(961\) −11.6061 −0.374391
\(962\) 0 0
\(963\) 26.9364 + 26.9364i 0.868012 + 0.868012i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6.93623 + 6.93623i 0.223054 + 0.223054i 0.809783 0.586729i \(-0.199585\pi\)
−0.586729 + 0.809783i \(0.699585\pi\)
\(968\) 0 0
\(969\) −18.3842 14.2020i −0.590586 0.456235i
\(970\) 0 0
\(971\) 1.19779i 0.0384389i 0.999815 + 0.0192194i \(0.00611811\pi\)
−0.999815 + 0.0192194i \(0.993882\pi\)
\(972\) 0 0
\(973\) 18.3242 18.3242i 0.587448 0.587448i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −35.2894 + 35.2894i −1.12901 + 1.12901i −0.138669 + 0.990339i \(0.544282\pi\)
−0.990339 + 0.138669i \(0.955718\pi\)
\(978\) 0 0
\(979\) 13.0547 0.417229
\(980\) 0 0
\(981\) 22.5160i 0.718880i
\(982\) 0 0
\(983\) −27.3807 27.3807i −0.873309 0.873309i 0.119522 0.992832i \(-0.461864\pi\)
−0.992832 + 0.119522i \(0.961864\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −39.1438 + 39.1438i −1.24596 + 1.24596i
\(988\) 0 0
\(989\) 17.3939i 0.553093i
\(990\) 0 0
\(991\) 31.4389i 0.998689i 0.866403 + 0.499345i \(0.166426\pi\)
−0.866403 + 0.499345i \(0.833574\pi\)
\(992\) 0 0
\(993\) −61.7253 + 61.7253i −1.95879 + 1.95879i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.08665 + 8.08665i 0.256107 + 0.256107i 0.823469 0.567362i \(-0.192036\pi\)
−0.567362 + 0.823469i \(0.692036\pi\)
\(998\) 0 0
\(999\) 11.3031i 0.357613i
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 1900.2.l.c.1557.2 yes 16
5.2 odd 4 inner 1900.2.l.c.493.1 16
5.3 odd 4 inner 1900.2.l.c.493.8 yes 16
5.4 even 2 inner 1900.2.l.c.1557.7 yes 16
19.18 odd 2 inner 1900.2.l.c.1557.8 yes 16
95.18 even 4 inner 1900.2.l.c.493.2 yes 16
95.37 even 4 inner 1900.2.l.c.493.7 yes 16
95.94 odd 2 inner 1900.2.l.c.1557.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.l.c.493.1 16 5.2 odd 4 inner
1900.2.l.c.493.2 yes 16 95.18 even 4 inner
1900.2.l.c.493.7 yes 16 95.37 even 4 inner
1900.2.l.c.493.8 yes 16 5.3 odd 4 inner
1900.2.l.c.1557.1 yes 16 95.94 odd 2 inner
1900.2.l.c.1557.2 yes 16 1.1 even 1 trivial
1900.2.l.c.1557.7 yes 16 5.4 even 2 inner
1900.2.l.c.1557.8 yes 16 19.18 odd 2 inner