Properties

Label 1183.2.c.d.337.4
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not 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: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.4
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.d.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} +1.41421 q^{3} -1.58579i q^{5} +2.00000i q^{6} +1.00000i q^{7} +2.82843i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421i q^{2} +1.41421 q^{3} -1.58579i q^{5} +2.00000i q^{6} +1.00000i q^{7} +2.82843i q^{8} -1.00000 q^{9} +2.24264 q^{10} +4.24264i q^{11} -1.41421 q^{14} -2.24264i q^{15} -4.00000 q^{16} -1.41421 q^{17} -1.41421i q^{18} +7.24264i q^{19} +1.41421i q^{21} -6.00000 q^{22} +5.82843 q^{23} +4.00000i q^{24} +2.48528 q^{25} -5.65685 q^{27} +0.171573 q^{29} +3.17157 q^{30} -3.24264i q^{31} +6.00000i q^{33} -2.00000i q^{34} +1.58579 q^{35} +2.24264i q^{37} -10.2426 q^{38} +4.48528 q^{40} -8.82843i q^{41} -2.00000 q^{42} +5.00000 q^{43} +1.58579i q^{45} +8.24264i q^{46} +1.58579i q^{47} -5.65685 q^{48} -1.00000 q^{49} +3.51472i q^{50} -2.00000 q^{51} -0.171573 q^{53} -8.00000i q^{54} +6.72792 q^{55} -2.82843 q^{56} +10.2426i q^{57} +0.242641i q^{58} +0.343146i q^{59} +6.00000 q^{61} +4.58579 q^{62} -1.00000i q^{63} -8.00000 q^{64} -8.48528 q^{66} +14.4853i q^{67} +8.24264 q^{69} +2.24264i q^{70} +13.0711i q^{71} -2.82843i q^{72} -9.24264i q^{73} -3.17157 q^{74} +3.51472 q^{75} -4.24264 q^{77} +15.4853 q^{79} +6.34315i q^{80} -5.00000 q^{81} +12.4853 q^{82} -13.2426i q^{83} +2.24264i q^{85} +7.07107i q^{86} +0.242641 q^{87} -12.0000 q^{88} +1.58579i q^{89} -2.24264 q^{90} -4.58579i q^{93} -2.24264 q^{94} +11.4853 q^{95} -11.7279i q^{97} -1.41421i q^{98} -4.24264i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 8 q^{10} - 16 q^{16} - 24 q^{22} + 12 q^{23} - 24 q^{25} + 12 q^{29} + 24 q^{30} + 12 q^{35} - 24 q^{38} - 16 q^{40} - 8 q^{42} + 20 q^{43} - 4 q^{49} - 8 q^{51} - 12 q^{53} - 24 q^{55} + 24 q^{61} + 24 q^{62} - 32 q^{64} + 16 q^{69} - 24 q^{74} + 48 q^{75} + 28 q^{79} - 20 q^{81} + 16 q^{82} - 16 q^{87} - 48 q^{88} + 8 q^{90} + 8 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) − 1.58579i − 0.709185i −0.935021 0.354593i \(-0.884620\pi\)
0.935021 0.354593i \(-0.115380\pi\)
\(6\) 2.00000i 0.816497i
\(7\) 1.00000i 0.377964i
\(8\) 2.82843i 1.00000i
\(9\) −1.00000 −0.333333
\(10\) 2.24264 0.709185
\(11\) 4.24264i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −1.41421 −0.377964
\(15\) − 2.24264i − 0.579047i
\(16\) −4.00000 −1.00000
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) − 1.41421i − 0.333333i
\(19\) 7.24264i 1.66158i 0.556589 + 0.830788i \(0.312110\pi\)
−0.556589 + 0.830788i \(0.687890\pi\)
\(20\) 0 0
\(21\) 1.41421i 0.308607i
\(22\) −6.00000 −1.27920
\(23\) 5.82843 1.21531 0.607656 0.794201i \(-0.292110\pi\)
0.607656 + 0.794201i \(0.292110\pi\)
\(24\) 4.00000i 0.816497i
\(25\) 2.48528 0.497056
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 0.171573 0.0318603 0.0159301 0.999873i \(-0.494929\pi\)
0.0159301 + 0.999873i \(0.494929\pi\)
\(30\) 3.17157 0.579047
\(31\) − 3.24264i − 0.582395i −0.956663 0.291198i \(-0.905946\pi\)
0.956663 0.291198i \(-0.0940538\pi\)
\(32\) 0 0
\(33\) 6.00000i 1.04447i
\(34\) − 2.00000i − 0.342997i
\(35\) 1.58579 0.268047
\(36\) 0 0
\(37\) 2.24264i 0.368688i 0.982862 + 0.184344i \(0.0590160\pi\)
−0.982862 + 0.184344i \(0.940984\pi\)
\(38\) −10.2426 −1.66158
\(39\) 0 0
\(40\) 4.48528 0.709185
\(41\) − 8.82843i − 1.37877i −0.724396 0.689384i \(-0.757881\pi\)
0.724396 0.689384i \(-0.242119\pi\)
\(42\) −2.00000 −0.308607
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 1.58579i 0.236395i
\(46\) 8.24264i 1.21531i
\(47\) 1.58579i 0.231311i 0.993289 + 0.115655i \(0.0368968\pi\)
−0.993289 + 0.115655i \(0.963103\pi\)
\(48\) −5.65685 −0.816497
\(49\) −1.00000 −0.142857
\(50\) 3.51472i 0.497056i
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −0.171573 −0.0235673 −0.0117837 0.999931i \(-0.503751\pi\)
−0.0117837 + 0.999931i \(0.503751\pi\)
\(54\) − 8.00000i − 1.08866i
\(55\) 6.72792 0.907193
\(56\) −2.82843 −0.377964
\(57\) 10.2426i 1.35667i
\(58\) 0.242641i 0.0318603i
\(59\) 0.343146i 0.0446738i 0.999751 + 0.0223369i \(0.00711064\pi\)
−0.999751 + 0.0223369i \(0.992889\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 4.58579 0.582395
\(63\) − 1.00000i − 0.125988i
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) −8.48528 −1.04447
\(67\) 14.4853i 1.76966i 0.465915 + 0.884829i \(0.345725\pi\)
−0.465915 + 0.884829i \(0.654275\pi\)
\(68\) 0 0
\(69\) 8.24264 0.992297
\(70\) 2.24264i 0.268047i
\(71\) 13.0711i 1.55125i 0.631194 + 0.775625i \(0.282565\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(72\) − 2.82843i − 0.333333i
\(73\) − 9.24264i − 1.08177i −0.841097 0.540885i \(-0.818090\pi\)
0.841097 0.540885i \(-0.181910\pi\)
\(74\) −3.17157 −0.368688
\(75\) 3.51472 0.405845
\(76\) 0 0
\(77\) −4.24264 −0.483494
\(78\) 0 0
\(79\) 15.4853 1.74223 0.871115 0.491079i \(-0.163397\pi\)
0.871115 + 0.491079i \(0.163397\pi\)
\(80\) 6.34315i 0.709185i
\(81\) −5.00000 −0.555556
\(82\) 12.4853 1.37877
\(83\) − 13.2426i − 1.45357i −0.686866 0.726784i \(-0.741014\pi\)
0.686866 0.726784i \(-0.258986\pi\)
\(84\) 0 0
\(85\) 2.24264i 0.243249i
\(86\) 7.07107i 0.762493i
\(87\) 0.242641 0.0260138
\(88\) −12.0000 −1.27920
\(89\) 1.58579i 0.168093i 0.996462 + 0.0840465i \(0.0267844\pi\)
−0.996462 + 0.0840465i \(0.973216\pi\)
\(90\) −2.24264 −0.236395
\(91\) 0 0
\(92\) 0 0
\(93\) − 4.58579i − 0.475524i
\(94\) −2.24264 −0.231311
\(95\) 11.4853 1.17837
\(96\) 0 0
\(97\) − 11.7279i − 1.19079i −0.803433 0.595395i \(-0.796996\pi\)
0.803433 0.595395i \(-0.203004\pi\)
\(98\) − 1.41421i − 0.142857i
\(99\) − 4.24264i − 0.426401i
\(100\) 0 0
\(101\) −10.2426 −1.01918 −0.509590 0.860417i \(-0.670203\pi\)
−0.509590 + 0.860417i \(0.670203\pi\)
\(102\) − 2.82843i − 0.280056i
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 2.24264 0.218859
\(106\) − 0.242641i − 0.0235673i
\(107\) −20.1421 −1.94721 −0.973607 0.228232i \(-0.926706\pi\)
−0.973607 + 0.228232i \(0.926706\pi\)
\(108\) 0 0
\(109\) − 16.7279i − 1.60224i −0.598501 0.801122i \(-0.704237\pi\)
0.598501 0.801122i \(-0.295763\pi\)
\(110\) 9.51472i 0.907193i
\(111\) 3.17157i 0.301032i
\(112\) − 4.00000i − 0.377964i
\(113\) 2.31371 0.217655 0.108828 0.994061i \(-0.465290\pi\)
0.108828 + 0.994061i \(0.465290\pi\)
\(114\) −14.4853 −1.35667
\(115\) − 9.24264i − 0.861881i
\(116\) 0 0
\(117\) 0 0
\(118\) −0.485281 −0.0446738
\(119\) − 1.41421i − 0.129641i
\(120\) 6.34315 0.579047
\(121\) −7.00000 −0.636364
\(122\) 8.48528i 0.768221i
\(123\) − 12.4853i − 1.12576i
\(124\) 0 0
\(125\) − 11.8701i − 1.06169i
\(126\) 1.41421 0.125988
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) − 11.3137i − 1.00000i
\(129\) 7.07107 0.622573
\(130\) 0 0
\(131\) −2.82843 −0.247121 −0.123560 0.992337i \(-0.539431\pi\)
−0.123560 + 0.992337i \(0.539431\pi\)
\(132\) 0 0
\(133\) −7.24264 −0.628017
\(134\) −20.4853 −1.76966
\(135\) 8.97056i 0.772063i
\(136\) − 4.00000i − 0.342997i
\(137\) − 4.58579i − 0.391790i −0.980625 0.195895i \(-0.937239\pi\)
0.980625 0.195895i \(-0.0627612\pi\)
\(138\) 11.6569i 0.992297i
\(139\) 6.24264 0.529494 0.264747 0.964318i \(-0.414712\pi\)
0.264747 + 0.964318i \(0.414712\pi\)
\(140\) 0 0
\(141\) 2.24264i 0.188864i
\(142\) −18.4853 −1.55125
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) − 0.272078i − 0.0225948i
\(146\) 13.0711 1.08177
\(147\) −1.41421 −0.116642
\(148\) 0 0
\(149\) − 16.2426i − 1.33065i −0.746554 0.665324i \(-0.768293\pi\)
0.746554 0.665324i \(-0.231707\pi\)
\(150\) 4.97056i 0.405845i
\(151\) − 9.75736i − 0.794043i −0.917809 0.397021i \(-0.870044\pi\)
0.917809 0.397021i \(-0.129956\pi\)
\(152\) −20.4853 −1.66158
\(153\) 1.41421 0.114332
\(154\) − 6.00000i − 0.483494i
\(155\) −5.14214 −0.413026
\(156\) 0 0
\(157\) −3.75736 −0.299870 −0.149935 0.988696i \(-0.547906\pi\)
−0.149935 + 0.988696i \(0.547906\pi\)
\(158\) 21.8995i 1.74223i
\(159\) −0.242641 −0.0192427
\(160\) 0 0
\(161\) 5.82843i 0.459344i
\(162\) − 7.07107i − 0.555556i
\(163\) 8.48528i 0.664619i 0.943170 + 0.332309i \(0.107828\pi\)
−0.943170 + 0.332309i \(0.892172\pi\)
\(164\) 0 0
\(165\) 9.51472 0.740720
\(166\) 18.7279 1.45357
\(167\) 15.3848i 1.19051i 0.803537 + 0.595255i \(0.202949\pi\)
−0.803537 + 0.595255i \(0.797051\pi\)
\(168\) −4.00000 −0.308607
\(169\) 0 0
\(170\) −3.17157 −0.243249
\(171\) − 7.24264i − 0.553859i
\(172\) 0 0
\(173\) −24.7279 −1.88003 −0.940015 0.341134i \(-0.889189\pi\)
−0.940015 + 0.341134i \(0.889189\pi\)
\(174\) 0.343146i 0.0260138i
\(175\) 2.48528i 0.187870i
\(176\) − 16.9706i − 1.27920i
\(177\) 0.485281i 0.0364760i
\(178\) −2.24264 −0.168093
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) 18.7279 1.39204 0.696018 0.718025i \(-0.254953\pi\)
0.696018 + 0.718025i \(0.254953\pi\)
\(182\) 0 0
\(183\) 8.48528 0.627250
\(184\) 16.4853i 1.21531i
\(185\) 3.55635 0.261468
\(186\) 6.48528 0.475524
\(187\) − 6.00000i − 0.438763i
\(188\) 0 0
\(189\) − 5.65685i − 0.411476i
\(190\) 16.2426i 1.17837i
\(191\) 15.1716 1.09778 0.548888 0.835896i \(-0.315051\pi\)
0.548888 + 0.835896i \(0.315051\pi\)
\(192\) −11.3137 −0.816497
\(193\) − 14.4853i − 1.04267i −0.853351 0.521337i \(-0.825434\pi\)
0.853351 0.521337i \(-0.174566\pi\)
\(194\) 16.5858 1.19079
\(195\) 0 0
\(196\) 0 0
\(197\) − 12.3431i − 0.879413i −0.898142 0.439706i \(-0.855083\pi\)
0.898142 0.439706i \(-0.144917\pi\)
\(198\) 6.00000 0.426401
\(199\) 3.75736 0.266352 0.133176 0.991092i \(-0.457482\pi\)
0.133176 + 0.991092i \(0.457482\pi\)
\(200\) 7.02944i 0.497056i
\(201\) 20.4853i 1.44492i
\(202\) − 14.4853i − 1.01918i
\(203\) 0.171573i 0.0120421i
\(204\) 0 0
\(205\) −14.0000 −0.977802
\(206\) 11.3137i 0.788263i
\(207\) −5.82843 −0.405104
\(208\) 0 0
\(209\) −30.7279 −2.12549
\(210\) 3.17157i 0.218859i
\(211\) −15.9706 −1.09946 −0.549729 0.835343i \(-0.685269\pi\)
−0.549729 + 0.835343i \(0.685269\pi\)
\(212\) 0 0
\(213\) 18.4853i 1.26659i
\(214\) − 28.4853i − 1.94721i
\(215\) − 7.92893i − 0.540749i
\(216\) − 16.0000i − 1.08866i
\(217\) 3.24264 0.220125
\(218\) 23.6569 1.60224
\(219\) − 13.0711i − 0.883261i
\(220\) 0 0
\(221\) 0 0
\(222\) −4.48528 −0.301032
\(223\) − 0.757359i − 0.0507165i −0.999678 0.0253583i \(-0.991927\pi\)
0.999678 0.0253583i \(-0.00807265\pi\)
\(224\) 0 0
\(225\) −2.48528 −0.165685
\(226\) 3.27208i 0.217655i
\(227\) − 26.8284i − 1.78067i −0.455311 0.890333i \(-0.650472\pi\)
0.455311 0.890333i \(-0.349528\pi\)
\(228\) 0 0
\(229\) 29.4558i 1.94650i 0.229755 + 0.973248i \(0.426207\pi\)
−0.229755 + 0.973248i \(0.573793\pi\)
\(230\) 13.0711 0.861881
\(231\) −6.00000 −0.394771
\(232\) 0.485281i 0.0318603i
\(233\) 14.6569 0.960202 0.480101 0.877213i \(-0.340600\pi\)
0.480101 + 0.877213i \(0.340600\pi\)
\(234\) 0 0
\(235\) 2.51472 0.164042
\(236\) 0 0
\(237\) 21.8995 1.42253
\(238\) 2.00000 0.129641
\(239\) − 3.51472i − 0.227348i −0.993518 0.113674i \(-0.963738\pi\)
0.993518 0.113674i \(-0.0362620\pi\)
\(240\) 8.97056i 0.579047i
\(241\) − 20.2132i − 1.30205i −0.759058 0.651023i \(-0.774340\pi\)
0.759058 0.651023i \(-0.225660\pi\)
\(242\) − 9.89949i − 0.636364i
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 1.58579i 0.101312i
\(246\) 17.6569 1.12576
\(247\) 0 0
\(248\) 9.17157 0.582395
\(249\) − 18.7279i − 1.18683i
\(250\) 16.7868 1.06169
\(251\) −16.5858 −1.04689 −0.523443 0.852061i \(-0.675353\pi\)
−0.523443 + 0.852061i \(0.675353\pi\)
\(252\) 0 0
\(253\) 24.7279i 1.55463i
\(254\) − 2.82843i − 0.177471i
\(255\) 3.17157i 0.198612i
\(256\) 0 0
\(257\) 19.4142 1.21103 0.605513 0.795836i \(-0.292968\pi\)
0.605513 + 0.795836i \(0.292968\pi\)
\(258\) 10.0000i 0.622573i
\(259\) −2.24264 −0.139351
\(260\) 0 0
\(261\) −0.171573 −0.0106201
\(262\) − 4.00000i − 0.247121i
\(263\) 1.97056 0.121510 0.0607551 0.998153i \(-0.480649\pi\)
0.0607551 + 0.998153i \(0.480649\pi\)
\(264\) −16.9706 −1.04447
\(265\) 0.272078i 0.0167136i
\(266\) − 10.2426i − 0.628017i
\(267\) 2.24264i 0.137247i
\(268\) 0 0
\(269\) 9.17157 0.559201 0.279600 0.960116i \(-0.409798\pi\)
0.279600 + 0.960116i \(0.409798\pi\)
\(270\) −12.6863 −0.772063
\(271\) 20.0000i 1.21491i 0.794353 + 0.607457i \(0.207810\pi\)
−0.794353 + 0.607457i \(0.792190\pi\)
\(272\) 5.65685 0.342997
\(273\) 0 0
\(274\) 6.48528 0.391790
\(275\) 10.5442i 0.635837i
\(276\) 0 0
\(277\) 7.48528 0.449747 0.224873 0.974388i \(-0.427803\pi\)
0.224873 + 0.974388i \(0.427803\pi\)
\(278\) 8.82843i 0.529494i
\(279\) 3.24264i 0.194132i
\(280\) 4.48528i 0.268047i
\(281\) 15.5563i 0.928014i 0.885832 + 0.464007i \(0.153589\pi\)
−0.885832 + 0.464007i \(0.846411\pi\)
\(282\) −3.17157 −0.188864
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) 0 0
\(285\) 16.2426 0.962131
\(286\) 0 0
\(287\) 8.82843 0.521126
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0.384776 0.0225948
\(291\) − 16.5858i − 0.972276i
\(292\) 0 0
\(293\) 15.3848i 0.898788i 0.893334 + 0.449394i \(0.148360\pi\)
−0.893334 + 0.449394i \(0.851640\pi\)
\(294\) − 2.00000i − 0.116642i
\(295\) 0.544156 0.0316820
\(296\) −6.34315 −0.368688
\(297\) − 24.0000i − 1.39262i
\(298\) 22.9706 1.33065
\(299\) 0 0
\(300\) 0 0
\(301\) 5.00000i 0.288195i
\(302\) 13.7990 0.794043
\(303\) −14.4853 −0.832158
\(304\) − 28.9706i − 1.66158i
\(305\) − 9.51472i − 0.544811i
\(306\) 2.00000i 0.114332i
\(307\) 13.2426i 0.755797i 0.925847 + 0.377899i \(0.123353\pi\)
−0.925847 + 0.377899i \(0.876647\pi\)
\(308\) 0 0
\(309\) 11.3137 0.643614
\(310\) − 7.27208i − 0.413026i
\(311\) 7.41421 0.420421 0.210211 0.977656i \(-0.432585\pi\)
0.210211 + 0.977656i \(0.432585\pi\)
\(312\) 0 0
\(313\) 23.2132 1.31209 0.656044 0.754723i \(-0.272229\pi\)
0.656044 + 0.754723i \(0.272229\pi\)
\(314\) − 5.31371i − 0.299870i
\(315\) −1.58579 −0.0893489
\(316\) 0 0
\(317\) 11.3137i 0.635441i 0.948184 + 0.317721i \(0.102917\pi\)
−0.948184 + 0.317721i \(0.897083\pi\)
\(318\) − 0.343146i − 0.0192427i
\(319\) 0.727922i 0.0407558i
\(320\) 12.6863i 0.709185i
\(321\) −28.4853 −1.58989
\(322\) −8.24264 −0.459344
\(323\) − 10.2426i − 0.569916i
\(324\) 0 0
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) − 23.6569i − 1.30823i
\(328\) 24.9706 1.37877
\(329\) −1.58579 −0.0874272
\(330\) 13.4558i 0.740720i
\(331\) 18.0000i 0.989369i 0.869072 + 0.494685i \(0.164716\pi\)
−0.869072 + 0.494685i \(0.835284\pi\)
\(332\) 0 0
\(333\) − 2.24264i − 0.122896i
\(334\) −21.7574 −1.19051
\(335\) 22.9706 1.25502
\(336\) − 5.65685i − 0.308607i
\(337\) 33.0000 1.79762 0.898812 0.438334i \(-0.144431\pi\)
0.898812 + 0.438334i \(0.144431\pi\)
\(338\) 0 0
\(339\) 3.27208 0.177715
\(340\) 0 0
\(341\) 13.7574 0.745003
\(342\) 10.2426 0.553859
\(343\) − 1.00000i − 0.0539949i
\(344\) 14.1421i 0.762493i
\(345\) − 13.0711i − 0.703723i
\(346\) − 34.9706i − 1.88003i
\(347\) −5.65685 −0.303676 −0.151838 0.988405i \(-0.548519\pi\)
−0.151838 + 0.988405i \(0.548519\pi\)
\(348\) 0 0
\(349\) − 25.7279i − 1.37718i −0.725149 0.688592i \(-0.758229\pi\)
0.725149 0.688592i \(-0.241771\pi\)
\(350\) −3.51472 −0.187870
\(351\) 0 0
\(352\) 0 0
\(353\) 8.48528i 0.451626i 0.974171 + 0.225813i \(0.0725038\pi\)
−0.974171 + 0.225813i \(0.927496\pi\)
\(354\) −0.686292 −0.0364760
\(355\) 20.7279 1.10012
\(356\) 0 0
\(357\) − 2.00000i − 0.105851i
\(358\) 12.7279i 0.672692i
\(359\) − 27.8995i − 1.47248i −0.676721 0.736240i \(-0.736600\pi\)
0.676721 0.736240i \(-0.263400\pi\)
\(360\) −4.48528 −0.236395
\(361\) −33.4558 −1.76083
\(362\) 26.4853i 1.39204i
\(363\) −9.89949 −0.519589
\(364\) 0 0
\(365\) −14.6569 −0.767175
\(366\) 12.0000i 0.627250i
\(367\) 10.2426 0.534661 0.267331 0.963605i \(-0.413858\pi\)
0.267331 + 0.963605i \(0.413858\pi\)
\(368\) −23.3137 −1.21531
\(369\) 8.82843i 0.459590i
\(370\) 5.02944i 0.261468i
\(371\) − 0.171573i − 0.00890762i
\(372\) 0 0
\(373\) 8.48528 0.439351 0.219676 0.975573i \(-0.429500\pi\)
0.219676 + 0.975573i \(0.429500\pi\)
\(374\) 8.48528 0.438763
\(375\) − 16.7868i − 0.866866i
\(376\) −4.48528 −0.231311
\(377\) 0 0
\(378\) 8.00000 0.411476
\(379\) − 23.7574i − 1.22033i −0.792273 0.610167i \(-0.791102\pi\)
0.792273 0.610167i \(-0.208898\pi\)
\(380\) 0 0
\(381\) −2.82843 −0.144905
\(382\) 21.4558i 1.09778i
\(383\) 20.4853i 1.04675i 0.852103 + 0.523374i \(0.175327\pi\)
−0.852103 + 0.523374i \(0.824673\pi\)
\(384\) − 16.0000i − 0.816497i
\(385\) 6.72792i 0.342887i
\(386\) 20.4853 1.04267
\(387\) −5.00000 −0.254164
\(388\) 0 0
\(389\) −29.6569 −1.50366 −0.751831 0.659356i \(-0.770829\pi\)
−0.751831 + 0.659356i \(0.770829\pi\)
\(390\) 0 0
\(391\) −8.24264 −0.416848
\(392\) − 2.82843i − 0.142857i
\(393\) −4.00000 −0.201773
\(394\) 17.4558 0.879413
\(395\) − 24.5563i − 1.23556i
\(396\) 0 0
\(397\) − 18.2132i − 0.914094i −0.889442 0.457047i \(-0.848907\pi\)
0.889442 0.457047i \(-0.151093\pi\)
\(398\) 5.31371i 0.266352i
\(399\) −10.2426 −0.512773
\(400\) −9.94113 −0.497056
\(401\) 6.34315i 0.316762i 0.987378 + 0.158381i \(0.0506274\pi\)
−0.987378 + 0.158381i \(0.949373\pi\)
\(402\) −28.9706 −1.44492
\(403\) 0 0
\(404\) 0 0
\(405\) 7.92893i 0.393992i
\(406\) −0.242641 −0.0120421
\(407\) −9.51472 −0.471627
\(408\) − 5.65685i − 0.280056i
\(409\) − 3.24264i − 0.160338i −0.996781 0.0801691i \(-0.974454\pi\)
0.996781 0.0801691i \(-0.0255460\pi\)
\(410\) − 19.7990i − 0.977802i
\(411\) − 6.48528i − 0.319895i
\(412\) 0 0
\(413\) −0.343146 −0.0168851
\(414\) − 8.24264i − 0.405104i
\(415\) −21.0000 −1.03085
\(416\) 0 0
\(417\) 8.82843 0.432330
\(418\) − 43.4558i − 2.12549i
\(419\) −20.8701 −1.01957 −0.509785 0.860302i \(-0.670275\pi\)
−0.509785 + 0.860302i \(0.670275\pi\)
\(420\) 0 0
\(421\) − 6.72792i − 0.327899i −0.986469 0.163949i \(-0.947577\pi\)
0.986469 0.163949i \(-0.0524234\pi\)
\(422\) − 22.5858i − 1.09946i
\(423\) − 1.58579i − 0.0771036i
\(424\) − 0.485281i − 0.0235673i
\(425\) −3.51472 −0.170489
\(426\) −26.1421 −1.26659
\(427\) 6.00000i 0.290360i
\(428\) 0 0
\(429\) 0 0
\(430\) 11.2132 0.540749
\(431\) 12.3431i 0.594548i 0.954792 + 0.297274i \(0.0960775\pi\)
−0.954792 + 0.297274i \(0.903922\pi\)
\(432\) 22.6274 1.08866
\(433\) 24.9706 1.20001 0.600004 0.799997i \(-0.295166\pi\)
0.600004 + 0.799997i \(0.295166\pi\)
\(434\) 4.58579i 0.220125i
\(435\) − 0.384776i − 0.0184486i
\(436\) 0 0
\(437\) 42.2132i 2.01933i
\(438\) 18.4853 0.883261
\(439\) 34.4853 1.64589 0.822946 0.568119i \(-0.192329\pi\)
0.822946 + 0.568119i \(0.192329\pi\)
\(440\) 19.0294i 0.907193i
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −3.68629 −0.175141 −0.0875705 0.996158i \(-0.527910\pi\)
−0.0875705 + 0.996158i \(0.527910\pi\)
\(444\) 0 0
\(445\) 2.51472 0.119209
\(446\) 1.07107 0.0507165
\(447\) − 22.9706i − 1.08647i
\(448\) − 8.00000i − 0.377964i
\(449\) 21.1716i 0.999148i 0.866271 + 0.499574i \(0.166510\pi\)
−0.866271 + 0.499574i \(0.833490\pi\)
\(450\) − 3.51472i − 0.165685i
\(451\) 37.4558 1.76373
\(452\) 0 0
\(453\) − 13.7990i − 0.648333i
\(454\) 37.9411 1.78067
\(455\) 0 0
\(456\) −28.9706 −1.35667
\(457\) 7.21320i 0.337419i 0.985666 + 0.168710i \(0.0539600\pi\)
−0.985666 + 0.168710i \(0.946040\pi\)
\(458\) −41.6569 −1.94650
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) − 8.82843i − 0.411181i −0.978638 0.205590i \(-0.934089\pi\)
0.978638 0.205590i \(-0.0659115\pi\)
\(462\) − 8.48528i − 0.394771i
\(463\) − 4.24264i − 0.197172i −0.995129 0.0985861i \(-0.968568\pi\)
0.995129 0.0985861i \(-0.0314320\pi\)
\(464\) −0.686292 −0.0318603
\(465\) −7.27208 −0.337235
\(466\) 20.7279i 0.960202i
\(467\) 3.89949 0.180447 0.0902236 0.995922i \(-0.471242\pi\)
0.0902236 + 0.995922i \(0.471242\pi\)
\(468\) 0 0
\(469\) −14.4853 −0.668868
\(470\) 3.55635i 0.164042i
\(471\) −5.31371 −0.244843
\(472\) −0.970563 −0.0446738
\(473\) 21.2132i 0.975384i
\(474\) 30.9706i 1.42253i
\(475\) 18.0000i 0.825897i
\(476\) 0 0
\(477\) 0.171573 0.00785578
\(478\) 4.97056 0.227348
\(479\) − 6.21320i − 0.283889i −0.989875 0.141944i \(-0.954665\pi\)
0.989875 0.141944i \(-0.0453354\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 28.5858 1.30205
\(483\) 8.24264i 0.375053i
\(484\) 0 0
\(485\) −18.5980 −0.844491
\(486\) 14.0000i 0.635053i
\(487\) 11.4558i 0.519114i 0.965728 + 0.259557i \(0.0835765\pi\)
−0.965728 + 0.259557i \(0.916423\pi\)
\(488\) 16.9706i 0.768221i
\(489\) 12.0000i 0.542659i
\(490\) −2.24264 −0.101312
\(491\) −16.6274 −0.750385 −0.375192 0.926947i \(-0.622423\pi\)
−0.375192 + 0.926947i \(0.622423\pi\)
\(492\) 0 0
\(493\) −0.242641 −0.0109280
\(494\) 0 0
\(495\) −6.72792 −0.302398
\(496\) 12.9706i 0.582395i
\(497\) −13.0711 −0.586318
\(498\) 26.4853 1.18683
\(499\) 13.2721i 0.594140i 0.954856 + 0.297070i \(0.0960094\pi\)
−0.954856 + 0.297070i \(0.903991\pi\)
\(500\) 0 0
\(501\) 21.7574i 0.972047i
\(502\) − 23.4558i − 1.04689i
\(503\) −28.6274 −1.27643 −0.638217 0.769857i \(-0.720328\pi\)
−0.638217 + 0.769857i \(0.720328\pi\)
\(504\) 2.82843 0.125988
\(505\) 16.2426i 0.722788i
\(506\) −34.9706 −1.55463
\(507\) 0 0
\(508\) 0 0
\(509\) 5.10051i 0.226076i 0.993591 + 0.113038i \(0.0360582\pi\)
−0.993591 + 0.113038i \(0.963942\pi\)
\(510\) −4.48528 −0.198612
\(511\) 9.24264 0.408870
\(512\) − 22.6274i − 1.00000i
\(513\) − 40.9706i − 1.80889i
\(514\) 27.4558i 1.21103i
\(515\) − 12.6863i − 0.559025i
\(516\) 0 0
\(517\) −6.72792 −0.295894
\(518\) − 3.17157i − 0.139351i
\(519\) −34.9706 −1.53504
\(520\) 0 0
\(521\) −6.34315 −0.277898 −0.138949 0.990300i \(-0.544372\pi\)
−0.138949 + 0.990300i \(0.544372\pi\)
\(522\) − 0.242641i − 0.0106201i
\(523\) 30.9706 1.35425 0.677124 0.735869i \(-0.263226\pi\)
0.677124 + 0.735869i \(0.263226\pi\)
\(524\) 0 0
\(525\) 3.51472i 0.153395i
\(526\) 2.78680i 0.121510i
\(527\) 4.58579i 0.199760i
\(528\) − 24.0000i − 1.04447i
\(529\) 10.9706 0.476981
\(530\) −0.384776 −0.0167136
\(531\) − 0.343146i − 0.0148913i
\(532\) 0 0
\(533\) 0 0
\(534\) −3.17157 −0.137247
\(535\) 31.9411i 1.38094i
\(536\) −40.9706 −1.76966
\(537\) 12.7279 0.549250
\(538\) 12.9706i 0.559201i
\(539\) − 4.24264i − 0.182743i
\(540\) 0 0
\(541\) − 7.21320i − 0.310120i −0.987905 0.155060i \(-0.950443\pi\)
0.987905 0.155060i \(-0.0495571\pi\)
\(542\) −28.2843 −1.21491
\(543\) 26.4853 1.13659
\(544\) 0 0
\(545\) −26.5269 −1.13629
\(546\) 0 0
\(547\) 35.4853 1.51724 0.758621 0.651533i \(-0.225874\pi\)
0.758621 + 0.651533i \(0.225874\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) −14.9117 −0.635837
\(551\) 1.24264i 0.0529383i
\(552\) 23.3137i 0.992297i
\(553\) 15.4853i 0.658501i
\(554\) 10.5858i 0.449747i
\(555\) 5.02944 0.213488
\(556\) 0 0
\(557\) 12.0000i 0.508456i 0.967144 + 0.254228i \(0.0818214\pi\)
−0.967144 + 0.254228i \(0.918179\pi\)
\(558\) −4.58579 −0.194132
\(559\) 0 0
\(560\) −6.34315 −0.268047
\(561\) − 8.48528i − 0.358249i
\(562\) −22.0000 −0.928014
\(563\) 6.34315 0.267332 0.133666 0.991026i \(-0.457325\pi\)
0.133666 + 0.991026i \(0.457325\pi\)
\(564\) 0 0
\(565\) − 3.66905i − 0.154358i
\(566\) 12.0000i 0.504398i
\(567\) − 5.00000i − 0.209980i
\(568\) −36.9706 −1.55125
\(569\) −5.14214 −0.215570 −0.107785 0.994174i \(-0.534376\pi\)
−0.107785 + 0.994174i \(0.534376\pi\)
\(570\) 22.9706i 0.962131i
\(571\) −42.4558 −1.77672 −0.888361 0.459146i \(-0.848156\pi\)
−0.888361 + 0.459146i \(0.848156\pi\)
\(572\) 0 0
\(573\) 21.4558 0.896331
\(574\) 12.4853i 0.521126i
\(575\) 14.4853 0.604078
\(576\) 8.00000 0.333333
\(577\) 6.97056i 0.290188i 0.989418 + 0.145094i \(0.0463485\pi\)
−0.989418 + 0.145094i \(0.953651\pi\)
\(578\) − 21.2132i − 0.882353i
\(579\) − 20.4853i − 0.851339i
\(580\) 0 0
\(581\) 13.2426 0.549397
\(582\) 23.4558 0.972276
\(583\) − 0.727922i − 0.0301475i
\(584\) 26.1421 1.08177
\(585\) 0 0
\(586\) −21.7574 −0.898788
\(587\) 6.55635i 0.270609i 0.990804 + 0.135305i \(0.0432013\pi\)
−0.990804 + 0.135305i \(0.956799\pi\)
\(588\) 0 0
\(589\) 23.4853 0.967694
\(590\) 0.769553i 0.0316820i
\(591\) − 17.4558i − 0.718037i
\(592\) − 8.97056i − 0.368688i
\(593\) − 0.556349i − 0.0228465i −0.999935 0.0114233i \(-0.996364\pi\)
0.999935 0.0114233i \(-0.00363622\pi\)
\(594\) 33.9411 1.39262
\(595\) −2.24264 −0.0919393
\(596\) 0 0
\(597\) 5.31371 0.217476
\(598\) 0 0
\(599\) −28.7990 −1.17669 −0.588347 0.808608i \(-0.700221\pi\)
−0.588347 + 0.808608i \(0.700221\pi\)
\(600\) 9.94113i 0.405845i
\(601\) −38.9706 −1.58964 −0.794821 0.606844i \(-0.792435\pi\)
−0.794821 + 0.606844i \(0.792435\pi\)
\(602\) −7.07107 −0.288195
\(603\) − 14.4853i − 0.589886i
\(604\) 0 0
\(605\) 11.1005i 0.451300i
\(606\) − 20.4853i − 0.832158i
\(607\) −26.7279 −1.08485 −0.542426 0.840103i \(-0.682494\pi\)
−0.542426 + 0.840103i \(0.682494\pi\)
\(608\) 0 0
\(609\) 0.242641i 0.00983230i
\(610\) 13.4558 0.544811
\(611\) 0 0
\(612\) 0 0
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) −18.7279 −0.755797
\(615\) −19.7990 −0.798372
\(616\) − 12.0000i − 0.483494i
\(617\) − 12.3431i − 0.496916i −0.968643 0.248458i \(-0.920076\pi\)
0.968643 0.248458i \(-0.0799239\pi\)
\(618\) 16.0000i 0.643614i
\(619\) 0.970563i 0.0390102i 0.999810 + 0.0195051i \(0.00620906\pi\)
−0.999810 + 0.0195051i \(0.993791\pi\)
\(620\) 0 0
\(621\) −32.9706 −1.32306
\(622\) 10.4853i 0.420421i
\(623\) −1.58579 −0.0635332
\(624\) 0 0
\(625\) −6.39697 −0.255879
\(626\) 32.8284i 1.31209i
\(627\) −43.4558 −1.73546
\(628\) 0 0
\(629\) − 3.17157i − 0.126459i
\(630\) − 2.24264i − 0.0893489i
\(631\) 2.00000i 0.0796187i 0.999207 + 0.0398094i \(0.0126751\pi\)
−0.999207 + 0.0398094i \(0.987325\pi\)
\(632\) 43.7990i 1.74223i
\(633\) −22.5858 −0.897704
\(634\) −16.0000 −0.635441
\(635\) 3.17157i 0.125860i
\(636\) 0 0
\(637\) 0 0
\(638\) −1.02944 −0.0407558
\(639\) − 13.0711i − 0.517083i
\(640\) −17.9411 −0.709185
\(641\) −15.3431 −0.606018 −0.303009 0.952988i \(-0.597991\pi\)
−0.303009 + 0.952988i \(0.597991\pi\)
\(642\) − 40.2843i − 1.58989i
\(643\) 12.4853i 0.492371i 0.969223 + 0.246186i \(0.0791773\pi\)
−0.969223 + 0.246186i \(0.920823\pi\)
\(644\) 0 0
\(645\) − 11.2132i − 0.441519i
\(646\) 14.4853 0.569916
\(647\) −14.5269 −0.571112 −0.285556 0.958362i \(-0.592178\pi\)
−0.285556 + 0.958362i \(0.592178\pi\)
\(648\) − 14.1421i − 0.555556i
\(649\) −1.45584 −0.0571469
\(650\) 0 0
\(651\) 4.58579 0.179731
\(652\) 0 0
\(653\) 17.3137 0.677538 0.338769 0.940870i \(-0.389990\pi\)
0.338769 + 0.940870i \(0.389990\pi\)
\(654\) 33.4558 1.30823
\(655\) 4.48528i 0.175254i
\(656\) 35.3137i 1.37877i
\(657\) 9.24264i 0.360590i
\(658\) − 2.24264i − 0.0874272i
\(659\) −15.3431 −0.597684 −0.298842 0.954303i \(-0.596600\pi\)
−0.298842 + 0.954303i \(0.596600\pi\)
\(660\) 0 0
\(661\) − 10.7574i − 0.418413i −0.977872 0.209206i \(-0.932912\pi\)
0.977872 0.209206i \(-0.0670880\pi\)
\(662\) −25.4558 −0.989369
\(663\) 0 0
\(664\) 37.4558 1.45357
\(665\) 11.4853i 0.445380i
\(666\) 3.17157 0.122896
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) − 1.07107i − 0.0414099i
\(670\) 32.4853i 1.25502i
\(671\) 25.4558i 0.982712i
\(672\) 0 0
\(673\) 26.9411 1.03850 0.519252 0.854621i \(-0.326211\pi\)
0.519252 + 0.854621i \(0.326211\pi\)
\(674\) 46.6690i 1.79762i
\(675\) −14.0589 −0.541126
\(676\) 0 0
\(677\) −41.3553 −1.58941 −0.794707 0.606993i \(-0.792376\pi\)
−0.794707 + 0.606993i \(0.792376\pi\)
\(678\) 4.62742i 0.177715i
\(679\) 11.7279 0.450076
\(680\) −6.34315 −0.243249
\(681\) − 37.9411i − 1.45391i
\(682\) 19.4558i 0.745003i
\(683\) − 21.1716i − 0.810108i −0.914293 0.405054i \(-0.867253\pi\)
0.914293 0.405054i \(-0.132747\pi\)
\(684\) 0 0
\(685\) −7.27208 −0.277852
\(686\) 1.41421 0.0539949
\(687\) 41.6569i 1.58931i
\(688\) −20.0000 −0.762493
\(689\) 0 0
\(690\) 18.4853 0.703723
\(691\) − 28.6985i − 1.09174i −0.837869 0.545871i \(-0.816199\pi\)
0.837869 0.545871i \(-0.183801\pi\)
\(692\) 0 0
\(693\) 4.24264 0.161165
\(694\) − 8.00000i − 0.303676i
\(695\) − 9.89949i − 0.375509i
\(696\) 0.686292i 0.0260138i
\(697\) 12.4853i 0.472914i
\(698\) 36.3848 1.37718
\(699\) 20.7279 0.784002
\(700\) 0 0
\(701\) 10.7990 0.407872 0.203936 0.978984i \(-0.434627\pi\)
0.203936 + 0.978984i \(0.434627\pi\)
\(702\) 0 0
\(703\) −16.2426 −0.612603
\(704\) − 33.9411i − 1.27920i
\(705\) 3.55635 0.133940
\(706\) −12.0000 −0.451626
\(707\) − 10.2426i − 0.385214i
\(708\) 0 0
\(709\) 0.727922i 0.0273377i 0.999907 + 0.0136688i \(0.00435106\pi\)
−0.999907 + 0.0136688i \(0.995649\pi\)
\(710\) 29.3137i 1.10012i
\(711\) −15.4853 −0.580743
\(712\) −4.48528 −0.168093
\(713\) − 18.8995i − 0.707792i
\(714\) 2.82843 0.105851
\(715\) 0 0
\(716\) 0 0
\(717\) − 4.97056i − 0.185629i
\(718\) 39.4558 1.47248
\(719\) 1.75736 0.0655384 0.0327692 0.999463i \(-0.489567\pi\)
0.0327692 + 0.999463i \(0.489567\pi\)
\(720\) − 6.34315i − 0.236395i
\(721\) 8.00000i 0.297936i
\(722\) − 47.3137i − 1.76083i
\(723\) − 28.5858i − 1.06312i
\(724\) 0 0
\(725\) 0.426407 0.0158364
\(726\) − 14.0000i − 0.519589i
\(727\) −42.0000 −1.55769 −0.778847 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) − 20.7279i − 0.767175i
\(731\) −7.07107 −0.261533
\(732\) 0 0
\(733\) 16.6985i 0.616773i 0.951261 + 0.308386i \(0.0997889\pi\)
−0.951261 + 0.308386i \(0.900211\pi\)
\(734\) 14.4853i 0.534661i
\(735\) 2.24264i 0.0827210i
\(736\) 0 0
\(737\) −61.4558 −2.26376
\(738\) −12.4853 −0.459590
\(739\) − 17.6985i − 0.651049i −0.945534 0.325525i \(-0.894459\pi\)
0.945534 0.325525i \(-0.105541\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.242641 0.00890762
\(743\) 29.6569i 1.08800i 0.839084 + 0.544002i \(0.183092\pi\)
−0.839084 + 0.544002i \(0.816908\pi\)
\(744\) 12.9706 0.475524
\(745\) −25.7574 −0.943677
\(746\) 12.0000i 0.439351i
\(747\) 13.2426i 0.484523i
\(748\) 0 0
\(749\) − 20.1421i − 0.735978i
\(750\) 23.7401 0.866866
\(751\) −15.4853 −0.565066 −0.282533 0.959258i \(-0.591175\pi\)
−0.282533 + 0.959258i \(0.591175\pi\)
\(752\) − 6.34315i − 0.231311i
\(753\) −23.4558 −0.854778
\(754\) 0 0
\(755\) −15.4731 −0.563123
\(756\) 0 0
\(757\) 21.4853 0.780896 0.390448 0.920625i \(-0.372320\pi\)
0.390448 + 0.920625i \(0.372320\pi\)
\(758\) 33.5980 1.22033
\(759\) 34.9706i 1.26935i
\(760\) 32.4853i 1.17837i
\(761\) 34.7574i 1.25995i 0.776614 + 0.629977i \(0.216936\pi\)
−0.776614 + 0.629977i \(0.783064\pi\)
\(762\) − 4.00000i − 0.144905i
\(763\) 16.7279 0.605591
\(764\) 0 0
\(765\) − 2.24264i − 0.0810828i
\(766\) −28.9706 −1.04675
\(767\) 0 0
\(768\) 0 0
\(769\) − 52.2132i − 1.88286i −0.337214 0.941428i \(-0.609484\pi\)
0.337214 0.941428i \(-0.390516\pi\)
\(770\) −9.51472 −0.342887
\(771\) 27.4558 0.988798
\(772\) 0 0
\(773\) 32.8284i 1.18076i 0.807127 + 0.590378i \(0.201021\pi\)
−0.807127 + 0.590378i \(0.798979\pi\)
\(774\) − 7.07107i − 0.254164i
\(775\) − 8.05887i − 0.289483i
\(776\) 33.1716 1.19079
\(777\) −3.17157 −0.113780
\(778\) − 41.9411i − 1.50366i
\(779\) 63.9411 2.29093
\(780\) 0 0
\(781\) −55.4558 −1.98437
\(782\) − 11.6569i − 0.416848i
\(783\) −0.970563 −0.0346851
\(784\) 4.00000 0.142857
\(785\) 5.95837i 0.212663i
\(786\) − 5.65685i − 0.201773i
\(787\) − 24.7574i − 0.882505i −0.897383 0.441252i \(-0.854534\pi\)
0.897383 0.441252i \(-0.145466\pi\)
\(788\) 0 0
\(789\) 2.78680 0.0992126
\(790\) 34.7279 1.23556
\(791\) 2.31371i 0.0822660i
\(792\) 12.0000 0.426401
\(793\) 0 0
\(794\) 25.7574 0.914094
\(795\) 0.384776i 0.0136466i
\(796\) 0 0
\(797\) 24.3431 0.862278 0.431139 0.902285i \(-0.358112\pi\)
0.431139 + 0.902285i \(0.358112\pi\)
\(798\) − 14.4853i − 0.512773i
\(799\) − 2.24264i − 0.0793389i
\(800\) 0 0
\(801\) − 1.58579i − 0.0560310i
\(802\) −8.97056 −0.316762
\(803\) 39.2132 1.38380
\(804\) 0 0
\(805\) 9.24264 0.325760
\(806\) 0 0
\(807\) 12.9706 0.456585
\(808\) − 28.9706i − 1.01918i
\(809\) 17.4853 0.614750 0.307375 0.951589i \(-0.400549\pi\)
0.307375 + 0.951589i \(0.400549\pi\)
\(810\) −11.2132 −0.393992
\(811\) 21.9411i 0.770457i 0.922821 + 0.385229i \(0.125877\pi\)
−0.922821 + 0.385229i \(0.874123\pi\)
\(812\) 0 0
\(813\) 28.2843i 0.991973i
\(814\) − 13.4558i − 0.471627i
\(815\) 13.4558 0.471338
\(816\) 8.00000 0.280056
\(817\) 36.2132i 1.26694i
\(818\) 4.58579 0.160338
\(819\) 0 0
\(820\) 0 0
\(821\) 20.1421i 0.702965i 0.936194 + 0.351483i \(0.114322\pi\)
−0.936194 + 0.351483i \(0.885678\pi\)
\(822\) 9.17157 0.319895
\(823\) −10.4853 −0.365494 −0.182747 0.983160i \(-0.558499\pi\)
−0.182747 + 0.983160i \(0.558499\pi\)
\(824\) 22.6274i 0.788263i
\(825\) 14.9117i 0.519158i
\(826\) − 0.485281i − 0.0168851i
\(827\) − 49.4558i − 1.71975i −0.510506 0.859874i \(-0.670542\pi\)
0.510506 0.859874i \(-0.329458\pi\)
\(828\) 0 0
\(829\) −50.7279 −1.76185 −0.880927 0.473253i \(-0.843080\pi\)
−0.880927 + 0.473253i \(0.843080\pi\)
\(830\) − 29.6985i − 1.03085i
\(831\) 10.5858 0.367217
\(832\) 0 0
\(833\) 1.41421 0.0489996
\(834\) 12.4853i 0.432330i
\(835\) 24.3970 0.844292
\(836\) 0 0
\(837\) 18.3431i 0.634032i
\(838\) − 29.5147i − 1.01957i
\(839\) − 33.1716i − 1.14521i −0.819831 0.572605i \(-0.805933\pi\)
0.819831 0.572605i \(-0.194067\pi\)
\(840\) 6.34315i 0.218859i
\(841\) −28.9706 −0.998985
\(842\) 9.51472 0.327899
\(843\) 22.0000i 0.757720i
\(844\) 0 0
\(845\) 0 0
\(846\) 2.24264 0.0771036
\(847\) − 7.00000i − 0.240523i
\(848\) 0.686292 0.0235673
\(849\) 12.0000 0.411839
\(850\) − 4.97056i − 0.170489i
\(851\) 13.0711i 0.448070i
\(852\) 0 0
\(853\) 39.7279i 1.36026i 0.733092 + 0.680129i \(0.238076\pi\)
−0.733092 + 0.680129i \(0.761924\pi\)
\(854\) −8.48528 −0.290360
\(855\) −11.4853 −0.392788
\(856\) − 56.9706i − 1.94721i
\(857\) 54.7696 1.87089 0.935446 0.353469i \(-0.114998\pi\)
0.935446 + 0.353469i \(0.114998\pi\)
\(858\) 0 0
\(859\) 30.9706 1.05670 0.528351 0.849026i \(-0.322811\pi\)
0.528351 + 0.849026i \(0.322811\pi\)
\(860\) 0 0
\(861\) 12.4853 0.425497
\(862\) −17.4558 −0.594548
\(863\) − 28.2843i − 0.962808i −0.876499 0.481404i \(-0.840127\pi\)
0.876499 0.481404i \(-0.159873\pi\)
\(864\) 0 0
\(865\) 39.2132i 1.33329i
\(866\) 35.3137i 1.20001i
\(867\) −21.2132 −0.720438
\(868\) 0 0
\(869\) 65.6985i 2.22867i
\(870\) 0.544156 0.0184486
\(871\) 0 0
\(872\) 47.3137 1.60224
\(873\) 11.7279i 0.396930i
\(874\) −59.6985 −2.01933
\(875\) 11.8701 0.401281
\(876\) 0 0
\(877\) 10.2426i 0.345869i 0.984933 + 0.172935i \(0.0553250\pi\)
−0.984933 + 0.172935i \(0.944675\pi\)
\(878\) 48.7696i 1.64589i
\(879\) 21.7574i 0.733858i
\(880\) −26.9117 −0.907193
\(881\) −13.1127 −0.441778 −0.220889 0.975299i \(-0.570896\pi\)
−0.220889 + 0.975299i \(0.570896\pi\)
\(882\) 1.41421i 0.0476190i
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 0 0
\(885\) 0.769553 0.0258682
\(886\) − 5.21320i − 0.175141i
\(887\) 19.1127 0.641742 0.320871 0.947123i \(-0.396024\pi\)
0.320871 + 0.947123i \(0.396024\pi\)
\(888\) −8.97056 −0.301032
\(889\) − 2.00000i − 0.0670778i
\(890\) 3.55635i 0.119209i
\(891\) − 21.2132i − 0.710669i
\(892\) 0 0
\(893\) −11.4853 −0.384340
\(894\) 32.4853 1.08647
\(895\) − 14.2721i − 0.477063i
\(896\) 11.3137 0.377964
\(897\) 0 0
\(898\) −29.9411 −0.999148
\(899\) − 0.556349i − 0.0185553i
\(900\) 0 0
\(901\) 0.242641 0.00808353
\(902\) 52.9706i 1.76373i
\(903\) 7.07107i 0.235310i
\(904\) 6.54416i 0.217655i
\(905\) − 29.6985i − 0.987211i
\(906\) 19.5147 0.648333
\(907\) −1.97056 −0.0654315 −0.0327157 0.999465i \(-0.510416\pi\)
−0.0327157 + 0.999465i \(0.510416\pi\)
\(908\) 0 0
\(909\) 10.2426 0.339727
\(910\) 0 0
\(911\) −43.9706 −1.45681 −0.728405 0.685147i \(-0.759738\pi\)
−0.728405 + 0.685147i \(0.759738\pi\)
\(912\) − 40.9706i − 1.35667i
\(913\) 56.1838 1.85941
\(914\) −10.2010 −0.337419
\(915\) − 13.4558i − 0.444836i
\(916\) 0 0
\(917\) − 2.82843i − 0.0934029i
\(918\) 11.3137i 0.373408i
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 26.1421 0.861881
\(921\) 18.7279i 0.617106i
\(922\) 12.4853 0.411181
\(923\) 0 0
\(924\) 0 0
\(925\) 5.57359i 0.183259i
\(926\) 6.00000 0.197172
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) − 36.8995i − 1.21063i −0.795985 0.605317i \(-0.793047\pi\)
0.795985 0.605317i \(-0.206953\pi\)
\(930\) − 10.2843i − 0.337235i
\(931\) − 7.24264i − 0.237368i
\(932\) 0 0
\(933\) 10.4853 0.343273
\(934\) 5.51472i 0.180447i
\(935\) −9.51472 −0.311165
\(936\) 0 0
\(937\) 45.2132 1.47705 0.738525 0.674226i \(-0.235522\pi\)
0.738525 + 0.674226i \(0.235522\pi\)
\(938\) − 20.4853i − 0.668868i
\(939\) 32.8284 1.07132
\(940\) 0 0
\(941\) − 40.0711i − 1.30628i −0.757237 0.653140i \(-0.773451\pi\)
0.757237 0.653140i \(-0.226549\pi\)
\(942\) − 7.51472i − 0.244843i
\(943\) − 51.4558i − 1.67563i
\(944\) − 1.37258i − 0.0446738i
\(945\) −8.97056 −0.291812
\(946\) −30.0000 −0.975384
\(947\) 15.1716i 0.493010i 0.969142 + 0.246505i \(0.0792822\pi\)
−0.969142 + 0.246505i \(0.920718\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −25.4558 −0.825897
\(951\) 16.0000i 0.518836i
\(952\) 4.00000 0.129641
\(953\) −11.1421 −0.360929 −0.180465 0.983581i \(-0.557760\pi\)
−0.180465 + 0.983581i \(0.557760\pi\)
\(954\) 0.242641i 0.00785578i
\(955\) − 24.0589i − 0.778527i
\(956\) 0 0
\(957\) 1.02944i 0.0332770i
\(958\) 8.78680 0.283889
\(959\) 4.58579 0.148083
\(960\) 17.9411i 0.579047i
\(961\) 20.4853 0.660816
\(962\) 0 0
\(963\) 20.1421 0.649071
\(964\) 0 0
\(965\) −22.9706 −0.739449
\(966\) −11.6569 −0.375053
\(967\) − 17.6985i − 0.569145i −0.958655 0.284572i \(-0.908148\pi\)
0.958655 0.284572i \(-0.0918516\pi\)
\(968\) − 19.7990i − 0.636364i
\(969\) − 14.4853i − 0.465334i
\(970\) − 26.3015i − 0.844491i
\(971\) 43.4558 1.39456 0.697282 0.716797i \(-0.254392\pi\)
0.697282 + 0.716797i \(0.254392\pi\)
\(972\) 0 0
\(973\) 6.24264i 0.200130i
\(974\) −16.2010 −0.519114
\(975\) 0 0
\(976\) −24.0000 −0.768221
\(977\) − 24.0416i − 0.769160i −0.923092 0.384580i \(-0.874346\pi\)
0.923092 0.384580i \(-0.125654\pi\)
\(978\) −16.9706 −0.542659
\(979\) −6.72792 −0.215025
\(980\) 0 0
\(981\) 16.7279i 0.534081i
\(982\) − 23.5147i − 0.750385i
\(983\) − 15.0416i − 0.479754i −0.970803 0.239877i \(-0.922893\pi\)
0.970803 0.239877i \(-0.0771070\pi\)
\(984\) 35.3137 1.12576
\(985\) −19.5736 −0.623667
\(986\) − 0.343146i − 0.0109280i
\(987\) −2.24264 −0.0713840
\(988\) 0 0
\(989\) 29.1421 0.926666
\(990\) − 9.51472i − 0.302398i
\(991\) 1.02944 0.0327012 0.0163506 0.999866i \(-0.494795\pi\)
0.0163506 + 0.999866i \(0.494795\pi\)
\(992\) 0 0
\(993\) 25.4558i 0.807817i
\(994\) − 18.4853i − 0.586318i
\(995\) − 5.95837i − 0.188893i
\(996\) 0 0
\(997\) −11.5147 −0.364675 −0.182337 0.983236i \(-0.558366\pi\)
−0.182337 + 0.983236i \(0.558366\pi\)
\(998\) −18.7696 −0.594140
\(999\) − 12.6863i − 0.401377i
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 1183.2.c.d.337.4 4
13.5 odd 4 1183.2.a.d.1.2 2
13.8 odd 4 91.2.a.c.1.1 2
13.12 even 2 inner 1183.2.c.d.337.2 4
39.8 even 4 819.2.a.h.1.2 2
52.47 even 4 1456.2.a.q.1.1 2
65.34 odd 4 2275.2.a.j.1.2 2
91.34 even 4 637.2.a.g.1.1 2
91.47 even 12 637.2.e.g.508.2 4
91.60 odd 12 637.2.e.f.79.2 4
91.73 even 12 637.2.e.g.79.2 4
91.83 even 4 8281.2.a.v.1.2 2
91.86 odd 12 637.2.e.f.508.2 4
104.21 odd 4 5824.2.a.bl.1.1 2
104.99 even 4 5824.2.a.bk.1.2 2
273.125 odd 4 5733.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.c.1.1 2 13.8 odd 4
637.2.a.g.1.1 2 91.34 even 4
637.2.e.f.79.2 4 91.60 odd 12
637.2.e.f.508.2 4 91.86 odd 12
637.2.e.g.79.2 4 91.73 even 12
637.2.e.g.508.2 4 91.47 even 12
819.2.a.h.1.2 2 39.8 even 4
1183.2.a.d.1.2 2 13.5 odd 4
1183.2.c.d.337.2 4 13.12 even 2 inner
1183.2.c.d.337.4 4 1.1 even 1 trivial
1456.2.a.q.1.1 2 52.47 even 4
2275.2.a.j.1.2 2 65.34 odd 4
5733.2.a.s.1.2 2 273.125 odd 4
5824.2.a.bk.1.2 2 104.99 even 4
5824.2.a.bl.1.1 2 104.21 odd 4
8281.2.a.v.1.2 2 91.83 even 4