Properties

Label 2240.2.g.j.449.1
Level $2240$
Weight $2$
Character 2240.449
Analytic conductor $17.886$
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] = [2240,2,Mod(449,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("2240.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 2240.449
Dual form 2240.2.g.j.449.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.44949i q^{3} +(-2.22474 + 0.224745i) q^{5} -1.00000i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-2.44949i q^{3} +(-2.22474 + 0.224745i) q^{5} -1.00000i q^{7} -3.00000 q^{9} +4.89898 q^{11} +4.44949i q^{13} +(0.550510 + 5.44949i) q^{15} +2.00000i q^{17} +1.55051 q^{19} -2.44949 q^{21} -2.89898i q^{23} +(4.89898 - 1.00000i) q^{25} +6.89898 q^{29} +8.89898 q^{31} -12.0000i q^{33} +(0.224745 + 2.22474i) q^{35} -2.00000i q^{37} +10.8990 q^{39} -1.10102 q^{41} -0.898979i q^{43} +(6.67423 - 0.674235i) q^{45} +8.89898i q^{47} -1.00000 q^{49} +4.89898 q^{51} -10.8990i q^{53} +(-10.8990 + 1.10102i) q^{55} -3.79796i q^{57} -1.55051 q^{59} -3.55051 q^{61} +3.00000i q^{63} +(-1.00000 - 9.89898i) q^{65} +8.00000i q^{67} -7.10102 q^{69} -1.10102 q^{71} -2.89898i q^{73} +(-2.44949 - 12.0000i) q^{75} -4.89898i q^{77} -6.89898 q^{79} -9.00000 q^{81} -2.44949i q^{83} +(-0.449490 - 4.44949i) q^{85} -16.8990i q^{87} +10.0000 q^{89} +4.44949 q^{91} -21.7980i q^{93} +(-3.44949 + 0.348469i) q^{95} +15.7980i q^{97} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 12 q^{9} + 12 q^{15} + 16 q^{19} + 8 q^{29} + 16 q^{31} - 4 q^{35} + 24 q^{39} - 24 q^{41} + 12 q^{45} - 4 q^{49} - 24 q^{55} - 16 q^{59} - 24 q^{61} - 4 q^{65} - 48 q^{69} - 24 q^{71} - 8 q^{79} - 36 q^{81} + 8 q^{85} + 40 q^{89} + 8 q^{91} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.44949i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(4\) 0 0
\(5\) −2.22474 + 0.224745i −0.994936 + 0.100509i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 0 0
\(13\) 4.44949i 1.23407i 0.786937 + 0.617033i \(0.211666\pi\)
−0.786937 + 0.617033i \(0.788334\pi\)
\(14\) 0 0
\(15\) 0.550510 + 5.44949i 0.142141 + 1.40705i
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 1.55051 0.355711 0.177856 0.984057i \(-0.443084\pi\)
0.177856 + 0.984057i \(0.443084\pi\)
\(20\) 0 0
\(21\) −2.44949 −0.534522
\(22\) 0 0
\(23\) 2.89898i 0.604479i −0.953232 0.302240i \(-0.902266\pi\)
0.953232 0.302240i \(-0.0977342\pi\)
\(24\) 0 0
\(25\) 4.89898 1.00000i 0.979796 0.200000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.89898 1.28111 0.640554 0.767913i \(-0.278705\pi\)
0.640554 + 0.767913i \(0.278705\pi\)
\(30\) 0 0
\(31\) 8.89898 1.59830 0.799152 0.601129i \(-0.205282\pi\)
0.799152 + 0.601129i \(0.205282\pi\)
\(32\) 0 0
\(33\) 12.0000i 2.08893i
\(34\) 0 0
\(35\) 0.224745 + 2.22474i 0.0379888 + 0.376051i
\(36\) 0 0
\(37\) 2.00000i 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 10.8990 1.74523
\(40\) 0 0
\(41\) −1.10102 −0.171951 −0.0859753 0.996297i \(-0.527401\pi\)
−0.0859753 + 0.996297i \(0.527401\pi\)
\(42\) 0 0
\(43\) 0.898979i 0.137093i −0.997648 0.0685465i \(-0.978164\pi\)
0.997648 0.0685465i \(-0.0218362\pi\)
\(44\) 0 0
\(45\) 6.67423 0.674235i 0.994936 0.100509i
\(46\) 0 0
\(47\) 8.89898i 1.29805i 0.760767 + 0.649025i \(0.224823\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 4.89898 0.685994
\(52\) 0 0
\(53\) 10.8990i 1.49709i −0.663084 0.748545i \(-0.730753\pi\)
0.663084 0.748545i \(-0.269247\pi\)
\(54\) 0 0
\(55\) −10.8990 + 1.10102i −1.46962 + 0.148462i
\(56\) 0 0
\(57\) 3.79796i 0.503052i
\(58\) 0 0
\(59\) −1.55051 −0.201859 −0.100930 0.994894i \(-0.532182\pi\)
−0.100930 + 0.994894i \(0.532182\pi\)
\(60\) 0 0
\(61\) −3.55051 −0.454596 −0.227298 0.973825i \(-0.572989\pi\)
−0.227298 + 0.973825i \(0.572989\pi\)
\(62\) 0 0
\(63\) 3.00000i 0.377964i
\(64\) 0 0
\(65\) −1.00000 9.89898i −0.124035 1.22782i
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) −7.10102 −0.854862
\(70\) 0 0
\(71\) −1.10102 −0.130667 −0.0653335 0.997863i \(-0.520811\pi\)
−0.0653335 + 0.997863i \(0.520811\pi\)
\(72\) 0 0
\(73\) 2.89898i 0.339300i −0.985504 0.169650i \(-0.945736\pi\)
0.985504 0.169650i \(-0.0542637\pi\)
\(74\) 0 0
\(75\) −2.44949 12.0000i −0.282843 1.38564i
\(76\) 0 0
\(77\) 4.89898i 0.558291i
\(78\) 0 0
\(79\) −6.89898 −0.776196 −0.388098 0.921618i \(-0.626868\pi\)
−0.388098 + 0.921618i \(0.626868\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 2.44949i 0.268866i −0.990923 0.134433i \(-0.957079\pi\)
0.990923 0.134433i \(-0.0429214\pi\)
\(84\) 0 0
\(85\) −0.449490 4.44949i −0.0487540 0.482615i
\(86\) 0 0
\(87\) 16.8990i 1.81176i
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 4.44949 0.466433
\(92\) 0 0
\(93\) 21.7980i 2.26034i
\(94\) 0 0
\(95\) −3.44949 + 0.348469i −0.353910 + 0.0357522i
\(96\) 0 0
\(97\) 15.7980i 1.60404i 0.597297 + 0.802020i \(0.296241\pi\)
−0.597297 + 0.802020i \(0.703759\pi\)
\(98\) 0 0
\(99\) −14.6969 −1.47710
\(100\) 0 0
\(101\) −3.55051 −0.353289 −0.176644 0.984275i \(-0.556524\pi\)
−0.176644 + 0.984275i \(0.556524\pi\)
\(102\) 0 0
\(103\) 12.8990i 1.27097i −0.772111 0.635487i \(-0.780799\pi\)
0.772111 0.635487i \(-0.219201\pi\)
\(104\) 0 0
\(105\) 5.44949 0.550510i 0.531816 0.0537243i
\(106\) 0 0
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 0 0
\(109\) 6.89898 0.660802 0.330401 0.943841i \(-0.392816\pi\)
0.330401 + 0.943841i \(0.392816\pi\)
\(110\) 0 0
\(111\) −4.89898 −0.464991
\(112\) 0 0
\(113\) 19.7980i 1.86244i −0.364464 0.931218i \(-0.618748\pi\)
0.364464 0.931218i \(-0.381252\pi\)
\(114\) 0 0
\(115\) 0.651531 + 6.44949i 0.0607556 + 0.601418i
\(116\) 0 0
\(117\) 13.3485i 1.23407i
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) 2.69694i 0.243175i
\(124\) 0 0
\(125\) −10.6742 + 3.32577i −0.954733 + 0.297465i
\(126\) 0 0
\(127\) 14.8990i 1.32207i −0.750355 0.661035i \(-0.770117\pi\)
0.750355 0.661035i \(-0.229883\pi\)
\(128\) 0 0
\(129\) −2.20204 −0.193879
\(130\) 0 0
\(131\) 6.44949 0.563495 0.281747 0.959489i \(-0.409086\pi\)
0.281747 + 0.959489i \(0.409086\pi\)
\(132\) 0 0
\(133\) 1.55051i 0.134446i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.79796i 0.153610i −0.997046 0.0768050i \(-0.975528\pi\)
0.997046 0.0768050i \(-0.0244719\pi\)
\(138\) 0 0
\(139\) 1.55051 0.131513 0.0657563 0.997836i \(-0.479054\pi\)
0.0657563 + 0.997836i \(0.479054\pi\)
\(140\) 0 0
\(141\) 21.7980 1.83572
\(142\) 0 0
\(143\) 21.7980i 1.82284i
\(144\) 0 0
\(145\) −15.3485 + 1.55051i −1.27462 + 0.128763i
\(146\) 0 0
\(147\) 2.44949i 0.202031i
\(148\) 0 0
\(149\) −3.79796 −0.311141 −0.155570 0.987825i \(-0.549722\pi\)
−0.155570 + 0.987825i \(0.549722\pi\)
\(150\) 0 0
\(151\) 19.5959 1.59469 0.797347 0.603522i \(-0.206236\pi\)
0.797347 + 0.603522i \(0.206236\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) −19.7980 + 2.00000i −1.59021 + 0.160644i
\(156\) 0 0
\(157\) 3.55051i 0.283362i −0.989912 0.141681i \(-0.954749\pi\)
0.989912 0.141681i \(-0.0452507\pi\)
\(158\) 0 0
\(159\) −26.6969 −2.11720
\(160\) 0 0
\(161\) −2.89898 −0.228472
\(162\) 0 0
\(163\) 7.10102i 0.556195i −0.960553 0.278097i \(-0.910296\pi\)
0.960553 0.278097i \(-0.0897038\pi\)
\(164\) 0 0
\(165\) 2.69694 + 26.6969i 0.209956 + 2.07835i
\(166\) 0 0
\(167\) 4.89898i 0.379094i −0.981872 0.189547i \(-0.939298\pi\)
0.981872 0.189547i \(-0.0607020\pi\)
\(168\) 0 0
\(169\) −6.79796 −0.522920
\(170\) 0 0
\(171\) −4.65153 −0.355711
\(172\) 0 0
\(173\) 6.24745i 0.474985i −0.971389 0.237492i \(-0.923675\pi\)
0.971389 0.237492i \(-0.0763255\pi\)
\(174\) 0 0
\(175\) −1.00000 4.89898i −0.0755929 0.370328i
\(176\) 0 0
\(177\) 3.79796i 0.285472i
\(178\) 0 0
\(179\) 13.7980 1.03131 0.515654 0.856797i \(-0.327549\pi\)
0.515654 + 0.856797i \(0.327549\pi\)
\(180\) 0 0
\(181\) 10.2474 0.761687 0.380843 0.924640i \(-0.375634\pi\)
0.380843 + 0.924640i \(0.375634\pi\)
\(182\) 0 0
\(183\) 8.69694i 0.642896i
\(184\) 0 0
\(185\) 0.449490 + 4.44949i 0.0330471 + 0.327133i
\(186\) 0 0
\(187\) 9.79796i 0.716498i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.6969 0.918718 0.459359 0.888251i \(-0.348079\pi\)
0.459359 + 0.888251i \(0.348079\pi\)
\(192\) 0 0
\(193\) 21.5959i 1.55451i 0.629187 + 0.777254i \(0.283388\pi\)
−0.629187 + 0.777254i \(0.716612\pi\)
\(194\) 0 0
\(195\) −24.2474 + 2.44949i −1.73640 + 0.175412i
\(196\) 0 0
\(197\) 18.8990i 1.34650i −0.739417 0.673248i \(-0.764899\pi\)
0.739417 0.673248i \(-0.235101\pi\)
\(198\) 0 0
\(199\) −16.8990 −1.19794 −0.598968 0.800773i \(-0.704423\pi\)
−0.598968 + 0.800773i \(0.704423\pi\)
\(200\) 0 0
\(201\) 19.5959 1.38219
\(202\) 0 0
\(203\) 6.89898i 0.484213i
\(204\) 0 0
\(205\) 2.44949 0.247449i 0.171080 0.0172826i
\(206\) 0 0
\(207\) 8.69694i 0.604479i
\(208\) 0 0
\(209\) 7.59592 0.525421
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 2.69694i 0.184791i
\(214\) 0 0
\(215\) 0.202041 + 2.00000i 0.0137791 + 0.136399i
\(216\) 0 0
\(217\) 8.89898i 0.604102i
\(218\) 0 0
\(219\) −7.10102 −0.479842
\(220\) 0 0
\(221\) −8.89898 −0.598610
\(222\) 0 0
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 0 0
\(225\) −14.6969 + 3.00000i −0.979796 + 0.200000i
\(226\) 0 0
\(227\) 7.34847i 0.487735i −0.969809 0.243868i \(-0.921584\pi\)
0.969809 0.243868i \(-0.0784162\pi\)
\(228\) 0 0
\(229\) −19.1464 −1.26523 −0.632616 0.774466i \(-0.718019\pi\)
−0.632616 + 0.774466i \(0.718019\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) 29.7980i 1.95213i −0.217481 0.976065i \(-0.569784\pi\)
0.217481 0.976065i \(-0.430216\pi\)
\(234\) 0 0
\(235\) −2.00000 19.7980i −0.130466 1.29148i
\(236\) 0 0
\(237\) 16.8990i 1.09771i
\(238\) 0 0
\(239\) 6.20204 0.401177 0.200588 0.979676i \(-0.435715\pi\)
0.200588 + 0.979676i \(0.435715\pi\)
\(240\) 0 0
\(241\) −8.69694 −0.560219 −0.280110 0.959968i \(-0.590371\pi\)
−0.280110 + 0.959968i \(0.590371\pi\)
\(242\) 0 0
\(243\) 22.0454i 1.41421i
\(244\) 0 0
\(245\) 2.22474 0.224745i 0.142134 0.0143584i
\(246\) 0 0
\(247\) 6.89898i 0.438972i
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 6.44949 0.407088 0.203544 0.979066i \(-0.434754\pi\)
0.203544 + 0.979066i \(0.434754\pi\)
\(252\) 0 0
\(253\) 14.2020i 0.892875i
\(254\) 0 0
\(255\) −10.8990 + 1.10102i −0.682521 + 0.0689486i
\(256\) 0 0
\(257\) 8.69694i 0.542500i −0.962509 0.271250i \(-0.912563\pi\)
0.962509 0.271250i \(-0.0874370\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −20.6969 −1.28111
\(262\) 0 0
\(263\) 9.79796i 0.604168i −0.953281 0.302084i \(-0.902318\pi\)
0.953281 0.302084i \(-0.0976823\pi\)
\(264\) 0 0
\(265\) 2.44949 + 24.2474i 0.150471 + 1.48951i
\(266\) 0 0
\(267\) 24.4949i 1.49906i
\(268\) 0 0
\(269\) 19.1464 1.16738 0.583689 0.811977i \(-0.301609\pi\)
0.583689 + 0.811977i \(0.301609\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 10.8990i 0.659636i
\(274\) 0 0
\(275\) 24.0000 4.89898i 1.44725 0.295420i
\(276\) 0 0
\(277\) 14.8990i 0.895193i 0.894236 + 0.447596i \(0.147720\pi\)
−0.894236 + 0.447596i \(0.852280\pi\)
\(278\) 0 0
\(279\) −26.6969 −1.59830
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 3.75255i 0.223066i 0.993761 + 0.111533i \(0.0355761\pi\)
−0.993761 + 0.111533i \(0.964424\pi\)
\(284\) 0 0
\(285\) 0.853572 + 8.44949i 0.0505612 + 0.500505i
\(286\) 0 0
\(287\) 1.10102i 0.0649912i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 38.6969 2.26845
\(292\) 0 0
\(293\) 18.2474i 1.06603i 0.846107 + 0.533014i \(0.178941\pi\)
−0.846107 + 0.533014i \(0.821059\pi\)
\(294\) 0 0
\(295\) 3.44949 0.348469i 0.200837 0.0202887i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.8990 0.745967
\(300\) 0 0
\(301\) −0.898979 −0.0518163
\(302\) 0 0
\(303\) 8.69694i 0.499626i
\(304\) 0 0
\(305\) 7.89898 0.797959i 0.452294 0.0456910i
\(306\) 0 0
\(307\) 20.2474i 1.15558i 0.816184 + 0.577791i \(0.196085\pi\)
−0.816184 + 0.577791i \(0.803915\pi\)
\(308\) 0 0
\(309\) −31.5959 −1.79743
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 21.5959i 1.22067i 0.792142 + 0.610337i \(0.208966\pi\)
−0.792142 + 0.610337i \(0.791034\pi\)
\(314\) 0 0
\(315\) −0.674235 6.67423i −0.0379888 0.376051i
\(316\) 0 0
\(317\) 22.4949i 1.26344i 0.775197 + 0.631720i \(0.217651\pi\)
−0.775197 + 0.631720i \(0.782349\pi\)
\(318\) 0 0
\(319\) 33.7980 1.89232
\(320\) 0 0
\(321\) 19.5959 1.09374
\(322\) 0 0
\(323\) 3.10102i 0.172545i
\(324\) 0 0
\(325\) 4.44949 + 21.7980i 0.246813 + 1.20913i
\(326\) 0 0
\(327\) 16.8990i 0.934516i
\(328\) 0 0
\(329\) 8.89898 0.490617
\(330\) 0 0
\(331\) 18.6969 1.02768 0.513838 0.857887i \(-0.328223\pi\)
0.513838 + 0.857887i \(0.328223\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) −1.79796 17.7980i −0.0982330 0.972406i
\(336\) 0 0
\(337\) 9.59592i 0.522723i 0.965241 + 0.261361i \(0.0841715\pi\)
−0.965241 + 0.261361i \(0.915829\pi\)
\(338\) 0 0
\(339\) −48.4949 −2.63388
\(340\) 0 0
\(341\) 43.5959 2.36085
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 15.7980 1.59592i 0.850534 0.0859213i
\(346\) 0 0
\(347\) 28.8990i 1.55138i −0.631115 0.775689i \(-0.717402\pi\)
0.631115 0.775689i \(-0.282598\pi\)
\(348\) 0 0
\(349\) −8.44949 −0.452291 −0.226145 0.974094i \(-0.572612\pi\)
−0.226145 + 0.974094i \(0.572612\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.8990i 1.21879i −0.792867 0.609395i \(-0.791412\pi\)
0.792867 0.609395i \(-0.208588\pi\)
\(354\) 0 0
\(355\) 2.44949 0.247449i 0.130005 0.0131332i
\(356\) 0 0
\(357\) 4.89898i 0.259281i
\(358\) 0 0
\(359\) 27.5959 1.45646 0.728228 0.685334i \(-0.240344\pi\)
0.728228 + 0.685334i \(0.240344\pi\)
\(360\) 0 0
\(361\) −16.5959 −0.873469
\(362\) 0 0
\(363\) 31.8434i 1.67134i
\(364\) 0 0
\(365\) 0.651531 + 6.44949i 0.0341027 + 0.337582i
\(366\) 0 0
\(367\) 32.0000i 1.67039i 0.549957 + 0.835193i \(0.314644\pi\)
−0.549957 + 0.835193i \(0.685356\pi\)
\(368\) 0 0
\(369\) 3.30306 0.171951
\(370\) 0 0
\(371\) −10.8990 −0.565847
\(372\) 0 0
\(373\) 4.69694i 0.243198i −0.992579 0.121599i \(-0.961198\pi\)
0.992579 0.121599i \(-0.0388022\pi\)
\(374\) 0 0
\(375\) 8.14643 + 26.1464i 0.420680 + 1.35020i
\(376\) 0 0
\(377\) 30.6969i 1.58097i
\(378\) 0 0
\(379\) 30.6969 1.57680 0.788398 0.615166i \(-0.210911\pi\)
0.788398 + 0.615166i \(0.210911\pi\)
\(380\) 0 0
\(381\) −36.4949 −1.86969
\(382\) 0 0
\(383\) 7.10102i 0.362845i 0.983405 + 0.181423i \(0.0580702\pi\)
−0.983405 + 0.181423i \(0.941930\pi\)
\(384\) 0 0
\(385\) 1.10102 + 10.8990i 0.0561132 + 0.555463i
\(386\) 0 0
\(387\) 2.69694i 0.137093i
\(388\) 0 0
\(389\) 13.1010 0.664248 0.332124 0.943236i \(-0.392235\pi\)
0.332124 + 0.943236i \(0.392235\pi\)
\(390\) 0 0
\(391\) 5.79796 0.293215
\(392\) 0 0
\(393\) 15.7980i 0.796902i
\(394\) 0 0
\(395\) 15.3485 1.55051i 0.772265 0.0780146i
\(396\) 0 0
\(397\) 2.65153i 0.133077i 0.997784 + 0.0665383i \(0.0211954\pi\)
−0.997784 + 0.0665383i \(0.978805\pi\)
\(398\) 0 0
\(399\) −3.79796 −0.190136
\(400\) 0 0
\(401\) −29.3939 −1.46786 −0.733930 0.679225i \(-0.762316\pi\)
−0.733930 + 0.679225i \(0.762316\pi\)
\(402\) 0 0
\(403\) 39.5959i 1.97241i
\(404\) 0 0
\(405\) 20.0227 2.02270i 0.994936 0.100509i
\(406\) 0 0
\(407\) 9.79796i 0.485667i
\(408\) 0 0
\(409\) −34.4949 −1.70566 −0.852831 0.522186i \(-0.825117\pi\)
−0.852831 + 0.522186i \(0.825117\pi\)
\(410\) 0 0
\(411\) −4.40408 −0.217237
\(412\) 0 0
\(413\) 1.55051i 0.0762956i
\(414\) 0 0
\(415\) 0.550510 + 5.44949i 0.0270235 + 0.267505i
\(416\) 0 0
\(417\) 3.79796i 0.185987i
\(418\) 0 0
\(419\) −1.55051 −0.0757474 −0.0378737 0.999283i \(-0.512058\pi\)
−0.0378737 + 0.999283i \(0.512058\pi\)
\(420\) 0 0
\(421\) 4.20204 0.204795 0.102397 0.994744i \(-0.467349\pi\)
0.102397 + 0.994744i \(0.467349\pi\)
\(422\) 0 0
\(423\) 26.6969i 1.29805i
\(424\) 0 0
\(425\) 2.00000 + 9.79796i 0.0970143 + 0.475271i
\(426\) 0 0
\(427\) 3.55051i 0.171821i
\(428\) 0 0
\(429\) 53.3939 2.57788
\(430\) 0 0
\(431\) −1.79796 −0.0866046 −0.0433023 0.999062i \(-0.513788\pi\)
−0.0433023 + 0.999062i \(0.513788\pi\)
\(432\) 0 0
\(433\) 0.202041i 0.00970947i 0.999988 + 0.00485474i \(0.00154532\pi\)
−0.999988 + 0.00485474i \(0.998455\pi\)
\(434\) 0 0
\(435\) 3.79796 + 37.5959i 0.182098 + 1.80259i
\(436\) 0 0
\(437\) 4.49490i 0.215020i
\(438\) 0 0
\(439\) 21.3939 1.02107 0.510537 0.859856i \(-0.329447\pi\)
0.510537 + 0.859856i \(0.329447\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 9.79796i 0.465515i 0.972535 + 0.232758i \(0.0747749\pi\)
−0.972535 + 0.232758i \(0.925225\pi\)
\(444\) 0 0
\(445\) −22.2474 + 2.24745i −1.05463 + 0.106539i
\(446\) 0 0
\(447\) 9.30306i 0.440020i
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −5.39388 −0.253988
\(452\) 0 0
\(453\) 48.0000i 2.25524i
\(454\) 0 0
\(455\) −9.89898 + 1.00000i −0.464071 + 0.0468807i
\(456\) 0 0
\(457\) 29.5959i 1.38444i 0.721687 + 0.692219i \(0.243367\pi\)
−0.721687 + 0.692219i \(0.756633\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −17.3485 −0.807999 −0.403999 0.914759i \(-0.632380\pi\)
−0.403999 + 0.914759i \(0.632380\pi\)
\(462\) 0 0
\(463\) 3.59592i 0.167116i −0.996503 0.0835582i \(-0.973372\pi\)
0.996503 0.0835582i \(-0.0266285\pi\)
\(464\) 0 0
\(465\) 4.89898 + 48.4949i 0.227185 + 2.24890i
\(466\) 0 0
\(467\) 10.4495i 0.483545i −0.970333 0.241772i \(-0.922271\pi\)
0.970333 0.241772i \(-0.0777287\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −8.69694 −0.400734
\(472\) 0 0
\(473\) 4.40408i 0.202500i
\(474\) 0 0
\(475\) 7.59592 1.55051i 0.348525 0.0711423i
\(476\) 0 0
\(477\) 32.6969i 1.49709i
\(478\) 0 0
\(479\) −9.30306 −0.425068 −0.212534 0.977154i \(-0.568172\pi\)
−0.212534 + 0.977154i \(0.568172\pi\)
\(480\) 0 0
\(481\) 8.89898 0.405759
\(482\) 0 0
\(483\) 7.10102i 0.323108i
\(484\) 0 0
\(485\) −3.55051 35.1464i −0.161220 1.59592i
\(486\) 0 0
\(487\) 7.30306i 0.330933i −0.986215 0.165467i \(-0.947087\pi\)
0.986215 0.165467i \(-0.0529130\pi\)
\(488\) 0 0
\(489\) −17.3939 −0.786578
\(490\) 0 0
\(491\) −19.5959 −0.884351 −0.442176 0.896928i \(-0.645793\pi\)
−0.442176 + 0.896928i \(0.645793\pi\)
\(492\) 0 0
\(493\) 13.7980i 0.621429i
\(494\) 0 0
\(495\) 32.6969 3.30306i 1.46962 0.148462i
\(496\) 0 0
\(497\) 1.10102i 0.0493875i
\(498\) 0 0
\(499\) 6.20204 0.277641 0.138821 0.990318i \(-0.455669\pi\)
0.138821 + 0.990318i \(0.455669\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 4.00000i 0.178351i 0.996016 + 0.0891756i \(0.0284232\pi\)
−0.996016 + 0.0891756i \(0.971577\pi\)
\(504\) 0 0
\(505\) 7.89898 0.797959i 0.351500 0.0355087i
\(506\) 0 0
\(507\) 16.6515i 0.739520i
\(508\) 0 0
\(509\) −31.5505 −1.39845 −0.699226 0.714901i \(-0.746472\pi\)
−0.699226 + 0.714901i \(0.746472\pi\)
\(510\) 0 0
\(511\) −2.89898 −0.128243
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.89898 + 28.6969i 0.127744 + 1.26454i
\(516\) 0 0
\(517\) 43.5959i 1.91735i
\(518\) 0 0
\(519\) −15.3031 −0.671730
\(520\) 0 0
\(521\) 32.6969 1.43248 0.716239 0.697855i \(-0.245862\pi\)
0.716239 + 0.697855i \(0.245862\pi\)
\(522\) 0 0
\(523\) 33.1464i 1.44939i −0.689069 0.724696i \(-0.741980\pi\)
0.689069 0.724696i \(-0.258020\pi\)
\(524\) 0 0
\(525\) −12.0000 + 2.44949i −0.523723 + 0.106904i
\(526\) 0 0
\(527\) 17.7980i 0.775291i
\(528\) 0 0
\(529\) 14.5959 0.634605
\(530\) 0 0
\(531\) 4.65153 0.201859
\(532\) 0 0
\(533\) 4.89898i 0.212198i
\(534\) 0 0
\(535\) −1.79796 17.7980i −0.0777325 0.769473i
\(536\) 0 0
\(537\) 33.7980i 1.45849i
\(538\) 0 0
\(539\) −4.89898 −0.211014
\(540\) 0 0
\(541\) −9.59592 −0.412561 −0.206280 0.978493i \(-0.566136\pi\)
−0.206280 + 0.978493i \(0.566136\pi\)
\(542\) 0 0
\(543\) 25.1010i 1.07719i
\(544\) 0 0
\(545\) −15.3485 + 1.55051i −0.657456 + 0.0664166i
\(546\) 0 0
\(547\) 18.6969i 0.799423i 0.916641 + 0.399712i \(0.130890\pi\)
−0.916641 + 0.399712i \(0.869110\pi\)
\(548\) 0 0
\(549\) 10.6515 0.454596
\(550\) 0 0
\(551\) 10.6969 0.455705
\(552\) 0 0
\(553\) 6.89898i 0.293374i
\(554\) 0 0
\(555\) 10.8990 1.10102i 0.462636 0.0467357i
\(556\) 0 0
\(557\) 12.6969i 0.537987i −0.963142 0.268993i \(-0.913309\pi\)
0.963142 0.268993i \(-0.0866909\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 30.0454i 1.26626i −0.774044 0.633131i \(-0.781769\pi\)
0.774044 0.633131i \(-0.218231\pi\)
\(564\) 0 0
\(565\) 4.44949 + 44.0454i 0.187191 + 1.85300i
\(566\) 0 0
\(567\) 9.00000i 0.377964i
\(568\) 0 0
\(569\) −33.7980 −1.41688 −0.708442 0.705769i \(-0.750602\pi\)
−0.708442 + 0.705769i \(0.750602\pi\)
\(570\) 0 0
\(571\) 11.1010 0.464563 0.232282 0.972649i \(-0.425381\pi\)
0.232282 + 0.972649i \(0.425381\pi\)
\(572\) 0 0
\(573\) 31.1010i 1.29926i
\(574\) 0 0
\(575\) −2.89898 14.2020i −0.120896 0.592266i
\(576\) 0 0
\(577\) 2.49490i 0.103864i −0.998651 0.0519320i \(-0.983462\pi\)
0.998651 0.0519320i \(-0.0165379\pi\)
\(578\) 0 0
\(579\) 52.8990 2.19841
\(580\) 0 0
\(581\) −2.44949 −0.101622
\(582\) 0 0
\(583\) 53.3939i 2.21135i
\(584\) 0 0
\(585\) 3.00000 + 29.6969i 0.124035 + 1.22782i
\(586\) 0 0
\(587\) 1.14643i 0.0473182i −0.999720 0.0236591i \(-0.992468\pi\)
0.999720 0.0236591i \(-0.00753162\pi\)
\(588\) 0 0
\(589\) 13.7980 0.568535
\(590\) 0 0
\(591\) −46.2929 −1.90423
\(592\) 0 0
\(593\) 10.8990i 0.447567i 0.974639 + 0.223784i \(0.0718409\pi\)
−0.974639 + 0.223784i \(0.928159\pi\)
\(594\) 0 0
\(595\) −4.44949 + 0.449490i −0.182411 + 0.0184273i
\(596\) 0 0
\(597\) 41.3939i 1.69414i
\(598\) 0 0
\(599\) 13.1010 0.535293 0.267647 0.963517i \(-0.413754\pi\)
0.267647 + 0.963517i \(0.413754\pi\)
\(600\) 0 0
\(601\) −39.3939 −1.60691 −0.803455 0.595366i \(-0.797007\pi\)
−0.803455 + 0.595366i \(0.797007\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) −28.9217 + 2.92168i −1.17583 + 0.118783i
\(606\) 0 0
\(607\) 33.3939i 1.35542i 0.735331 + 0.677708i \(0.237027\pi\)
−0.735331 + 0.677708i \(0.762973\pi\)
\(608\) 0 0
\(609\) −16.8990 −0.684781
\(610\) 0 0
\(611\) −39.5959 −1.60188
\(612\) 0 0
\(613\) 27.7980i 1.12275i −0.827562 0.561374i \(-0.810273\pi\)
0.827562 0.561374i \(-0.189727\pi\)
\(614\) 0 0
\(615\) −0.606123 6.00000i −0.0244412 0.241943i
\(616\) 0 0
\(617\) 29.5959i 1.19149i 0.803175 + 0.595743i \(0.203142\pi\)
−0.803175 + 0.595743i \(0.796858\pi\)
\(618\) 0 0
\(619\) −41.5505 −1.67006 −0.835028 0.550207i \(-0.814549\pi\)
−0.835028 + 0.550207i \(0.814549\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 23.0000 9.79796i 0.920000 0.391918i
\(626\) 0 0
\(627\) 18.6061i 0.743057i
\(628\) 0 0
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −42.4949 −1.69170 −0.845848 0.533425i \(-0.820905\pi\)
−0.845848 + 0.533425i \(0.820905\pi\)
\(632\) 0 0
\(633\) 29.3939i 1.16830i
\(634\) 0 0
\(635\) 3.34847 + 33.1464i 0.132880 + 1.31538i
\(636\) 0 0
\(637\) 4.44949i 0.176295i
\(638\) 0 0
\(639\) 3.30306 0.130667
\(640\) 0 0
\(641\) 25.7980 1.01896 0.509479 0.860483i \(-0.329838\pi\)
0.509479 + 0.860483i \(0.329838\pi\)
\(642\) 0 0
\(643\) 25.1464i 0.991678i 0.868414 + 0.495839i \(0.165139\pi\)
−0.868414 + 0.495839i \(0.834861\pi\)
\(644\) 0 0
\(645\) 4.89898 0.494897i 0.192897 0.0194866i
\(646\) 0 0
\(647\) 46.2929i 1.81996i −0.414652 0.909980i \(-0.636097\pi\)
0.414652 0.909980i \(-0.363903\pi\)
\(648\) 0 0
\(649\) −7.59592 −0.298166
\(650\) 0 0
\(651\) −21.7980 −0.854329
\(652\) 0 0
\(653\) 20.2020i 0.790567i −0.918559 0.395283i \(-0.870646\pi\)
0.918559 0.395283i \(-0.129354\pi\)
\(654\) 0 0
\(655\) −14.3485 + 1.44949i −0.560641 + 0.0566363i
\(656\) 0 0
\(657\) 8.69694i 0.339300i
\(658\) 0 0
\(659\) 16.8990 0.658291 0.329145 0.944279i \(-0.393239\pi\)
0.329145 + 0.944279i \(0.393239\pi\)
\(660\) 0 0
\(661\) 40.9444 1.59255 0.796276 0.604933i \(-0.206800\pi\)
0.796276 + 0.604933i \(0.206800\pi\)
\(662\) 0 0
\(663\) 21.7980i 0.846563i
\(664\) 0 0
\(665\) 0.348469 + 3.44949i 0.0135131 + 0.133765i
\(666\) 0 0
\(667\) 20.0000i 0.774403i
\(668\) 0 0
\(669\) 9.79796 0.378811
\(670\) 0 0
\(671\) −17.3939 −0.671483
\(672\) 0 0
\(673\) 17.7980i 0.686061i 0.939324 + 0.343030i \(0.111453\pi\)
−0.939324 + 0.343030i \(0.888547\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 36.4495i 1.40087i 0.713717 + 0.700434i \(0.247010\pi\)
−0.713717 + 0.700434i \(0.752990\pi\)
\(678\) 0 0
\(679\) 15.7980 0.606270
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) 3.59592i 0.137594i 0.997631 + 0.0687970i \(0.0219161\pi\)
−0.997631 + 0.0687970i \(0.978084\pi\)
\(684\) 0 0
\(685\) 0.404082 + 4.00000i 0.0154392 + 0.152832i
\(686\) 0 0
\(687\) 46.8990i 1.78931i
\(688\) 0 0
\(689\) 48.4949 1.84751
\(690\) 0 0
\(691\) −21.1464 −0.804448 −0.402224 0.915541i \(-0.631763\pi\)
−0.402224 + 0.915541i \(0.631763\pi\)
\(692\) 0 0
\(693\) 14.6969i 0.558291i
\(694\) 0 0
\(695\) −3.44949 + 0.348469i −0.130847 + 0.0132182i
\(696\) 0 0
\(697\) 2.20204i 0.0834083i
\(698\) 0 0
\(699\) −72.9898 −2.76073
\(700\) 0 0
\(701\) −11.3031 −0.426911 −0.213455 0.976953i \(-0.568472\pi\)
−0.213455 + 0.976953i \(0.568472\pi\)
\(702\) 0 0
\(703\) 3.10102i 0.116957i
\(704\) 0 0
\(705\) −48.4949 + 4.89898i −1.82642 + 0.184506i
\(706\) 0 0
\(707\) 3.55051i 0.133531i
\(708\) 0 0
\(709\) −28.2929 −1.06256 −0.531280 0.847196i \(-0.678289\pi\)
−0.531280 + 0.847196i \(0.678289\pi\)
\(710\) 0 0
\(711\) 20.6969 0.776196
\(712\) 0 0
\(713\) 25.7980i 0.966141i
\(714\) 0 0
\(715\) −4.89898 48.4949i −0.183211 1.81361i
\(716\) 0 0
\(717\) 15.1918i 0.567350i
\(718\) 0 0
\(719\) 4.49490 0.167631 0.0838157 0.996481i \(-0.473289\pi\)
0.0838157 + 0.996481i \(0.473289\pi\)
\(720\) 0 0
\(721\) −12.8990 −0.480383
\(722\) 0 0
\(723\) 21.3031i 0.792269i
\(724\) 0 0
\(725\) 33.7980 6.89898i 1.25522 0.256222i
\(726\) 0 0
\(727\) 22.6969i 0.841783i 0.907111 + 0.420891i \(0.138283\pi\)
−0.907111 + 0.420891i \(0.861717\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 1.79796 0.0664999
\(732\) 0 0
\(733\) 39.6413i 1.46419i 0.681205 + 0.732093i \(0.261456\pi\)
−0.681205 + 0.732093i \(0.738544\pi\)
\(734\) 0 0
\(735\) −0.550510 5.44949i −0.0203059 0.201007i
\(736\) 0 0
\(737\) 39.1918i 1.44365i
\(738\) 0 0
\(739\) 4.49490 0.165347 0.0826737 0.996577i \(-0.473654\pi\)
0.0826737 + 0.996577i \(0.473654\pi\)
\(740\) 0 0
\(741\) 16.8990 0.620800
\(742\) 0 0
\(743\) 44.6969i 1.63977i 0.572527 + 0.819886i \(0.305963\pi\)
−0.572527 + 0.819886i \(0.694037\pi\)
\(744\) 0 0
\(745\) 8.44949 0.853572i 0.309565 0.0312725i
\(746\) 0 0
\(747\) 7.34847i 0.268866i
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −41.7980 −1.52523 −0.762615 0.646853i \(-0.776085\pi\)
−0.762615 + 0.646853i \(0.776085\pi\)
\(752\) 0 0
\(753\) 15.7980i 0.575710i
\(754\) 0 0
\(755\) −43.5959 + 4.40408i −1.58662 + 0.160281i
\(756\) 0 0
\(757\) 51.7980i 1.88263i 0.337531 + 0.941314i \(0.390408\pi\)
−0.337531 + 0.941314i \(0.609592\pi\)
\(758\) 0 0
\(759\) −34.7878 −1.26272
\(760\) 0 0
\(761\) −21.1010 −0.764911 −0.382456 0.923974i \(-0.624922\pi\)
−0.382456 + 0.923974i \(0.624922\pi\)
\(762\) 0 0
\(763\) 6.89898i 0.249760i
\(764\) 0 0
\(765\) 1.34847 + 13.3485i 0.0487540 + 0.482615i
\(766\) 0 0
\(767\) 6.89898i 0.249108i
\(768\) 0 0
\(769\) 40.6969 1.46757 0.733785 0.679382i \(-0.237752\pi\)
0.733785 + 0.679382i \(0.237752\pi\)
\(770\) 0 0
\(771\) −21.3031 −0.767211
\(772\) 0 0
\(773\) 1.34847i 0.0485011i 0.999706 + 0.0242505i \(0.00771994\pi\)
−0.999706 + 0.0242505i \(0.992280\pi\)
\(774\) 0 0
\(775\) 43.5959 8.89898i 1.56601 0.319661i
\(776\) 0 0
\(777\) 4.89898i 0.175750i
\(778\) 0 0
\(779\) −1.70714 −0.0611648
\(780\) 0 0
\(781\) −5.39388 −0.193008
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.797959 + 7.89898i 0.0284804 + 0.281927i
\(786\) 0 0
\(787\) 50.4495i 1.79833i −0.437610 0.899165i \(-0.644175\pi\)
0.437610 0.899165i \(-0.355825\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −19.7980 −0.703934
\(792\) 0 0
\(793\) 15.7980i 0.561002i
\(794\) 0 0
\(795\) 59.3939 6.00000i 2.10648 0.212798i
\(796\) 0 0
\(797\) 0.944387i 0.0334519i 0.999860 + 0.0167260i \(0.00532429\pi\)
−0.999860 + 0.0167260i \(0.994676\pi\)
\(798\) 0 0
\(799\) −17.7980 −0.629647
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 14.2020i 0.501179i
\(804\) 0 0
\(805\) 6.44949 0.651531i 0.227315 0.0229634i
\(806\) 0 0
\(807\) 46.8990i 1.65092i
\(808\) 0 0
\(809\) −47.5959 −1.67338 −0.836692 0.547674i \(-0.815513\pi\)
−0.836692 + 0.547674i \(0.815513\pi\)
\(810\) 0 0
\(811\) −14.9444 −0.524768 −0.262384 0.964963i \(-0.584509\pi\)
−0.262384 + 0.964963i \(0.584509\pi\)
\(812\) 0 0
\(813\) 29.3939i 1.03089i
\(814\) 0 0
\(815\) 1.59592 + 15.7980i 0.0559026 + 0.553378i
\(816\) 0 0
\(817\) 1.39388i 0.0487656i
\(818\) 0 0
\(819\) −13.3485 −0.466433
\(820\) 0 0
\(821\) −8.20204 −0.286253 −0.143127 0.989704i \(-0.545716\pi\)
−0.143127 + 0.989704i \(0.545716\pi\)
\(822\) 0 0
\(823\) 39.1918i 1.36614i 0.730352 + 0.683071i \(0.239356\pi\)
−0.730352 + 0.683071i \(0.760644\pi\)
\(824\) 0 0
\(825\) −12.0000 58.7878i −0.417786 2.04673i
\(826\) 0 0
\(827\) 15.5959i 0.542323i 0.962534 + 0.271162i \(0.0874078\pi\)
−0.962534 + 0.271162i \(0.912592\pi\)
\(828\) 0 0
\(829\) −43.6413 −1.51573 −0.757863 0.652414i \(-0.773756\pi\)
−0.757863 + 0.652414i \(0.773756\pi\)
\(830\) 0 0
\(831\) 36.4949 1.26599
\(832\) 0 0
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) 1.10102 + 10.8990i 0.0381024 + 0.377175i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −36.8990 −1.27389 −0.636947 0.770907i \(-0.719803\pi\)
−0.636947 + 0.770907i \(0.719803\pi\)
\(840\) 0 0
\(841\) 18.5959 0.641239
\(842\) 0 0
\(843\) 44.0908i 1.51857i
\(844\) 0 0
\(845\) 15.1237 1.52781i 0.520272 0.0525581i
\(846\) 0 0
\(847\) 13.0000i 0.446685i
\(848\) 0 0
\(849\) 9.19184 0.315463
\(850\) 0 0
\(851\) −5.79796 −0.198751
\(852\) 0 0
\(853\) 33.8434i 1.15877i −0.815052 0.579387i \(-0.803292\pi\)
0.815052 0.579387i \(-0.196708\pi\)
\(854\) 0 0
\(855\) 10.3485 1.04541i 0.353910 0.0357522i
\(856\) 0 0
\(857\) 53.1918i 1.81700i −0.417886 0.908499i \(-0.637229\pi\)
0.417886 0.908499i \(-0.362771\pi\)
\(858\) 0 0
\(859\) 53.6413 1.83022 0.915109 0.403206i \(-0.132104\pi\)
0.915109 + 0.403206i \(0.132104\pi\)
\(860\) 0 0
\(861\) 2.69694 0.0919114
\(862\) 0 0
\(863\) 45.3939i 1.54523i 0.634878 + 0.772613i \(0.281051\pi\)
−0.634878 + 0.772613i \(0.718949\pi\)
\(864\) 0 0
\(865\) 1.40408 + 13.8990i 0.0477402 + 0.472579i
\(866\) 0 0
\(867\) 31.8434i 1.08146i
\(868\) 0 0
\(869\) −33.7980 −1.14652
\(870\) 0 0
\(871\) −35.5959 −1.20612
\(872\) 0 0
\(873\) 47.3939i 1.60404i
\(874\) 0 0
\(875\) 3.32577 + 10.6742i 0.112431 + 0.360855i
\(876\) 0 0
\(877\) 39.3939i 1.33024i 0.746738 + 0.665118i \(0.231619\pi\)
−0.746738 + 0.665118i \(0.768381\pi\)
\(878\) 0 0
\(879\) 44.6969 1.50759
\(880\) 0 0
\(881\) 8.20204 0.276334 0.138167 0.990409i \(-0.455879\pi\)
0.138167 + 0.990409i \(0.455879\pi\)
\(882\) 0 0
\(883\) 22.2020i 0.747158i 0.927598 + 0.373579i \(0.121870\pi\)
−0.927598 + 0.373579i \(0.878130\pi\)
\(884\) 0 0
\(885\) −0.853572 8.44949i −0.0286925 0.284026i
\(886\) 0 0
\(887\) 2.69694i 0.0905543i 0.998974 + 0.0452772i \(0.0144171\pi\)
−0.998974 + 0.0452772i \(0.985583\pi\)
\(888\) 0 0
\(889\) −14.8990 −0.499696
\(890\) 0 0
\(891\) −44.0908 −1.47710
\(892\) 0 0
\(893\) 13.7980i 0.461731i
\(894\) 0 0
\(895\) −30.6969 + 3.10102i −1.02609 + 0.103656i
\(896\) 0 0
\(897\) 31.5959i 1.05496i
\(898\) 0 0
\(899\) 61.3939 2.04760
\(900\) 0 0
\(901\) 21.7980 0.726195
\(902\) 0 0
\(903\) 2.20204i 0.0732793i
\(904\) 0 0
\(905\) −22.7980 + 2.30306i −0.757830 + 0.0765564i
\(906\) 0 0
\(907\) 41.7980i 1.38788i 0.720034 + 0.693939i \(0.244126\pi\)
−0.720034 + 0.693939i \(0.755874\pi\)
\(908\) 0 0
\(909\) 10.6515 0.353289
\(910\) 0 0
\(911\) −35.5959 −1.17935 −0.589673 0.807642i \(-0.700743\pi\)
−0.589673 + 0.807642i \(0.700743\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) 0 0
\(915\) −1.95459 19.3485i −0.0646168 0.639641i
\(916\) 0 0
\(917\) 6.44949i 0.212981i
\(918\) 0 0
\(919\) 26.8990 0.887315 0.443658 0.896196i \(-0.353681\pi\)
0.443658 + 0.896196i \(0.353681\pi\)
\(920\) 0 0
\(921\) 49.5959 1.63424
\(922\) 0 0
\(923\) 4.89898i 0.161252i
\(924\) 0 0
\(925\) −2.00000 9.79796i −0.0657596 0.322155i
\(926\) 0 0
\(927\) 38.6969i 1.27097i
\(928\) 0 0
\(929\) 28.2929 0.928259 0.464129 0.885767i \(-0.346367\pi\)
0.464129 + 0.885767i \(0.346367\pi\)
\(930\) 0 0
\(931\) −1.55051 −0.0508159
\(932\) 0 0
\(933\) 29.3939i 0.962312i
\(934\) 0 0
\(935\) −2.20204 21.7980i −0.0720144 0.712869i
\(936\) 0 0
\(937\) 41.1010i 1.34271i −0.741135 0.671356i \(-0.765712\pi\)
0.741135 0.671356i \(-0.234288\pi\)
\(938\) 0 0
\(939\) 52.8990 1.72629
\(940\) 0 0
\(941\) 19.5505 0.637328 0.318664 0.947868i \(-0.396766\pi\)
0.318664 + 0.947868i \(0.396766\pi\)
\(942\) 0 0
\(943\) 3.19184i 0.103940i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 44.0908i 1.43276i −0.697711 0.716379i \(-0.745798\pi\)
0.697711 0.716379i \(-0.254202\pi\)
\(948\) 0 0
\(949\) 12.8990 0.418719
\(950\) 0 0
\(951\) 55.1010 1.78677
\(952\) 0 0
\(953\) 2.20204i 0.0713311i −0.999364 0.0356656i \(-0.988645\pi\)
0.999364 0.0356656i \(-0.0113551\pi\)
\(954\) 0 0
\(955\) −28.2474 + 2.85357i −0.914066 + 0.0923394i
\(956\) 0 0
\(957\) 82.7878i 2.67615i
\(958\) 0 0
\(959\) −1.79796 −0.0580591
\(960\) 0 0
\(961\) 48.1918 1.55458
\(962\) 0 0
\(963\) 24.0000i 0.773389i
\(964\) 0 0
\(965\) −4.85357 48.0454i −0.156242 1.54664i
\(966\) 0 0
\(967\) 36.2929i 1.16710i −0.812077 0.583550i \(-0.801663\pi\)
0.812077 0.583550i \(-0.198337\pi\)
\(968\) 0 0
\(969\) 7.59592 0.244016
\(970\) 0 0
\(971\) 9.55051 0.306490 0.153245 0.988188i \(-0.451028\pi\)
0.153245 + 0.988188i \(0.451028\pi\)
\(972\) 0 0
\(973\) 1.55051i 0.0497071i
\(974\) 0 0
\(975\) 53.3939 10.8990i 1.70997 0.349047i
\(976\) 0 0
\(977\) 29.3939i 0.940393i −0.882562 0.470197i \(-0.844183\pi\)
0.882562 0.470197i \(-0.155817\pi\)
\(978\) 0 0
\(979\) 48.9898 1.56572
\(980\) 0 0
\(981\) −20.6969 −0.660802
\(982\) 0 0
\(983\) 13.3031i 0.424302i 0.977237 + 0.212151i \(0.0680468\pi\)
−0.977237 + 0.212151i \(0.931953\pi\)
\(984\) 0 0
\(985\) 4.24745 + 42.0454i 0.135335 + 1.33968i
\(986\) 0 0
\(987\) 21.7980i 0.693837i
\(988\) 0 0
\(989\) −2.60612 −0.0828699
\(990\) 0 0
\(991\) 31.3031 0.994375 0.497187 0.867643i \(-0.334366\pi\)
0.497187 + 0.867643i \(0.334366\pi\)
\(992\) 0 0
\(993\) 45.7980i 1.45335i
\(994\) 0 0
\(995\) 37.5959 3.79796i 1.19187 0.120403i
\(996\) 0 0
\(997\) 57.3485i 1.81624i −0.418705 0.908122i \(-0.637516\pi\)
0.418705 0.908122i \(-0.362484\pi\)
\(998\) 0 0
\(999\) 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 2240.2.g.j.449.1 4
4.3 odd 2 2240.2.g.i.449.3 4
5.4 even 2 inner 2240.2.g.j.449.3 4
8.3 odd 2 560.2.g.e.449.2 4
8.5 even 2 70.2.c.a.29.2 4
20.19 odd 2 2240.2.g.i.449.1 4
24.5 odd 2 630.2.g.g.379.3 4
24.11 even 2 5040.2.t.t.1009.2 4
40.3 even 4 2800.2.a.bm.1.2 2
40.13 odd 4 350.2.a.g.1.1 2
40.19 odd 2 560.2.g.e.449.4 4
40.27 even 4 2800.2.a.bl.1.1 2
40.29 even 2 70.2.c.a.29.3 yes 4
40.37 odd 4 350.2.a.h.1.2 2
56.5 odd 6 490.2.i.f.459.2 8
56.13 odd 2 490.2.c.e.99.1 4
56.37 even 6 490.2.i.c.459.1 8
56.45 odd 6 490.2.i.f.79.3 8
56.53 even 6 490.2.i.c.79.4 8
120.29 odd 2 630.2.g.g.379.1 4
120.53 even 4 3150.2.a.bt.1.2 2
120.59 even 2 5040.2.t.t.1009.1 4
120.77 even 4 3150.2.a.bs.1.2 2
280.13 even 4 2450.2.a.bl.1.2 2
280.69 odd 2 490.2.c.e.99.4 4
280.109 even 6 490.2.i.c.79.1 8
280.149 even 6 490.2.i.c.459.4 8
280.229 odd 6 490.2.i.f.459.3 8
280.237 even 4 2450.2.a.bq.1.1 2
280.269 odd 6 490.2.i.f.79.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.c.a.29.2 4 8.5 even 2
70.2.c.a.29.3 yes 4 40.29 even 2
350.2.a.g.1.1 2 40.13 odd 4
350.2.a.h.1.2 2 40.37 odd 4
490.2.c.e.99.1 4 56.13 odd 2
490.2.c.e.99.4 4 280.69 odd 2
490.2.i.c.79.1 8 280.109 even 6
490.2.i.c.79.4 8 56.53 even 6
490.2.i.c.459.1 8 56.37 even 6
490.2.i.c.459.4 8 280.149 even 6
490.2.i.f.79.2 8 280.269 odd 6
490.2.i.f.79.3 8 56.45 odd 6
490.2.i.f.459.2 8 56.5 odd 6
490.2.i.f.459.3 8 280.229 odd 6
560.2.g.e.449.2 4 8.3 odd 2
560.2.g.e.449.4 4 40.19 odd 2
630.2.g.g.379.1 4 120.29 odd 2
630.2.g.g.379.3 4 24.5 odd 2
2240.2.g.i.449.1 4 20.19 odd 2
2240.2.g.i.449.3 4 4.3 odd 2
2240.2.g.j.449.1 4 1.1 even 1 trivial
2240.2.g.j.449.3 4 5.4 even 2 inner
2450.2.a.bl.1.2 2 280.13 even 4
2450.2.a.bq.1.1 2 280.237 even 4
2800.2.a.bl.1.1 2 40.27 even 4
2800.2.a.bm.1.2 2 40.3 even 4
3150.2.a.bs.1.2 2 120.77 even 4
3150.2.a.bt.1.2 2 120.53 even 4
5040.2.t.t.1009.1 4 120.59 even 2
5040.2.t.t.1009.2 4 24.11 even 2