Properties

Label 630.2.g.g.379.1
Level $630$
Weight $2$
Character 630.379
Analytic conductor $5.031$
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] = [630,2,Mod(379,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("630.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.03057532734\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 379.1
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 630.379
Dual form 630.2.g.g.379.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-2.22474 - 0.224745i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-2.22474 - 0.224745i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +(-0.224745 + 2.22474i) q^{10} +4.89898 q^{11} +4.44949i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.55051 q^{19} +(2.22474 + 0.224745i) q^{20} -4.89898i q^{22} -2.89898i q^{23} +(4.89898 + 1.00000i) q^{25} +4.44949 q^{26} -1.00000i q^{28} +6.89898 q^{29} +8.89898 q^{31} -1.00000i q^{32} +2.00000 q^{34} +(0.224745 - 2.22474i) q^{35} -2.00000i q^{37} +1.55051i q^{38} +(0.224745 - 2.22474i) q^{40} +1.10102 q^{41} -0.898979i q^{43} -4.89898 q^{44} -2.89898 q^{46} +8.89898i q^{47} -1.00000 q^{49} +(1.00000 - 4.89898i) q^{50} -4.44949i q^{52} +10.8990i q^{53} +(-10.8990 - 1.10102i) q^{55} -1.00000 q^{56} -6.89898i q^{58} -1.55051 q^{59} +3.55051 q^{61} -8.89898i q^{62} -1.00000 q^{64} +(1.00000 - 9.89898i) q^{65} +8.00000i q^{67} -2.00000i q^{68} +(-2.22474 - 0.224745i) q^{70} +1.10102 q^{71} +2.89898i q^{73} -2.00000 q^{74} +1.55051 q^{76} +4.89898i q^{77} -6.89898 q^{79} +(-2.22474 - 0.224745i) q^{80} -1.10102i q^{82} +2.44949i q^{83} +(0.449490 - 4.44949i) q^{85} -0.898979 q^{86} +4.89898i q^{88} -10.0000 q^{89} -4.44949 q^{91} +2.89898i q^{92} +8.89898 q^{94} +(3.44949 + 0.348469i) q^{95} -15.7980i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{5} + 4 q^{10} + 4 q^{14} + 4 q^{16} - 16 q^{19} + 4 q^{20} + 8 q^{26} + 8 q^{29} + 16 q^{31} + 8 q^{34} - 4 q^{35} - 4 q^{40} + 24 q^{41} + 8 q^{46} - 4 q^{49} + 4 q^{50} - 24 q^{55} - 4 q^{56} - 16 q^{59} + 24 q^{61} - 4 q^{64} + 4 q^{65} - 4 q^{70} + 24 q^{71} - 8 q^{74} + 16 q^{76} - 8 q^{79} - 4 q^{80} - 8 q^{85} + 16 q^{86} - 40 q^{89} - 8 q^{91} + 16 q^{94} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −2.22474 0.224745i −0.994936 0.100509i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −0.224745 + 2.22474i −0.0710706 + 0.703526i
\(11\) 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 \(0.788334\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 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 \(-0.556916\pi\)
−0.177856 + 0.984057i \(0.556916\pi\)
\(20\) 2.22474 + 0.224745i 0.497468 + 0.0502545i
\(21\) 0 0
\(22\) 4.89898i 1.04447i
\(23\) 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
\(26\) 4.44949 0.872617
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) 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 \(0.205282\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 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 \(-0.0525685\pi\)
\(38\) 1.55051i 0.251526i
\(39\) 0 0
\(40\) 0.224745 2.22474i 0.0355353 0.351763i
\(41\) 1.10102 0.171951 0.0859753 0.996297i \(-0.472599\pi\)
0.0859753 + 0.996297i \(0.472599\pi\)
\(42\) 0 0
\(43\) 0.898979i 0.137093i −0.997648 0.0685465i \(-0.978164\pi\)
0.997648 0.0685465i \(-0.0218362\pi\)
\(44\) −4.89898 −0.738549
\(45\) 0 0
\(46\) −2.89898 −0.427431
\(47\) 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
\(50\) 1.00000 4.89898i 0.141421 0.692820i
\(51\) 0 0
\(52\) 4.44949i 0.617033i
\(53\) 10.8990i 1.49709i 0.663084 + 0.748545i \(0.269247\pi\)
−0.663084 + 0.748545i \(0.730753\pi\)
\(54\) 0 0
\(55\) −10.8990 1.10102i −1.46962 0.148462i
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.89898i 0.905880i
\(59\) −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 \(-0.427011\pi\)
0.227298 + 0.973825i \(0.427011\pi\)
\(62\) 8.89898i 1.13017i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 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 \(0.837479\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 0 0
\(70\) −2.22474 0.224745i −0.265908 0.0268622i
\(71\) 1.10102 0.130667 0.0653335 0.997863i \(-0.479189\pi\)
0.0653335 + 0.997863i \(0.479189\pi\)
\(72\) 0 0
\(73\) 2.89898i 0.339300i 0.985504 + 0.169650i \(0.0542637\pi\)
−0.985504 + 0.169650i \(0.945736\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 1.55051 0.177856
\(77\) 4.89898i 0.558291i
\(78\) 0 0
\(79\) −6.89898 −0.776196 −0.388098 0.921618i \(-0.626868\pi\)
−0.388098 + 0.921618i \(0.626868\pi\)
\(80\) −2.22474 0.224745i −0.248734 0.0251272i
\(81\) 0 0
\(82\) 1.10102i 0.121587i
\(83\) 2.44949i 0.268866i 0.990923 + 0.134433i \(0.0429214\pi\)
−0.990923 + 0.134433i \(0.957079\pi\)
\(84\) 0 0
\(85\) 0.449490 4.44949i 0.0487540 0.482615i
\(86\) −0.898979 −0.0969395
\(87\) 0 0
\(88\) 4.89898i 0.522233i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −4.44949 −0.466433
\(92\) 2.89898i 0.302240i
\(93\) 0 0
\(94\) 8.89898 0.917860
\(95\) 3.44949 + 0.348469i 0.353910 + 0.0357522i
\(96\) 0 0
\(97\) 15.7980i 1.60404i −0.597297 0.802020i \(-0.703759\pi\)
0.597297 0.802020i \(-0.296241\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) −4.89898 1.00000i −0.489898 0.100000i
\(101\) −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 \(0.219201\pi\)
−0.772111 + 0.635487i \(0.780799\pi\)
\(104\) −4.44949 −0.436308
\(105\) 0 0
\(106\) 10.8990 1.05860
\(107\) 8.00000i 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) −6.89898 −0.660802 −0.330401 0.943841i \(-0.607184\pi\)
−0.330401 + 0.943841i \(0.607184\pi\)
\(110\) −1.10102 + 10.8990i −0.104978 + 1.03918i
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) 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
\(116\) −6.89898 −0.640554
\(117\) 0 0
\(118\) 1.55051i 0.142736i
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 3.55051i 0.321448i
\(123\) 0 0
\(124\) −8.89898 −0.799152
\(125\) −10.6742 3.32577i −0.954733 0.297465i
\(126\) 0 0
\(127\) 14.8990i 1.32207i 0.750355 + 0.661035i \(0.229883\pi\)
−0.750355 + 0.661035i \(0.770117\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −9.89898 1.00000i −0.868198 0.0877058i
\(131\) 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
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 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 \(-0.520946\pi\)
−0.0657563 + 0.997836i \(0.520946\pi\)
\(140\) −0.224745 + 2.22474i −0.0189944 + 0.188025i
\(141\) 0 0
\(142\) 1.10102i 0.0923956i
\(143\) 21.7980i 1.82284i
\(144\) 0 0
\(145\) −15.3485 1.55051i −1.27462 0.128763i
\(146\) 2.89898 0.239921
\(147\) 0 0
\(148\) 2.00000i 0.164399i
\(149\) −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 \(0.206236\pi\)
\(152\) 1.55051i 0.125763i
\(153\) 0 0
\(154\) 4.89898 0.394771
\(155\) −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 \(-0.0452507\pi\)
\(158\) 6.89898i 0.548853i
\(159\) 0 0
\(160\) −0.224745 + 2.22474i −0.0177676 + 0.175882i
\(161\) 2.89898 0.228472
\(162\) 0 0
\(163\) 7.10102i 0.556195i −0.960553 0.278097i \(-0.910296\pi\)
0.960553 0.278097i \(-0.0897038\pi\)
\(164\) −1.10102 −0.0859753
\(165\) 0 0
\(166\) 2.44949 0.190117
\(167\) 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
\(170\) −4.44949 0.449490i −0.341260 0.0344743i
\(171\) 0 0
\(172\) 0.898979i 0.0685465i
\(173\) 6.24745i 0.474985i 0.971389 + 0.237492i \(0.0763255\pi\)
−0.971389 + 0.237492i \(0.923675\pi\)
\(174\) 0 0
\(175\) −1.00000 + 4.89898i −0.0755929 + 0.370328i
\(176\) 4.89898 0.369274
\(177\) 0 0
\(178\) 10.0000i 0.749532i
\(179\) 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 \(-0.624366\pi\)
−0.380843 + 0.924640i \(0.624366\pi\)
\(182\) 4.44949i 0.329818i
\(183\) 0 0
\(184\) 2.89898 0.213716
\(185\) −0.449490 + 4.44949i −0.0330471 + 0.327133i
\(186\) 0 0
\(187\) 9.79796i 0.716498i
\(188\) 8.89898i 0.649025i
\(189\) 0 0
\(190\) 0.348469 3.44949i 0.0252806 0.250252i
\(191\) −12.6969 −0.918718 −0.459359 0.888251i \(-0.651921\pi\)
−0.459359 + 0.888251i \(0.651921\pi\)
\(192\) 0 0
\(193\) 21.5959i 1.55451i −0.629187 0.777254i \(-0.716612\pi\)
0.629187 0.777254i \(-0.283388\pi\)
\(194\) −15.7980 −1.13423
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.8990i 1.34650i 0.739417 + 0.673248i \(0.235101\pi\)
−0.739417 + 0.673248i \(0.764899\pi\)
\(198\) 0 0
\(199\) −16.8990 −1.19794 −0.598968 0.800773i \(-0.704423\pi\)
−0.598968 + 0.800773i \(0.704423\pi\)
\(200\) −1.00000 + 4.89898i −0.0707107 + 0.346410i
\(201\) 0 0
\(202\) 3.55051i 0.249813i
\(203\) 6.89898i 0.484213i
\(204\) 0 0
\(205\) −2.44949 0.247449i −0.171080 0.0172826i
\(206\) 12.8990 0.898714
\(207\) 0 0
\(208\) 4.44949i 0.308517i
\(209\) −7.59592 −0.525421
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 10.8990i 0.748545i
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −0.202041 + 2.00000i −0.0137791 + 0.136399i
\(216\) 0 0
\(217\) 8.89898i 0.604102i
\(218\) 6.89898i 0.467258i
\(219\) 0 0
\(220\) 10.8990 + 1.10102i 0.734809 + 0.0742308i
\(221\) −8.89898 −0.598610
\(222\) 0 0
\(223\) 4.00000i 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −19.7980 −1.31694
\(227\) 7.34847i 0.487735i 0.969809 + 0.243868i \(0.0784162\pi\)
−0.969809 + 0.243868i \(0.921584\pi\)
\(228\) 0 0
\(229\) 19.1464 1.26523 0.632616 0.774466i \(-0.281981\pi\)
0.632616 + 0.774466i \(0.281981\pi\)
\(230\) 6.44949 + 0.651531i 0.425267 + 0.0429607i
\(231\) 0 0
\(232\) 6.89898i 0.452940i
\(233\) 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
\(236\) 1.55051 0.100930
\(237\) 0 0
\(238\) 2.00000i 0.129641i
\(239\) −6.20204 −0.401177 −0.200588 0.979676i \(-0.564285\pi\)
−0.200588 + 0.979676i \(0.564285\pi\)
\(240\) 0 0
\(241\) −8.69694 −0.560219 −0.280110 0.959968i \(-0.590371\pi\)
−0.280110 + 0.959968i \(0.590371\pi\)
\(242\) 13.0000i 0.835672i
\(243\) 0 0
\(244\) −3.55051 −0.227298
\(245\) 2.22474 + 0.224745i 0.142134 + 0.0143584i
\(246\) 0 0
\(247\) 6.89898i 0.438972i
\(248\) 8.89898i 0.565086i
\(249\) 0 0
\(250\) −3.32577 + 10.6742i −0.210340 + 0.675098i
\(251\) 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
\(254\) 14.8990 0.934845
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 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
\(260\) −1.00000 + 9.89898i −0.0620174 + 0.613909i
\(261\) 0 0
\(262\) 6.44949i 0.398451i
\(263\) 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
\(266\) −1.55051 −0.0950679
\(267\) 0 0
\(268\) 8.00000i 0.488678i
\(269\) 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 \(0.381249\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) −1.79796 −0.108619
\(275\) 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 \(0.852280\pi\)
\(278\) 1.55051i 0.0929934i
\(279\) 0 0
\(280\) 2.22474 + 0.224745i 0.132954 + 0.0134311i
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 3.75255i 0.223066i 0.993761 + 0.111533i \(0.0355761\pi\)
−0.993761 + 0.111533i \(0.964424\pi\)
\(284\) −1.10102 −0.0653335
\(285\) 0 0
\(286\) 21.7980 1.28894
\(287\) 1.10102i 0.0649912i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) −1.55051 + 15.3485i −0.0910491 + 0.901293i
\(291\) 0 0
\(292\) 2.89898i 0.169650i
\(293\) 18.2474i 1.06603i −0.846107 0.533014i \(-0.821059\pi\)
0.846107 0.533014i \(-0.178941\pi\)
\(294\) 0 0
\(295\) 3.44949 + 0.348469i 0.200837 + 0.0202887i
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 3.79796i 0.220010i
\(299\) 12.8990 0.745967
\(300\) 0 0
\(301\) 0.898979 0.0518163
\(302\) 19.5959i 1.12762i
\(303\) 0 0
\(304\) −1.55051 −0.0889279
\(305\) −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 \(0.803915\pi\)
\(308\) 4.89898i 0.279145i
\(309\) 0 0
\(310\) −2.00000 + 19.7980i −0.113592 + 1.12445i
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 21.5959i 1.22067i −0.792142 0.610337i \(-0.791034\pi\)
0.792142 0.610337i \(-0.208966\pi\)
\(314\) −3.55051 −0.200367
\(315\) 0 0
\(316\) 6.89898 0.388098
\(317\) 22.4949i 1.26344i −0.775197 0.631720i \(-0.782349\pi\)
0.775197 0.631720i \(-0.217651\pi\)
\(318\) 0 0
\(319\) 33.7980 1.89232
\(320\) 2.22474 + 0.224745i 0.124367 + 0.0125636i
\(321\) 0 0
\(322\) 2.89898i 0.161554i
\(323\) 3.10102i 0.172545i
\(324\) 0 0
\(325\) −4.44949 + 21.7980i −0.246813 + 1.20913i
\(326\) −7.10102 −0.393289
\(327\) 0 0
\(328\) 1.10102i 0.0607937i
\(329\) −8.89898 −0.490617
\(330\) 0 0
\(331\) −18.6969 −1.02768 −0.513838 0.857887i \(-0.671777\pi\)
−0.513838 + 0.857887i \(0.671777\pi\)
\(332\) 2.44949i 0.134433i
\(333\) 0 0
\(334\) −4.89898 −0.268060
\(335\) 1.79796 17.7980i 0.0982330 0.972406i
\(336\) 0 0
\(337\) 9.59592i 0.522723i −0.965241 0.261361i \(-0.915829\pi\)
0.965241 0.261361i \(-0.0841715\pi\)
\(338\) 6.79796i 0.369760i
\(339\) 0 0
\(340\) −0.449490 + 4.44949i −0.0243770 + 0.241307i
\(341\) 43.5959 2.36085
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0.898979 0.0484697
\(345\) 0 0
\(346\) 6.24745 0.335865
\(347\) 28.8990i 1.55138i 0.631115 + 0.775689i \(0.282598\pi\)
−0.631115 + 0.775689i \(0.717402\pi\)
\(348\) 0 0
\(349\) 8.44949 0.452291 0.226145 0.974094i \(-0.427388\pi\)
0.226145 + 0.974094i \(0.427388\pi\)
\(350\) 4.89898 + 1.00000i 0.261861 + 0.0534522i
\(351\) 0 0
\(352\) 4.89898i 0.261116i
\(353\) 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
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 13.7980i 0.729245i
\(359\) −27.5959 −1.45646 −0.728228 0.685334i \(-0.759656\pi\)
−0.728228 + 0.685334i \(0.759656\pi\)
\(360\) 0 0
\(361\) −16.5959 −0.873469
\(362\) 10.2474i 0.538594i
\(363\) 0 0
\(364\) 4.44949 0.233217
\(365\) 0.651531 6.44949i 0.0341027 0.337582i
\(366\) 0 0
\(367\) 32.0000i 1.67039i −0.549957 0.835193i \(-0.685356\pi\)
0.549957 0.835193i \(-0.314644\pi\)
\(368\) 2.89898i 0.151120i
\(369\) 0 0
\(370\) 4.44949 + 0.449490i 0.231318 + 0.0233679i
\(371\) −10.8990 −0.565847
\(372\) 0 0
\(373\) 4.69694i 0.243198i −0.992579 0.121599i \(-0.961198\pi\)
0.992579 0.121599i \(-0.0388022\pi\)
\(374\) 9.79796 0.506640
\(375\) 0 0
\(376\) −8.89898 −0.458930
\(377\) 30.6969i 1.58097i
\(378\) 0 0
\(379\) −30.6969 −1.57680 −0.788398 0.615166i \(-0.789089\pi\)
−0.788398 + 0.615166i \(0.789089\pi\)
\(380\) −3.44949 0.348469i −0.176955 0.0178761i
\(381\) 0 0
\(382\) 12.6969i 0.649632i
\(383\) 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
\(386\) −21.5959 −1.09920
\(387\) 0 0
\(388\) 15.7980i 0.802020i
\(389\) 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
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) 18.8990 0.952117
\(395\) 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 \(0.978805\pi\)
\(398\) 16.8990i 0.847069i
\(399\) 0 0
\(400\) 4.89898 + 1.00000i 0.244949 + 0.0500000i
\(401\) 29.3939 1.46786 0.733930 0.679225i \(-0.237684\pi\)
0.733930 + 0.679225i \(0.237684\pi\)
\(402\) 0 0
\(403\) 39.5959i 1.97241i
\(404\) 3.55051 0.176644
\(405\) 0 0
\(406\) 6.89898 0.342391
\(407\) 9.79796i 0.485667i
\(408\) 0 0
\(409\) −34.4949 −1.70566 −0.852831 0.522186i \(-0.825117\pi\)
−0.852831 + 0.522186i \(0.825117\pi\)
\(410\) −0.247449 + 2.44949i −0.0122206 + 0.120972i
\(411\) 0 0
\(412\) 12.8990i 0.635487i
\(413\) 1.55051i 0.0762956i
\(414\) 0 0
\(415\) 0.550510 5.44949i 0.0270235 0.267505i
\(416\) 4.44949 0.218154
\(417\) 0 0
\(418\) 7.59592i 0.371528i
\(419\) −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 \(-0.532651\pi\)
−0.102397 + 0.994744i \(0.532651\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 0 0
\(424\) −10.8990 −0.529301
\(425\) −2.00000 + 9.79796i −0.0970143 + 0.475271i
\(426\) 0 0
\(427\) 3.55051i 0.171821i
\(428\) 8.00000i 0.386695i
\(429\) 0 0
\(430\) 2.00000 + 0.202041i 0.0964486 + 0.00974328i
\(431\) 1.79796 0.0866046 0.0433023 0.999062i \(-0.486212\pi\)
0.0433023 + 0.999062i \(0.486212\pi\)
\(432\) 0 0
\(433\) 0.202041i 0.00970947i −0.999988 0.00485474i \(-0.998455\pi\)
0.999988 0.00485474i \(-0.00154532\pi\)
\(434\) 8.89898 0.427165
\(435\) 0 0
\(436\) 6.89898 0.330401
\(437\) 4.49490i 0.215020i
\(438\) 0 0
\(439\) 21.3939 1.02107 0.510537 0.859856i \(-0.329447\pi\)
0.510537 + 0.859856i \(0.329447\pi\)
\(440\) 1.10102 10.8990i 0.0524891 0.519588i
\(441\) 0 0
\(442\) 8.89898i 0.423281i
\(443\) 9.79796i 0.465515i −0.972535 0.232758i \(-0.925225\pi\)
0.972535 0.232758i \(-0.0747749\pi\)
\(444\) 0 0
\(445\) 22.2474 + 2.24745i 1.05463 + 0.106539i
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 5.39388 0.253988
\(452\) 19.7980i 0.931218i
\(453\) 0 0
\(454\) 7.34847 0.344881
\(455\) 9.89898 + 1.00000i 0.464071 + 0.0468807i
\(456\) 0 0
\(457\) 29.5959i 1.38444i −0.721687 0.692219i \(-0.756633\pi\)
0.721687 0.692219i \(-0.243367\pi\)
\(458\) 19.1464i 0.894654i
\(459\) 0 0
\(460\) 0.651531 6.44949i 0.0303778 0.300709i
\(461\) −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 \(0.0266285\pi\)
−0.996503 + 0.0835582i \(0.973372\pi\)
\(464\) 6.89898 0.320277
\(465\) 0 0
\(466\) −29.7980 −1.38036
\(467\) 10.4495i 0.483545i 0.970333 + 0.241772i \(0.0777287\pi\)
−0.970333 + 0.241772i \(0.922271\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −19.7980 2.00000i −0.913212 0.0922531i
\(471\) 0 0
\(472\) 1.55051i 0.0713680i
\(473\) 4.40408i 0.202500i
\(474\) 0 0
\(475\) −7.59592 1.55051i −0.348525 0.0711423i
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 6.20204i 0.283675i
\(479\) 9.30306 0.425068 0.212534 0.977154i \(-0.431828\pi\)
0.212534 + 0.977154i \(0.431828\pi\)
\(480\) 0 0
\(481\) 8.89898 0.405759
\(482\) 8.69694i 0.396135i
\(483\) 0 0
\(484\) −13.0000 −0.590909
\(485\) −3.55051 + 35.1464i −0.161220 + 1.59592i
\(486\) 0 0
\(487\) 7.30306i 0.330933i 0.986215 + 0.165467i \(0.0529130\pi\)
−0.986215 + 0.165467i \(0.947087\pi\)
\(488\) 3.55051i 0.160724i
\(489\) 0 0
\(490\) 0.224745 2.22474i 0.0101529 0.100504i
\(491\) −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
\(494\) −6.89898 −0.310400
\(495\) 0 0
\(496\) 8.89898 0.399576
\(497\) 1.10102i 0.0493875i
\(498\) 0 0
\(499\) −6.20204 −0.277641 −0.138821 0.990318i \(-0.544331\pi\)
−0.138821 + 0.990318i \(0.544331\pi\)
\(500\) 10.6742 + 3.32577i 0.477366 + 0.148733i
\(501\) 0 0
\(502\) 6.44949i 0.287855i
\(503\) 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
\(506\) −14.2020 −0.631358
\(507\) 0 0
\(508\) 14.8990i 0.661035i
\(509\) −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
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −8.69694 −0.383606
\(515\) 2.89898 28.6969i 0.127744 1.26454i
\(516\) 0 0
\(517\) 43.5959i 1.91735i
\(518\) 2.00000i 0.0878750i
\(519\) 0 0
\(520\) 9.89898 + 1.00000i 0.434099 + 0.0438529i
\(521\) −32.6969 −1.43248 −0.716239 0.697855i \(-0.754138\pi\)
−0.716239 + 0.697855i \(0.754138\pi\)
\(522\) 0 0
\(523\) 33.1464i 1.44939i −0.689069 0.724696i \(-0.741980\pi\)
0.689069 0.724696i \(-0.258020\pi\)
\(524\) −6.44949 −0.281747
\(525\) 0 0
\(526\) −9.79796 −0.427211
\(527\) 17.7980i 0.775291i
\(528\) 0 0
\(529\) 14.5959 0.634605
\(530\) −24.2474 2.44949i −1.05324 0.106399i
\(531\) 0 0
\(532\) 1.55051i 0.0672231i
\(533\) 4.89898i 0.212198i
\(534\) 0 0
\(535\) −1.79796 + 17.7980i −0.0777325 + 0.769473i
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 19.1464i 0.825461i
\(539\) −4.89898 −0.211014
\(540\) 0 0
\(541\) 9.59592 0.412561 0.206280 0.978493i \(-0.433864\pi\)
0.206280 + 0.978493i \(0.433864\pi\)
\(542\) 12.0000i 0.515444i
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) 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 \(0.869110\pi\)
\(548\) 1.79796i 0.0768050i
\(549\) 0 0
\(550\) 4.89898 24.0000i 0.208893 1.02336i
\(551\) −10.6969 −0.455705
\(552\) 0 0
\(553\) 6.89898i 0.293374i
\(554\) 14.8990 0.632997
\(555\) 0 0
\(556\) 1.55051 0.0657563
\(557\) 12.6969i 0.537987i 0.963142 + 0.268993i \(0.0866909\pi\)
−0.963142 + 0.268993i \(0.913309\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0.224745 2.22474i 0.00949720 0.0940126i
\(561\) 0 0
\(562\) 18.0000i 0.759284i
\(563\) 30.0454i 1.26626i 0.774044 + 0.633131i \(0.218231\pi\)
−0.774044 + 0.633131i \(0.781769\pi\)
\(564\) 0 0
\(565\) −4.44949 + 44.0454i −0.187191 + 1.85300i
\(566\) 3.75255 0.157731
\(567\) 0 0
\(568\) 1.10102i 0.0461978i
\(569\) 33.7980 1.41688 0.708442 0.705769i \(-0.249398\pi\)
0.708442 + 0.705769i \(0.249398\pi\)
\(570\) 0 0
\(571\) −11.1010 −0.464563 −0.232282 0.972649i \(-0.574619\pi\)
−0.232282 + 0.972649i \(0.574619\pi\)
\(572\) 21.7980i 0.911418i
\(573\) 0 0
\(574\) 1.10102 0.0459557
\(575\) 2.89898 14.2020i 0.120896 0.592266i
\(576\) 0 0
\(577\) 2.49490i 0.103864i 0.998651 + 0.0519320i \(0.0165379\pi\)
−0.998651 + 0.0519320i \(0.983462\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 0 0
\(580\) 15.3485 + 1.55051i 0.637310 + 0.0643814i
\(581\) −2.44949 −0.101622
\(582\) 0 0
\(583\) 53.3939i 2.21135i
\(584\) −2.89898 −0.119961
\(585\) 0 0
\(586\) −18.2474 −0.753795
\(587\) 1.14643i 0.0473182i 0.999720 + 0.0236591i \(0.00753162\pi\)
−0.999720 + 0.0236591i \(0.992468\pi\)
\(588\) 0 0
\(589\) −13.7980 −0.568535
\(590\) 0.348469 3.44949i 0.0143463 0.142013i
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) 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
\(596\) 3.79796 0.155570
\(597\) 0 0
\(598\) 12.8990i 0.527478i
\(599\) −13.1010 −0.535293 −0.267647 0.963517i \(-0.586246\pi\)
−0.267647 + 0.963517i \(0.586246\pi\)
\(600\) 0 0
\(601\) −39.3939 −1.60691 −0.803455 0.595366i \(-0.797007\pi\)
−0.803455 + 0.595366i \(0.797007\pi\)
\(602\) 0.898979i 0.0366397i
\(603\) 0 0
\(604\) −19.5959 −0.797347
\(605\) −28.9217 2.92168i −1.17583 0.118783i
\(606\) 0 0
\(607\) 33.3939i 1.35542i −0.735331 0.677708i \(-0.762973\pi\)
0.735331 0.677708i \(-0.237027\pi\)
\(608\) 1.55051i 0.0628815i
\(609\) 0 0
\(610\) −0.797959 + 7.89898i −0.0323084 + 0.319820i
\(611\) −39.5959 −1.60188
\(612\) 0 0
\(613\) 27.7980i 1.12275i −0.827562 0.561374i \(-0.810273\pi\)
0.827562 0.561374i \(-0.189727\pi\)
\(614\) 20.2474 0.817121
\(615\) 0 0
\(616\) −4.89898 −0.197386
\(617\) 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 \(-0.185451\pi\)
0.835028 + 0.550207i \(0.185451\pi\)
\(620\) 19.7980 + 2.00000i 0.795105 + 0.0803219i
\(621\) 0 0
\(622\) 12.0000i 0.481156i
\(623\) 10.0000i 0.400642i
\(624\) 0 0
\(625\) 23.0000 + 9.79796i 0.920000 + 0.391918i
\(626\) −21.5959 −0.863146
\(627\) 0 0
\(628\) 3.55051i 0.141681i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −42.4949 −1.69170 −0.845848 0.533425i \(-0.820905\pi\)
−0.845848 + 0.533425i \(0.820905\pi\)
\(632\) 6.89898i 0.274427i
\(633\) 0 0
\(634\) −22.4949 −0.893387
\(635\) 3.34847 33.1464i 0.132880 1.31538i
\(636\) 0 0
\(637\) 4.44949i 0.176295i
\(638\) 33.7980i 1.33807i
\(639\) 0 0
\(640\) 0.224745 2.22474i 0.00888382 0.0879408i
\(641\) −25.7980 −1.01896 −0.509479 0.860483i \(-0.670162\pi\)
−0.509479 + 0.860483i \(0.670162\pi\)
\(642\) 0 0
\(643\) 25.1464i 0.991678i 0.868414 + 0.495839i \(0.165139\pi\)
−0.868414 + 0.495839i \(0.834861\pi\)
\(644\) −2.89898 −0.114236
\(645\) 0 0
\(646\) −3.10102 −0.122008
\(647\) 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
\(650\) 21.7980 + 4.44949i 0.854986 + 0.174523i
\(651\) 0 0
\(652\) 7.10102i 0.278097i
\(653\) 20.2020i 0.790567i 0.918559 + 0.395283i \(0.129354\pi\)
−0.918559 + 0.395283i \(0.870646\pi\)
\(654\) 0 0
\(655\) −14.3485 1.44949i −0.560641 0.0566363i
\(656\) 1.10102 0.0429876
\(657\) 0 0
\(658\) 8.89898i 0.346918i
\(659\) 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 \(-0.793200\pi\)
−0.796276 + 0.604933i \(0.793200\pi\)
\(662\) 18.6969i 0.726677i
\(663\) 0 0
\(664\) −2.44949 −0.0950586
\(665\) −0.348469 + 3.44949i −0.0135131 + 0.133765i
\(666\) 0 0
\(667\) 20.0000i 0.774403i
\(668\) 4.89898i 0.189547i
\(669\) 0 0
\(670\) −17.7980 1.79796i −0.687595 0.0694612i
\(671\) 17.3939 0.671483
\(672\) 0 0
\(673\) 17.7980i 0.686061i −0.939324 0.343030i \(-0.888547\pi\)
0.939324 0.343030i \(-0.111453\pi\)
\(674\) −9.59592 −0.369621
\(675\) 0 0
\(676\) 6.79796 0.261460
\(677\) 36.4495i 1.40087i −0.713717 0.700434i \(-0.752990\pi\)
0.713717 0.700434i \(-0.247010\pi\)
\(678\) 0 0
\(679\) 15.7980 0.606270
\(680\) 4.44949 + 0.449490i 0.170630 + 0.0172371i
\(681\) 0 0
\(682\) 43.5959i 1.66937i
\(683\) 3.59592i 0.137594i −0.997631 0.0687970i \(-0.978084\pi\)
0.997631 0.0687970i \(-0.0219161\pi\)
\(684\) 0 0
\(685\) −0.404082 + 4.00000i −0.0154392 + 0.152832i
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 0.898979i 0.0342733i
\(689\) −48.4949 −1.84751
\(690\) 0 0
\(691\) 21.1464 0.804448 0.402224 0.915541i \(-0.368237\pi\)
0.402224 + 0.915541i \(0.368237\pi\)
\(692\) 6.24745i 0.237492i
\(693\) 0 0
\(694\) 28.8990 1.09699
\(695\) 3.44949 + 0.348469i 0.130847 + 0.0132182i
\(696\) 0 0
\(697\) 2.20204i 0.0834083i
\(698\) 8.44949i 0.319818i
\(699\) 0 0
\(700\) 1.00000 4.89898i 0.0377964 0.185164i
\(701\) −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
\(704\) −4.89898 −0.184637
\(705\) 0 0
\(706\) −22.8990 −0.861814
\(707\) 3.55051i 0.133531i
\(708\) 0 0
\(709\) 28.2929 1.06256 0.531280 0.847196i \(-0.321711\pi\)
0.531280 + 0.847196i \(0.321711\pi\)
\(710\) −0.247449 + 2.44949i −0.00928658 + 0.0919277i
\(711\) 0 0
\(712\) 10.0000i 0.374766i
\(713\) 25.7980i 0.966141i
\(714\) 0 0
\(715\) 4.89898 48.4949i 0.183211 1.81361i
\(716\) −13.7980 −0.515654
\(717\) 0 0
\(718\) 27.5959i 1.02987i
\(719\) −4.49490 −0.167631 −0.0838157 0.996481i \(-0.526711\pi\)
−0.0838157 + 0.996481i \(0.526711\pi\)
\(720\) 0 0
\(721\) −12.8990 −0.480383
\(722\) 16.5959i 0.617636i
\(723\) 0 0
\(724\) 10.2474 0.380843
\(725\) 33.7980 + 6.89898i 1.25522 + 0.256222i
\(726\) 0 0
\(727\) 22.6969i 0.841783i −0.907111 0.420891i \(-0.861717\pi\)
0.907111 0.420891i \(-0.138283\pi\)
\(728\) 4.44949i 0.164909i
\(729\) 0 0
\(730\) −6.44949 0.651531i −0.238706 0.0241142i
\(731\) 1.79796 0.0664999
\(732\) 0 0
\(733\) 39.6413i 1.46419i 0.681205 + 0.732093i \(0.261456\pi\)
−0.681205 + 0.732093i \(0.738544\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −2.89898 −0.106858
\(737\) 39.1918i 1.44365i
\(738\) 0 0
\(739\) −4.49490 −0.165347 −0.0826737 0.996577i \(-0.526346\pi\)
−0.0826737 + 0.996577i \(0.526346\pi\)
\(740\) 0.449490 4.44949i 0.0165236 0.163566i
\(741\) 0 0
\(742\) 10.8990i 0.400114i
\(743\) 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
\(746\) −4.69694 −0.171967
\(747\) 0 0
\(748\) 9.79796i 0.358249i
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −41.7980 −1.52523 −0.762615 0.646853i \(-0.776085\pi\)
−0.762615 + 0.646853i \(0.776085\pi\)
\(752\) 8.89898i 0.324512i
\(753\) 0 0
\(754\) 30.6969 1.11792
\(755\) −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 \(0.609592\pi\)
\(758\) 30.6969i 1.11496i
\(759\) 0 0
\(760\) −0.348469 + 3.44949i −0.0126403 + 0.125126i
\(761\) 21.1010 0.764911 0.382456 0.923974i \(-0.375078\pi\)
0.382456 + 0.923974i \(0.375078\pi\)
\(762\) 0 0
\(763\) 6.89898i 0.249760i
\(764\) 12.6969 0.459359
\(765\) 0 0
\(766\) 7.10102 0.256570
\(767\) 6.89898i 0.249108i
\(768\) 0 0
\(769\) 40.6969 1.46757 0.733785 0.679382i \(-0.237752\pi\)
0.733785 + 0.679382i \(0.237752\pi\)
\(770\) −10.8990 1.10102i −0.392772 0.0396780i
\(771\) 0 0
\(772\) 21.5959i 0.777254i
\(773\) 1.34847i 0.0485011i −0.999706 0.0242505i \(-0.992280\pi\)
0.999706 0.0242505i \(-0.00771994\pi\)
\(774\) 0 0
\(775\) 43.5959 + 8.89898i 1.56601 + 0.319661i
\(776\) 15.7980 0.567114
\(777\) 0 0
\(778\) 13.1010i 0.469694i
\(779\) −1.70714 −0.0611648
\(780\) 0 0
\(781\) 5.39388 0.193008
\(782\) 5.79796i 0.207335i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) −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 \(-0.355825\pi\)
\(788\) 18.8990i 0.673248i
\(789\) 0 0
\(790\) 1.55051 15.3485i 0.0551647 0.546074i
\(791\) 19.7980 0.703934
\(792\) 0 0
\(793\) 15.7980i 0.561002i
\(794\) 2.65153 0.0940993
\(795\) 0 0
\(796\) 16.8990 0.598968
\(797\) 0.944387i 0.0334519i −0.999860 0.0167260i \(-0.994676\pi\)
0.999860 0.0167260i \(-0.00532429\pi\)
\(798\) 0 0
\(799\) −17.7980 −0.629647
\(800\) 1.00000 4.89898i 0.0353553 0.173205i
\(801\) 0 0
\(802\) 29.3939i 1.03793i
\(803\) 14.2020i 0.501179i
\(804\) 0 0
\(805\) −6.44949 0.651531i −0.227315 0.0229634i
\(806\) 39.5959 1.39471
\(807\) 0 0
\(808\) 3.55051i 0.124907i
\(809\) 47.5959 1.67338 0.836692 0.547674i \(-0.184487\pi\)
0.836692 + 0.547674i \(0.184487\pi\)
\(810\) 0 0
\(811\) 14.9444 0.524768 0.262384 0.964963i \(-0.415491\pi\)
0.262384 + 0.964963i \(0.415491\pi\)
\(812\) 6.89898i 0.242107i
\(813\) 0 0
\(814\) −9.79796 −0.343418
\(815\) −1.59592 + 15.7980i −0.0559026 + 0.553378i
\(816\) 0 0
\(817\) 1.39388i 0.0487656i
\(818\) 34.4949i 1.20609i
\(819\) 0 0
\(820\) 2.44949 + 0.247449i 0.0855399 + 0.00864128i
\(821\) −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 \(-0.760644\pi\)
0.730352 0.683071i \(-0.239356\pi\)
\(824\) −12.8990 −0.449357
\(825\) 0 0
\(826\) −1.55051 −0.0539492
\(827\) 15.5959i 0.542323i −0.962534 0.271162i \(-0.912592\pi\)
0.962534 0.271162i \(-0.0874078\pi\)
\(828\) 0 0
\(829\) 43.6413 1.51573 0.757863 0.652414i \(-0.226244\pi\)
0.757863 + 0.652414i \(0.226244\pi\)
\(830\) −5.44949 0.550510i −0.189155 0.0191085i
\(831\) 0 0
\(832\) 4.44949i 0.154258i
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) −1.10102 + 10.8990i −0.0381024 + 0.377175i
\(836\) 7.59592 0.262710
\(837\) 0 0
\(838\) 1.55051i 0.0535615i
\(839\) 36.8990 1.27389 0.636947 0.770907i \(-0.280197\pi\)
0.636947 + 0.770907i \(0.280197\pi\)
\(840\) 0 0
\(841\) 18.5959 0.641239
\(842\) 4.20204i 0.144812i
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 15.1237 + 1.52781i 0.520272 + 0.0525581i
\(846\) 0 0
\(847\) 13.0000i 0.446685i
\(848\) 10.8990i 0.374272i
\(849\) 0 0
\(850\) 9.79796 + 2.00000i 0.336067 + 0.0685994i
\(851\) −5.79796 −0.198751
\(852\) 0 0
\(853\) 33.8434i 1.15877i −0.815052 0.579387i \(-0.803292\pi\)
0.815052 0.579387i \(-0.196708\pi\)
\(854\) 3.55051 0.121496
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) 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 \(-0.867896\pi\)
−0.915109 + 0.403206i \(0.867896\pi\)
\(860\) 0.202041 2.00000i 0.00688954 0.0681994i
\(861\) 0 0
\(862\) 1.79796i 0.0612387i
\(863\) 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
\(866\) −0.202041 −0.00686563
\(867\) 0 0
\(868\) 8.89898i 0.302051i
\(869\) −33.7980 −1.14652
\(870\) 0 0
\(871\) −35.5959 −1.20612
\(872\) 6.89898i 0.233629i
\(873\) 0 0
\(874\) 4.49490 0.152042
\(875\) 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 \(0.768381\pi\)
\(878\) 21.3939i 0.722008i
\(879\) 0 0
\(880\) −10.8990 1.10102i −0.367405 0.0371154i
\(881\) −8.20204 −0.276334 −0.138167 0.990409i \(-0.544121\pi\)
−0.138167 + 0.990409i \(0.544121\pi\)
\(882\) 0 0
\(883\) 22.2020i 0.747158i 0.927598 + 0.373579i \(0.121870\pi\)
−0.927598 + 0.373579i \(0.878130\pi\)
\(884\) 8.89898 0.299305
\(885\) 0 0
\(886\) −9.79796 −0.329169
\(887\) 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
\(890\) 2.24745 22.2474i 0.0753347 0.745736i
\(891\) 0 0
\(892\) 4.00000i 0.133930i
\(893\) 13.7980i 0.461731i
\(894\) 0 0
\(895\) −30.6969 3.10102i −1.02609 0.103656i
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 10.0000i 0.333704i
\(899\) 61.3939 2.04760
\(900\) 0 0
\(901\) −21.7980 −0.726195
\(902\) 5.39388i 0.179596i
\(903\) 0 0
\(904\) 19.7980 0.658470
\(905\) 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 \(0.755874\pi\)
\(908\) 7.34847i 0.243868i
\(909\) 0 0
\(910\) 1.00000 9.89898i 0.0331497 0.328148i
\(911\) 35.5959 1.17935 0.589673 0.807642i \(-0.299257\pi\)
0.589673 + 0.807642i \(0.299257\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) −29.5959 −0.978946
\(915\) 0 0
\(916\) −19.1464 −0.632616
\(917\) 6.44949i 0.212981i
\(918\) 0 0
\(919\) 26.8990 0.887315 0.443658 0.896196i \(-0.353681\pi\)
0.443658 + 0.896196i \(0.353681\pi\)
\(920\) −6.44949 0.651531i −0.212633 0.0214803i
\(921\) 0 0
\(922\) 17.3485i 0.571341i
\(923\) 4.89898i 0.161252i
\(924\) 0 0
\(925\) 2.00000 9.79796i 0.0657596 0.322155i
\(926\) 3.59592 0.118169
\(927\) 0 0
\(928\) 6.89898i 0.226470i
\(929\) −28.2929 −0.928259 −0.464129 0.885767i \(-0.653633\pi\)
−0.464129 + 0.885767i \(0.653633\pi\)
\(930\) 0 0
\(931\) 1.55051 0.0508159
\(932\) 29.7980i 0.976065i
\(933\) 0 0
\(934\) 10.4495 0.341918
\(935\) 2.20204 21.7980i 0.0720144 0.712869i
\(936\) 0 0
\(937\) 41.1010i 1.34271i 0.741135 + 0.671356i \(0.234288\pi\)
−0.741135 + 0.671356i \(0.765712\pi\)
\(938\) 8.00000i 0.261209i
\(939\) 0 0
\(940\) −2.00000 + 19.7980i −0.0652328 + 0.645738i
\(941\) 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
\(944\) −1.55051 −0.0504648
\(945\) 0 0
\(946\) −4.40408 −0.143189
\(947\) 44.0908i 1.43276i 0.697711 + 0.716379i \(0.254202\pi\)
−0.697711 + 0.716379i \(0.745798\pi\)
\(948\) 0 0
\(949\) −12.8990 −0.418719
\(950\) −1.55051 + 7.59592i −0.0503052 + 0.246444i
\(951\) 0 0
\(952\) 2.00000i 0.0648204i
\(953\) 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
\(956\) 6.20204 0.200588
\(957\) 0 0
\(958\) 9.30306i 0.300568i
\(959\) 1.79796 0.0580591
\(960\) 0 0
\(961\) 48.1918 1.55458
\(962\) 8.89898i 0.286915i
\(963\) 0 0
\(964\) 8.69694 0.280110
\(965\) −4.85357 + 48.0454i −0.156242 + 1.54664i
\(966\) 0 0
\(967\) 36.2929i 1.16710i 0.812077 + 0.583550i \(0.198337\pi\)
−0.812077 + 0.583550i \(0.801663\pi\)
\(968\) 13.0000i 0.417836i
\(969\) 0 0
\(970\) 35.1464 + 3.55051i 1.12848 + 0.114000i
\(971\) 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
\(974\) 7.30306 0.234005
\(975\) 0 0
\(976\) 3.55051 0.113649
\(977\) 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
\(980\) −2.22474 0.224745i −0.0710669 0.00717921i
\(981\) 0 0
\(982\) 19.5959i 0.625331i
\(983\) 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
\(986\) 13.7980 0.439417
\(987\) 0 0
\(988\) 6.89898i 0.219486i
\(989\) −2.60612 −0.0828699
\(990\) 0 0
\(991\) 31.3031 0.994375 0.497187 0.867643i \(-0.334366\pi\)
0.497187 + 0.867643i \(0.334366\pi\)
\(992\) 8.89898i 0.282543i
\(993\) 0 0
\(994\) 1.10102 0.0349223
\(995\) 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 \(-0.362484\pi\)
\(998\) 6.20204i 0.196322i
\(999\) 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 630.2.g.g.379.1 4
3.2 odd 2 70.2.c.a.29.3 yes 4
4.3 odd 2 5040.2.t.t.1009.1 4
5.2 odd 4 3150.2.a.bt.1.2 2
5.3 odd 4 3150.2.a.bs.1.2 2
5.4 even 2 inner 630.2.g.g.379.3 4
12.11 even 2 560.2.g.e.449.4 4
15.2 even 4 350.2.a.g.1.1 2
15.8 even 4 350.2.a.h.1.2 2
15.14 odd 2 70.2.c.a.29.2 4
20.19 odd 2 5040.2.t.t.1009.2 4
21.2 odd 6 490.2.i.c.459.4 8
21.5 even 6 490.2.i.f.459.3 8
21.11 odd 6 490.2.i.c.79.1 8
21.17 even 6 490.2.i.f.79.2 8
21.20 even 2 490.2.c.e.99.4 4
24.5 odd 2 2240.2.g.j.449.3 4
24.11 even 2 2240.2.g.i.449.1 4
60.23 odd 4 2800.2.a.bl.1.1 2
60.47 odd 4 2800.2.a.bm.1.2 2
60.59 even 2 560.2.g.e.449.2 4
105.44 odd 6 490.2.i.c.459.1 8
105.59 even 6 490.2.i.f.79.3 8
105.62 odd 4 2450.2.a.bl.1.2 2
105.74 odd 6 490.2.i.c.79.4 8
105.83 odd 4 2450.2.a.bq.1.1 2
105.89 even 6 490.2.i.f.459.2 8
105.104 even 2 490.2.c.e.99.1 4
120.29 odd 2 2240.2.g.j.449.1 4
120.59 even 2 2240.2.g.i.449.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.c.a.29.2 4 15.14 odd 2
70.2.c.a.29.3 yes 4 3.2 odd 2
350.2.a.g.1.1 2 15.2 even 4
350.2.a.h.1.2 2 15.8 even 4
490.2.c.e.99.1 4 105.104 even 2
490.2.c.e.99.4 4 21.20 even 2
490.2.i.c.79.1 8 21.11 odd 6
490.2.i.c.79.4 8 105.74 odd 6
490.2.i.c.459.1 8 105.44 odd 6
490.2.i.c.459.4 8 21.2 odd 6
490.2.i.f.79.2 8 21.17 even 6
490.2.i.f.79.3 8 105.59 even 6
490.2.i.f.459.2 8 105.89 even 6
490.2.i.f.459.3 8 21.5 even 6
560.2.g.e.449.2 4 60.59 even 2
560.2.g.e.449.4 4 12.11 even 2
630.2.g.g.379.1 4 1.1 even 1 trivial
630.2.g.g.379.3 4 5.4 even 2 inner
2240.2.g.i.449.1 4 24.11 even 2
2240.2.g.i.449.3 4 120.59 even 2
2240.2.g.j.449.1 4 120.29 odd 2
2240.2.g.j.449.3 4 24.5 odd 2
2450.2.a.bl.1.2 2 105.62 odd 4
2450.2.a.bq.1.1 2 105.83 odd 4
2800.2.a.bl.1.1 2 60.23 odd 4
2800.2.a.bm.1.2 2 60.47 odd 4
3150.2.a.bs.1.2 2 5.3 odd 4
3150.2.a.bt.1.2 2 5.2 odd 4
5040.2.t.t.1009.1 4 4.3 odd 2
5040.2.t.t.1009.2 4 20.19 odd 2