Properties

Label 1150.4.b.f.599.2
Level $1150$
Weight $4$
Character 1150.599
Analytic conductor $67.852$
Analytic rank $0$
Dimension $2$
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] = [1150,4,Mod(599,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.599");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.f.599.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -1.00000i q^{3} -4.00000 q^{4} +2.00000 q^{6} -18.0000i q^{7} -8.00000i q^{8} +26.0000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -1.00000i q^{3} -4.00000 q^{4} +2.00000 q^{6} -18.0000i q^{7} -8.00000i q^{8} +26.0000 q^{9} -32.0000 q^{11} +4.00000i q^{12} +47.0000i q^{13} +36.0000 q^{14} +16.0000 q^{16} +20.0000i q^{17} +52.0000i q^{18} -36.0000 q^{19} -18.0000 q^{21} -64.0000i q^{22} +23.0000i q^{23} -8.00000 q^{24} -94.0000 q^{26} -53.0000i q^{27} +72.0000i q^{28} +27.0000 q^{29} -33.0000 q^{31} +32.0000i q^{32} +32.0000i q^{33} -40.0000 q^{34} -104.000 q^{36} +56.0000i q^{37} -72.0000i q^{38} +47.0000 q^{39} -157.000 q^{41} -36.0000i q^{42} -18.0000i q^{43} +128.000 q^{44} -46.0000 q^{46} +65.0000i q^{47} -16.0000i q^{48} +19.0000 q^{49} +20.0000 q^{51} -188.000i q^{52} +14.0000i q^{53} +106.000 q^{54} -144.000 q^{56} +36.0000i q^{57} +54.0000i q^{58} +744.000 q^{59} +552.000 q^{61} -66.0000i q^{62} -468.000i q^{63} -64.0000 q^{64} -64.0000 q^{66} -156.000i q^{67} -80.0000i q^{68} +23.0000 q^{69} +699.000 q^{71} -208.000i q^{72} +609.000i q^{73} -112.000 q^{74} +144.000 q^{76} +576.000i q^{77} +94.0000i q^{78} +644.000 q^{79} +649.000 q^{81} -314.000i q^{82} -512.000i q^{83} +72.0000 q^{84} +36.0000 q^{86} -27.0000i q^{87} +256.000i q^{88} +102.000 q^{89} +846.000 q^{91} -92.0000i q^{92} +33.0000i q^{93} -130.000 q^{94} +32.0000 q^{96} +578.000i q^{97} +38.0000i q^{98} -832.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 64 q^{11} + 72 q^{14} + 32 q^{16} - 72 q^{19} - 36 q^{21} - 16 q^{24} - 188 q^{26} + 54 q^{29} - 66 q^{31} - 80 q^{34} - 208 q^{36} + 94 q^{39} - 314 q^{41} + 256 q^{44} - 92 q^{46} + 38 q^{49} + 40 q^{51} + 212 q^{54} - 288 q^{56} + 1488 q^{59} + 1104 q^{61} - 128 q^{64} - 128 q^{66} + 46 q^{69} + 1398 q^{71} - 224 q^{74} + 288 q^{76} + 1288 q^{79} + 1298 q^{81} + 144 q^{84} + 72 q^{86} + 204 q^{89} + 1692 q^{91} - 260 q^{94} + 64 q^{96} - 1664 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) − 1.00000i − 0.192450i −0.995360 0.0962250i \(-0.969323\pi\)
0.995360 0.0962250i \(-0.0306768\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.136083
\(7\) − 18.0000i − 0.971909i −0.873984 0.485954i \(-0.838472\pi\)
0.873984 0.485954i \(-0.161528\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 26.0000 0.962963
\(10\) 0 0
\(11\) −32.0000 −0.877124 −0.438562 0.898701i \(-0.644512\pi\)
−0.438562 + 0.898701i \(0.644512\pi\)
\(12\) 4.00000i 0.0962250i
\(13\) 47.0000i 1.00273i 0.865237 + 0.501364i \(0.167168\pi\)
−0.865237 + 0.501364i \(0.832832\pi\)
\(14\) 36.0000 0.687243
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 20.0000i 0.285336i 0.989771 + 0.142668i \(0.0455681\pi\)
−0.989771 + 0.142668i \(0.954432\pi\)
\(18\) 52.0000i 0.680918i
\(19\) −36.0000 −0.434682 −0.217341 0.976096i \(-0.569738\pi\)
−0.217341 + 0.976096i \(0.569738\pi\)
\(20\) 0 0
\(21\) −18.0000 −0.187044
\(22\) − 64.0000i − 0.620220i
\(23\) 23.0000i 0.208514i
\(24\) −8.00000 −0.0680414
\(25\) 0 0
\(26\) −94.0000 −0.709035
\(27\) − 53.0000i − 0.377772i
\(28\) 72.0000i 0.485954i
\(29\) 27.0000 0.172889 0.0864444 0.996257i \(-0.472450\pi\)
0.0864444 + 0.996257i \(0.472450\pi\)
\(30\) 0 0
\(31\) −33.0000 −0.191193 −0.0955964 0.995420i \(-0.530476\pi\)
−0.0955964 + 0.995420i \(0.530476\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 32.0000i 0.168803i
\(34\) −40.0000 −0.201763
\(35\) 0 0
\(36\) −104.000 −0.481481
\(37\) 56.0000i 0.248820i 0.992231 + 0.124410i \(0.0397038\pi\)
−0.992231 + 0.124410i \(0.960296\pi\)
\(38\) − 72.0000i − 0.307367i
\(39\) 47.0000 0.192975
\(40\) 0 0
\(41\) −157.000 −0.598031 −0.299016 0.954248i \(-0.596658\pi\)
−0.299016 + 0.954248i \(0.596658\pi\)
\(42\) − 36.0000i − 0.132260i
\(43\) − 18.0000i − 0.0638366i −0.999490 0.0319183i \(-0.989838\pi\)
0.999490 0.0319183i \(-0.0101616\pi\)
\(44\) 128.000 0.438562
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) 65.0000i 0.201728i 0.994900 + 0.100864i \(0.0321607\pi\)
−0.994900 + 0.100864i \(0.967839\pi\)
\(48\) − 16.0000i − 0.0481125i
\(49\) 19.0000 0.0553936
\(50\) 0 0
\(51\) 20.0000 0.0549129
\(52\) − 188.000i − 0.501364i
\(53\) 14.0000i 0.0362839i 0.999835 + 0.0181420i \(0.00577508\pi\)
−0.999835 + 0.0181420i \(0.994225\pi\)
\(54\) 106.000 0.267125
\(55\) 0 0
\(56\) −144.000 −0.343622
\(57\) 36.0000i 0.0836547i
\(58\) 54.0000i 0.122251i
\(59\) 744.000 1.64170 0.820852 0.571141i \(-0.193499\pi\)
0.820852 + 0.571141i \(0.193499\pi\)
\(60\) 0 0
\(61\) 552.000 1.15863 0.579314 0.815104i \(-0.303320\pi\)
0.579314 + 0.815104i \(0.303320\pi\)
\(62\) − 66.0000i − 0.135194i
\(63\) − 468.000i − 0.935912i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −64.0000 −0.119361
\(67\) − 156.000i − 0.284454i −0.989834 0.142227i \(-0.954574\pi\)
0.989834 0.142227i \(-0.0454263\pi\)
\(68\) − 80.0000i − 0.142668i
\(69\) 23.0000 0.0401286
\(70\) 0 0
\(71\) 699.000 1.16839 0.584197 0.811612i \(-0.301409\pi\)
0.584197 + 0.811612i \(0.301409\pi\)
\(72\) − 208.000i − 0.340459i
\(73\) 609.000i 0.976412i 0.872728 + 0.488206i \(0.162348\pi\)
−0.872728 + 0.488206i \(0.837652\pi\)
\(74\) −112.000 −0.175942
\(75\) 0 0
\(76\) 144.000 0.217341
\(77\) 576.000i 0.852484i
\(78\) 94.0000i 0.136454i
\(79\) 644.000 0.917160 0.458580 0.888653i \(-0.348358\pi\)
0.458580 + 0.888653i \(0.348358\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) − 314.000i − 0.422872i
\(83\) − 512.000i − 0.677100i −0.940948 0.338550i \(-0.890064\pi\)
0.940948 0.338550i \(-0.109936\pi\)
\(84\) 72.0000 0.0935220
\(85\) 0 0
\(86\) 36.0000 0.0451393
\(87\) − 27.0000i − 0.0332725i
\(88\) 256.000i 0.310110i
\(89\) 102.000 0.121483 0.0607415 0.998154i \(-0.480653\pi\)
0.0607415 + 0.998154i \(0.480653\pi\)
\(90\) 0 0
\(91\) 846.000 0.974559
\(92\) − 92.0000i − 0.104257i
\(93\) 33.0000i 0.0367951i
\(94\) −130.000 −0.142643
\(95\) 0 0
\(96\) 32.0000 0.0340207
\(97\) 578.000i 0.605021i 0.953146 + 0.302510i \(0.0978247\pi\)
−0.953146 + 0.302510i \(0.902175\pi\)
\(98\) 38.0000i 0.0391692i
\(99\) −832.000 −0.844638
\(100\) 0 0
\(101\) −6.00000 −0.00591111 −0.00295556 0.999996i \(-0.500941\pi\)
−0.00295556 + 0.999996i \(0.500941\pi\)
\(102\) 40.0000i 0.0388293i
\(103\) 160.000i 0.153061i 0.997067 + 0.0765304i \(0.0243842\pi\)
−0.997067 + 0.0765304i \(0.975616\pi\)
\(104\) 376.000 0.354518
\(105\) 0 0
\(106\) −28.0000 −0.0256566
\(107\) 380.000i 0.343327i 0.985156 + 0.171663i \(0.0549142\pi\)
−0.985156 + 0.171663i \(0.945086\pi\)
\(108\) 212.000i 0.188886i
\(109\) −250.000 −0.219685 −0.109842 0.993949i \(-0.535035\pi\)
−0.109842 + 0.993949i \(0.535035\pi\)
\(110\) 0 0
\(111\) 56.0000 0.0478854
\(112\) − 288.000i − 0.242977i
\(113\) 390.000i 0.324674i 0.986735 + 0.162337i \(0.0519031\pi\)
−0.986735 + 0.162337i \(0.948097\pi\)
\(114\) −72.0000 −0.0591528
\(115\) 0 0
\(116\) −108.000 −0.0864444
\(117\) 1222.00i 0.965589i
\(118\) 1488.00i 1.16086i
\(119\) 360.000 0.277321
\(120\) 0 0
\(121\) −307.000 −0.230654
\(122\) 1104.00i 0.819274i
\(123\) 157.000i 0.115091i
\(124\) 132.000 0.0955964
\(125\) 0 0
\(126\) 936.000 0.661790
\(127\) − 769.000i − 0.537305i −0.963237 0.268652i \(-0.913422\pi\)
0.963237 0.268652i \(-0.0865783\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) −18.0000 −0.0122854
\(130\) 0 0
\(131\) −213.000 −0.142060 −0.0710301 0.997474i \(-0.522629\pi\)
−0.0710301 + 0.997474i \(0.522629\pi\)
\(132\) − 128.000i − 0.0844013i
\(133\) 648.000i 0.422472i
\(134\) 312.000 0.201140
\(135\) 0 0
\(136\) 160.000 0.100882
\(137\) 2836.00i 1.76858i 0.466936 + 0.884291i \(0.345358\pi\)
−0.466936 + 0.884291i \(0.654642\pi\)
\(138\) 46.0000i 0.0283752i
\(139\) 1631.00 0.995249 0.497625 0.867393i \(-0.334206\pi\)
0.497625 + 0.867393i \(0.334206\pi\)
\(140\) 0 0
\(141\) 65.0000 0.0388226
\(142\) 1398.00i 0.826180i
\(143\) − 1504.00i − 0.879516i
\(144\) 416.000 0.240741
\(145\) 0 0
\(146\) −1218.00 −0.690427
\(147\) − 19.0000i − 0.0106605i
\(148\) − 224.000i − 0.124410i
\(149\) 1966.00 1.08095 0.540473 0.841361i \(-0.318245\pi\)
0.540473 + 0.841361i \(0.318245\pi\)
\(150\) 0 0
\(151\) 35.0000 0.0188626 0.00943132 0.999956i \(-0.496998\pi\)
0.00943132 + 0.999956i \(0.496998\pi\)
\(152\) 288.000i 0.153683i
\(153\) 520.000i 0.274768i
\(154\) −1152.00 −0.602797
\(155\) 0 0
\(156\) −188.000 −0.0964875
\(157\) 1702.00i 0.865187i 0.901589 + 0.432594i \(0.142401\pi\)
−0.901589 + 0.432594i \(0.857599\pi\)
\(158\) 1288.00i 0.648530i
\(159\) 14.0000 0.00698284
\(160\) 0 0
\(161\) 414.000 0.202657
\(162\) 1298.00i 0.629509i
\(163\) 2045.00i 0.982680i 0.870968 + 0.491340i \(0.163493\pi\)
−0.870968 + 0.491340i \(0.836507\pi\)
\(164\) 628.000 0.299016
\(165\) 0 0
\(166\) 1024.00 0.478782
\(167\) 1016.00i 0.470781i 0.971901 + 0.235391i \(0.0756369\pi\)
−0.971901 + 0.235391i \(0.924363\pi\)
\(168\) 144.000i 0.0661300i
\(169\) −12.0000 −0.00546199
\(170\) 0 0
\(171\) −936.000 −0.418583
\(172\) 72.0000i 0.0319183i
\(173\) − 598.000i − 0.262804i −0.991329 0.131402i \(-0.958052\pi\)
0.991329 0.131402i \(-0.0419479\pi\)
\(174\) 54.0000 0.0235272
\(175\) 0 0
\(176\) −512.000 −0.219281
\(177\) − 744.000i − 0.315946i
\(178\) 204.000i 0.0859014i
\(179\) 4607.00 1.92371 0.961853 0.273567i \(-0.0882036\pi\)
0.961853 + 0.273567i \(0.0882036\pi\)
\(180\) 0 0
\(181\) −1212.00 −0.497720 −0.248860 0.968540i \(-0.580056\pi\)
−0.248860 + 0.968540i \(0.580056\pi\)
\(182\) 1692.00i 0.689117i
\(183\) − 552.000i − 0.222978i
\(184\) 184.000 0.0737210
\(185\) 0 0
\(186\) −66.0000 −0.0260180
\(187\) − 640.000i − 0.250275i
\(188\) − 260.000i − 0.100864i
\(189\) −954.000 −0.367160
\(190\) 0 0
\(191\) −1058.00 −0.400807 −0.200404 0.979713i \(-0.564225\pi\)
−0.200404 + 0.979713i \(0.564225\pi\)
\(192\) 64.0000i 0.0240563i
\(193\) − 1047.00i − 0.390491i −0.980754 0.195245i \(-0.937450\pi\)
0.980754 0.195245i \(-0.0625503\pi\)
\(194\) −1156.00 −0.427814
\(195\) 0 0
\(196\) −76.0000 −0.0276968
\(197\) 251.000i 0.0907767i 0.998969 + 0.0453883i \(0.0144525\pi\)
−0.998969 + 0.0453883i \(0.985547\pi\)
\(198\) − 1664.00i − 0.597249i
\(199\) 3508.00 1.24963 0.624813 0.780775i \(-0.285175\pi\)
0.624813 + 0.780775i \(0.285175\pi\)
\(200\) 0 0
\(201\) −156.000 −0.0547432
\(202\) − 12.0000i − 0.00417979i
\(203\) − 486.000i − 0.168032i
\(204\) −80.0000 −0.0274565
\(205\) 0 0
\(206\) −320.000 −0.108230
\(207\) 598.000i 0.200792i
\(208\) 752.000i 0.250682i
\(209\) 1152.00 0.381270
\(210\) 0 0
\(211\) −3296.00 −1.07538 −0.537692 0.843141i \(-0.680704\pi\)
−0.537692 + 0.843141i \(0.680704\pi\)
\(212\) − 56.0000i − 0.0181420i
\(213\) − 699.000i − 0.224858i
\(214\) −760.000 −0.242769
\(215\) 0 0
\(216\) −424.000 −0.133563
\(217\) 594.000i 0.185822i
\(218\) − 500.000i − 0.155341i
\(219\) 609.000 0.187911
\(220\) 0 0
\(221\) −940.000 −0.286114
\(222\) 112.000i 0.0338601i
\(223\) 2720.00i 0.816792i 0.912805 + 0.408396i \(0.133912\pi\)
−0.912805 + 0.408396i \(0.866088\pi\)
\(224\) 576.000 0.171811
\(225\) 0 0
\(226\) −780.000 −0.229579
\(227\) − 4134.00i − 1.20874i −0.796705 0.604368i \(-0.793426\pi\)
0.796705 0.604368i \(-0.206574\pi\)
\(228\) − 144.000i − 0.0418273i
\(229\) 4510.00 1.30144 0.650719 0.759319i \(-0.274468\pi\)
0.650719 + 0.759319i \(0.274468\pi\)
\(230\) 0 0
\(231\) 576.000 0.164061
\(232\) − 216.000i − 0.0611254i
\(233\) 5003.00i 1.40668i 0.710852 + 0.703342i \(0.248310\pi\)
−0.710852 + 0.703342i \(0.751690\pi\)
\(234\) −2444.00 −0.682775
\(235\) 0 0
\(236\) −2976.00 −0.820852
\(237\) − 644.000i − 0.176508i
\(238\) 720.000i 0.196095i
\(239\) 6309.00 1.70751 0.853756 0.520674i \(-0.174319\pi\)
0.853756 + 0.520674i \(0.174319\pi\)
\(240\) 0 0
\(241\) 3038.00 0.812012 0.406006 0.913871i \(-0.366921\pi\)
0.406006 + 0.913871i \(0.366921\pi\)
\(242\) − 614.000i − 0.163097i
\(243\) − 2080.00i − 0.549103i
\(244\) −2208.00 −0.579314
\(245\) 0 0
\(246\) −314.000 −0.0813817
\(247\) − 1692.00i − 0.435868i
\(248\) 264.000i 0.0675968i
\(249\) −512.000 −0.130308
\(250\) 0 0
\(251\) −1332.00 −0.334961 −0.167480 0.985875i \(-0.553563\pi\)
−0.167480 + 0.985875i \(0.553563\pi\)
\(252\) 1872.00i 0.467956i
\(253\) − 736.000i − 0.182893i
\(254\) 1538.00 0.379932
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 3301.00i 0.801209i 0.916251 + 0.400605i \(0.131200\pi\)
−0.916251 + 0.400605i \(0.868800\pi\)
\(258\) − 36.0000i − 0.00868706i
\(259\) 1008.00 0.241830
\(260\) 0 0
\(261\) 702.000 0.166485
\(262\) − 426.000i − 0.100452i
\(263\) − 2072.00i − 0.485798i −0.970052 0.242899i \(-0.921902\pi\)
0.970052 0.242899i \(-0.0780984\pi\)
\(264\) 256.000 0.0596807
\(265\) 0 0
\(266\) −1296.00 −0.298733
\(267\) − 102.000i − 0.0233794i
\(268\) 624.000i 0.142227i
\(269\) −5721.00 −1.29671 −0.648356 0.761337i \(-0.724543\pi\)
−0.648356 + 0.761337i \(0.724543\pi\)
\(270\) 0 0
\(271\) −5900.00 −1.32251 −0.661254 0.750162i \(-0.729975\pi\)
−0.661254 + 0.750162i \(0.729975\pi\)
\(272\) 320.000i 0.0713340i
\(273\) − 846.000i − 0.187554i
\(274\) −5672.00 −1.25058
\(275\) 0 0
\(276\) −92.0000 −0.0200643
\(277\) 6371.00i 1.38194i 0.722885 + 0.690968i \(0.242815\pi\)
−0.722885 + 0.690968i \(0.757185\pi\)
\(278\) 3262.00i 0.703747i
\(279\) −858.000 −0.184112
\(280\) 0 0
\(281\) 3190.00 0.677222 0.338611 0.940926i \(-0.390043\pi\)
0.338611 + 0.940926i \(0.390043\pi\)
\(282\) 130.000i 0.0274517i
\(283\) 4226.00i 0.887667i 0.896109 + 0.443833i \(0.146382\pi\)
−0.896109 + 0.443833i \(0.853618\pi\)
\(284\) −2796.00 −0.584197
\(285\) 0 0
\(286\) 3008.00 0.621912
\(287\) 2826.00i 0.581232i
\(288\) 832.000i 0.170229i
\(289\) 4513.00 0.918583
\(290\) 0 0
\(291\) 578.000 0.116436
\(292\) − 2436.00i − 0.488206i
\(293\) − 6048.00i − 1.20590i −0.797780 0.602949i \(-0.793992\pi\)
0.797780 0.602949i \(-0.206008\pi\)
\(294\) 38.0000 0.00753811
\(295\) 0 0
\(296\) 448.000 0.0879712
\(297\) 1696.00i 0.331353i
\(298\) 3932.00i 0.764344i
\(299\) −1081.00 −0.209083
\(300\) 0 0
\(301\) −324.000 −0.0620434
\(302\) 70.0000i 0.0133379i
\(303\) 6.00000i 0.00113759i
\(304\) −576.000 −0.108671
\(305\) 0 0
\(306\) −1040.00 −0.194290
\(307\) 8628.00i 1.60399i 0.597328 + 0.801997i \(0.296229\pi\)
−0.597328 + 0.801997i \(0.703771\pi\)
\(308\) − 2304.00i − 0.426242i
\(309\) 160.000 0.0294566
\(310\) 0 0
\(311\) 8247.00 1.50368 0.751840 0.659346i \(-0.229167\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(312\) − 376.000i − 0.0682269i
\(313\) − 2620.00i − 0.473135i −0.971615 0.236567i \(-0.923978\pi\)
0.971615 0.236567i \(-0.0760224\pi\)
\(314\) −3404.00 −0.611780
\(315\) 0 0
\(316\) −2576.00 −0.458580
\(317\) 9906.00i 1.75513i 0.479457 + 0.877565i \(0.340834\pi\)
−0.479457 + 0.877565i \(0.659166\pi\)
\(318\) 28.0000i 0.00493762i
\(319\) −864.000 −0.151645
\(320\) 0 0
\(321\) 380.000 0.0660733
\(322\) 828.000i 0.143300i
\(323\) − 720.000i − 0.124031i
\(324\) −2596.00 −0.445130
\(325\) 0 0
\(326\) −4090.00 −0.694859
\(327\) 250.000i 0.0422784i
\(328\) 1256.00i 0.211436i
\(329\) 1170.00 0.196061
\(330\) 0 0
\(331\) −8115.00 −1.34756 −0.673778 0.738934i \(-0.735329\pi\)
−0.673778 + 0.738934i \(0.735329\pi\)
\(332\) 2048.00i 0.338550i
\(333\) 1456.00i 0.239605i
\(334\) −2032.00 −0.332892
\(335\) 0 0
\(336\) −288.000 −0.0467610
\(337\) − 7586.00i − 1.22622i −0.789998 0.613109i \(-0.789918\pi\)
0.789998 0.613109i \(-0.210082\pi\)
\(338\) − 24.0000i − 0.00386221i
\(339\) 390.000 0.0624835
\(340\) 0 0
\(341\) 1056.00 0.167700
\(342\) − 1872.00i − 0.295983i
\(343\) − 6516.00i − 1.02575i
\(344\) −144.000 −0.0225697
\(345\) 0 0
\(346\) 1196.00 0.185831
\(347\) 1356.00i 0.209781i 0.994484 + 0.104890i \(0.0334492\pi\)
−0.994484 + 0.104890i \(0.966551\pi\)
\(348\) 108.000i 0.0166362i
\(349\) −6649.00 −1.01981 −0.509904 0.860231i \(-0.670319\pi\)
−0.509904 + 0.860231i \(0.670319\pi\)
\(350\) 0 0
\(351\) 2491.00 0.378803
\(352\) − 1024.00i − 0.155055i
\(353\) − 10691.0i − 1.61197i −0.591938 0.805984i \(-0.701637\pi\)
0.591938 0.805984i \(-0.298363\pi\)
\(354\) 1488.00 0.223408
\(355\) 0 0
\(356\) −408.000 −0.0607415
\(357\) − 360.000i − 0.0533704i
\(358\) 9214.00i 1.36027i
\(359\) 6420.00 0.943829 0.471915 0.881644i \(-0.343563\pi\)
0.471915 + 0.881644i \(0.343563\pi\)
\(360\) 0 0
\(361\) −5563.00 −0.811051
\(362\) − 2424.00i − 0.351941i
\(363\) 307.000i 0.0443893i
\(364\) −3384.00 −0.487280
\(365\) 0 0
\(366\) 1104.00 0.157669
\(367\) − 524.000i − 0.0745302i −0.999305 0.0372651i \(-0.988135\pi\)
0.999305 0.0372651i \(-0.0118646\pi\)
\(368\) 368.000i 0.0521286i
\(369\) −4082.00 −0.575882
\(370\) 0 0
\(371\) 252.000 0.0352647
\(372\) − 132.000i − 0.0183975i
\(373\) − 5566.00i − 0.772645i −0.922364 0.386322i \(-0.873745\pi\)
0.922364 0.386322i \(-0.126255\pi\)
\(374\) 1280.00 0.176971
\(375\) 0 0
\(376\) 520.000 0.0713217
\(377\) 1269.00i 0.173360i
\(378\) − 1908.00i − 0.259622i
\(379\) −2240.00 −0.303591 −0.151796 0.988412i \(-0.548506\pi\)
−0.151796 + 0.988412i \(0.548506\pi\)
\(380\) 0 0
\(381\) −769.000 −0.103404
\(382\) − 2116.00i − 0.283414i
\(383\) − 8778.00i − 1.17111i −0.810633 0.585555i \(-0.800877\pi\)
0.810633 0.585555i \(-0.199123\pi\)
\(384\) −128.000 −0.0170103
\(385\) 0 0
\(386\) 2094.00 0.276119
\(387\) − 468.000i − 0.0614723i
\(388\) − 2312.00i − 0.302510i
\(389\) −4056.00 −0.528656 −0.264328 0.964433i \(-0.585150\pi\)
−0.264328 + 0.964433i \(0.585150\pi\)
\(390\) 0 0
\(391\) −460.000 −0.0594967
\(392\) − 152.000i − 0.0195846i
\(393\) 213.000i 0.0273395i
\(394\) −502.000 −0.0641888
\(395\) 0 0
\(396\) 3328.00 0.422319
\(397\) − 9151.00i − 1.15687i −0.815730 0.578433i \(-0.803665\pi\)
0.815730 0.578433i \(-0.196335\pi\)
\(398\) 7016.00i 0.883619i
\(399\) 648.000 0.0813047
\(400\) 0 0
\(401\) 15930.0 1.98381 0.991903 0.126997i \(-0.0405340\pi\)
0.991903 + 0.126997i \(0.0405340\pi\)
\(402\) − 312.000i − 0.0387093i
\(403\) − 1551.00i − 0.191714i
\(404\) 24.0000 0.00295556
\(405\) 0 0
\(406\) 972.000 0.118817
\(407\) − 1792.00i − 0.218246i
\(408\) − 160.000i − 0.0194147i
\(409\) 5891.00 0.712203 0.356102 0.934447i \(-0.384106\pi\)
0.356102 + 0.934447i \(0.384106\pi\)
\(410\) 0 0
\(411\) 2836.00 0.340364
\(412\) − 640.000i − 0.0765304i
\(413\) − 13392.0i − 1.59559i
\(414\) −1196.00 −0.141981
\(415\) 0 0
\(416\) −1504.00 −0.177259
\(417\) − 1631.00i − 0.191536i
\(418\) 2304.00i 0.269599i
\(419\) −15282.0 −1.78180 −0.890900 0.454199i \(-0.849926\pi\)
−0.890900 + 0.454199i \(0.849926\pi\)
\(420\) 0 0
\(421\) −10934.0 −1.26577 −0.632887 0.774244i \(-0.718130\pi\)
−0.632887 + 0.774244i \(0.718130\pi\)
\(422\) − 6592.00i − 0.760411i
\(423\) 1690.00i 0.194257i
\(424\) 112.000 0.0128283
\(425\) 0 0
\(426\) 1398.00 0.158998
\(427\) − 9936.00i − 1.12608i
\(428\) − 1520.00i − 0.171663i
\(429\) −1504.00 −0.169263
\(430\) 0 0
\(431\) −2794.00 −0.312256 −0.156128 0.987737i \(-0.549901\pi\)
−0.156128 + 0.987737i \(0.549901\pi\)
\(432\) − 848.000i − 0.0944431i
\(433\) 15062.0i 1.67167i 0.548980 + 0.835835i \(0.315016\pi\)
−0.548980 + 0.835835i \(0.684984\pi\)
\(434\) −1188.00 −0.131396
\(435\) 0 0
\(436\) 1000.00 0.109842
\(437\) − 828.000i − 0.0906376i
\(438\) 1218.00i 0.132873i
\(439\) −261.000 −0.0283755 −0.0141878 0.999899i \(-0.504516\pi\)
−0.0141878 + 0.999899i \(0.504516\pi\)
\(440\) 0 0
\(441\) 494.000 0.0533420
\(442\) − 1880.00i − 0.202313i
\(443\) 7083.00i 0.759647i 0.925059 + 0.379823i \(0.124015\pi\)
−0.925059 + 0.379823i \(0.875985\pi\)
\(444\) −224.000 −0.0239427
\(445\) 0 0
\(446\) −5440.00 −0.577559
\(447\) − 1966.00i − 0.208028i
\(448\) 1152.00i 0.121489i
\(449\) 10370.0 1.08996 0.544978 0.838450i \(-0.316538\pi\)
0.544978 + 0.838450i \(0.316538\pi\)
\(450\) 0 0
\(451\) 5024.00 0.524547
\(452\) − 1560.00i − 0.162337i
\(453\) − 35.0000i − 0.00363012i
\(454\) 8268.00 0.854706
\(455\) 0 0
\(456\) 288.000 0.0295764
\(457\) − 10496.0i − 1.07436i −0.843468 0.537180i \(-0.819490\pi\)
0.843468 0.537180i \(-0.180510\pi\)
\(458\) 9020.00i 0.920255i
\(459\) 1060.00 0.107792
\(460\) 0 0
\(461\) 18021.0 1.82065 0.910327 0.413889i \(-0.135830\pi\)
0.910327 + 0.413889i \(0.135830\pi\)
\(462\) 1152.00i 0.116008i
\(463\) 17188.0i 1.72526i 0.505838 + 0.862629i \(0.331183\pi\)
−0.505838 + 0.862629i \(0.668817\pi\)
\(464\) 432.000 0.0432222
\(465\) 0 0
\(466\) −10006.0 −0.994676
\(467\) − 15246.0i − 1.51071i −0.655317 0.755354i \(-0.727465\pi\)
0.655317 0.755354i \(-0.272535\pi\)
\(468\) − 4888.00i − 0.482795i
\(469\) −2808.00 −0.276464
\(470\) 0 0
\(471\) 1702.00 0.166505
\(472\) − 5952.00i − 0.580430i
\(473\) 576.000i 0.0559926i
\(474\) 1288.00 0.124810
\(475\) 0 0
\(476\) −1440.00 −0.138660
\(477\) 364.000i 0.0349401i
\(478\) 12618.0i 1.20739i
\(479\) 8556.00 0.816145 0.408073 0.912949i \(-0.366201\pi\)
0.408073 + 0.912949i \(0.366201\pi\)
\(480\) 0 0
\(481\) −2632.00 −0.249499
\(482\) 6076.00i 0.574179i
\(483\) − 414.000i − 0.0390014i
\(484\) 1228.00 0.115327
\(485\) 0 0
\(486\) 4160.00 0.388275
\(487\) − 1805.00i − 0.167951i −0.996468 0.0839757i \(-0.973238\pi\)
0.996468 0.0839757i \(-0.0267618\pi\)
\(488\) − 4416.00i − 0.409637i
\(489\) 2045.00 0.189117
\(490\) 0 0
\(491\) 5245.00 0.482085 0.241042 0.970515i \(-0.422511\pi\)
0.241042 + 0.970515i \(0.422511\pi\)
\(492\) − 628.000i − 0.0575456i
\(493\) 540.000i 0.0493314i
\(494\) 3384.00 0.308205
\(495\) 0 0
\(496\) −528.000 −0.0477982
\(497\) − 12582.0i − 1.13557i
\(498\) − 1024.00i − 0.0921416i
\(499\) −9027.00 −0.809828 −0.404914 0.914355i \(-0.632698\pi\)
−0.404914 + 0.914355i \(0.632698\pi\)
\(500\) 0 0
\(501\) 1016.00 0.0906019
\(502\) − 2664.00i − 0.236853i
\(503\) − 3522.00i − 0.312203i −0.987741 0.156102i \(-0.950107\pi\)
0.987741 0.156102i \(-0.0498927\pi\)
\(504\) −3744.00 −0.330895
\(505\) 0 0
\(506\) 1472.00 0.129325
\(507\) 12.0000i 0.00105116i
\(508\) 3076.00i 0.268652i
\(509\) −3949.00 −0.343883 −0.171941 0.985107i \(-0.555004\pi\)
−0.171941 + 0.985107i \(0.555004\pi\)
\(510\) 0 0
\(511\) 10962.0 0.948983
\(512\) 512.000i 0.0441942i
\(513\) 1908.00i 0.164211i
\(514\) −6602.00 −0.566540
\(515\) 0 0
\(516\) 72.0000 0.00614268
\(517\) − 2080.00i − 0.176941i
\(518\) 2016.00i 0.171000i
\(519\) −598.000 −0.0505767
\(520\) 0 0
\(521\) 3236.00 0.272115 0.136057 0.990701i \(-0.456557\pi\)
0.136057 + 0.990701i \(0.456557\pi\)
\(522\) 1404.00i 0.117723i
\(523\) − 12394.0i − 1.03624i −0.855309 0.518118i \(-0.826633\pi\)
0.855309 0.518118i \(-0.173367\pi\)
\(524\) 852.000 0.0710301
\(525\) 0 0
\(526\) 4144.00 0.343511
\(527\) − 660.000i − 0.0545542i
\(528\) 512.000i 0.0422006i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) 19344.0 1.58090
\(532\) − 2592.00i − 0.211236i
\(533\) − 7379.00i − 0.599662i
\(534\) 204.000 0.0165317
\(535\) 0 0
\(536\) −1248.00 −0.100570
\(537\) − 4607.00i − 0.370217i
\(538\) − 11442.0i − 0.916914i
\(539\) −608.000 −0.0485870
\(540\) 0 0
\(541\) −7159.00 −0.568927 −0.284463 0.958687i \(-0.591815\pi\)
−0.284463 + 0.958687i \(0.591815\pi\)
\(542\) − 11800.0i − 0.935154i
\(543\) 1212.00i 0.0957862i
\(544\) −640.000 −0.0504408
\(545\) 0 0
\(546\) 1692.00 0.132621
\(547\) − 19761.0i − 1.54464i −0.635232 0.772321i \(-0.719096\pi\)
0.635232 0.772321i \(-0.280904\pi\)
\(548\) − 11344.0i − 0.884291i
\(549\) 14352.0 1.11572
\(550\) 0 0
\(551\) −972.000 −0.0751517
\(552\) − 184.000i − 0.0141876i
\(553\) − 11592.0i − 0.891396i
\(554\) −12742.0 −0.977176
\(555\) 0 0
\(556\) −6524.00 −0.497625
\(557\) 18010.0i 1.37003i 0.728528 + 0.685016i \(0.240205\pi\)
−0.728528 + 0.685016i \(0.759795\pi\)
\(558\) − 1716.00i − 0.130187i
\(559\) 846.000 0.0640107
\(560\) 0 0
\(561\) −640.000 −0.0481655
\(562\) 6380.00i 0.478868i
\(563\) 2648.00i 0.198224i 0.995076 + 0.0991118i \(0.0316001\pi\)
−0.995076 + 0.0991118i \(0.968400\pi\)
\(564\) −260.000 −0.0194113
\(565\) 0 0
\(566\) −8452.00 −0.627675
\(567\) − 11682.0i − 0.865252i
\(568\) − 5592.00i − 0.413090i
\(569\) 1566.00 0.115378 0.0576890 0.998335i \(-0.481627\pi\)
0.0576890 + 0.998335i \(0.481627\pi\)
\(570\) 0 0
\(571\) 2864.00 0.209903 0.104952 0.994477i \(-0.466531\pi\)
0.104952 + 0.994477i \(0.466531\pi\)
\(572\) 6016.00i 0.439758i
\(573\) 1058.00i 0.0771354i
\(574\) −5652.00 −0.410993
\(575\) 0 0
\(576\) −1664.00 −0.120370
\(577\) − 929.000i − 0.0670273i −0.999438 0.0335137i \(-0.989330\pi\)
0.999438 0.0335137i \(-0.0106697\pi\)
\(578\) 9026.00i 0.649537i
\(579\) −1047.00 −0.0751500
\(580\) 0 0
\(581\) −9216.00 −0.658079
\(582\) 1156.00i 0.0823329i
\(583\) − 448.000i − 0.0318255i
\(584\) 4872.00 0.345214
\(585\) 0 0
\(586\) 12096.0 0.852698
\(587\) − 19499.0i − 1.37106i −0.728046 0.685528i \(-0.759571\pi\)
0.728046 0.685528i \(-0.240429\pi\)
\(588\) 76.0000i 0.00533025i
\(589\) 1188.00 0.0831081
\(590\) 0 0
\(591\) 251.000 0.0174700
\(592\) 896.000i 0.0622050i
\(593\) − 6570.00i − 0.454971i −0.973782 0.227485i \(-0.926950\pi\)
0.973782 0.227485i \(-0.0730504\pi\)
\(594\) −3392.00 −0.234302
\(595\) 0 0
\(596\) −7864.00 −0.540473
\(597\) − 3508.00i − 0.240491i
\(598\) − 2162.00i − 0.147844i
\(599\) −1880.00 −0.128238 −0.0641191 0.997942i \(-0.520424\pi\)
−0.0641191 + 0.997942i \(0.520424\pi\)
\(600\) 0 0
\(601\) 3701.00 0.251193 0.125596 0.992081i \(-0.459916\pi\)
0.125596 + 0.992081i \(0.459916\pi\)
\(602\) − 648.000i − 0.0438713i
\(603\) − 4056.00i − 0.273919i
\(604\) −140.000 −0.00943132
\(605\) 0 0
\(606\) −12.0000 −0.000804400 0
\(607\) − 3080.00i − 0.205953i −0.994684 0.102976i \(-0.967163\pi\)
0.994684 0.102976i \(-0.0328366\pi\)
\(608\) − 1152.00i − 0.0768417i
\(609\) −486.000 −0.0323378
\(610\) 0 0
\(611\) −3055.00 −0.202278
\(612\) − 2080.00i − 0.137384i
\(613\) − 24004.0i − 1.58159i −0.612083 0.790793i \(-0.709668\pi\)
0.612083 0.790793i \(-0.290332\pi\)
\(614\) −17256.0 −1.13419
\(615\) 0 0
\(616\) 4608.00 0.301399
\(617\) 780.000i 0.0508940i 0.999676 + 0.0254470i \(0.00810091\pi\)
−0.999676 + 0.0254470i \(0.991899\pi\)
\(618\) 320.000i 0.0208289i
\(619\) −21892.0 −1.42151 −0.710754 0.703440i \(-0.751646\pi\)
−0.710754 + 0.703440i \(0.751646\pi\)
\(620\) 0 0
\(621\) 1219.00 0.0787710
\(622\) 16494.0i 1.06326i
\(623\) − 1836.00i − 0.118070i
\(624\) 752.000 0.0482437
\(625\) 0 0
\(626\) 5240.00 0.334557
\(627\) − 1152.00i − 0.0733755i
\(628\) − 6808.00i − 0.432594i
\(629\) −1120.00 −0.0709973
\(630\) 0 0
\(631\) 8050.00 0.507869 0.253935 0.967221i \(-0.418275\pi\)
0.253935 + 0.967221i \(0.418275\pi\)
\(632\) − 5152.00i − 0.324265i
\(633\) 3296.00i 0.206958i
\(634\) −19812.0 −1.24106
\(635\) 0 0
\(636\) −56.0000 −0.00349142
\(637\) 893.000i 0.0555447i
\(638\) − 1728.00i − 0.107229i
\(639\) 18174.0 1.12512
\(640\) 0 0
\(641\) −25890.0 −1.59531 −0.797655 0.603114i \(-0.793926\pi\)
−0.797655 + 0.603114i \(0.793926\pi\)
\(642\) 760.000i 0.0467209i
\(643\) − 4774.00i − 0.292797i −0.989226 0.146398i \(-0.953232\pi\)
0.989226 0.146398i \(-0.0467681\pi\)
\(644\) −1656.00 −0.101328
\(645\) 0 0
\(646\) 1440.00 0.0877029
\(647\) 3349.00i 0.203497i 0.994810 + 0.101749i \(0.0324438\pi\)
−0.994810 + 0.101749i \(0.967556\pi\)
\(648\) − 5192.00i − 0.314755i
\(649\) −23808.0 −1.43998
\(650\) 0 0
\(651\) 594.000 0.0357614
\(652\) − 8180.00i − 0.491340i
\(653\) − 24813.0i − 1.48699i −0.668739 0.743497i \(-0.733166\pi\)
0.668739 0.743497i \(-0.266834\pi\)
\(654\) −500.000 −0.0298953
\(655\) 0 0
\(656\) −2512.00 −0.149508
\(657\) 15834.0i 0.940248i
\(658\) 2340.00i 0.138636i
\(659\) −18180.0 −1.07465 −0.537323 0.843376i \(-0.680565\pi\)
−0.537323 + 0.843376i \(0.680565\pi\)
\(660\) 0 0
\(661\) −29250.0 −1.72117 −0.860585 0.509307i \(-0.829902\pi\)
−0.860585 + 0.509307i \(0.829902\pi\)
\(662\) − 16230.0i − 0.952865i
\(663\) 940.000i 0.0550627i
\(664\) −4096.00 −0.239391
\(665\) 0 0
\(666\) −2912.00 −0.169426
\(667\) 621.000i 0.0360498i
\(668\) − 4064.00i − 0.235391i
\(669\) 2720.00 0.157192
\(670\) 0 0
\(671\) −17664.0 −1.01626
\(672\) − 576.000i − 0.0330650i
\(673\) 23027.0i 1.31891i 0.751745 + 0.659454i \(0.229213\pi\)
−0.751745 + 0.659454i \(0.770787\pi\)
\(674\) 15172.0 0.867068
\(675\) 0 0
\(676\) 48.0000 0.00273100
\(677\) 20106.0i 1.14141i 0.821154 + 0.570706i \(0.193331\pi\)
−0.821154 + 0.570706i \(0.806669\pi\)
\(678\) 780.000i 0.0441825i
\(679\) 10404.0 0.588025
\(680\) 0 0
\(681\) −4134.00 −0.232621
\(682\) 2112.00i 0.118582i
\(683\) 18745.0i 1.05016i 0.851054 + 0.525079i \(0.175964\pi\)
−0.851054 + 0.525079i \(0.824036\pi\)
\(684\) 3744.00 0.209292
\(685\) 0 0
\(686\) 13032.0 0.725312
\(687\) − 4510.00i − 0.250462i
\(688\) − 288.000i − 0.0159592i
\(689\) −658.000 −0.0363829
\(690\) 0 0
\(691\) 24424.0 1.34462 0.672310 0.740270i \(-0.265302\pi\)
0.672310 + 0.740270i \(0.265302\pi\)
\(692\) 2392.00i 0.131402i
\(693\) 14976.0i 0.820911i
\(694\) −2712.00 −0.148337
\(695\) 0 0
\(696\) −216.000 −0.0117636
\(697\) − 3140.00i − 0.170640i
\(698\) − 13298.0i − 0.721113i
\(699\) 5003.00 0.270717
\(700\) 0 0
\(701\) −27278.0 −1.46972 −0.734862 0.678217i \(-0.762753\pi\)
−0.734862 + 0.678217i \(0.762753\pi\)
\(702\) 4982.00i 0.267854i
\(703\) − 2016.00i − 0.108158i
\(704\) 2048.00 0.109640
\(705\) 0 0
\(706\) 21382.0 1.13983
\(707\) 108.000i 0.00574506i
\(708\) 2976.00i 0.157973i
\(709\) −12214.0 −0.646977 −0.323488 0.946232i \(-0.604856\pi\)
−0.323488 + 0.946232i \(0.604856\pi\)
\(710\) 0 0
\(711\) 16744.0 0.883191
\(712\) − 816.000i − 0.0429507i
\(713\) − 759.000i − 0.0398664i
\(714\) 720.000 0.0377385
\(715\) 0 0
\(716\) −18428.0 −0.961853
\(717\) − 6309.00i − 0.328611i
\(718\) 12840.0i 0.667388i
\(719\) −12932.0 −0.670768 −0.335384 0.942082i \(-0.608866\pi\)
−0.335384 + 0.942082i \(0.608866\pi\)
\(720\) 0 0
\(721\) 2880.00 0.148761
\(722\) − 11126.0i − 0.573500i
\(723\) − 3038.00i − 0.156272i
\(724\) 4848.00 0.248860
\(725\) 0 0
\(726\) −614.000 −0.0313880
\(727\) 10046.0i 0.512497i 0.966611 + 0.256249i \(0.0824866\pi\)
−0.966611 + 0.256249i \(0.917513\pi\)
\(728\) − 6768.00i − 0.344559i
\(729\) 15443.0 0.784586
\(730\) 0 0
\(731\) 360.000 0.0182149
\(732\) 2208.00i 0.111489i
\(733\) 5924.00i 0.298510i 0.988799 + 0.149255i \(0.0476876\pi\)
−0.988799 + 0.149255i \(0.952312\pi\)
\(734\) 1048.00 0.0527008
\(735\) 0 0
\(736\) −736.000 −0.0368605
\(737\) 4992.00i 0.249502i
\(738\) − 8164.00i − 0.407210i
\(739\) −829.000 −0.0412656 −0.0206328 0.999787i \(-0.506568\pi\)
−0.0206328 + 0.999787i \(0.506568\pi\)
\(740\) 0 0
\(741\) −1692.00 −0.0838828
\(742\) 504.000i 0.0249359i
\(743\) − 7072.00i − 0.349188i −0.984641 0.174594i \(-0.944139\pi\)
0.984641 0.174594i \(-0.0558613\pi\)
\(744\) 264.000 0.0130090
\(745\) 0 0
\(746\) 11132.0 0.546342
\(747\) − 13312.0i − 0.652022i
\(748\) 2560.00i 0.125138i
\(749\) 6840.00 0.333682
\(750\) 0 0
\(751\) 16234.0 0.788798 0.394399 0.918939i \(-0.370953\pi\)
0.394399 + 0.918939i \(0.370953\pi\)
\(752\) 1040.00i 0.0504320i
\(753\) 1332.00i 0.0644632i
\(754\) −2538.00 −0.122584
\(755\) 0 0
\(756\) 3816.00 0.183580
\(757\) 9128.00i 0.438260i 0.975696 + 0.219130i \(0.0703219\pi\)
−0.975696 + 0.219130i \(0.929678\pi\)
\(758\) − 4480.00i − 0.214671i
\(759\) −736.000 −0.0351978
\(760\) 0 0
\(761\) 165.000 0.00785972 0.00392986 0.999992i \(-0.498749\pi\)
0.00392986 + 0.999992i \(0.498749\pi\)
\(762\) − 1538.00i − 0.0731179i
\(763\) 4500.00i 0.213514i
\(764\) 4232.00 0.200404
\(765\) 0 0
\(766\) 17556.0 0.828099
\(767\) 34968.0i 1.64618i
\(768\) − 256.000i − 0.0120281i
\(769\) 20834.0 0.976974 0.488487 0.872571i \(-0.337549\pi\)
0.488487 + 0.872571i \(0.337549\pi\)
\(770\) 0 0
\(771\) 3301.00 0.154193
\(772\) 4188.00i 0.195245i
\(773\) 31782.0i 1.47881i 0.673262 + 0.739404i \(0.264893\pi\)
−0.673262 + 0.739404i \(0.735107\pi\)
\(774\) 936.000 0.0434675
\(775\) 0 0
\(776\) 4624.00 0.213907
\(777\) − 1008.00i − 0.0465403i
\(778\) − 8112.00i − 0.373817i
\(779\) 5652.00 0.259954
\(780\) 0 0
\(781\) −22368.0 −1.02483
\(782\) − 920.000i − 0.0420705i
\(783\) − 1431.00i − 0.0653126i
\(784\) 304.000 0.0138484
\(785\) 0 0
\(786\) −426.000 −0.0193320
\(787\) 33104.0i 1.49940i 0.661776 + 0.749701i \(0.269803\pi\)
−0.661776 + 0.749701i \(0.730197\pi\)
\(788\) − 1004.00i − 0.0453883i
\(789\) −2072.00 −0.0934920
\(790\) 0 0
\(791\) 7020.00 0.315553
\(792\) 6656.00i 0.298625i
\(793\) 25944.0i 1.16179i
\(794\) 18302.0 0.818027
\(795\) 0 0
\(796\) −14032.0 −0.624813
\(797\) 4736.00i 0.210486i 0.994447 + 0.105243i \(0.0335621\pi\)
−0.994447 + 0.105243i \(0.966438\pi\)
\(798\) 1296.00i 0.0574911i
\(799\) −1300.00 −0.0575603
\(800\) 0 0
\(801\) 2652.00 0.116984
\(802\) 31860.0i 1.40276i
\(803\) − 19488.0i − 0.856434i
\(804\) 624.000 0.0273716
\(805\) 0 0
\(806\) 3102.00 0.135562
\(807\) 5721.00i 0.249552i
\(808\) 48.0000i 0.00208989i
\(809\) 7470.00 0.324637 0.162318 0.986738i \(-0.448103\pi\)
0.162318 + 0.986738i \(0.448103\pi\)
\(810\) 0 0
\(811\) 19919.0 0.862455 0.431227 0.902243i \(-0.358081\pi\)
0.431227 + 0.902243i \(0.358081\pi\)
\(812\) 1944.00i 0.0840160i
\(813\) 5900.00i 0.254517i
\(814\) 3584.00 0.154323
\(815\) 0 0
\(816\) 320.000 0.0137282
\(817\) 648.000i 0.0277487i
\(818\) 11782.0i 0.503604i
\(819\) 21996.0 0.938465
\(820\) 0 0
\(821\) −22694.0 −0.964709 −0.482354 0.875976i \(-0.660218\pi\)
−0.482354 + 0.875976i \(0.660218\pi\)
\(822\) 5672.00i 0.240674i
\(823\) 31907.0i 1.35141i 0.737173 + 0.675704i \(0.236160\pi\)
−0.737173 + 0.675704i \(0.763840\pi\)
\(824\) 1280.00 0.0541152
\(825\) 0 0
\(826\) 26784.0 1.12825
\(827\) − 15236.0i − 0.640638i −0.947310 0.320319i \(-0.896210\pi\)
0.947310 0.320319i \(-0.103790\pi\)
\(828\) − 2392.00i − 0.100396i
\(829\) −27286.0 −1.14316 −0.571581 0.820545i \(-0.693670\pi\)
−0.571581 + 0.820545i \(0.693670\pi\)
\(830\) 0 0
\(831\) 6371.00 0.265954
\(832\) − 3008.00i − 0.125341i
\(833\) 380.000i 0.0158058i
\(834\) 3262.00 0.135436
\(835\) 0 0
\(836\) −4608.00 −0.190635
\(837\) 1749.00i 0.0722273i
\(838\) − 30564.0i − 1.25992i
\(839\) −23054.0 −0.948644 −0.474322 0.880351i \(-0.657307\pi\)
−0.474322 + 0.880351i \(0.657307\pi\)
\(840\) 0 0
\(841\) −23660.0 −0.970109
\(842\) − 21868.0i − 0.895037i
\(843\) − 3190.00i − 0.130331i
\(844\) 13184.0 0.537692
\(845\) 0 0
\(846\) −3380.00 −0.137360
\(847\) 5526.00i 0.224174i
\(848\) 224.000i 0.00907098i
\(849\) 4226.00 0.170832
\(850\) 0 0
\(851\) −1288.00 −0.0518826
\(852\) 2796.00i 0.112429i
\(853\) 34506.0i 1.38507i 0.721385 + 0.692534i \(0.243506\pi\)
−0.721385 + 0.692534i \(0.756494\pi\)
\(854\) 19872.0 0.796260
\(855\) 0 0
\(856\) 3040.00 0.121384
\(857\) − 22263.0i − 0.887386i −0.896179 0.443693i \(-0.853668\pi\)
0.896179 0.443693i \(-0.146332\pi\)
\(858\) − 3008.00i − 0.119687i
\(859\) −12851.0 −0.510443 −0.255221 0.966883i \(-0.582148\pi\)
−0.255221 + 0.966883i \(0.582148\pi\)
\(860\) 0 0
\(861\) 2826.00 0.111858
\(862\) − 5588.00i − 0.220798i
\(863\) − 15723.0i − 0.620182i −0.950707 0.310091i \(-0.899640\pi\)
0.950707 0.310091i \(-0.100360\pi\)
\(864\) 1696.00 0.0667814
\(865\) 0 0
\(866\) −30124.0 −1.18205
\(867\) − 4513.00i − 0.176781i
\(868\) − 2376.00i − 0.0929109i
\(869\) −20608.0 −0.804463
\(870\) 0 0
\(871\) 7332.00 0.285230
\(872\) 2000.00i 0.0776704i
\(873\) 15028.0i 0.582613i
\(874\) 1656.00 0.0640904
\(875\) 0 0
\(876\) −2436.00 −0.0939553
\(877\) − 886.000i − 0.0341141i −0.999855 0.0170571i \(-0.994570\pi\)
0.999855 0.0170571i \(-0.00542970\pi\)
\(878\) − 522.000i − 0.0200645i
\(879\) −6048.00 −0.232075
\(880\) 0 0
\(881\) −37120.0 −1.41953 −0.709764 0.704439i \(-0.751199\pi\)
−0.709764 + 0.704439i \(0.751199\pi\)
\(882\) 988.000i 0.0377185i
\(883\) 7524.00i 0.286753i 0.989668 + 0.143376i \(0.0457960\pi\)
−0.989668 + 0.143376i \(0.954204\pi\)
\(884\) 3760.00 0.143057
\(885\) 0 0
\(886\) −14166.0 −0.537151
\(887\) 9221.00i 0.349054i 0.984652 + 0.174527i \(0.0558396\pi\)
−0.984652 + 0.174527i \(0.944160\pi\)
\(888\) − 448.000i − 0.0169301i
\(889\) −13842.0 −0.522211
\(890\) 0 0
\(891\) −20768.0 −0.780869
\(892\) − 10880.0i − 0.408396i
\(893\) − 2340.00i − 0.0876877i
\(894\) 3932.00 0.147098
\(895\) 0 0
\(896\) −2304.00 −0.0859054
\(897\) 1081.00i 0.0402381i
\(898\) 20740.0i 0.770716i
\(899\) −891.000 −0.0330551
\(900\) 0 0
\(901\) −280.000 −0.0103531
\(902\) 10048.0i 0.370911i
\(903\) 324.000i 0.0119402i
\(904\) 3120.00 0.114789
\(905\) 0 0
\(906\) 70.0000 0.00256688
\(907\) 29116.0i 1.06591i 0.846143 + 0.532955i \(0.178919\pi\)
−0.846143 + 0.532955i \(0.821081\pi\)
\(908\) 16536.0i 0.604368i
\(909\) −156.000 −0.00569218
\(910\) 0 0
\(911\) 11440.0 0.416053 0.208026 0.978123i \(-0.433296\pi\)
0.208026 + 0.978123i \(0.433296\pi\)
\(912\) 576.000i 0.0209137i
\(913\) 16384.0i 0.593901i
\(914\) 20992.0 0.759687
\(915\) 0 0
\(916\) −18040.0 −0.650719
\(917\) 3834.00i 0.138070i
\(918\) 2120.00i 0.0762205i
\(919\) −2958.00 −0.106176 −0.0530878 0.998590i \(-0.516906\pi\)
−0.0530878 + 0.998590i \(0.516906\pi\)
\(920\) 0 0
\(921\) 8628.00 0.308689
\(922\) 36042.0i 1.28740i
\(923\) 32853.0i 1.17158i
\(924\) −2304.00 −0.0820303
\(925\) 0 0
\(926\) −34376.0 −1.21994
\(927\) 4160.00i 0.147392i
\(928\) 864.000i 0.0305627i
\(929\) −20907.0 −0.738360 −0.369180 0.929358i \(-0.620361\pi\)
−0.369180 + 0.929358i \(0.620361\pi\)
\(930\) 0 0
\(931\) −684.000 −0.0240786
\(932\) − 20012.0i − 0.703342i
\(933\) − 8247.00i − 0.289383i
\(934\) 30492.0 1.06823
\(935\) 0 0
\(936\) 9776.00 0.341387
\(937\) 9748.00i 0.339865i 0.985456 + 0.169932i \(0.0543549\pi\)
−0.985456 + 0.169932i \(0.945645\pi\)
\(938\) − 5616.00i − 0.195489i
\(939\) −2620.00 −0.0910548
\(940\) 0 0
\(941\) 19624.0 0.679834 0.339917 0.940455i \(-0.389601\pi\)
0.339917 + 0.940455i \(0.389601\pi\)
\(942\) 3404.00i 0.117737i
\(943\) − 3611.00i − 0.124698i
\(944\) 11904.0 0.410426
\(945\) 0 0
\(946\) −1152.00 −0.0395928
\(947\) − 41859.0i − 1.43636i −0.695856 0.718181i \(-0.744975\pi\)
0.695856 0.718181i \(-0.255025\pi\)
\(948\) 2576.00i 0.0882538i
\(949\) −28623.0 −0.979075
\(950\) 0 0
\(951\) 9906.00 0.337775
\(952\) − 2880.00i − 0.0980476i
\(953\) − 29226.0i − 0.993413i −0.867918 0.496707i \(-0.834542\pi\)
0.867918 0.496707i \(-0.165458\pi\)
\(954\) −728.000 −0.0247064
\(955\) 0 0
\(956\) −25236.0 −0.853756
\(957\) 864.000i 0.0291841i
\(958\) 17112.0i 0.577102i
\(959\) 51048.0 1.71890
\(960\) 0 0
\(961\) −28702.0 −0.963445
\(962\) − 5264.00i − 0.176422i
\(963\) 9880.00i 0.330611i
\(964\) −12152.0 −0.406006
\(965\) 0 0
\(966\) 828.000 0.0275781
\(967\) − 29849.0i − 0.992636i −0.868141 0.496318i \(-0.834685\pi\)
0.868141 0.496318i \(-0.165315\pi\)
\(968\) 2456.00i 0.0815484i
\(969\) −720.000 −0.0238697
\(970\) 0 0
\(971\) −9390.00 −0.310339 −0.155170 0.987888i \(-0.549592\pi\)
−0.155170 + 0.987888i \(0.549592\pi\)
\(972\) 8320.00i 0.274552i
\(973\) − 29358.0i − 0.967291i
\(974\) 3610.00 0.118760
\(975\) 0 0
\(976\) 8832.00 0.289657
\(977\) 33536.0i 1.09817i 0.835767 + 0.549085i \(0.185024\pi\)
−0.835767 + 0.549085i \(0.814976\pi\)
\(978\) 4090.00i 0.133726i
\(979\) −3264.00 −0.106556
\(980\) 0 0
\(981\) −6500.00 −0.211548
\(982\) 10490.0i 0.340885i
\(983\) − 28994.0i − 0.940758i −0.882465 0.470379i \(-0.844117\pi\)
0.882465 0.470379i \(-0.155883\pi\)
\(984\) 1256.00 0.0406909
\(985\) 0 0
\(986\) −1080.00 −0.0348826
\(987\) − 1170.00i − 0.0377320i
\(988\) 6768.00i 0.217934i
\(989\) 414.000 0.0133109
\(990\) 0 0
\(991\) 11272.0 0.361319 0.180659 0.983546i \(-0.442177\pi\)
0.180659 + 0.983546i \(0.442177\pi\)
\(992\) − 1056.00i − 0.0337984i
\(993\) 8115.00i 0.259337i
\(994\) 25164.0 0.802971
\(995\) 0 0
\(996\) 2048.00 0.0651540
\(997\) 61186.0i 1.94361i 0.235784 + 0.971805i \(0.424234\pi\)
−0.235784 + 0.971805i \(0.575766\pi\)
\(998\) − 18054.0i − 0.572635i
\(999\) 2968.00 0.0939974
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 1150.4.b.f.599.2 2
5.2 odd 4 1150.4.a.b.1.1 1
5.3 odd 4 230.4.a.e.1.1 1
5.4 even 2 inner 1150.4.b.f.599.1 2
15.8 even 4 2070.4.a.e.1.1 1
20.3 even 4 1840.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.e.1.1 1 5.3 odd 4
1150.4.a.b.1.1 1 5.2 odd 4
1150.4.b.f.599.1 2 5.4 even 2 inner
1150.4.b.f.599.2 2 1.1 even 1 trivial
1840.4.a.d.1.1 1 20.3 even 4
2070.4.a.e.1.1 1 15.8 even 4