Properties

Label 1150.4.b.o.599.8
Level $1150$
Weight $4$
Character 1150.599
Analytic conductor $67.852$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 124x^{6} + 4272x^{4} + 28129x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.8
Root \(8.04090i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.o.599.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} +5.04090i q^{3} -4.00000 q^{4} -10.0818 q^{6} +5.03071i q^{7} -8.00000i q^{8} +1.58932 q^{9} -5.58219 q^{11} -20.1636i q^{12} -62.7277i q^{13} -10.0614 q^{14} +16.0000 q^{16} -19.7435i q^{17} +3.17864i q^{18} -158.545 q^{19} -25.3593 q^{21} -11.1644i q^{22} +23.0000i q^{23} +40.3272 q^{24} +125.455 q^{26} +144.116i q^{27} -20.1228i q^{28} +35.5033 q^{29} +282.041 q^{31} +32.0000i q^{32} -28.1393i q^{33} +39.4870 q^{34} -6.35728 q^{36} -139.981i q^{37} -317.090i q^{38} +316.204 q^{39} +227.680 q^{41} -50.7186i q^{42} -436.962i q^{43} +22.3288 q^{44} -46.0000 q^{46} +90.2701i q^{47} +80.6544i q^{48} +317.692 q^{49} +99.5250 q^{51} +250.911i q^{52} -330.183i q^{53} -288.232 q^{54} +40.2457 q^{56} -799.209i q^{57} +71.0066i q^{58} +796.203 q^{59} -568.580 q^{61} +564.081i q^{62} +7.99541i q^{63} -64.0000 q^{64} +56.2785 q^{66} +85.1419i q^{67} +78.9740i q^{68} -115.941 q^{69} -369.578 q^{71} -12.7146i q^{72} +310.188i q^{73} +279.963 q^{74} +634.180 q^{76} -28.0824i q^{77} +632.408i q^{78} +1325.46 q^{79} -683.562 q^{81} +455.360i q^{82} +158.806i q^{83} +101.437 q^{84} +873.924 q^{86} +178.969i q^{87} +44.6575i q^{88} +1233.89 q^{89} +315.565 q^{91} -92.0000i q^{92} +1421.74i q^{93} -180.540 q^{94} -161.309 q^{96} -106.389i q^{97} +635.384i q^{98} -8.87189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 56 q^{6} - 128 q^{9} + 42 q^{11} - 32 q^{14} + 128 q^{16} - 346 q^{19} - 240 q^{21} - 224 q^{24} + 280 q^{26} + 236 q^{29} + 34 q^{31} - 224 q^{34} + 512 q^{36} + 442 q^{39} + 278 q^{41}+ \cdots + 5490 q^{99}+O(q^{100}) \) 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
\(3\) 5.04090i 0.970122i 0.874480 + 0.485061i \(0.161203\pi\)
−0.874480 + 0.485061i \(0.838797\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −10.0818 −0.685980
\(7\) 5.03071i 0.271633i 0.990734 + 0.135816i \(0.0433657\pi\)
−0.990734 + 0.135816i \(0.956634\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 1.58932 0.0588637
\(10\) 0 0
\(11\) −5.58219 −0.153008 −0.0765042 0.997069i \(-0.524376\pi\)
−0.0765042 + 0.997069i \(0.524376\pi\)
\(12\) − 20.1636i − 0.485061i
\(13\) − 62.7277i − 1.33827i −0.743140 0.669136i \(-0.766664\pi\)
0.743140 0.669136i \(-0.233336\pi\)
\(14\) −10.0614 −0.192073
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 19.7435i − 0.281677i −0.990033 0.140838i \(-0.955020\pi\)
0.990033 0.140838i \(-0.0449798\pi\)
\(18\) 3.17864i 0.0416229i
\(19\) −158.545 −1.91435 −0.957177 0.289505i \(-0.906509\pi\)
−0.957177 + 0.289505i \(0.906509\pi\)
\(20\) 0 0
\(21\) −25.3593 −0.263517
\(22\) − 11.1644i − 0.108193i
\(23\) 23.0000i 0.208514i
\(24\) 40.3272 0.342990
\(25\) 0 0
\(26\) 125.455 0.946301
\(27\) 144.116i 1.02723i
\(28\) − 20.1228i − 0.135816i
\(29\) 35.5033 0.227338 0.113669 0.993519i \(-0.463740\pi\)
0.113669 + 0.993519i \(0.463740\pi\)
\(30\) 0 0
\(31\) 282.041 1.63406 0.817032 0.576592i \(-0.195618\pi\)
0.817032 + 0.576592i \(0.195618\pi\)
\(32\) 32.0000i 0.176777i
\(33\) − 28.1393i − 0.148437i
\(34\) 39.4870 0.199175
\(35\) 0 0
\(36\) −6.35728 −0.0294319
\(37\) − 139.981i − 0.621967i −0.950415 0.310984i \(-0.899342\pi\)
0.950415 0.310984i \(-0.100658\pi\)
\(38\) − 317.090i − 1.35365i
\(39\) 316.204 1.29829
\(40\) 0 0
\(41\) 227.680 0.867260 0.433630 0.901091i \(-0.357232\pi\)
0.433630 + 0.901091i \(0.357232\pi\)
\(42\) − 50.7186i − 0.186335i
\(43\) − 436.962i − 1.54968i −0.632159 0.774838i \(-0.717831\pi\)
0.632159 0.774838i \(-0.282169\pi\)
\(44\) 22.3288 0.0765042
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) 90.2701i 0.280154i 0.990141 + 0.140077i \(0.0447350\pi\)
−0.990141 + 0.140077i \(0.955265\pi\)
\(48\) 80.6544i 0.242530i
\(49\) 317.692 0.926216
\(50\) 0 0
\(51\) 99.5250 0.273261
\(52\) 250.911i 0.669136i
\(53\) − 330.183i − 0.855737i −0.903841 0.427869i \(-0.859265\pi\)
0.903841 0.427869i \(-0.140735\pi\)
\(54\) −288.232 −0.726359
\(55\) 0 0
\(56\) 40.2457 0.0960367
\(57\) − 799.209i − 1.85716i
\(58\) 71.0066i 0.160752i
\(59\) 796.203 1.75689 0.878447 0.477839i \(-0.158580\pi\)
0.878447 + 0.477839i \(0.158580\pi\)
\(60\) 0 0
\(61\) −568.580 −1.19343 −0.596715 0.802453i \(-0.703528\pi\)
−0.596715 + 0.802453i \(0.703528\pi\)
\(62\) 564.081i 1.15546i
\(63\) 7.99541i 0.0159893i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 56.2785 0.104961
\(67\) 85.1419i 0.155250i 0.996983 + 0.0776250i \(0.0247337\pi\)
−0.996983 + 0.0776250i \(0.975266\pi\)
\(68\) 78.9740i 0.140838i
\(69\) −115.941 −0.202284
\(70\) 0 0
\(71\) −369.578 −0.617758 −0.308879 0.951101i \(-0.599954\pi\)
−0.308879 + 0.951101i \(0.599954\pi\)
\(72\) − 12.7146i − 0.0208115i
\(73\) 310.188i 0.497325i 0.968590 + 0.248662i \(0.0799910\pi\)
−0.968590 + 0.248662i \(0.920009\pi\)
\(74\) 279.963 0.439797
\(75\) 0 0
\(76\) 634.180 0.957177
\(77\) − 28.0824i − 0.0415621i
\(78\) 632.408i 0.918027i
\(79\) 1325.46 1.88766 0.943832 0.330427i \(-0.107193\pi\)
0.943832 + 0.330427i \(0.107193\pi\)
\(80\) 0 0
\(81\) −683.562 −0.937671
\(82\) 455.360i 0.613246i
\(83\) 158.806i 0.210014i 0.994471 + 0.105007i \(0.0334865\pi\)
−0.994471 + 0.105007i \(0.966513\pi\)
\(84\) 101.437 0.131758
\(85\) 0 0
\(86\) 873.924 1.09579
\(87\) 178.969i 0.220545i
\(88\) 44.6575i 0.0540967i
\(89\) 1233.89 1.46957 0.734787 0.678298i \(-0.237282\pi\)
0.734787 + 0.678298i \(0.237282\pi\)
\(90\) 0 0
\(91\) 315.565 0.363519
\(92\) − 92.0000i − 0.104257i
\(93\) 1421.74i 1.58524i
\(94\) −180.540 −0.198099
\(95\) 0 0
\(96\) −161.309 −0.171495
\(97\) − 106.389i − 0.111363i −0.998449 0.0556814i \(-0.982267\pi\)
0.998449 0.0556814i \(-0.0177331\pi\)
\(98\) 635.384i 0.654933i
\(99\) −8.87189 −0.00900665
\(100\) 0 0
\(101\) 642.676 0.633155 0.316578 0.948567i \(-0.397466\pi\)
0.316578 + 0.948567i \(0.397466\pi\)
\(102\) 199.050i 0.193224i
\(103\) − 1621.99i − 1.55165i −0.630951 0.775823i \(-0.717335\pi\)
0.630951 0.775823i \(-0.282665\pi\)
\(104\) −501.822 −0.473150
\(105\) 0 0
\(106\) 660.365 0.605098
\(107\) 1490.86i 1.34698i 0.739196 + 0.673490i \(0.235205\pi\)
−0.739196 + 0.673490i \(0.764795\pi\)
\(108\) − 576.464i − 0.513613i
\(109\) −1204.00 −1.05800 −0.529001 0.848621i \(-0.677433\pi\)
−0.529001 + 0.848621i \(0.677433\pi\)
\(110\) 0 0
\(111\) 705.632 0.603384
\(112\) 80.4914i 0.0679082i
\(113\) − 276.771i − 0.230411i −0.993342 0.115206i \(-0.963247\pi\)
0.993342 0.115206i \(-0.0367527\pi\)
\(114\) 1598.42 1.31321
\(115\) 0 0
\(116\) −142.013 −0.113669
\(117\) − 99.6944i − 0.0787757i
\(118\) 1592.41i 1.24231i
\(119\) 99.3239 0.0765126
\(120\) 0 0
\(121\) −1299.84 −0.976588
\(122\) − 1137.16i − 0.843883i
\(123\) 1147.71i 0.841348i
\(124\) −1128.16 −0.817032
\(125\) 0 0
\(126\) −15.9908 −0.0113062
\(127\) 1552.79i 1.08494i 0.840075 + 0.542471i \(0.182511\pi\)
−0.840075 + 0.542471i \(0.817489\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) 2202.68 1.50338
\(130\) 0 0
\(131\) −313.873 −0.209337 −0.104669 0.994507i \(-0.533378\pi\)
−0.104669 + 0.994507i \(0.533378\pi\)
\(132\) 112.557i 0.0742184i
\(133\) − 797.594i − 0.520001i
\(134\) −170.284 −0.109778
\(135\) 0 0
\(136\) −157.948 −0.0995877
\(137\) − 1066.09i − 0.664835i −0.943132 0.332417i \(-0.892136\pi\)
0.943132 0.332417i \(-0.107864\pi\)
\(138\) − 231.881i − 0.143037i
\(139\) 1594.79 0.973152 0.486576 0.873638i \(-0.338246\pi\)
0.486576 + 0.873638i \(0.338246\pi\)
\(140\) 0 0
\(141\) −455.043 −0.271784
\(142\) − 739.155i − 0.436821i
\(143\) 350.158i 0.204767i
\(144\) 25.4291 0.0147159
\(145\) 0 0
\(146\) −620.375 −0.351662
\(147\) 1601.45i 0.898542i
\(148\) 559.925i 0.310984i
\(149\) 1096.91 0.603102 0.301551 0.953450i \(-0.402496\pi\)
0.301551 + 0.953450i \(0.402496\pi\)
\(150\) 0 0
\(151\) 2734.41 1.47366 0.736832 0.676076i \(-0.236321\pi\)
0.736832 + 0.676076i \(0.236321\pi\)
\(152\) 1268.36i 0.676826i
\(153\) − 31.3788i − 0.0165805i
\(154\) 56.1647 0.0293889
\(155\) 0 0
\(156\) −1264.82 −0.649143
\(157\) − 1439.63i − 0.731813i −0.930652 0.365907i \(-0.880759\pi\)
0.930652 0.365907i \(-0.119241\pi\)
\(158\) 2650.91i 1.33478i
\(159\) 1664.42 0.830169
\(160\) 0 0
\(161\) −115.706 −0.0566394
\(162\) − 1367.12i − 0.663034i
\(163\) 2995.82i 1.43957i 0.694195 + 0.719787i \(0.255760\pi\)
−0.694195 + 0.719787i \(0.744240\pi\)
\(164\) −910.721 −0.433630
\(165\) 0 0
\(166\) −317.611 −0.148503
\(167\) − 351.532i − 0.162888i −0.996678 0.0814442i \(-0.974047\pi\)
0.996678 0.0814442i \(-0.0259532\pi\)
\(168\) 202.874i 0.0931673i
\(169\) −1737.76 −0.790971
\(170\) 0 0
\(171\) −251.979 −0.112686
\(172\) 1747.85i 0.774838i
\(173\) 3701.77i 1.62682i 0.581690 + 0.813411i \(0.302392\pi\)
−0.581690 + 0.813411i \(0.697608\pi\)
\(174\) −357.937 −0.155949
\(175\) 0 0
\(176\) −89.3150 −0.0382521
\(177\) 4013.58i 1.70440i
\(178\) 2467.78i 1.03915i
\(179\) −3584.09 −1.49658 −0.748290 0.663372i \(-0.769125\pi\)
−0.748290 + 0.663372i \(0.769125\pi\)
\(180\) 0 0
\(181\) 783.672 0.321822 0.160911 0.986969i \(-0.448557\pi\)
0.160911 + 0.986969i \(0.448557\pi\)
\(182\) 631.130i 0.257046i
\(183\) − 2866.16i − 1.15777i
\(184\) 184.000 0.0737210
\(185\) 0 0
\(186\) −2843.48 −1.12093
\(187\) 110.212i 0.0430989i
\(188\) − 361.080i − 0.140077i
\(189\) −725.005 −0.279029
\(190\) 0 0
\(191\) 1491.95 0.565201 0.282601 0.959238i \(-0.408803\pi\)
0.282601 + 0.959238i \(0.408803\pi\)
\(192\) − 322.618i − 0.121265i
\(193\) − 3091.09i − 1.15286i −0.817147 0.576430i \(-0.804446\pi\)
0.817147 0.576430i \(-0.195554\pi\)
\(194\) 212.778 0.0787454
\(195\) 0 0
\(196\) −1270.77 −0.463108
\(197\) 2820.30i 1.01999i 0.860178 + 0.509995i \(0.170353\pi\)
−0.860178 + 0.509995i \(0.829647\pi\)
\(198\) − 17.7438i − 0.00636866i
\(199\) −1316.11 −0.468825 −0.234413 0.972137i \(-0.575317\pi\)
−0.234413 + 0.972137i \(0.575317\pi\)
\(200\) 0 0
\(201\) −429.192 −0.150611
\(202\) 1285.35i 0.447708i
\(203\) 178.607i 0.0617524i
\(204\) −398.100 −0.136630
\(205\) 0 0
\(206\) 3243.98 1.09718
\(207\) 36.5544i 0.0122739i
\(208\) − 1003.64i − 0.334568i
\(209\) 885.028 0.292912
\(210\) 0 0
\(211\) −3399.19 −1.10905 −0.554525 0.832167i \(-0.687100\pi\)
−0.554525 + 0.832167i \(0.687100\pi\)
\(212\) 1320.73i 0.427869i
\(213\) − 1863.00i − 0.599300i
\(214\) −2981.72 −0.952458
\(215\) 0 0
\(216\) 1152.93 0.363180
\(217\) 1418.86i 0.443865i
\(218\) − 2408.00i − 0.748121i
\(219\) −1563.62 −0.482466
\(220\) 0 0
\(221\) −1238.46 −0.376960
\(222\) 1411.26i 0.426657i
\(223\) − 864.660i − 0.259650i −0.991537 0.129825i \(-0.958558\pi\)
0.991537 0.129825i \(-0.0414416\pi\)
\(224\) −160.983 −0.0480184
\(225\) 0 0
\(226\) 553.543 0.162925
\(227\) − 1979.40i − 0.578754i −0.957215 0.289377i \(-0.906552\pi\)
0.957215 0.289377i \(-0.0934481\pi\)
\(228\) 3196.84i 0.928578i
\(229\) 3113.53 0.898462 0.449231 0.893416i \(-0.351698\pi\)
0.449231 + 0.893416i \(0.351698\pi\)
\(230\) 0 0
\(231\) 141.560 0.0403203
\(232\) − 284.026i − 0.0803761i
\(233\) − 6018.65i − 1.69225i −0.532983 0.846126i \(-0.678929\pi\)
0.532983 0.846126i \(-0.321071\pi\)
\(234\) 199.389 0.0557028
\(235\) 0 0
\(236\) −3184.81 −0.878447
\(237\) 6681.49i 1.83126i
\(238\) 198.648i 0.0541026i
\(239\) 4224.88 1.14345 0.571725 0.820445i \(-0.306274\pi\)
0.571725 + 0.820445i \(0.306274\pi\)
\(240\) 0 0
\(241\) 1394.24 0.372658 0.186329 0.982487i \(-0.440341\pi\)
0.186329 + 0.982487i \(0.440341\pi\)
\(242\) − 2599.68i − 0.690552i
\(243\) 445.360i 0.117571i
\(244\) 2274.32 0.596715
\(245\) 0 0
\(246\) −2295.43 −0.594923
\(247\) 9945.16i 2.56192i
\(248\) − 2256.32i − 0.577729i
\(249\) −800.523 −0.203739
\(250\) 0 0
\(251\) −5968.83 −1.50099 −0.750497 0.660874i \(-0.770186\pi\)
−0.750497 + 0.660874i \(0.770186\pi\)
\(252\) − 31.9816i − 0.00799466i
\(253\) − 128.390i − 0.0319045i
\(254\) −3105.57 −0.767169
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 1636.06i − 0.397100i −0.980091 0.198550i \(-0.936377\pi\)
0.980091 0.198550i \(-0.0636232\pi\)
\(258\) 4405.37i 1.06305i
\(259\) 704.206 0.168947
\(260\) 0 0
\(261\) 56.4261 0.0133820
\(262\) − 627.745i − 0.148024i
\(263\) − 2995.55i − 0.702333i −0.936313 0.351166i \(-0.885785\pi\)
0.936313 0.351166i \(-0.114215\pi\)
\(264\) −225.114 −0.0524803
\(265\) 0 0
\(266\) 1595.19 0.367696
\(267\) 6219.92i 1.42567i
\(268\) − 340.568i − 0.0776250i
\(269\) 5190.17 1.17640 0.588198 0.808717i \(-0.299838\pi\)
0.588198 + 0.808717i \(0.299838\pi\)
\(270\) 0 0
\(271\) 6720.43 1.50641 0.753205 0.657786i \(-0.228507\pi\)
0.753205 + 0.657786i \(0.228507\pi\)
\(272\) − 315.896i − 0.0704192i
\(273\) 1590.73i 0.352657i
\(274\) 2132.18 0.470109
\(275\) 0 0
\(276\) 463.763 0.101142
\(277\) − 6114.74i − 1.32635i −0.748464 0.663176i \(-0.769208\pi\)
0.748464 0.663176i \(-0.230792\pi\)
\(278\) 3189.58i 0.688123i
\(279\) 448.253 0.0961871
\(280\) 0 0
\(281\) −3549.07 −0.753450 −0.376725 0.926325i \(-0.622950\pi\)
−0.376725 + 0.926325i \(0.622950\pi\)
\(282\) − 910.085i − 0.192180i
\(283\) − 3103.11i − 0.651806i −0.945403 0.325903i \(-0.894332\pi\)
0.945403 0.325903i \(-0.105668\pi\)
\(284\) 1478.31 0.308879
\(285\) 0 0
\(286\) −700.316 −0.144792
\(287\) 1145.39i 0.235576i
\(288\) 50.8583i 0.0104057i
\(289\) 4523.19 0.920658
\(290\) 0 0
\(291\) 536.297 0.108035
\(292\) − 1240.75i − 0.248662i
\(293\) − 1218.51i − 0.242957i −0.992594 0.121478i \(-0.961237\pi\)
0.992594 0.121478i \(-0.0387635\pi\)
\(294\) −3202.91 −0.635365
\(295\) 0 0
\(296\) −1119.85 −0.219899
\(297\) − 804.482i − 0.157174i
\(298\) 2193.82i 0.426458i
\(299\) 1442.74 0.279049
\(300\) 0 0
\(301\) 2198.23 0.420943
\(302\) 5468.82i 1.04204i
\(303\) 3239.67i 0.614238i
\(304\) −2536.72 −0.478588
\(305\) 0 0
\(306\) 62.7575 0.0117242
\(307\) 2564.45i 0.476745i 0.971174 + 0.238373i \(0.0766140\pi\)
−0.971174 + 0.238373i \(0.923386\pi\)
\(308\) 112.329i 0.0207811i
\(309\) 8176.30 1.50529
\(310\) 0 0
\(311\) −92.6253 −0.0168884 −0.00844421 0.999964i \(-0.502688\pi\)
−0.00844421 + 0.999964i \(0.502688\pi\)
\(312\) − 2529.63i − 0.459014i
\(313\) 4276.44i 0.772263i 0.922444 + 0.386132i \(0.126189\pi\)
−0.922444 + 0.386132i \(0.873811\pi\)
\(314\) 2879.25 0.517470
\(315\) 0 0
\(316\) −5301.82 −0.943832
\(317\) − 2964.24i − 0.525199i −0.964905 0.262600i \(-0.915420\pi\)
0.964905 0.262600i \(-0.0845798\pi\)
\(318\) 3328.84i 0.587018i
\(319\) −198.186 −0.0347846
\(320\) 0 0
\(321\) −7515.27 −1.30673
\(322\) − 231.413i − 0.0400501i
\(323\) 3130.23i 0.539229i
\(324\) 2734.25 0.468836
\(325\) 0 0
\(326\) −5991.63 −1.01793
\(327\) − 6069.24i − 1.02639i
\(328\) − 1821.44i − 0.306623i
\(329\) −454.123 −0.0760991
\(330\) 0 0
\(331\) −1348.81 −0.223979 −0.111989 0.993709i \(-0.535722\pi\)
−0.111989 + 0.993709i \(0.535722\pi\)
\(332\) − 635.223i − 0.105007i
\(333\) − 222.475i − 0.0366113i
\(334\) 703.064 0.115180
\(335\) 0 0
\(336\) −405.749 −0.0658792
\(337\) − 5764.30i − 0.931755i −0.884849 0.465878i \(-0.845739\pi\)
0.884849 0.465878i \(-0.154261\pi\)
\(338\) − 3475.53i − 0.559301i
\(339\) 1395.18 0.223527
\(340\) 0 0
\(341\) −1574.40 −0.250026
\(342\) − 503.958i − 0.0796810i
\(343\) 3323.75i 0.523223i
\(344\) −3495.70 −0.547894
\(345\) 0 0
\(346\) −7403.53 −1.15034
\(347\) 4066.16i 0.629057i 0.949248 + 0.314528i \(0.101846\pi\)
−0.949248 + 0.314528i \(0.898154\pi\)
\(348\) − 715.875i − 0.110273i
\(349\) −8407.82 −1.28957 −0.644786 0.764363i \(-0.723053\pi\)
−0.644786 + 0.764363i \(0.723053\pi\)
\(350\) 0 0
\(351\) 9040.06 1.37471
\(352\) − 178.630i − 0.0270483i
\(353\) − 256.068i − 0.0386095i −0.999814 0.0193047i \(-0.993855\pi\)
0.999814 0.0193047i \(-0.00614527\pi\)
\(354\) −8027.16 −1.20519
\(355\) 0 0
\(356\) −4935.56 −0.734787
\(357\) 500.682i 0.0742266i
\(358\) − 7168.19i − 1.05824i
\(359\) 12052.5 1.77189 0.885943 0.463794i \(-0.153512\pi\)
0.885943 + 0.463794i \(0.153512\pi\)
\(360\) 0 0
\(361\) 18277.5 2.66475
\(362\) 1567.34i 0.227563i
\(363\) − 6552.36i − 0.947410i
\(364\) −1262.26 −0.181759
\(365\) 0 0
\(366\) 5732.31 0.818669
\(367\) − 9245.40i − 1.31500i −0.753453 0.657502i \(-0.771613\pi\)
0.753453 0.657502i \(-0.228387\pi\)
\(368\) 368.000i 0.0521286i
\(369\) 361.857 0.0510502
\(370\) 0 0
\(371\) 1661.05 0.232446
\(372\) − 5686.96i − 0.792621i
\(373\) 869.265i 0.120667i 0.998178 + 0.0603335i \(0.0192164\pi\)
−0.998178 + 0.0603335i \(0.980784\pi\)
\(374\) −220.424 −0.0304755
\(375\) 0 0
\(376\) 722.161 0.0990495
\(377\) − 2227.04i − 0.304240i
\(378\) − 1450.01i − 0.197303i
\(379\) −1278.76 −0.173313 −0.0866566 0.996238i \(-0.527618\pi\)
−0.0866566 + 0.996238i \(0.527618\pi\)
\(380\) 0 0
\(381\) −7827.44 −1.05252
\(382\) 2983.89i 0.399658i
\(383\) 5343.75i 0.712932i 0.934308 + 0.356466i \(0.116018\pi\)
−0.934308 + 0.356466i \(0.883982\pi\)
\(384\) 645.235 0.0857475
\(385\) 0 0
\(386\) 6182.19 0.815194
\(387\) − 694.473i − 0.0912198i
\(388\) 425.557i 0.0556814i
\(389\) 708.346 0.0923254 0.0461627 0.998934i \(-0.485301\pi\)
0.0461627 + 0.998934i \(0.485301\pi\)
\(390\) 0 0
\(391\) 454.101 0.0587336
\(392\) − 2541.54i − 0.327467i
\(393\) − 1582.20i − 0.203083i
\(394\) −5640.60 −0.721241
\(395\) 0 0
\(396\) 35.4875 0.00450332
\(397\) − 5924.01i − 0.748911i −0.927245 0.374455i \(-0.877830\pi\)
0.927245 0.374455i \(-0.122170\pi\)
\(398\) − 2632.21i − 0.331510i
\(399\) 4020.59 0.504464
\(400\) 0 0
\(401\) −11393.6 −1.41888 −0.709441 0.704765i \(-0.751052\pi\)
−0.709441 + 0.704765i \(0.751052\pi\)
\(402\) − 858.384i − 0.106498i
\(403\) − 17691.8i − 2.18682i
\(404\) −2570.71 −0.316578
\(405\) 0 0
\(406\) −357.214 −0.0436656
\(407\) 781.402i 0.0951663i
\(408\) − 796.200i − 0.0966122i
\(409\) −13804.4 −1.66891 −0.834457 0.551073i \(-0.814218\pi\)
−0.834457 + 0.551073i \(0.814218\pi\)
\(410\) 0 0
\(411\) 5374.06 0.644971
\(412\) 6487.96i 0.775823i
\(413\) 4005.47i 0.477230i
\(414\) −73.1087 −0.00867898
\(415\) 0 0
\(416\) 2007.29 0.236575
\(417\) 8039.17i 0.944076i
\(418\) 1770.06i 0.207120i
\(419\) 14795.8 1.72512 0.862558 0.505958i \(-0.168861\pi\)
0.862558 + 0.505958i \(0.168861\pi\)
\(420\) 0 0
\(421\) 5804.66 0.671976 0.335988 0.941866i \(-0.390930\pi\)
0.335988 + 0.941866i \(0.390930\pi\)
\(422\) − 6798.37i − 0.784217i
\(423\) 143.468i 0.0164909i
\(424\) −2641.46 −0.302549
\(425\) 0 0
\(426\) 3726.01 0.423769
\(427\) − 2860.36i − 0.324175i
\(428\) − 5963.44i − 0.673490i
\(429\) −1765.11 −0.198649
\(430\) 0 0
\(431\) 15268.2 1.70636 0.853181 0.521615i \(-0.174670\pi\)
0.853181 + 0.521615i \(0.174670\pi\)
\(432\) 2305.85i 0.256807i
\(433\) − 2250.62i − 0.249788i −0.992170 0.124894i \(-0.960141\pi\)
0.992170 0.124894i \(-0.0398591\pi\)
\(434\) −2837.73 −0.313860
\(435\) 0 0
\(436\) 4816.00 0.529001
\(437\) − 3646.53i − 0.399170i
\(438\) − 3127.25i − 0.341155i
\(439\) −10366.2 −1.12700 −0.563499 0.826117i \(-0.690545\pi\)
−0.563499 + 0.826117i \(0.690545\pi\)
\(440\) 0 0
\(441\) 504.914 0.0545205
\(442\) − 2476.93i − 0.266551i
\(443\) 3881.11i 0.416246i 0.978103 + 0.208123i \(0.0667355\pi\)
−0.978103 + 0.208123i \(0.933264\pi\)
\(444\) −2822.53 −0.301692
\(445\) 0 0
\(446\) 1729.32 0.183600
\(447\) 5529.41i 0.585083i
\(448\) − 321.965i − 0.0339541i
\(449\) 10146.5 1.06647 0.533233 0.845968i \(-0.320977\pi\)
0.533233 + 0.845968i \(0.320977\pi\)
\(450\) 0 0
\(451\) −1270.95 −0.132698
\(452\) 1107.09i 0.115206i
\(453\) 13783.9i 1.42963i
\(454\) 3958.79 0.409241
\(455\) 0 0
\(456\) −6393.68 −0.656604
\(457\) − 4728.87i − 0.484042i −0.970271 0.242021i \(-0.922190\pi\)
0.970271 0.242021i \(-0.0778103\pi\)
\(458\) 6227.06i 0.635308i
\(459\) 2845.35 0.289346
\(460\) 0 0
\(461\) 8380.69 0.846698 0.423349 0.905967i \(-0.360854\pi\)
0.423349 + 0.905967i \(0.360854\pi\)
\(462\) 283.121i 0.0285108i
\(463\) − 6845.17i − 0.687089i −0.939136 0.343545i \(-0.888372\pi\)
0.939136 0.343545i \(-0.111628\pi\)
\(464\) 568.053 0.0568345
\(465\) 0 0
\(466\) 12037.3 1.19660
\(467\) − 14884.0i − 1.47484i −0.675436 0.737418i \(-0.736045\pi\)
0.675436 0.737418i \(-0.263955\pi\)
\(468\) 398.778i 0.0393878i
\(469\) −428.324 −0.0421710
\(470\) 0 0
\(471\) 7257.01 0.709948
\(472\) − 6369.62i − 0.621156i
\(473\) 2439.20i 0.237114i
\(474\) −13363.0 −1.29490
\(475\) 0 0
\(476\) −397.295 −0.0382563
\(477\) − 524.766i − 0.0503719i
\(478\) 8449.76i 0.808542i
\(479\) −6048.69 −0.576976 −0.288488 0.957483i \(-0.593153\pi\)
−0.288488 + 0.957483i \(0.593153\pi\)
\(480\) 0 0
\(481\) −8780.71 −0.832361
\(482\) 2788.47i 0.263509i
\(483\) − 583.264i − 0.0549471i
\(484\) 5199.36 0.488294
\(485\) 0 0
\(486\) −890.720 −0.0831355
\(487\) 3199.35i 0.297693i 0.988860 + 0.148846i \(0.0475560\pi\)
−0.988860 + 0.148846i \(0.952444\pi\)
\(488\) 4548.64i 0.421941i
\(489\) −15101.6 −1.39656
\(490\) 0 0
\(491\) −1614.41 −0.148385 −0.0741926 0.997244i \(-0.523638\pi\)
−0.0741926 + 0.997244i \(0.523638\pi\)
\(492\) − 4590.85i − 0.420674i
\(493\) − 700.960i − 0.0640358i
\(494\) −19890.3 −1.81155
\(495\) 0 0
\(496\) 4512.65 0.408516
\(497\) − 1859.24i − 0.167803i
\(498\) − 1601.05i − 0.144066i
\(499\) 5392.04 0.483729 0.241864 0.970310i \(-0.422241\pi\)
0.241864 + 0.970310i \(0.422241\pi\)
\(500\) 0 0
\(501\) 1772.04 0.158022
\(502\) − 11937.7i − 1.06136i
\(503\) − 12625.4i − 1.11917i −0.828775 0.559583i \(-0.810961\pi\)
0.828775 0.559583i \(-0.189039\pi\)
\(504\) 63.9633 0.00565308
\(505\) 0 0
\(506\) 256.781 0.0225599
\(507\) − 8759.89i − 0.767338i
\(508\) − 6211.15i − 0.542471i
\(509\) −7065.76 −0.615294 −0.307647 0.951501i \(-0.599541\pi\)
−0.307647 + 0.951501i \(0.599541\pi\)
\(510\) 0 0
\(511\) −1560.46 −0.135090
\(512\) 512.000i 0.0441942i
\(513\) − 22848.9i − 1.96647i
\(514\) 3272.12 0.280792
\(515\) 0 0
\(516\) −8810.73 −0.751688
\(517\) − 503.905i − 0.0428660i
\(518\) 1408.41i 0.119463i
\(519\) −18660.2 −1.57822
\(520\) 0 0
\(521\) −8399.20 −0.706287 −0.353144 0.935569i \(-0.614887\pi\)
−0.353144 + 0.935569i \(0.614887\pi\)
\(522\) 112.852i 0.00946247i
\(523\) 7002.06i 0.585428i 0.956200 + 0.292714i \(0.0945584\pi\)
−0.956200 + 0.292714i \(0.905442\pi\)
\(524\) 1255.49 0.104669
\(525\) 0 0
\(526\) 5991.10 0.496624
\(527\) − 5568.47i − 0.460278i
\(528\) − 450.228i − 0.0371092i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) 1265.42 0.103417
\(532\) 3190.37i 0.260001i
\(533\) − 14281.9i − 1.16063i
\(534\) −12439.8 −1.00810
\(535\) 0 0
\(536\) 681.136 0.0548891
\(537\) − 18067.1i − 1.45186i
\(538\) 10380.3i 0.831838i
\(539\) −1773.42 −0.141719
\(540\) 0 0
\(541\) 3338.13 0.265282 0.132641 0.991164i \(-0.457654\pi\)
0.132641 + 0.991164i \(0.457654\pi\)
\(542\) 13440.9i 1.06519i
\(543\) 3950.41i 0.312207i
\(544\) 631.792 0.0497939
\(545\) 0 0
\(546\) −3181.46 −0.249366
\(547\) 9609.09i 0.751106i 0.926801 + 0.375553i \(0.122547\pi\)
−0.926801 + 0.375553i \(0.877453\pi\)
\(548\) 4264.37i 0.332417i
\(549\) −903.656 −0.0702497
\(550\) 0 0
\(551\) −5628.87 −0.435205
\(552\) 927.526i 0.0715183i
\(553\) 6667.98i 0.512751i
\(554\) 12229.5 0.937872
\(555\) 0 0
\(556\) −6379.15 −0.486576
\(557\) − 11931.1i − 0.907608i −0.891102 0.453804i \(-0.850067\pi\)
0.891102 0.453804i \(-0.149933\pi\)
\(558\) 896.506i 0.0680146i
\(559\) −27409.6 −2.07389
\(560\) 0 0
\(561\) −555.567 −0.0418112
\(562\) − 7098.13i − 0.532770i
\(563\) 10082.4i 0.754746i 0.926061 + 0.377373i \(0.123173\pi\)
−0.926061 + 0.377373i \(0.876827\pi\)
\(564\) 1820.17 0.135892
\(565\) 0 0
\(566\) 6206.23 0.460896
\(567\) − 3438.80i − 0.254702i
\(568\) 2956.62i 0.218410i
\(569\) 18710.5 1.37853 0.689267 0.724507i \(-0.257933\pi\)
0.689267 + 0.724507i \(0.257933\pi\)
\(570\) 0 0
\(571\) 13510.1 0.990159 0.495079 0.868848i \(-0.335139\pi\)
0.495079 + 0.868848i \(0.335139\pi\)
\(572\) − 1400.63i − 0.102383i
\(573\) 7520.75i 0.548314i
\(574\) −2290.79 −0.166578
\(575\) 0 0
\(576\) −101.717 −0.00735797
\(577\) 11582.3i 0.835662i 0.908525 + 0.417831i \(0.137210\pi\)
−0.908525 + 0.417831i \(0.862790\pi\)
\(578\) 9046.39i 0.651004i
\(579\) 15581.9 1.11841
\(580\) 0 0
\(581\) −798.905 −0.0570468
\(582\) 1072.59i 0.0763926i
\(583\) 1843.14i 0.130935i
\(584\) 2481.50 0.175831
\(585\) 0 0
\(586\) 2437.03 0.171796
\(587\) − 6019.34i − 0.423245i −0.977351 0.211623i \(-0.932125\pi\)
0.977351 0.211623i \(-0.0678747\pi\)
\(588\) − 6405.81i − 0.449271i
\(589\) −44716.1 −3.12818
\(590\) 0 0
\(591\) −14216.8 −0.989514
\(592\) − 2239.70i − 0.155492i
\(593\) − 26299.3i − 1.82122i −0.413270 0.910608i \(-0.635613\pi\)
0.413270 0.910608i \(-0.364387\pi\)
\(594\) 1608.96 0.111139
\(595\) 0 0
\(596\) −4387.63 −0.301551
\(597\) − 6634.36i − 0.454818i
\(598\) 2885.47i 0.197317i
\(599\) 27368.0 1.86682 0.933409 0.358813i \(-0.116819\pi\)
0.933409 + 0.358813i \(0.116819\pi\)
\(600\) 0 0
\(601\) −3472.33 −0.235673 −0.117836 0.993033i \(-0.537596\pi\)
−0.117836 + 0.993033i \(0.537596\pi\)
\(602\) 4396.46i 0.297652i
\(603\) 135.318i 0.00913859i
\(604\) −10937.6 −0.736832
\(605\) 0 0
\(606\) −6479.33 −0.434332
\(607\) 5061.10i 0.338425i 0.985580 + 0.169212i \(0.0541224\pi\)
−0.985580 + 0.169212i \(0.945878\pi\)
\(608\) − 5073.44i − 0.338413i
\(609\) −900.339 −0.0599074
\(610\) 0 0
\(611\) 5662.44 0.374922
\(612\) 125.515i 0.00829027i
\(613\) 28272.6i 1.86283i 0.363953 + 0.931417i \(0.381427\pi\)
−0.363953 + 0.931417i \(0.618573\pi\)
\(614\) −5128.90 −0.337110
\(615\) 0 0
\(616\) −224.659 −0.0146944
\(617\) − 2222.47i − 0.145013i −0.997368 0.0725066i \(-0.976900\pi\)
0.997368 0.0725066i \(-0.0230999\pi\)
\(618\) 16352.6i 1.06440i
\(619\) 10498.2 0.681679 0.340840 0.940121i \(-0.389289\pi\)
0.340840 + 0.940121i \(0.389289\pi\)
\(620\) 0 0
\(621\) −3314.67 −0.214192
\(622\) − 185.251i − 0.0119419i
\(623\) 6207.34i 0.399185i
\(624\) 5059.27 0.324572
\(625\) 0 0
\(626\) −8552.87 −0.546073
\(627\) 4461.34i 0.284160i
\(628\) 5758.51i 0.365907i
\(629\) −2763.72 −0.175194
\(630\) 0 0
\(631\) −11402.2 −0.719355 −0.359678 0.933077i \(-0.617113\pi\)
−0.359678 + 0.933077i \(0.617113\pi\)
\(632\) − 10603.6i − 0.667390i
\(633\) − 17135.0i − 1.07591i
\(634\) 5928.47 0.371372
\(635\) 0 0
\(636\) −6657.67 −0.415085
\(637\) − 19928.1i − 1.23953i
\(638\) − 396.372i − 0.0245964i
\(639\) −587.377 −0.0363635
\(640\) 0 0
\(641\) 26128.3 1.60999 0.804996 0.593280i \(-0.202167\pi\)
0.804996 + 0.593280i \(0.202167\pi\)
\(642\) − 15030.5i − 0.924001i
\(643\) − 863.502i − 0.0529599i −0.999649 0.0264799i \(-0.991570\pi\)
0.999649 0.0264799i \(-0.00842981\pi\)
\(644\) 462.825 0.0283197
\(645\) 0 0
\(646\) −6260.47 −0.381292
\(647\) − 5477.43i − 0.332828i −0.986056 0.166414i \(-0.946781\pi\)
0.986056 0.166414i \(-0.0532188\pi\)
\(648\) 5468.50i 0.331517i
\(649\) −4444.55 −0.268820
\(650\) 0 0
\(651\) −7152.36 −0.430604
\(652\) − 11983.3i − 0.719787i
\(653\) − 19233.2i − 1.15261i −0.817235 0.576305i \(-0.804494\pi\)
0.817235 0.576305i \(-0.195506\pi\)
\(654\) 12138.5 0.725768
\(655\) 0 0
\(656\) 3642.88 0.216815
\(657\) 492.987i 0.0292744i
\(658\) − 908.245i − 0.0538102i
\(659\) −9952.51 −0.588307 −0.294154 0.955758i \(-0.595038\pi\)
−0.294154 + 0.955758i \(0.595038\pi\)
\(660\) 0 0
\(661\) 3047.96 0.179352 0.0896762 0.995971i \(-0.471417\pi\)
0.0896762 + 0.995971i \(0.471417\pi\)
\(662\) − 2697.61i − 0.158377i
\(663\) − 6242.98i − 0.365697i
\(664\) 1270.45 0.0742513
\(665\) 0 0
\(666\) 444.950 0.0258881
\(667\) 816.576i 0.0474032i
\(668\) 1406.13i 0.0814442i
\(669\) 4358.67 0.251892
\(670\) 0 0
\(671\) 3173.92 0.182605
\(672\) − 811.498i − 0.0465836i
\(673\) − 16245.9i − 0.930508i −0.885177 0.465254i \(-0.845963\pi\)
0.885177 0.465254i \(-0.154037\pi\)
\(674\) 11528.6 0.658850
\(675\) 0 0
\(676\) 6951.05 0.395486
\(677\) − 30056.6i − 1.70631i −0.521659 0.853154i \(-0.674687\pi\)
0.521659 0.853154i \(-0.325313\pi\)
\(678\) 2790.35i 0.158057i
\(679\) 535.213 0.0302498
\(680\) 0 0
\(681\) 9977.94 0.561462
\(682\) − 3148.81i − 0.176795i
\(683\) − 10064.8i − 0.563866i −0.959434 0.281933i \(-0.909024\pi\)
0.959434 0.281933i \(-0.0909755\pi\)
\(684\) 1007.92 0.0563430
\(685\) 0 0
\(686\) −6647.50 −0.369975
\(687\) 15695.0i 0.871617i
\(688\) − 6991.39i − 0.387419i
\(689\) −20711.6 −1.14521
\(690\) 0 0
\(691\) 10260.7 0.564886 0.282443 0.959284i \(-0.408855\pi\)
0.282443 + 0.959284i \(0.408855\pi\)
\(692\) − 14807.1i − 0.813411i
\(693\) − 44.6319i − 0.00244650i
\(694\) −8132.31 −0.444810
\(695\) 0 0
\(696\) 1431.75 0.0779746
\(697\) − 4495.21i − 0.244287i
\(698\) − 16815.6i − 0.911865i
\(699\) 30339.4 1.64169
\(700\) 0 0
\(701\) −19815.6 −1.06766 −0.533828 0.845593i \(-0.679247\pi\)
−0.533828 + 0.845593i \(0.679247\pi\)
\(702\) 18080.1i 0.972066i
\(703\) 22193.3i 1.19067i
\(704\) 357.260 0.0191261
\(705\) 0 0
\(706\) 512.136 0.0273010
\(707\) 3233.12i 0.171986i
\(708\) − 16054.3i − 0.852201i
\(709\) 30533.5 1.61736 0.808681 0.588247i \(-0.200182\pi\)
0.808681 + 0.588247i \(0.200182\pi\)
\(710\) 0 0
\(711\) 2106.57 0.111115
\(712\) − 9871.12i − 0.519573i
\(713\) 6486.93i 0.340726i
\(714\) −1001.36 −0.0524861
\(715\) 0 0
\(716\) 14336.4 0.748290
\(717\) 21297.2i 1.10929i
\(718\) 24105.0i 1.25291i
\(719\) −9125.45 −0.473327 −0.236663 0.971592i \(-0.576054\pi\)
−0.236663 + 0.971592i \(0.576054\pi\)
\(720\) 0 0
\(721\) 8159.77 0.421478
\(722\) 36555.0i 1.88426i
\(723\) 7028.21i 0.361524i
\(724\) −3134.69 −0.160911
\(725\) 0 0
\(726\) 13104.7 0.669920
\(727\) 2635.53i 0.134452i 0.997738 + 0.0672259i \(0.0214148\pi\)
−0.997738 + 0.0672259i \(0.978585\pi\)
\(728\) − 2524.52i − 0.128523i
\(729\) −20701.2 −1.05173
\(730\) 0 0
\(731\) −8627.17 −0.436508
\(732\) 11464.6i 0.578886i
\(733\) − 10778.0i − 0.543102i −0.962424 0.271551i \(-0.912463\pi\)
0.962424 0.271551i \(-0.0875366\pi\)
\(734\) 18490.8 0.929848
\(735\) 0 0
\(736\) −736.000 −0.0368605
\(737\) − 475.278i − 0.0237545i
\(738\) 723.714i 0.0360979i
\(739\) 1758.72 0.0875449 0.0437725 0.999042i \(-0.486062\pi\)
0.0437725 + 0.999042i \(0.486062\pi\)
\(740\) 0 0
\(741\) −50132.6 −2.48538
\(742\) 3322.11i 0.164364i
\(743\) − 9737.97i − 0.480823i −0.970671 0.240412i \(-0.922718\pi\)
0.970671 0.240412i \(-0.0772824\pi\)
\(744\) 11373.9 0.560467
\(745\) 0 0
\(746\) −1738.53 −0.0853245
\(747\) 252.393i 0.0123622i
\(748\) − 440.848i − 0.0215495i
\(749\) −7500.08 −0.365884
\(750\) 0 0
\(751\) −23795.7 −1.15621 −0.578107 0.815961i \(-0.696208\pi\)
−0.578107 + 0.815961i \(0.696208\pi\)
\(752\) 1444.32i 0.0700385i
\(753\) − 30088.3i − 1.45615i
\(754\) 4454.08 0.215130
\(755\) 0 0
\(756\) 2900.02 0.139514
\(757\) 23647.6i 1.13538i 0.823241 + 0.567692i \(0.192163\pi\)
−0.823241 + 0.567692i \(0.807837\pi\)
\(758\) − 2557.53i − 0.122551i
\(759\) 647.203 0.0309512
\(760\) 0 0
\(761\) −4606.09 −0.219410 −0.109705 0.993964i \(-0.534991\pi\)
−0.109705 + 0.993964i \(0.534991\pi\)
\(762\) − 15654.9i − 0.744248i
\(763\) − 6056.98i − 0.287388i
\(764\) −5967.79 −0.282601
\(765\) 0 0
\(766\) −10687.5 −0.504119
\(767\) − 49944.0i − 2.35120i
\(768\) 1290.47i 0.0606326i
\(769\) −17760.9 −0.832867 −0.416433 0.909166i \(-0.636720\pi\)
−0.416433 + 0.909166i \(0.636720\pi\)
\(770\) 0 0
\(771\) 8247.22 0.385235
\(772\) 12364.4i 0.576430i
\(773\) − 7147.47i − 0.332570i −0.986078 0.166285i \(-0.946823\pi\)
0.986078 0.166285i \(-0.0531772\pi\)
\(774\) 1388.95 0.0645021
\(775\) 0 0
\(776\) −851.114 −0.0393727
\(777\) 3549.83i 0.163899i
\(778\) 1416.69i 0.0652839i
\(779\) −36097.6 −1.66024
\(780\) 0 0
\(781\) 2063.05 0.0945222
\(782\) 908.201i 0.0415310i
\(783\) 5116.59i 0.233528i
\(784\) 5083.07 0.231554
\(785\) 0 0
\(786\) 3164.40 0.143601
\(787\) − 31719.1i − 1.43668i −0.695694 0.718338i \(-0.744903\pi\)
0.695694 0.718338i \(-0.255097\pi\)
\(788\) − 11281.2i − 0.509995i
\(789\) 15100.3 0.681348
\(790\) 0 0
\(791\) 1392.36 0.0625872
\(792\) 70.9751i 0.00318433i
\(793\) 35665.7i 1.59713i
\(794\) 11848.0 0.529560
\(795\) 0 0
\(796\) 5264.42 0.234413
\(797\) 30828.3i 1.37013i 0.728481 + 0.685066i \(0.240227\pi\)
−0.728481 + 0.685066i \(0.759773\pi\)
\(798\) 8041.18i 0.356710i
\(799\) 1782.25 0.0789129
\(800\) 0 0
\(801\) 1961.05 0.0865046
\(802\) − 22787.3i − 1.00330i
\(803\) − 1731.53i − 0.0760949i
\(804\) 1716.77 0.0753057
\(805\) 0 0
\(806\) 35383.5 1.54632
\(807\) 26163.2i 1.14125i
\(808\) − 5141.41i − 0.223854i
\(809\) −36679.2 −1.59403 −0.797015 0.603959i \(-0.793589\pi\)
−0.797015 + 0.603959i \(0.793589\pi\)
\(810\) 0 0
\(811\) −12210.0 −0.528672 −0.264336 0.964431i \(-0.585153\pi\)
−0.264336 + 0.964431i \(0.585153\pi\)
\(812\) − 714.427i − 0.0308762i
\(813\) 33877.0i 1.46140i
\(814\) −1562.80 −0.0672927
\(815\) 0 0
\(816\) 1592.40 0.0683152
\(817\) 69278.2i 2.96663i
\(818\) − 27608.9i − 1.18010i
\(819\) 501.534 0.0213981
\(820\) 0 0
\(821\) −9996.47 −0.424944 −0.212472 0.977167i \(-0.568151\pi\)
−0.212472 + 0.977167i \(0.568151\pi\)
\(822\) 10748.1i 0.456063i
\(823\) 42868.8i 1.81569i 0.419306 + 0.907845i \(0.362273\pi\)
−0.419306 + 0.907845i \(0.637727\pi\)
\(824\) −12975.9 −0.548590
\(825\) 0 0
\(826\) −8010.93 −0.337453
\(827\) − 15833.4i − 0.665759i −0.942969 0.332880i \(-0.891980\pi\)
0.942969 0.332880i \(-0.108020\pi\)
\(828\) − 146.217i − 0.00613697i
\(829\) 17722.3 0.742484 0.371242 0.928536i \(-0.378932\pi\)
0.371242 + 0.928536i \(0.378932\pi\)
\(830\) 0 0
\(831\) 30823.8 1.28672
\(832\) 4014.57i 0.167284i
\(833\) − 6272.35i − 0.260893i
\(834\) −16078.3 −0.667563
\(835\) 0 0
\(836\) −3540.11 −0.146456
\(837\) 40646.5i 1.67855i
\(838\) 29591.7i 1.21984i
\(839\) 40528.4 1.66769 0.833847 0.551996i \(-0.186134\pi\)
0.833847 + 0.551996i \(0.186134\pi\)
\(840\) 0 0
\(841\) −23128.5 −0.948317
\(842\) 11609.3i 0.475159i
\(843\) − 17890.5i − 0.730938i
\(844\) 13596.7 0.554525
\(845\) 0 0
\(846\) −286.936 −0.0116608
\(847\) − 6539.11i − 0.265273i
\(848\) − 5282.92i − 0.213934i
\(849\) 15642.5 0.632331
\(850\) 0 0
\(851\) 3219.57 0.129689
\(852\) 7452.02i 0.299650i
\(853\) − 8116.97i − 0.325814i −0.986641 0.162907i \(-0.947913\pi\)
0.986641 0.162907i \(-0.0520871\pi\)
\(854\) 5720.73 0.229226
\(855\) 0 0
\(856\) 11926.9 0.476229
\(857\) 16285.5i 0.649126i 0.945864 + 0.324563i \(0.105217\pi\)
−0.945864 + 0.324563i \(0.894783\pi\)
\(858\) − 3530.22i − 0.140466i
\(859\) 26487.5 1.05209 0.526043 0.850458i \(-0.323675\pi\)
0.526043 + 0.850458i \(0.323675\pi\)
\(860\) 0 0
\(861\) −5773.81 −0.228538
\(862\) 30536.3i 1.20658i
\(863\) − 40396.2i − 1.59340i −0.604376 0.796699i \(-0.706578\pi\)
0.604376 0.796699i \(-0.293422\pi\)
\(864\) −4611.71 −0.181590
\(865\) 0 0
\(866\) 4501.25 0.176627
\(867\) 22801.0i 0.893151i
\(868\) − 5675.46i − 0.221933i
\(869\) −7398.94 −0.288828
\(870\) 0 0
\(871\) 5340.76 0.207767
\(872\) 9632.00i 0.374060i
\(873\) − 169.087i − 0.00655523i
\(874\) 7293.07 0.282256
\(875\) 0 0
\(876\) 6254.50 0.241233
\(877\) 10962.9i 0.422109i 0.977474 + 0.211054i \(0.0676897\pi\)
−0.977474 + 0.211054i \(0.932310\pi\)
\(878\) − 20732.4i − 0.796908i
\(879\) 6142.40 0.235698
\(880\) 0 0
\(881\) 23906.5 0.914222 0.457111 0.889410i \(-0.348884\pi\)
0.457111 + 0.889410i \(0.348884\pi\)
\(882\) 1009.83i 0.0385518i
\(883\) 30509.1i 1.16275i 0.813634 + 0.581377i \(0.197486\pi\)
−0.813634 + 0.581377i \(0.802514\pi\)
\(884\) 4953.86 0.188480
\(885\) 0 0
\(886\) −7762.22 −0.294331
\(887\) − 40034.3i − 1.51547i −0.652565 0.757733i \(-0.726307\pi\)
0.652565 0.757733i \(-0.273693\pi\)
\(888\) − 5645.06i − 0.213328i
\(889\) −7811.62 −0.294706
\(890\) 0 0
\(891\) 3815.77 0.143472
\(892\) 3458.64i 0.129825i
\(893\) − 14311.9i − 0.536314i
\(894\) −11058.8 −0.413716
\(895\) 0 0
\(896\) 643.931 0.0240092
\(897\) 7272.69i 0.270711i
\(898\) 20293.0i 0.754106i
\(899\) 10013.4 0.371485
\(900\) 0 0
\(901\) −6518.96 −0.241041
\(902\) − 2541.91i − 0.0938318i
\(903\) 11081.1i 0.408366i
\(904\) −2214.17 −0.0814626
\(905\) 0 0
\(906\) −27567.8 −1.01090
\(907\) − 2843.48i − 0.104097i −0.998645 0.0520486i \(-0.983425\pi\)
0.998645 0.0520486i \(-0.0165751\pi\)
\(908\) 7917.58i 0.289377i
\(909\) 1021.42 0.0372699
\(910\) 0 0
\(911\) 42268.6 1.53723 0.768617 0.639710i \(-0.220945\pi\)
0.768617 + 0.639710i \(0.220945\pi\)
\(912\) − 12787.4i − 0.464289i
\(913\) − 886.483i − 0.0321340i
\(914\) 9457.74 0.342269
\(915\) 0 0
\(916\) −12454.1 −0.449231
\(917\) − 1579.00i − 0.0568629i
\(918\) 5690.71i 0.204598i
\(919\) 12699.6 0.455844 0.227922 0.973679i \(-0.426807\pi\)
0.227922 + 0.973679i \(0.426807\pi\)
\(920\) 0 0
\(921\) −12927.1 −0.462501
\(922\) 16761.4i 0.598706i
\(923\) 23182.8i 0.826728i
\(924\) −566.242 −0.0201602
\(925\) 0 0
\(926\) 13690.3 0.485845
\(927\) − 2577.86i − 0.0913357i
\(928\) 1136.11i 0.0401880i
\(929\) 24889.6 0.879011 0.439505 0.898240i \(-0.355154\pi\)
0.439505 + 0.898240i \(0.355154\pi\)
\(930\) 0 0
\(931\) −50368.5 −1.77310
\(932\) 24074.6i 0.846126i
\(933\) − 466.915i − 0.0163838i
\(934\) 29768.0 1.04287
\(935\) 0 0
\(936\) −797.555 −0.0278514
\(937\) 9585.48i 0.334199i 0.985940 + 0.167099i \(0.0534401\pi\)
−0.985940 + 0.167099i \(0.946560\pi\)
\(938\) − 856.649i − 0.0298194i
\(939\) −21557.1 −0.749189
\(940\) 0 0
\(941\) −21146.9 −0.732592 −0.366296 0.930498i \(-0.619374\pi\)
−0.366296 + 0.930498i \(0.619374\pi\)
\(942\) 14514.0i 0.502009i
\(943\) 5236.65i 0.180836i
\(944\) 12739.2 0.439224
\(945\) 0 0
\(946\) −4878.41 −0.167665
\(947\) 48747.2i 1.67272i 0.548177 + 0.836362i \(0.315322\pi\)
−0.548177 + 0.836362i \(0.684678\pi\)
\(948\) − 26726.0i − 0.915632i
\(949\) 19457.3 0.665556
\(950\) 0 0
\(951\) 14942.4 0.509507
\(952\) − 794.591i − 0.0270513i
\(953\) − 24925.9i − 0.847249i −0.905838 0.423625i \(-0.860758\pi\)
0.905838 0.423625i \(-0.139242\pi\)
\(954\) 1049.53 0.0356183
\(955\) 0 0
\(956\) −16899.5 −0.571725
\(957\) − 999.036i − 0.0337453i
\(958\) − 12097.4i − 0.407984i
\(959\) 5363.20 0.180591
\(960\) 0 0
\(961\) 49755.9 1.67017
\(962\) − 17561.4i − 0.588568i
\(963\) 2369.45i 0.0792882i
\(964\) −5576.95 −0.186329
\(965\) 0 0
\(966\) 1166.53 0.0388534
\(967\) − 1881.92i − 0.0625838i −0.999510 0.0312919i \(-0.990038\pi\)
0.999510 0.0312919i \(-0.00996215\pi\)
\(968\) 10398.7i 0.345276i
\(969\) −15779.2 −0.523117
\(970\) 0 0
\(971\) −14637.3 −0.483763 −0.241881 0.970306i \(-0.577764\pi\)
−0.241881 + 0.970306i \(0.577764\pi\)
\(972\) − 1781.44i − 0.0587857i
\(973\) 8022.92i 0.264340i
\(974\) −6398.70 −0.210501
\(975\) 0 0
\(976\) −9097.28 −0.298358
\(977\) 42591.4i 1.39470i 0.716733 + 0.697348i \(0.245637\pi\)
−0.716733 + 0.697348i \(0.754363\pi\)
\(978\) − 30203.2i − 0.987518i
\(979\) −6887.81 −0.224857
\(980\) 0 0
\(981\) −1913.54 −0.0622780
\(982\) − 3228.81i − 0.104924i
\(983\) 48858.3i 1.58529i 0.609684 + 0.792644i \(0.291296\pi\)
−0.609684 + 0.792644i \(0.708704\pi\)
\(984\) 9181.71 0.297462
\(985\) 0 0
\(986\) 1401.92 0.0452801
\(987\) − 2289.19i − 0.0738254i
\(988\) − 39780.6i − 1.28096i
\(989\) 10050.1 0.323130
\(990\) 0 0
\(991\) −56522.1 −1.81179 −0.905895 0.423502i \(-0.860801\pi\)
−0.905895 + 0.423502i \(0.860801\pi\)
\(992\) 9025.30i 0.288864i
\(993\) − 6799.19i − 0.217287i
\(994\) 3718.48 0.118655
\(995\) 0 0
\(996\) 3202.09 0.101870
\(997\) − 20767.9i − 0.659706i −0.944032 0.329853i \(-0.893001\pi\)
0.944032 0.329853i \(-0.106999\pi\)
\(998\) 10784.1i 0.342048i
\(999\) 20173.5 0.638902
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.o.599.8 8
5.2 odd 4 1150.4.a.n.1.4 4
5.3 odd 4 230.4.a.j.1.1 4
5.4 even 2 inner 1150.4.b.o.599.1 8
15.8 even 4 2070.4.a.bg.1.3 4
20.3 even 4 1840.4.a.k.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.j.1.1 4 5.3 odd 4
1150.4.a.n.1.4 4 5.2 odd 4
1150.4.b.o.599.1 8 5.4 even 2 inner
1150.4.b.o.599.8 8 1.1 even 1 trivial
1840.4.a.k.1.4 4 20.3 even 4
2070.4.a.bg.1.3 4 15.8 even 4