Properties

Label 550.2.b.f.199.3
Level $550$
Weight $2$
Character 550.199
Analytic conductor $4.392$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [550,2,Mod(199,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("550.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) 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 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(-2.37228i\) of defining polynomial
Character \(\chi\) \(=\) 550.199
Dual form 550.2.b.f.199.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -3.37228i q^{3} -1.00000 q^{4} +3.37228 q^{6} -3.37228i q^{7} -1.00000i q^{8} -8.37228 q^{9} -1.00000 q^{11} +3.37228i q^{12} +2.00000i q^{13} +3.37228 q^{14} +1.00000 q^{16} -1.37228i q^{17} -8.37228i q^{18} -0.627719 q^{19} -11.3723 q^{21} -1.00000i q^{22} +2.74456i q^{23} -3.37228 q^{24} -2.00000 q^{26} +18.1168i q^{27} +3.37228i q^{28} -1.37228 q^{29} +3.37228 q^{31} +1.00000i q^{32} +3.37228i q^{33} +1.37228 q^{34} +8.37228 q^{36} -9.37228i q^{37} -0.627719i q^{38} +6.74456 q^{39} -11.4891 q^{41} -11.3723i q^{42} -4.00000i q^{43} +1.00000 q^{44} -2.74456 q^{46} -2.74456i q^{47} -3.37228i q^{48} -4.37228 q^{49} -4.62772 q^{51} -2.00000i q^{52} -4.11684i q^{53} -18.1168 q^{54} -3.37228 q^{56} +2.11684i q^{57} -1.37228i q^{58} +2.74456 q^{59} -5.37228 q^{61} +3.37228i q^{62} +28.2337i q^{63} -1.00000 q^{64} -3.37228 q^{66} -8.00000i q^{67} +1.37228i q^{68} +9.25544 q^{69} +10.1168 q^{71} +8.37228i q^{72} -15.4891i q^{73} +9.37228 q^{74} +0.627719 q^{76} +3.37228i q^{77} +6.74456i q^{78} +1.25544 q^{79} +35.9783 q^{81} -11.4891i q^{82} -2.74456i q^{83} +11.3723 q^{84} +4.00000 q^{86} +4.62772i q^{87} +1.00000i q^{88} +1.37228 q^{89} +6.74456 q^{91} -2.74456i q^{92} -11.3723i q^{93} +2.74456 q^{94} +3.37228 q^{96} +12.7446i q^{97} -4.37228i q^{98} +8.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{6} - 22 q^{9} - 4 q^{11} + 2 q^{14} + 4 q^{16} - 14 q^{19} - 34 q^{21} - 2 q^{24} - 8 q^{26} + 6 q^{29} + 2 q^{31} - 6 q^{34} + 22 q^{36} + 4 q^{39} + 4 q^{44} + 12 q^{46} - 6 q^{49}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 3.37228i − 1.94699i −0.228714 0.973494i \(-0.573452\pi\)
0.228714 0.973494i \(-0.426548\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 3.37228 1.37673
\(7\) − 3.37228i − 1.27460i −0.770615 0.637301i \(-0.780051\pi\)
0.770615 0.637301i \(-0.219949\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −8.37228 −2.79076
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 3.37228i 0.973494i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 3.37228 0.901280
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.37228i − 0.332827i −0.986056 0.166414i \(-0.946781\pi\)
0.986056 0.166414i \(-0.0532187\pi\)
\(18\) − 8.37228i − 1.97337i
\(19\) −0.627719 −0.144009 −0.0720043 0.997404i \(-0.522940\pi\)
−0.0720043 + 0.997404i \(0.522940\pi\)
\(20\) 0 0
\(21\) −11.3723 −2.48164
\(22\) − 1.00000i − 0.213201i
\(23\) 2.74456i 0.572281i 0.958188 + 0.286140i \(0.0923724\pi\)
−0.958188 + 0.286140i \(0.907628\pi\)
\(24\) −3.37228 −0.688364
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 18.1168i 3.48659i
\(28\) 3.37228i 0.637301i
\(29\) −1.37228 −0.254826 −0.127413 0.991850i \(-0.540667\pi\)
−0.127413 + 0.991850i \(0.540667\pi\)
\(30\) 0 0
\(31\) 3.37228 0.605680 0.302840 0.953041i \(-0.402065\pi\)
0.302840 + 0.953041i \(0.402065\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 3.37228i 0.587039i
\(34\) 1.37228 0.235344
\(35\) 0 0
\(36\) 8.37228 1.39538
\(37\) − 9.37228i − 1.54079i −0.637565 0.770397i \(-0.720058\pi\)
0.637565 0.770397i \(-0.279942\pi\)
\(38\) − 0.627719i − 0.101829i
\(39\) 6.74456 1.07999
\(40\) 0 0
\(41\) −11.4891 −1.79430 −0.897150 0.441726i \(-0.854366\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) − 11.3723i − 1.75478i
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −2.74456 −0.404664
\(47\) − 2.74456i − 0.400336i −0.979762 0.200168i \(-0.935851\pi\)
0.979762 0.200168i \(-0.0641487\pi\)
\(48\) − 3.37228i − 0.486747i
\(49\) −4.37228 −0.624612
\(50\) 0 0
\(51\) −4.62772 −0.648010
\(52\) − 2.00000i − 0.277350i
\(53\) − 4.11684i − 0.565492i −0.959195 0.282746i \(-0.908755\pi\)
0.959195 0.282746i \(-0.0912454\pi\)
\(54\) −18.1168 −2.46539
\(55\) 0 0
\(56\) −3.37228 −0.450640
\(57\) 2.11684i 0.280383i
\(58\) − 1.37228i − 0.180189i
\(59\) 2.74456 0.357312 0.178656 0.983912i \(-0.442825\pi\)
0.178656 + 0.983912i \(0.442825\pi\)
\(60\) 0 0
\(61\) −5.37228 −0.687850 −0.343925 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) 3.37228i 0.428280i
\(63\) 28.2337i 3.55711i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −3.37228 −0.415099
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 1.37228i 0.166414i
\(69\) 9.25544 1.11422
\(70\) 0 0
\(71\) 10.1168 1.20065 0.600324 0.799757i \(-0.295038\pi\)
0.600324 + 0.799757i \(0.295038\pi\)
\(72\) 8.37228i 0.986683i
\(73\) − 15.4891i − 1.81286i −0.422351 0.906432i \(-0.638795\pi\)
0.422351 0.906432i \(-0.361205\pi\)
\(74\) 9.37228 1.08951
\(75\) 0 0
\(76\) 0.627719 0.0720043
\(77\) 3.37228i 0.384307i
\(78\) 6.74456i 0.763671i
\(79\) 1.25544 0.141248 0.0706239 0.997503i \(-0.477501\pi\)
0.0706239 + 0.997503i \(0.477501\pi\)
\(80\) 0 0
\(81\) 35.9783 3.99758
\(82\) − 11.4891i − 1.26876i
\(83\) − 2.74456i − 0.301255i −0.988591 0.150627i \(-0.951871\pi\)
0.988591 0.150627i \(-0.0481294\pi\)
\(84\) 11.3723 1.24082
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 4.62772i 0.496144i
\(88\) 1.00000i 0.106600i
\(89\) 1.37228 0.145462 0.0727308 0.997352i \(-0.476829\pi\)
0.0727308 + 0.997352i \(0.476829\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) − 2.74456i − 0.286140i
\(93\) − 11.3723i − 1.17925i
\(94\) 2.74456 0.283080
\(95\) 0 0
\(96\) 3.37228 0.344182
\(97\) 12.7446i 1.29401i 0.762484 + 0.647007i \(0.223980\pi\)
−0.762484 + 0.647007i \(0.776020\pi\)
\(98\) − 4.37228i − 0.441667i
\(99\) 8.37228 0.841446
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) − 4.62772i − 0.458212i
\(103\) − 9.48913i − 0.934991i −0.883995 0.467496i \(-0.845156\pi\)
0.883995 0.467496i \(-0.154844\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 4.11684 0.399863
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) − 18.1168i − 1.74329i
\(109\) 15.4891 1.48359 0.741795 0.670627i \(-0.233975\pi\)
0.741795 + 0.670627i \(0.233975\pi\)
\(110\) 0 0
\(111\) −31.6060 −2.99991
\(112\) − 3.37228i − 0.318651i
\(113\) 3.25544i 0.306246i 0.988207 + 0.153123i \(0.0489330\pi\)
−0.988207 + 0.153123i \(0.951067\pi\)
\(114\) −2.11684 −0.198261
\(115\) 0 0
\(116\) 1.37228 0.127413
\(117\) − 16.7446i − 1.54804i
\(118\) 2.74456i 0.252657i
\(119\) −4.62772 −0.424222
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 5.37228i − 0.486383i
\(123\) 38.7446i 3.49348i
\(124\) −3.37228 −0.302840
\(125\) 0 0
\(126\) −28.2337 −2.51526
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −13.4891 −1.18765
\(130\) 0 0
\(131\) 22.1168 1.93236 0.966179 0.257873i \(-0.0830216\pi\)
0.966179 + 0.257873i \(0.0830216\pi\)
\(132\) − 3.37228i − 0.293519i
\(133\) 2.11684i 0.183554i
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −1.37228 −0.117672
\(137\) − 8.74456i − 0.747098i −0.927610 0.373549i \(-0.878141\pi\)
0.927610 0.373549i \(-0.121859\pi\)
\(138\) 9.25544i 0.787875i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −9.25544 −0.779448
\(142\) 10.1168i 0.848987i
\(143\) − 2.00000i − 0.167248i
\(144\) −8.37228 −0.697690
\(145\) 0 0
\(146\) 15.4891 1.28189
\(147\) 14.7446i 1.21611i
\(148\) 9.37228i 0.770397i
\(149\) −21.6060 −1.77003 −0.885015 0.465563i \(-0.845852\pi\)
−0.885015 + 0.465563i \(0.845852\pi\)
\(150\) 0 0
\(151\) −12.2337 −0.995563 −0.497782 0.867302i \(-0.665852\pi\)
−0.497782 + 0.867302i \(0.665852\pi\)
\(152\) 0.627719i 0.0509147i
\(153\) 11.4891i 0.928841i
\(154\) −3.37228 −0.271746
\(155\) 0 0
\(156\) −6.74456 −0.539997
\(157\) − 9.37228i − 0.747989i −0.927431 0.373995i \(-0.877988\pi\)
0.927431 0.373995i \(-0.122012\pi\)
\(158\) 1.25544i 0.0998772i
\(159\) −13.8832 −1.10101
\(160\) 0 0
\(161\) 9.25544 0.729431
\(162\) 35.9783i 2.82672i
\(163\) − 5.88316i − 0.460804i −0.973095 0.230402i \(-0.925996\pi\)
0.973095 0.230402i \(-0.0740042\pi\)
\(164\) 11.4891 0.897150
\(165\) 0 0
\(166\) 2.74456 0.213019
\(167\) 4.62772i 0.358104i 0.983840 + 0.179052i \(0.0573030\pi\)
−0.983840 + 0.179052i \(0.942697\pi\)
\(168\) 11.3723i 0.877391i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 5.25544 0.401893
\(172\) 4.00000i 0.304997i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) −4.62772 −0.350826
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) − 9.25544i − 0.695681i
\(178\) 1.37228i 0.102857i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 6.74456i 0.499940i
\(183\) 18.1168i 1.33924i
\(184\) 2.74456 0.202332
\(185\) 0 0
\(186\) 11.3723 0.833856
\(187\) 1.37228i 0.100351i
\(188\) 2.74456i 0.200168i
\(189\) 61.0951 4.44401
\(190\) 0 0
\(191\) −5.48913 −0.397179 −0.198590 0.980083i \(-0.563636\pi\)
−0.198590 + 0.980083i \(0.563636\pi\)
\(192\) 3.37228i 0.243373i
\(193\) 14.8614i 1.06975i 0.844932 + 0.534874i \(0.179641\pi\)
−0.844932 + 0.534874i \(0.820359\pi\)
\(194\) −12.7446 −0.915006
\(195\) 0 0
\(196\) 4.37228 0.312306
\(197\) 20.7446i 1.47799i 0.673712 + 0.738994i \(0.264699\pi\)
−0.673712 + 0.738994i \(0.735301\pi\)
\(198\) 8.37228i 0.594992i
\(199\) −18.1168 −1.28427 −0.642135 0.766592i \(-0.721951\pi\)
−0.642135 + 0.766592i \(0.721951\pi\)
\(200\) 0 0
\(201\) −26.9783 −1.90290
\(202\) 6.00000i 0.422159i
\(203\) 4.62772i 0.324802i
\(204\) 4.62772 0.324005
\(205\) 0 0
\(206\) 9.48913 0.661139
\(207\) − 22.9783i − 1.59710i
\(208\) 2.00000i 0.138675i
\(209\) 0.627719 0.0434202
\(210\) 0 0
\(211\) 6.11684 0.421101 0.210550 0.977583i \(-0.432474\pi\)
0.210550 + 0.977583i \(0.432474\pi\)
\(212\) 4.11684i 0.282746i
\(213\) − 34.1168i − 2.33765i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 18.1168 1.23270
\(217\) − 11.3723i − 0.772001i
\(218\) 15.4891i 1.04906i
\(219\) −52.2337 −3.52963
\(220\) 0 0
\(221\) 2.74456 0.184619
\(222\) − 31.6060i − 2.12125i
\(223\) − 18.7446i − 1.25523i −0.778524 0.627614i \(-0.784031\pi\)
0.778524 0.627614i \(-0.215969\pi\)
\(224\) 3.37228 0.225320
\(225\) 0 0
\(226\) −3.25544 −0.216548
\(227\) 2.74456i 0.182163i 0.995843 + 0.0910815i \(0.0290324\pi\)
−0.995843 + 0.0910815i \(0.970968\pi\)
\(228\) − 2.11684i − 0.140191i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 11.3723 0.748241
\(232\) 1.37228i 0.0900947i
\(233\) 1.37228i 0.0899011i 0.998989 + 0.0449506i \(0.0143130\pi\)
−0.998989 + 0.0449506i \(0.985687\pi\)
\(234\) 16.7446 1.09463
\(235\) 0 0
\(236\) −2.74456 −0.178656
\(237\) − 4.23369i − 0.275008i
\(238\) − 4.62772i − 0.299970i
\(239\) 14.7446 0.953746 0.476873 0.878972i \(-0.341770\pi\)
0.476873 + 0.878972i \(0.341770\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) − 66.9783i − 4.29666i
\(244\) 5.37228 0.343925
\(245\) 0 0
\(246\) −38.7446 −2.47026
\(247\) − 1.25544i − 0.0798816i
\(248\) − 3.37228i − 0.214140i
\(249\) −9.25544 −0.586540
\(250\) 0 0
\(251\) 2.74456 0.173235 0.0866176 0.996242i \(-0.472394\pi\)
0.0866176 + 0.996242i \(0.472394\pi\)
\(252\) − 28.2337i − 1.77856i
\(253\) − 2.74456i − 0.172549i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) − 13.4891i − 0.839796i
\(259\) −31.6060 −1.96390
\(260\) 0 0
\(261\) 11.4891 0.711159
\(262\) 22.1168i 1.36638i
\(263\) − 24.8614i − 1.53302i −0.642232 0.766510i \(-0.721992\pi\)
0.642232 0.766510i \(-0.278008\pi\)
\(264\) 3.37228 0.207550
\(265\) 0 0
\(266\) −2.11684 −0.129792
\(267\) − 4.62772i − 0.283212i
\(268\) 8.00000i 0.488678i
\(269\) 8.74456 0.533165 0.266583 0.963812i \(-0.414105\pi\)
0.266583 + 0.963812i \(0.414105\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) − 1.37228i − 0.0832068i
\(273\) − 22.7446i − 1.37656i
\(274\) 8.74456 0.528278
\(275\) 0 0
\(276\) −9.25544 −0.557112
\(277\) 12.7446i 0.765747i 0.923801 + 0.382873i \(0.125065\pi\)
−0.923801 + 0.382873i \(0.874935\pi\)
\(278\) 4.00000i 0.239904i
\(279\) −28.2337 −1.69031
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) − 9.25544i − 0.551153i
\(283\) 5.25544i 0.312403i 0.987725 + 0.156202i \(0.0499250\pi\)
−0.987725 + 0.156202i \(0.950075\pi\)
\(284\) −10.1168 −0.600324
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 38.7446i 2.28702i
\(288\) − 8.37228i − 0.493341i
\(289\) 15.1168 0.889226
\(290\) 0 0
\(291\) 42.9783 2.51943
\(292\) 15.4891i 0.906432i
\(293\) 23.4891i 1.37225i 0.727484 + 0.686125i \(0.240690\pi\)
−0.727484 + 0.686125i \(0.759310\pi\)
\(294\) −14.7446 −0.859920
\(295\) 0 0
\(296\) −9.37228 −0.544753
\(297\) − 18.1168i − 1.05125i
\(298\) − 21.6060i − 1.25160i
\(299\) −5.48913 −0.317444
\(300\) 0 0
\(301\) −13.4891 −0.777500
\(302\) − 12.2337i − 0.703970i
\(303\) − 20.2337i − 1.16240i
\(304\) −0.627719 −0.0360021
\(305\) 0 0
\(306\) −11.4891 −0.656790
\(307\) − 5.25544i − 0.299944i −0.988690 0.149972i \(-0.952082\pi\)
0.988690 0.149972i \(-0.0479183\pi\)
\(308\) − 3.37228i − 0.192154i
\(309\) −32.0000 −1.82042
\(310\) 0 0
\(311\) 19.3723 1.09850 0.549251 0.835658i \(-0.314913\pi\)
0.549251 + 0.835658i \(0.314913\pi\)
\(312\) − 6.74456i − 0.381836i
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) 9.37228 0.528908
\(315\) 0 0
\(316\) −1.25544 −0.0706239
\(317\) 24.3505i 1.36766i 0.729640 + 0.683831i \(0.239687\pi\)
−0.729640 + 0.683831i \(0.760313\pi\)
\(318\) − 13.8832i − 0.778529i
\(319\) 1.37228 0.0768330
\(320\) 0 0
\(321\) 40.4674 2.25867
\(322\) 9.25544i 0.515785i
\(323\) 0.861407i 0.0479299i
\(324\) −35.9783 −1.99879
\(325\) 0 0
\(326\) 5.88316 0.325838
\(327\) − 52.2337i − 2.88853i
\(328\) 11.4891i 0.634381i
\(329\) −9.25544 −0.510269
\(330\) 0 0
\(331\) 30.9783 1.70272 0.851359 0.524583i \(-0.175779\pi\)
0.851359 + 0.524583i \(0.175779\pi\)
\(332\) 2.74456i 0.150627i
\(333\) 78.4674i 4.29999i
\(334\) −4.62772 −0.253217
\(335\) 0 0
\(336\) −11.3723 −0.620409
\(337\) − 24.1168i − 1.31373i −0.754009 0.656864i \(-0.771883\pi\)
0.754009 0.656864i \(-0.228117\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 10.9783 0.596257
\(340\) 0 0
\(341\) −3.37228 −0.182619
\(342\) 5.25544i 0.284182i
\(343\) − 8.86141i − 0.478471i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 32.2337i 1.73040i 0.501431 + 0.865198i \(0.332807\pi\)
−0.501431 + 0.865198i \(0.667193\pi\)
\(348\) − 4.62772i − 0.248072i
\(349\) −19.4891 −1.04323 −0.521614 0.853181i \(-0.674670\pi\)
−0.521614 + 0.853181i \(0.674670\pi\)
\(350\) 0 0
\(351\) −36.2337 −1.93401
\(352\) − 1.00000i − 0.0533002i
\(353\) − 0.510875i − 0.0271911i −0.999908 0.0135956i \(-0.995672\pi\)
0.999908 0.0135956i \(-0.00432773\pi\)
\(354\) 9.25544 0.491921
\(355\) 0 0
\(356\) −1.37228 −0.0727308
\(357\) 15.6060i 0.825955i
\(358\) 12.0000i 0.634220i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −18.6060 −0.979262
\(362\) − 10.0000i − 0.525588i
\(363\) − 3.37228i − 0.176999i
\(364\) −6.74456 −0.353511
\(365\) 0 0
\(366\) −18.1168 −0.946983
\(367\) 4.00000i 0.208798i 0.994535 + 0.104399i \(0.0332919\pi\)
−0.994535 + 0.104399i \(0.966708\pi\)
\(368\) 2.74456i 0.143070i
\(369\) 96.1902 5.00746
\(370\) 0 0
\(371\) −13.8832 −0.720778
\(372\) 11.3723i 0.589625i
\(373\) 31.4891i 1.63045i 0.579148 + 0.815223i \(0.303385\pi\)
−0.579148 + 0.815223i \(0.696615\pi\)
\(374\) −1.37228 −0.0709590
\(375\) 0 0
\(376\) −2.74456 −0.141540
\(377\) − 2.74456i − 0.141352i
\(378\) 61.0951i 3.14239i
\(379\) 0.233688 0.0120037 0.00600187 0.999982i \(-0.498090\pi\)
0.00600187 + 0.999982i \(0.498090\pi\)
\(380\) 0 0
\(381\) −26.9783 −1.38214
\(382\) − 5.48913i − 0.280848i
\(383\) 32.2337i 1.64706i 0.567269 + 0.823532i \(0.308000\pi\)
−0.567269 + 0.823532i \(0.692000\pi\)
\(384\) −3.37228 −0.172091
\(385\) 0 0
\(386\) −14.8614 −0.756426
\(387\) 33.4891i 1.70235i
\(388\) − 12.7446i − 0.647007i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 3.76631 0.190471
\(392\) 4.37228i 0.220834i
\(393\) − 74.5842i − 3.76228i
\(394\) −20.7446 −1.04510
\(395\) 0 0
\(396\) −8.37228 −0.420723
\(397\) − 24.9783i − 1.25362i −0.779171 0.626811i \(-0.784360\pi\)
0.779171 0.626811i \(-0.215640\pi\)
\(398\) − 18.1168i − 0.908115i
\(399\) 7.13859 0.357377
\(400\) 0 0
\(401\) 13.3723 0.667780 0.333890 0.942612i \(-0.391639\pi\)
0.333890 + 0.942612i \(0.391639\pi\)
\(402\) − 26.9783i − 1.34555i
\(403\) 6.74456i 0.335971i
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −4.62772 −0.229670
\(407\) 9.37228i 0.464567i
\(408\) 4.62772i 0.229106i
\(409\) 1.76631 0.0873385 0.0436693 0.999046i \(-0.486095\pi\)
0.0436693 + 0.999046i \(0.486095\pi\)
\(410\) 0 0
\(411\) −29.4891 −1.45459
\(412\) 9.48913i 0.467496i
\(413\) − 9.25544i − 0.455430i
\(414\) 22.9783 1.12932
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) − 13.4891i − 0.660565i
\(418\) 0.627719i 0.0307027i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 10.2337 0.498759 0.249380 0.968406i \(-0.419773\pi\)
0.249380 + 0.968406i \(0.419773\pi\)
\(422\) 6.11684i 0.297763i
\(423\) 22.9783i 1.11724i
\(424\) −4.11684 −0.199932
\(425\) 0 0
\(426\) 34.1168 1.65297
\(427\) 18.1168i 0.876736i
\(428\) − 12.0000i − 0.580042i
\(429\) −6.74456 −0.325631
\(430\) 0 0
\(431\) −34.9783 −1.68484 −0.842422 0.538819i \(-0.818871\pi\)
−0.842422 + 0.538819i \(0.818871\pi\)
\(432\) 18.1168i 0.871647i
\(433\) 27.7228i 1.33227i 0.745830 + 0.666137i \(0.232053\pi\)
−0.745830 + 0.666137i \(0.767947\pi\)
\(434\) 11.3723 0.545887
\(435\) 0 0
\(436\) −15.4891 −0.741795
\(437\) − 1.72281i − 0.0824133i
\(438\) − 52.2337i − 2.49582i
\(439\) −18.9783 −0.905782 −0.452891 0.891566i \(-0.649607\pi\)
−0.452891 + 0.891566i \(0.649607\pi\)
\(440\) 0 0
\(441\) 36.6060 1.74314
\(442\) 2.74456i 0.130546i
\(443\) 29.4891i 1.40107i 0.713618 + 0.700535i \(0.247055\pi\)
−0.713618 + 0.700535i \(0.752945\pi\)
\(444\) 31.6060 1.49995
\(445\) 0 0
\(446\) 18.7446 0.887581
\(447\) 72.8614i 3.44623i
\(448\) 3.37228i 0.159325i
\(449\) −28.9783 −1.36757 −0.683784 0.729684i \(-0.739667\pi\)
−0.683784 + 0.729684i \(0.739667\pi\)
\(450\) 0 0
\(451\) 11.4891 0.541002
\(452\) − 3.25544i − 0.153123i
\(453\) 41.2554i 1.93835i
\(454\) −2.74456 −0.128809
\(455\) 0 0
\(456\) 2.11684 0.0991303
\(457\) 16.3505i 0.764846i 0.923987 + 0.382423i \(0.124910\pi\)
−0.923987 + 0.382423i \(0.875090\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 24.8614 1.16043
\(460\) 0 0
\(461\) 16.1168 0.750636 0.375318 0.926896i \(-0.377533\pi\)
0.375318 + 0.926896i \(0.377533\pi\)
\(462\) 11.3723i 0.529086i
\(463\) − 0.233688i − 0.0108604i −0.999985 0.00543020i \(-0.998272\pi\)
0.999985 0.00543020i \(-0.00172849\pi\)
\(464\) −1.37228 −0.0637066
\(465\) 0 0
\(466\) −1.37228 −0.0635697
\(467\) 19.3723i 0.896442i 0.893923 + 0.448221i \(0.147942\pi\)
−0.893923 + 0.448221i \(0.852058\pi\)
\(468\) 16.7446i 0.774018i
\(469\) −26.9783 −1.24574
\(470\) 0 0
\(471\) −31.6060 −1.45633
\(472\) − 2.74456i − 0.126329i
\(473\) 4.00000i 0.183920i
\(474\) 4.23369 0.194460
\(475\) 0 0
\(476\) 4.62772 0.212111
\(477\) 34.4674i 1.57815i
\(478\) 14.7446i 0.674401i
\(479\) −5.48913 −0.250805 −0.125402 0.992106i \(-0.540022\pi\)
−0.125402 + 0.992106i \(0.540022\pi\)
\(480\) 0 0
\(481\) 18.7446 0.854678
\(482\) − 22.0000i − 1.00207i
\(483\) − 31.2119i − 1.42019i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 66.9783 3.03820
\(487\) − 20.0000i − 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) 5.37228i 0.243192i
\(489\) −19.8397 −0.897180
\(490\) 0 0
\(491\) −7.37228 −0.332706 −0.166353 0.986066i \(-0.553199\pi\)
−0.166353 + 0.986066i \(0.553199\pi\)
\(492\) − 38.7446i − 1.74674i
\(493\) 1.88316i 0.0848131i
\(494\) 1.25544 0.0564848
\(495\) 0 0
\(496\) 3.37228 0.151420
\(497\) − 34.1168i − 1.53035i
\(498\) − 9.25544i − 0.414746i
\(499\) 33.4891 1.49918 0.749590 0.661903i \(-0.230251\pi\)
0.749590 + 0.661903i \(0.230251\pi\)
\(500\) 0 0
\(501\) 15.6060 0.697223
\(502\) 2.74456i 0.122496i
\(503\) − 34.9783i − 1.55960i −0.626027 0.779802i \(-0.715320\pi\)
0.626027 0.779802i \(-0.284680\pi\)
\(504\) 28.2337 1.25763
\(505\) 0 0
\(506\) 2.74456 0.122011
\(507\) − 30.3505i − 1.34791i
\(508\) 8.00000i 0.354943i
\(509\) −9.76631 −0.432884 −0.216442 0.976295i \(-0.569445\pi\)
−0.216442 + 0.976295i \(0.569445\pi\)
\(510\) 0 0
\(511\) −52.2337 −2.31068
\(512\) 1.00000i 0.0441942i
\(513\) − 11.3723i − 0.502098i
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 13.4891 0.593826
\(517\) 2.74456i 0.120706i
\(518\) − 31.6060i − 1.38869i
\(519\) −20.2337 −0.888160
\(520\) 0 0
\(521\) 12.5109 0.548111 0.274056 0.961714i \(-0.411635\pi\)
0.274056 + 0.961714i \(0.411635\pi\)
\(522\) 11.4891i 0.502865i
\(523\) 30.9783i 1.35458i 0.735714 + 0.677292i \(0.236847\pi\)
−0.735714 + 0.677292i \(0.763153\pi\)
\(524\) −22.1168 −0.966179
\(525\) 0 0
\(526\) 24.8614 1.08401
\(527\) − 4.62772i − 0.201587i
\(528\) 3.37228i 0.146760i
\(529\) 15.4674 0.672495
\(530\) 0 0
\(531\) −22.9783 −0.997171
\(532\) − 2.11684i − 0.0917768i
\(533\) − 22.9783i − 0.995299i
\(534\) 4.62772 0.200261
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) − 40.4674i − 1.74630i
\(538\) 8.74456i 0.377005i
\(539\) 4.37228 0.188327
\(540\) 0 0
\(541\) −20.1168 −0.864891 −0.432445 0.901660i \(-0.642349\pi\)
−0.432445 + 0.901660i \(0.642349\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) 33.7228i 1.44718i
\(544\) 1.37228 0.0588361
\(545\) 0 0
\(546\) 22.7446 0.973377
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 8.74456i 0.373549i
\(549\) 44.9783 1.91962
\(550\) 0 0
\(551\) 0.861407 0.0366972
\(552\) − 9.25544i − 0.393938i
\(553\) − 4.23369i − 0.180035i
\(554\) −12.7446 −0.541465
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) − 4.97825i − 0.210935i −0.994423 0.105468i \(-0.966366\pi\)
0.994423 0.105468i \(-0.0336339\pi\)
\(558\) − 28.2337i − 1.19523i
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 4.62772 0.195382
\(562\) 18.0000i 0.759284i
\(563\) − 8.23369i − 0.347009i −0.984833 0.173504i \(-0.944491\pi\)
0.984833 0.173504i \(-0.0555090\pi\)
\(564\) 9.25544 0.389724
\(565\) 0 0
\(566\) −5.25544 −0.220903
\(567\) − 121.329i − 5.09533i
\(568\) − 10.1168i − 0.424493i
\(569\) 15.2554 0.639541 0.319771 0.947495i \(-0.396394\pi\)
0.319771 + 0.947495i \(0.396394\pi\)
\(570\) 0 0
\(571\) 15.3723 0.643310 0.321655 0.946857i \(-0.395761\pi\)
0.321655 + 0.946857i \(0.395761\pi\)
\(572\) 2.00000i 0.0836242i
\(573\) 18.5109i 0.773303i
\(574\) −38.7446 −1.61717
\(575\) 0 0
\(576\) 8.37228 0.348845
\(577\) − 36.9783i − 1.53942i −0.638391 0.769712i \(-0.720400\pi\)
0.638391 0.769712i \(-0.279600\pi\)
\(578\) 15.1168i 0.628778i
\(579\) 50.1168 2.08278
\(580\) 0 0
\(581\) −9.25544 −0.383980
\(582\) 42.9783i 1.78151i
\(583\) 4.11684i 0.170502i
\(584\) −15.4891 −0.640945
\(585\) 0 0
\(586\) −23.4891 −0.970327
\(587\) − 24.8614i − 1.02614i −0.858347 0.513070i \(-0.828508\pi\)
0.858347 0.513070i \(-0.171492\pi\)
\(588\) − 14.7446i − 0.608056i
\(589\) −2.11684 −0.0872230
\(590\) 0 0
\(591\) 69.9565 2.87763
\(592\) − 9.37228i − 0.385198i
\(593\) − 12.5109i − 0.513760i −0.966443 0.256880i \(-0.917305\pi\)
0.966443 0.256880i \(-0.0826945\pi\)
\(594\) 18.1168 0.743343
\(595\) 0 0
\(596\) 21.6060 0.885015
\(597\) 61.0951i 2.50046i
\(598\) − 5.48913i − 0.224467i
\(599\) 39.6060 1.61826 0.809128 0.587632i \(-0.199940\pi\)
0.809128 + 0.587632i \(0.199940\pi\)
\(600\) 0 0
\(601\) −16.5109 −0.673493 −0.336746 0.941595i \(-0.609326\pi\)
−0.336746 + 0.941595i \(0.609326\pi\)
\(602\) − 13.4891i − 0.549776i
\(603\) 66.9783i 2.72757i
\(604\) 12.2337 0.497782
\(605\) 0 0
\(606\) 20.2337 0.821937
\(607\) 5.88316i 0.238790i 0.992847 + 0.119395i \(0.0380955\pi\)
−0.992847 + 0.119395i \(0.961905\pi\)
\(608\) − 0.627719i − 0.0254574i
\(609\) 15.6060 0.632386
\(610\) 0 0
\(611\) 5.48913 0.222066
\(612\) − 11.4891i − 0.464420i
\(613\) 20.5109i 0.828426i 0.910180 + 0.414213i \(0.135943\pi\)
−0.910180 + 0.414213i \(0.864057\pi\)
\(614\) 5.25544 0.212092
\(615\) 0 0
\(616\) 3.37228 0.135873
\(617\) 2.23369i 0.0899249i 0.998989 + 0.0449624i \(0.0143168\pi\)
−0.998989 + 0.0449624i \(0.985683\pi\)
\(618\) − 32.0000i − 1.28723i
\(619\) 44.4674 1.78729 0.893647 0.448770i \(-0.148138\pi\)
0.893647 + 0.448770i \(0.148138\pi\)
\(620\) 0 0
\(621\) −49.7228 −1.99531
\(622\) 19.3723i 0.776758i
\(623\) − 4.62772i − 0.185406i
\(624\) 6.74456 0.269999
\(625\) 0 0
\(626\) 22.0000 0.879297
\(627\) − 2.11684i − 0.0845386i
\(628\) 9.37228i 0.373995i
\(629\) −12.8614 −0.512818
\(630\) 0 0
\(631\) 42.1168 1.67665 0.838323 0.545175i \(-0.183537\pi\)
0.838323 + 0.545175i \(0.183537\pi\)
\(632\) − 1.25544i − 0.0499386i
\(633\) − 20.6277i − 0.819878i
\(634\) −24.3505 −0.967083
\(635\) 0 0
\(636\) 13.8832 0.550503
\(637\) − 8.74456i − 0.346472i
\(638\) 1.37228i 0.0543291i
\(639\) −84.7011 −3.35072
\(640\) 0 0
\(641\) −27.0951 −1.07019 −0.535096 0.844791i \(-0.679725\pi\)
−0.535096 + 0.844791i \(0.679725\pi\)
\(642\) 40.4674i 1.59712i
\(643\) − 5.88316i − 0.232009i −0.993249 0.116005i \(-0.962991\pi\)
0.993249 0.116005i \(-0.0370087\pi\)
\(644\) −9.25544 −0.364715
\(645\) 0 0
\(646\) −0.861407 −0.0338916
\(647\) 37.7228i 1.48304i 0.670933 + 0.741518i \(0.265894\pi\)
−0.670933 + 0.741518i \(0.734106\pi\)
\(648\) − 35.9783i − 1.41336i
\(649\) −2.74456 −0.107734
\(650\) 0 0
\(651\) −38.3505 −1.50308
\(652\) 5.88316i 0.230402i
\(653\) 10.6277i 0.415895i 0.978140 + 0.207947i \(0.0666783\pi\)
−0.978140 + 0.207947i \(0.933322\pi\)
\(654\) 52.2337 2.04250
\(655\) 0 0
\(656\) −11.4891 −0.448575
\(657\) 129.679i 5.05927i
\(658\) − 9.25544i − 0.360815i
\(659\) −12.8614 −0.501009 −0.250505 0.968115i \(-0.580597\pi\)
−0.250505 + 0.968115i \(0.580597\pi\)
\(660\) 0 0
\(661\) 8.51087 0.331035 0.165517 0.986207i \(-0.447071\pi\)
0.165517 + 0.986207i \(0.447071\pi\)
\(662\) 30.9783i 1.20400i
\(663\) − 9.25544i − 0.359451i
\(664\) −2.74456 −0.106510
\(665\) 0 0
\(666\) −78.4674 −3.04055
\(667\) − 3.76631i − 0.145832i
\(668\) − 4.62772i − 0.179052i
\(669\) −63.2119 −2.44391
\(670\) 0 0
\(671\) 5.37228 0.207395
\(672\) − 11.3723i − 0.438695i
\(673\) 14.8614i 0.572865i 0.958100 + 0.286433i \(0.0924694\pi\)
−0.958100 + 0.286433i \(0.907531\pi\)
\(674\) 24.1168 0.928946
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 3.25544i − 0.125117i −0.998041 0.0625583i \(-0.980074\pi\)
0.998041 0.0625583i \(-0.0199259\pi\)
\(678\) 10.9783i 0.421617i
\(679\) 42.9783 1.64935
\(680\) 0 0
\(681\) 9.25544 0.354669
\(682\) − 3.37228i − 0.129131i
\(683\) − 28.6277i − 1.09541i −0.836672 0.547705i \(-0.815502\pi\)
0.836672 0.547705i \(-0.184498\pi\)
\(684\) −5.25544 −0.200947
\(685\) 0 0
\(686\) 8.86141 0.338330
\(687\) − 33.7228i − 1.28661i
\(688\) − 4.00000i − 0.152499i
\(689\) 8.23369 0.313679
\(690\) 0 0
\(691\) 40.2337 1.53056 0.765281 0.643697i \(-0.222600\pi\)
0.765281 + 0.643697i \(0.222600\pi\)
\(692\) 6.00000i 0.228086i
\(693\) − 28.2337i − 1.07251i
\(694\) −32.2337 −1.22357
\(695\) 0 0
\(696\) 4.62772 0.175413
\(697\) 15.7663i 0.597192i
\(698\) − 19.4891i − 0.737674i
\(699\) 4.62772 0.175036
\(700\) 0 0
\(701\) −37.3723 −1.41153 −0.705766 0.708445i \(-0.749397\pi\)
−0.705766 + 0.708445i \(0.749397\pi\)
\(702\) − 36.2337i − 1.36755i
\(703\) 5.88316i 0.221887i
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 0.510875 0.0192270
\(707\) − 20.2337i − 0.760966i
\(708\) 9.25544i 0.347841i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −10.5109 −0.394189
\(712\) − 1.37228i − 0.0514284i
\(713\) 9.25544i 0.346619i
\(714\) −15.6060 −0.584039
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 49.7228i − 1.85693i
\(718\) 0 0
\(719\) −13.8832 −0.517754 −0.258877 0.965910i \(-0.583352\pi\)
−0.258877 + 0.965910i \(0.583352\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) − 18.6060i − 0.692442i
\(723\) 74.1902i 2.75916i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 3.37228 0.125157
\(727\) 24.2337i 0.898778i 0.893336 + 0.449389i \(0.148358\pi\)
−0.893336 + 0.449389i \(0.851642\pi\)
\(728\) − 6.74456i − 0.249970i
\(729\) −117.935 −4.36795
\(730\) 0 0
\(731\) −5.48913 −0.203023
\(732\) − 18.1168i − 0.669618i
\(733\) 46.2337i 1.70768i 0.520535 + 0.853840i \(0.325732\pi\)
−0.520535 + 0.853840i \(0.674268\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −2.74456 −0.101166
\(737\) 8.00000i 0.294684i
\(738\) 96.1902i 3.54081i
\(739\) 20.4674 0.752905 0.376452 0.926436i \(-0.377144\pi\)
0.376452 + 0.926436i \(0.377144\pi\)
\(740\) 0 0
\(741\) −4.23369 −0.155528
\(742\) − 13.8832i − 0.509667i
\(743\) − 4.62772i − 0.169775i −0.996391 0.0848873i \(-0.972947\pi\)
0.996391 0.0848873i \(-0.0270530\pi\)
\(744\) −11.3723 −0.416928
\(745\) 0 0
\(746\) −31.4891 −1.15290
\(747\) 22.9783i 0.840730i
\(748\) − 1.37228i − 0.0501756i
\(749\) 40.4674 1.47865
\(750\) 0 0
\(751\) 8.86141 0.323357 0.161679 0.986843i \(-0.448309\pi\)
0.161679 + 0.986843i \(0.448309\pi\)
\(752\) − 2.74456i − 0.100084i
\(753\) − 9.25544i − 0.337287i
\(754\) 2.74456 0.0999511
\(755\) 0 0
\(756\) −61.0951 −2.22201
\(757\) 20.9783i 0.762467i 0.924479 + 0.381234i \(0.124501\pi\)
−0.924479 + 0.381234i \(0.875499\pi\)
\(758\) 0.233688i 0.00848793i
\(759\) −9.25544 −0.335951
\(760\) 0 0
\(761\) 4.97825 0.180461 0.0902307 0.995921i \(-0.471240\pi\)
0.0902307 + 0.995921i \(0.471240\pi\)
\(762\) − 26.9783i − 0.977319i
\(763\) − 52.2337i − 1.89099i
\(764\) 5.48913 0.198590
\(765\) 0 0
\(766\) −32.2337 −1.16465
\(767\) 5.48913i 0.198201i
\(768\) − 3.37228i − 0.121687i
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) −60.7011 −2.18610
\(772\) − 14.8614i − 0.534874i
\(773\) − 33.6060i − 1.20872i −0.796710 0.604361i \(-0.793428\pi\)
0.796710 0.604361i \(-0.206572\pi\)
\(774\) −33.4891 −1.20374
\(775\) 0 0
\(776\) 12.7446 0.457503
\(777\) 106.584i 3.82369i
\(778\) − 6.00000i − 0.215110i
\(779\) 7.21194 0.258395
\(780\) 0 0
\(781\) −10.1168 −0.362009
\(782\) 3.76631i 0.134683i
\(783\) − 24.8614i − 0.888474i
\(784\) −4.37228 −0.156153
\(785\) 0 0
\(786\) 74.5842 2.66033
\(787\) 44.4674i 1.58509i 0.609813 + 0.792545i \(0.291245\pi\)
−0.609813 + 0.792545i \(0.708755\pi\)
\(788\) − 20.7446i − 0.738994i
\(789\) −83.8397 −2.98477
\(790\) 0 0
\(791\) 10.9783 0.390342
\(792\) − 8.37228i − 0.297496i
\(793\) − 10.7446i − 0.381551i
\(794\) 24.9783 0.886445
\(795\) 0 0
\(796\) 18.1168 0.642135
\(797\) − 11.4891i − 0.406966i −0.979079 0.203483i \(-0.934774\pi\)
0.979079 0.203483i \(-0.0652261\pi\)
\(798\) 7.13859i 0.252703i
\(799\) −3.76631 −0.133243
\(800\) 0 0
\(801\) −11.4891 −0.405948
\(802\) 13.3723i 0.472192i
\(803\) 15.4891i 0.546599i
\(804\) 26.9783 0.951450
\(805\) 0 0
\(806\) −6.74456 −0.237567
\(807\) − 29.4891i − 1.03807i
\(808\) − 6.00000i − 0.211079i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 44.8614 1.57530 0.787649 0.616125i \(-0.211298\pi\)
0.787649 + 0.616125i \(0.211298\pi\)
\(812\) − 4.62772i − 0.162401i
\(813\) 53.9565i 1.89234i
\(814\) −9.37228 −0.328498
\(815\) 0 0
\(816\) −4.62772 −0.162003
\(817\) 2.51087i 0.0878444i
\(818\) 1.76631i 0.0617577i
\(819\) −56.4674 −1.97313
\(820\) 0 0
\(821\) 11.4891 0.400973 0.200487 0.979696i \(-0.435748\pi\)
0.200487 + 0.979696i \(0.435748\pi\)
\(822\) − 29.4891i − 1.02855i
\(823\) − 28.0000i − 0.976019i −0.872838 0.488009i \(-0.837723\pi\)
0.872838 0.488009i \(-0.162277\pi\)
\(824\) −9.48913 −0.330569
\(825\) 0 0
\(826\) 9.25544 0.322038
\(827\) − 46.9783i − 1.63359i −0.576925 0.816797i \(-0.695748\pi\)
0.576925 0.816797i \(-0.304252\pi\)
\(828\) 22.9783i 0.798549i
\(829\) 24.7446 0.859414 0.429707 0.902968i \(-0.358617\pi\)
0.429707 + 0.902968i \(0.358617\pi\)
\(830\) 0 0
\(831\) 42.9783 1.49090
\(832\) − 2.00000i − 0.0693375i
\(833\) 6.00000i 0.207888i
\(834\) 13.4891 0.467090
\(835\) 0 0
\(836\) −0.627719 −0.0217101
\(837\) 61.0951i 2.11176i
\(838\) 12.0000i 0.414533i
\(839\) 10.9783 0.379011 0.189506 0.981880i \(-0.439311\pi\)
0.189506 + 0.981880i \(0.439311\pi\)
\(840\) 0 0
\(841\) −27.1168 −0.935064
\(842\) 10.2337i 0.352676i
\(843\) − 60.7011i − 2.09066i
\(844\) −6.11684 −0.210550
\(845\) 0 0
\(846\) −22.9783 −0.790009
\(847\) − 3.37228i − 0.115873i
\(848\) − 4.11684i − 0.141373i
\(849\) 17.7228 0.608245
\(850\) 0 0
\(851\) 25.7228 0.881767
\(852\) 34.1168i 1.16882i
\(853\) − 38.4674i − 1.31710i −0.752538 0.658549i \(-0.771171\pi\)
0.752538 0.658549i \(-0.228829\pi\)
\(854\) −18.1168 −0.619946
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) − 36.3505i − 1.24171i −0.783925 0.620855i \(-0.786785\pi\)
0.783925 0.620855i \(-0.213215\pi\)
\(858\) − 6.74456i − 0.230256i
\(859\) 42.7446 1.45843 0.729213 0.684287i \(-0.239886\pi\)
0.729213 + 0.684287i \(0.239886\pi\)
\(860\) 0 0
\(861\) 130.658 4.45280
\(862\) − 34.9783i − 1.19136i
\(863\) 21.2554i 0.723544i 0.932267 + 0.361772i \(0.117828\pi\)
−0.932267 + 0.361772i \(0.882172\pi\)
\(864\) −18.1168 −0.616348
\(865\) 0 0
\(866\) −27.7228 −0.942060
\(867\) − 50.9783i − 1.73131i
\(868\) 11.3723i 0.386000i
\(869\) −1.25544 −0.0425878
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) − 15.4891i − 0.524528i
\(873\) − 106.701i − 3.61128i
\(874\) 1.72281 0.0582750
\(875\) 0 0
\(876\) 52.2337 1.76481
\(877\) − 36.9783i − 1.24867i −0.781158 0.624333i \(-0.785371\pi\)
0.781158 0.624333i \(-0.214629\pi\)
\(878\) − 18.9783i − 0.640485i
\(879\) 79.2119 2.67175
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 36.6060i 1.23259i
\(883\) 3.37228i 0.113486i 0.998389 + 0.0567432i \(0.0180716\pi\)
−0.998389 + 0.0567432i \(0.981928\pi\)
\(884\) −2.74456 −0.0923096
\(885\) 0 0
\(886\) −29.4891 −0.990707
\(887\) 10.9783i 0.368614i 0.982869 + 0.184307i \(0.0590040\pi\)
−0.982869 + 0.184307i \(0.940996\pi\)
\(888\) 31.6060i 1.06063i
\(889\) −26.9783 −0.904821
\(890\) 0 0
\(891\) −35.9783 −1.20532
\(892\) 18.7446i 0.627614i
\(893\) 1.72281i 0.0576517i
\(894\) −72.8614 −2.43685
\(895\) 0 0
\(896\) −3.37228 −0.112660
\(897\) 18.5109i 0.618060i
\(898\) − 28.9783i − 0.967017i
\(899\) −4.62772 −0.154343
\(900\) 0 0
\(901\) −5.64947 −0.188211
\(902\) 11.4891i 0.382546i
\(903\) 45.4891i 1.51378i
\(904\) 3.25544 0.108274
\(905\) 0 0
\(906\) −41.2554 −1.37062
\(907\) 0.394031i 0.0130836i 0.999979 + 0.00654179i \(0.00208233\pi\)
−0.999979 + 0.00654179i \(0.997918\pi\)
\(908\) − 2.74456i − 0.0910815i
\(909\) −50.2337 −1.66615
\(910\) 0 0
\(911\) −8.39403 −0.278107 −0.139053 0.990285i \(-0.544406\pi\)
−0.139053 + 0.990285i \(0.544406\pi\)
\(912\) 2.11684i 0.0700957i
\(913\) 2.74456i 0.0908318i
\(914\) −16.3505 −0.540828
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 74.5842i − 2.46299i
\(918\) 24.8614i 0.820549i
\(919\) −18.9783 −0.626035 −0.313017 0.949747i \(-0.601340\pi\)
−0.313017 + 0.949747i \(0.601340\pi\)
\(920\) 0 0
\(921\) −17.7228 −0.583987
\(922\) 16.1168i 0.530780i
\(923\) 20.2337i 0.666000i
\(924\) −11.3723 −0.374121
\(925\) 0 0
\(926\) 0.233688 0.00767946
\(927\) 79.4456i 2.60934i
\(928\) − 1.37228i − 0.0450473i
\(929\) −24.3505 −0.798915 −0.399458 0.916752i \(-0.630801\pi\)
−0.399458 + 0.916752i \(0.630801\pi\)
\(930\) 0 0
\(931\) 2.74456 0.0899494
\(932\) − 1.37228i − 0.0449506i
\(933\) − 65.3288i − 2.13877i
\(934\) −19.3723 −0.633880
\(935\) 0 0
\(936\) −16.7446 −0.547313
\(937\) 28.5109i 0.931410i 0.884940 + 0.465705i \(0.154199\pi\)
−0.884940 + 0.465705i \(0.845801\pi\)
\(938\) − 26.9783i − 0.880871i
\(939\) −74.1902 −2.42111
\(940\) 0 0
\(941\) −15.0951 −0.492086 −0.246043 0.969259i \(-0.579130\pi\)
−0.246043 + 0.969259i \(0.579130\pi\)
\(942\) − 31.6060i − 1.02978i
\(943\) − 31.5326i − 1.02684i
\(944\) 2.74456 0.0893279
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) − 48.8614i − 1.58778i −0.608060 0.793891i \(-0.708052\pi\)
0.608060 0.793891i \(-0.291948\pi\)
\(948\) 4.23369i 0.137504i
\(949\) 30.9783 1.00560
\(950\) 0 0
\(951\) 82.1168 2.66282
\(952\) 4.62772i 0.149985i
\(953\) 40.1168i 1.29951i 0.760143 + 0.649756i \(0.225129\pi\)
−0.760143 + 0.649756i \(0.774871\pi\)
\(954\) −34.4674 −1.11592
\(955\) 0 0
\(956\) −14.7446 −0.476873
\(957\) − 4.62772i − 0.149593i
\(958\) − 5.48913i − 0.177346i
\(959\) −29.4891 −0.952254
\(960\) 0 0
\(961\) −19.6277 −0.633152
\(962\) 18.7446i 0.604349i
\(963\) − 100.467i − 3.23752i
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 31.2119 1.00423
\(967\) − 47.6060i − 1.53090i −0.643493 0.765452i \(-0.722515\pi\)
0.643493 0.765452i \(-0.277485\pi\)
\(968\) − 1.00000i − 0.0321412i
\(969\) 2.90491 0.0933190
\(970\) 0 0
\(971\) −1.02175 −0.0327895 −0.0163947 0.999866i \(-0.505219\pi\)
−0.0163947 + 0.999866i \(0.505219\pi\)
\(972\) 66.9783i 2.14833i
\(973\) − 13.4891i − 0.432442i
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −5.37228 −0.171963
\(977\) − 14.2337i − 0.455376i −0.973734 0.227688i \(-0.926883\pi\)
0.973734 0.227688i \(-0.0731166\pi\)
\(978\) − 19.8397i − 0.634402i
\(979\) −1.37228 −0.0438583
\(980\) 0 0
\(981\) −129.679 −4.14034
\(982\) − 7.37228i − 0.235259i
\(983\) − 13.7228i − 0.437690i −0.975760 0.218845i \(-0.929771\pi\)
0.975760 0.218845i \(-0.0702289\pi\)
\(984\) 38.7446 1.23513
\(985\) 0 0
\(986\) −1.88316 −0.0599719
\(987\) 31.2119i 0.993487i
\(988\) 1.25544i 0.0399408i
\(989\) 10.9783 0.349088
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 3.37228i 0.107070i
\(993\) − 104.467i − 3.31517i
\(994\) 34.1168 1.08212
\(995\) 0 0
\(996\) 9.25544 0.293270
\(997\) − 22.2337i − 0.704148i −0.935972 0.352074i \(-0.885477\pi\)
0.935972 0.352074i \(-0.114523\pi\)
\(998\) 33.4891i 1.06008i
\(999\) 169.796 5.37211
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 550.2.b.f.199.3 4
3.2 odd 2 4950.2.c.bc.199.1 4
4.3 odd 2 4400.2.b.p.4049.4 4
5.2 odd 4 110.2.a.d.1.1 2
5.3 odd 4 550.2.a.n.1.2 2
5.4 even 2 inner 550.2.b.f.199.2 4
15.2 even 4 990.2.a.m.1.2 2
15.8 even 4 4950.2.a.bw.1.1 2
15.14 odd 2 4950.2.c.bc.199.4 4
20.3 even 4 4400.2.a.bl.1.1 2
20.7 even 4 880.2.a.n.1.2 2
20.19 odd 2 4400.2.b.p.4049.1 4
35.27 even 4 5390.2.a.bp.1.2 2
40.27 even 4 3520.2.a.bj.1.1 2
40.37 odd 4 3520.2.a.bq.1.2 2
55.32 even 4 1210.2.a.r.1.1 2
55.43 even 4 6050.2.a.cb.1.2 2
60.47 odd 4 7920.2.a.bq.1.1 2
220.87 odd 4 9680.2.a.bt.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.1 2 5.2 odd 4
550.2.a.n.1.2 2 5.3 odd 4
550.2.b.f.199.2 4 5.4 even 2 inner
550.2.b.f.199.3 4 1.1 even 1 trivial
880.2.a.n.1.2 2 20.7 even 4
990.2.a.m.1.2 2 15.2 even 4
1210.2.a.r.1.1 2 55.32 even 4
3520.2.a.bj.1.1 2 40.27 even 4
3520.2.a.bq.1.2 2 40.37 odd 4
4400.2.a.bl.1.1 2 20.3 even 4
4400.2.b.p.4049.1 4 20.19 odd 2
4400.2.b.p.4049.4 4 4.3 odd 2
4950.2.a.bw.1.1 2 15.8 even 4
4950.2.c.bc.199.1 4 3.2 odd 2
4950.2.c.bc.199.4 4 15.14 odd 2
5390.2.a.bp.1.2 2 35.27 even 4
6050.2.a.cb.1.2 2 55.43 even 4
7920.2.a.bq.1.1 2 60.47 odd 4
9680.2.a.bt.1.2 2 220.87 odd 4