Properties

Label 1200.3.c.h.449.4
Level $1200$
Weight $3$
Character 1200.449
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(449,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.c (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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.4
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1200.449
Dual form 1200.3.c.h.449.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.82843 + 1.00000i) q^{3} -7.00000i q^{7} +(7.00000 + 5.65685i) q^{9} +O(q^{10})\) \(q+(2.82843 + 1.00000i) q^{3} -7.00000i q^{7} +(7.00000 + 5.65685i) q^{9} +8.48528i q^{11} -25.0000i q^{13} +25.4558 q^{17} -7.00000 q^{19} +(7.00000 - 19.7990i) q^{21} -25.4558 q^{23} +(14.1421 + 23.0000i) q^{27} -42.4264i q^{29} +7.00000 q^{31} +(-8.48528 + 24.0000i) q^{33} -2.00000i q^{37} +(25.0000 - 70.7107i) q^{39} +8.48528i q^{41} -41.0000i q^{43} +(72.0000 + 25.4558i) q^{51} +59.3970 q^{53} +(-19.7990 - 7.00000i) q^{57} -33.9411i q^{59} -1.00000 q^{61} +(39.5980 - 49.0000i) q^{63} +17.0000i q^{67} +(-72.0000 - 25.4558i) q^{69} -42.4264i q^{71} -70.0000i q^{73} +59.3970 q^{77} -58.0000 q^{79} +(17.0000 + 79.1960i) q^{81} +118.794 q^{83} +(42.4264 - 120.000i) q^{87} -135.765i q^{89} -175.000 q^{91} +(19.7990 + 7.00000i) q^{93} +49.0000i q^{97} +(-48.0000 + 59.3970i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{9} - 28 q^{19} + 28 q^{21} + 28 q^{31} + 100 q^{39} + 288 q^{51} - 4 q^{61} - 288 q^{69} - 232 q^{79} + 68 q^{81} - 700 q^{91} - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.82843 + 1.00000i 0.942809 + 0.333333i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(8\) 0 0
\(9\) 7.00000 + 5.65685i 0.777778 + 0.628539i
\(10\) 0 0
\(11\) 8.48528i 0.771389i 0.922627 + 0.385695i \(0.126038\pi\)
−0.922627 + 0.385695i \(0.873962\pi\)
\(12\) 0 0
\(13\) 25.0000i 1.92308i −0.274670 0.961538i \(-0.588569\pi\)
0.274670 0.961538i \(-0.411431\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 25.4558 1.49740 0.748701 0.662908i \(-0.230678\pi\)
0.748701 + 0.662908i \(0.230678\pi\)
\(18\) 0 0
\(19\) −7.00000 −0.368421 −0.184211 0.982887i \(-0.558973\pi\)
−0.184211 + 0.982887i \(0.558973\pi\)
\(20\) 0 0
\(21\) 7.00000 19.7990i 0.333333 0.942809i
\(22\) 0 0
\(23\) −25.4558 −1.10678 −0.553388 0.832924i \(-0.686665\pi\)
−0.553388 + 0.832924i \(0.686665\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 14.1421 + 23.0000i 0.523783 + 0.851852i
\(28\) 0 0
\(29\) 42.4264i 1.46298i −0.681852 0.731490i \(-0.738825\pi\)
0.681852 0.731490i \(-0.261175\pi\)
\(30\) 0 0
\(31\) 7.00000 0.225806 0.112903 0.993606i \(-0.463985\pi\)
0.112903 + 0.993606i \(0.463985\pi\)
\(32\) 0 0
\(33\) −8.48528 + 24.0000i −0.257130 + 0.727273i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.0540541i −0.999635 0.0270270i \(-0.991396\pi\)
0.999635 0.0270270i \(-0.00860402\pi\)
\(38\) 0 0
\(39\) 25.0000 70.7107i 0.641026 1.81309i
\(40\) 0 0
\(41\) 8.48528i 0.206958i 0.994632 + 0.103479i \(0.0329975\pi\)
−0.994632 + 0.103479i \(0.967003\pi\)
\(42\) 0 0
\(43\) 41.0000i 0.953488i −0.879042 0.476744i \(-0.841817\pi\)
0.879042 0.476744i \(-0.158183\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 72.0000 + 25.4558i 1.41176 + 0.499134i
\(52\) 0 0
\(53\) 59.3970 1.12070 0.560349 0.828257i \(-0.310667\pi\)
0.560349 + 0.828257i \(0.310667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −19.7990 7.00000i −0.347351 0.122807i
\(58\) 0 0
\(59\) 33.9411i 0.575273i −0.957740 0.287637i \(-0.907130\pi\)
0.957740 0.287637i \(-0.0928695\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.0163934 −0.00819672 0.999966i \(-0.502609\pi\)
−0.00819672 + 0.999966i \(0.502609\pi\)
\(62\) 0 0
\(63\) 39.5980 49.0000i 0.628539 0.777778i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 17.0000i 0.253731i 0.991920 + 0.126866i \(0.0404917\pi\)
−0.991920 + 0.126866i \(0.959508\pi\)
\(68\) 0 0
\(69\) −72.0000 25.4558i −1.04348 0.368925i
\(70\) 0 0
\(71\) 42.4264i 0.597555i −0.954323 0.298778i \(-0.903421\pi\)
0.954323 0.298778i \(-0.0965788\pi\)
\(72\) 0 0
\(73\) 70.0000i 0.958904i −0.877568 0.479452i \(-0.840835\pi\)
0.877568 0.479452i \(-0.159165\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 59.3970 0.771389
\(78\) 0 0
\(79\) −58.0000 −0.734177 −0.367089 0.930186i \(-0.619645\pi\)
−0.367089 + 0.930186i \(0.619645\pi\)
\(80\) 0 0
\(81\) 17.0000 + 79.1960i 0.209877 + 0.977728i
\(82\) 0 0
\(83\) 118.794 1.43125 0.715626 0.698484i \(-0.246141\pi\)
0.715626 + 0.698484i \(0.246141\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 42.4264 120.000i 0.487660 1.37931i
\(88\) 0 0
\(89\) 135.765i 1.52544i −0.646727 0.762722i \(-0.723863\pi\)
0.646727 0.762722i \(-0.276137\pi\)
\(90\) 0 0
\(91\) −175.000 −1.92308
\(92\) 0 0
\(93\) 19.7990 + 7.00000i 0.212892 + 0.0752688i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 49.0000i 0.505155i 0.967577 + 0.252577i \(0.0812782\pi\)
−0.967577 + 0.252577i \(0.918722\pi\)
\(98\) 0 0
\(99\) −48.0000 + 59.3970i −0.484848 + 0.599969i
\(100\) 0 0
\(101\) 59.3970i 0.588089i 0.955792 + 0.294044i \(0.0950014\pi\)
−0.955792 + 0.294044i \(0.904999\pi\)
\(102\) 0 0
\(103\) 154.000i 1.49515i 0.664180 + 0.747573i \(0.268781\pi\)
−0.664180 + 0.747573i \(0.731219\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 178.191 1.66534 0.832668 0.553773i \(-0.186812\pi\)
0.832668 + 0.553773i \(0.186812\pi\)
\(108\) 0 0
\(109\) 25.0000 0.229358 0.114679 0.993403i \(-0.463416\pi\)
0.114679 + 0.993403i \(0.463416\pi\)
\(110\) 0 0
\(111\) 2.00000 5.65685i 0.0180180 0.0509627i
\(112\) 0 0
\(113\) −16.9706 −0.150182 −0.0750910 0.997177i \(-0.523925\pi\)
−0.0750910 + 0.997177i \(0.523925\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 141.421 175.000i 1.20873 1.49573i
\(118\) 0 0
\(119\) 178.191i 1.49740i
\(120\) 0 0
\(121\) 49.0000 0.404959
\(122\) 0 0
\(123\) −8.48528 + 24.0000i −0.0689860 + 0.195122i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 34.0000i 0.267717i −0.991000 0.133858i \(-0.957263\pi\)
0.991000 0.133858i \(-0.0427367\pi\)
\(128\) 0 0
\(129\) 41.0000 115.966i 0.317829 0.898957i
\(130\) 0 0
\(131\) 195.161i 1.48978i 0.667186 + 0.744891i \(0.267499\pi\)
−0.667186 + 0.744891i \(0.732501\pi\)
\(132\) 0 0
\(133\) 49.0000i 0.368421i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −118.794 −0.867109 −0.433555 0.901127i \(-0.642741\pi\)
−0.433555 + 0.901127i \(0.642741\pi\)
\(138\) 0 0
\(139\) −154.000 −1.10791 −0.553957 0.832545i \(-0.686883\pi\)
−0.553957 + 0.832545i \(0.686883\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 212.132 1.48344
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 152.735i 1.02507i 0.858667 + 0.512534i \(0.171293\pi\)
−0.858667 + 0.512534i \(0.828707\pi\)
\(150\) 0 0
\(151\) 199.000 1.31788 0.658940 0.752195i \(-0.271005\pi\)
0.658940 + 0.752195i \(0.271005\pi\)
\(152\) 0 0
\(153\) 178.191 + 144.000i 1.16465 + 0.941176i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 145.000i 0.923567i 0.886993 + 0.461783i \(0.152790\pi\)
−0.886993 + 0.461783i \(0.847210\pi\)
\(158\) 0 0
\(159\) 168.000 + 59.3970i 1.05660 + 0.373566i
\(160\) 0 0
\(161\) 178.191i 1.10678i
\(162\) 0 0
\(163\) 161.000i 0.987730i −0.869539 0.493865i \(-0.835584\pi\)
0.869539 0.493865i \(-0.164416\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −110.309 −0.660531 −0.330265 0.943888i \(-0.607138\pi\)
−0.330265 + 0.943888i \(0.607138\pi\)
\(168\) 0 0
\(169\) −456.000 −2.69822
\(170\) 0 0
\(171\) −49.0000 39.5980i −0.286550 0.231567i
\(172\) 0 0
\(173\) 178.191 1.03001 0.515003 0.857189i \(-0.327791\pi\)
0.515003 + 0.857189i \(0.327791\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 33.9411 96.0000i 0.191758 0.542373i
\(178\) 0 0
\(179\) 118.794i 0.663653i −0.943340 0.331827i \(-0.892335\pi\)
0.943340 0.331827i \(-0.107665\pi\)
\(180\) 0 0
\(181\) −217.000 −1.19890 −0.599448 0.800414i \(-0.704613\pi\)
−0.599448 + 0.800414i \(0.704613\pi\)
\(182\) 0 0
\(183\) −2.82843 1.00000i −0.0154559 0.00546448i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 216.000i 1.15508i
\(188\) 0 0
\(189\) 161.000 98.9949i 0.851852 0.523783i
\(190\) 0 0
\(191\) 59.3970i 0.310979i 0.987838 + 0.155489i \(0.0496955\pi\)
−0.987838 + 0.155489i \(0.950305\pi\)
\(192\) 0 0
\(193\) 25.0000i 0.129534i −0.997900 0.0647668i \(-0.979370\pi\)
0.997900 0.0647668i \(-0.0206304\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 135.765 0.689160 0.344580 0.938757i \(-0.388021\pi\)
0.344580 + 0.938757i \(0.388021\pi\)
\(198\) 0 0
\(199\) −103.000 −0.517588 −0.258794 0.965933i \(-0.583325\pi\)
−0.258794 + 0.965933i \(0.583325\pi\)
\(200\) 0 0
\(201\) −17.0000 + 48.0833i −0.0845771 + 0.239220i
\(202\) 0 0
\(203\) −296.985 −1.46298
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −178.191 144.000i −0.860826 0.695652i
\(208\) 0 0
\(209\) 59.3970i 0.284196i
\(210\) 0 0
\(211\) 7.00000 0.0331754 0.0165877 0.999862i \(-0.494720\pi\)
0.0165877 + 0.999862i \(0.494720\pi\)
\(212\) 0 0
\(213\) 42.4264 120.000i 0.199185 0.563380i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 49.0000i 0.225806i
\(218\) 0 0
\(219\) 70.0000 197.990i 0.319635 0.904063i
\(220\) 0 0
\(221\) 636.396i 2.87962i
\(222\) 0 0
\(223\) 161.000i 0.721973i −0.932571 0.360987i \(-0.882440\pi\)
0.932571 0.360987i \(-0.117560\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 59.3970 0.261661 0.130830 0.991405i \(-0.458236\pi\)
0.130830 + 0.991405i \(0.458236\pi\)
\(228\) 0 0
\(229\) 97.0000 0.423581 0.211790 0.977315i \(-0.432071\pi\)
0.211790 + 0.977315i \(0.432071\pi\)
\(230\) 0 0
\(231\) 168.000 + 59.3970i 0.727273 + 0.257130i
\(232\) 0 0
\(233\) −263.044 −1.12894 −0.564472 0.825453i \(-0.690920\pi\)
−0.564472 + 0.825453i \(0.690920\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −164.049 58.0000i −0.692189 0.244726i
\(238\) 0 0
\(239\) 59.3970i 0.248523i 0.992250 + 0.124261i \(0.0396562\pi\)
−0.992250 + 0.124261i \(0.960344\pi\)
\(240\) 0 0
\(241\) 119.000 0.493776 0.246888 0.969044i \(-0.420592\pi\)
0.246888 + 0.969044i \(0.420592\pi\)
\(242\) 0 0
\(243\) −31.1127 + 241.000i −0.128036 + 0.991770i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 175.000i 0.708502i
\(248\) 0 0
\(249\) 336.000 + 118.794i 1.34940 + 0.477084i
\(250\) 0 0
\(251\) 288.500i 1.14940i −0.818364 0.574700i \(-0.805119\pi\)
0.818364 0.574700i \(-0.194881\pi\)
\(252\) 0 0
\(253\) 216.000i 0.853755i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 118.794 0.462233 0.231117 0.972926i \(-0.425762\pi\)
0.231117 + 0.972926i \(0.425762\pi\)
\(258\) 0 0
\(259\) −14.0000 −0.0540541
\(260\) 0 0
\(261\) 240.000 296.985i 0.919540 1.13787i
\(262\) 0 0
\(263\) −8.48528 −0.0322634 −0.0161317 0.999870i \(-0.505135\pi\)
−0.0161317 + 0.999870i \(0.505135\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 135.765 384.000i 0.508481 1.43820i
\(268\) 0 0
\(269\) 59.3970i 0.220807i −0.993887 0.110403i \(-0.964786\pi\)
0.993887 0.110403i \(-0.0352142\pi\)
\(270\) 0 0
\(271\) −470.000 −1.73432 −0.867159 0.498032i \(-0.834056\pi\)
−0.867159 + 0.498032i \(0.834056\pi\)
\(272\) 0 0
\(273\) −494.975 175.000i −1.81309 0.641026i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 217.000i 0.783394i 0.920094 + 0.391697i \(0.128112\pi\)
−0.920094 + 0.391697i \(0.871888\pi\)
\(278\) 0 0
\(279\) 49.0000 + 39.5980i 0.175627 + 0.141928i
\(280\) 0 0
\(281\) 517.602i 1.84200i 0.389562 + 0.921000i \(0.372626\pi\)
−0.389562 + 0.921000i \(0.627374\pi\)
\(282\) 0 0
\(283\) 65.0000i 0.229682i −0.993384 0.114841i \(-0.963364\pi\)
0.993384 0.114841i \(-0.0366359\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 59.3970 0.206958
\(288\) 0 0
\(289\) 359.000 1.24221
\(290\) 0 0
\(291\) −49.0000 + 138.593i −0.168385 + 0.476264i
\(292\) 0 0
\(293\) 169.706 0.579200 0.289600 0.957148i \(-0.406478\pi\)
0.289600 + 0.957148i \(0.406478\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −195.161 + 120.000i −0.657109 + 0.404040i
\(298\) 0 0
\(299\) 636.396i 2.12842i
\(300\) 0 0
\(301\) −287.000 −0.953488
\(302\) 0 0
\(303\) −59.3970 + 168.000i −0.196030 + 0.554455i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 521.000i 1.69707i 0.529141 + 0.848534i \(0.322514\pi\)
−0.529141 + 0.848534i \(0.677486\pi\)
\(308\) 0 0
\(309\) −154.000 + 435.578i −0.498382 + 1.40964i
\(310\) 0 0
\(311\) 33.9411i 0.109135i 0.998510 + 0.0545677i \(0.0173781\pi\)
−0.998510 + 0.0545677i \(0.982622\pi\)
\(312\) 0 0
\(313\) 119.000i 0.380192i 0.981766 + 0.190096i \(0.0608799\pi\)
−0.981766 + 0.190096i \(0.939120\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 152.735 0.481814 0.240907 0.970548i \(-0.422555\pi\)
0.240907 + 0.970548i \(0.422555\pi\)
\(318\) 0 0
\(319\) 360.000 1.12853
\(320\) 0 0
\(321\) 504.000 + 178.191i 1.57009 + 0.555112i
\(322\) 0 0
\(323\) −178.191 −0.551675
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 70.7107 + 25.0000i 0.216241 + 0.0764526i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 418.000 1.26284 0.631420 0.775441i \(-0.282472\pi\)
0.631420 + 0.775441i \(0.282472\pi\)
\(332\) 0 0
\(333\) 11.3137 14.0000i 0.0339751 0.0420420i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 553.000i 1.64095i 0.571683 + 0.820475i \(0.306291\pi\)
−0.571683 + 0.820475i \(0.693709\pi\)
\(338\) 0 0
\(339\) −48.0000 16.9706i −0.141593 0.0500607i
\(340\) 0 0
\(341\) 59.3970i 0.174185i
\(342\) 0 0
\(343\) 343.000i 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −644.881 −1.85845 −0.929224 0.369517i \(-0.879523\pi\)
−0.929224 + 0.369517i \(0.879523\pi\)
\(348\) 0 0
\(349\) −266.000 −0.762178 −0.381089 0.924538i \(-0.624451\pi\)
−0.381089 + 0.924538i \(0.624451\pi\)
\(350\) 0 0
\(351\) 575.000 353.553i 1.63818 1.00727i
\(352\) 0 0
\(353\) −415.779 −1.17784 −0.588922 0.808190i \(-0.700447\pi\)
−0.588922 + 0.808190i \(0.700447\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 178.191 504.000i 0.499134 1.41176i
\(358\) 0 0
\(359\) 330.926i 0.921799i 0.887452 + 0.460900i \(0.152473\pi\)
−0.887452 + 0.460900i \(0.847527\pi\)
\(360\) 0 0
\(361\) −312.000 −0.864266
\(362\) 0 0
\(363\) 138.593 + 49.0000i 0.381799 + 0.134986i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 103.000i 0.280654i −0.990105 0.140327i \(-0.955185\pi\)
0.990105 0.140327i \(-0.0448154\pi\)
\(368\) 0 0
\(369\) −48.0000 + 59.3970i −0.130081 + 0.160967i
\(370\) 0 0
\(371\) 415.779i 1.12070i
\(372\) 0 0
\(373\) 359.000i 0.962466i 0.876593 + 0.481233i \(0.159811\pi\)
−0.876593 + 0.481233i \(0.840189\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1060.66 −2.81342
\(378\) 0 0
\(379\) 377.000 0.994723 0.497361 0.867543i \(-0.334302\pi\)
0.497361 + 0.867543i \(0.334302\pi\)
\(380\) 0 0
\(381\) 34.0000 96.1665i 0.0892388 0.252406i
\(382\) 0 0
\(383\) −610.940 −1.59514 −0.797572 0.603224i \(-0.793883\pi\)
−0.797572 + 0.603224i \(0.793883\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 231.931 287.000i 0.599305 0.741602i
\(388\) 0 0
\(389\) 347.897i 0.894336i 0.894450 + 0.447168i \(0.147567\pi\)
−0.894450 + 0.447168i \(0.852433\pi\)
\(390\) 0 0
\(391\) −648.000 −1.65729
\(392\) 0 0
\(393\) −195.161 + 552.000i −0.496594 + 1.40458i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 239.000i 0.602015i −0.953622 0.301008i \(-0.902677\pi\)
0.953622 0.301008i \(-0.0973229\pi\)
\(398\) 0 0
\(399\) −49.0000 + 138.593i −0.122807 + 0.347351i
\(400\) 0 0
\(401\) 93.3381i 0.232763i −0.993205 0.116382i \(-0.962870\pi\)
0.993205 0.116382i \(-0.0371296\pi\)
\(402\) 0 0
\(403\) 175.000i 0.434243i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.9706 0.0416967
\(408\) 0 0
\(409\) −455.000 −1.11247 −0.556235 0.831025i \(-0.687754\pi\)
−0.556235 + 0.831025i \(0.687754\pi\)
\(410\) 0 0
\(411\) −336.000 118.794i −0.817518 0.289036i
\(412\) 0 0
\(413\) −237.588 −0.575273
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −435.578 154.000i −1.04455 0.369305i
\(418\) 0 0
\(419\) 296.985i 0.708794i −0.935095 0.354397i \(-0.884686\pi\)
0.935095 0.354397i \(-0.115314\pi\)
\(420\) 0 0
\(421\) −526.000 −1.24941 −0.624703 0.780862i \(-0.714780\pi\)
−0.624703 + 0.780862i \(0.714780\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 7.00000i 0.0163934i
\(428\) 0 0
\(429\) 600.000 + 212.132i 1.39860 + 0.494480i
\(430\) 0 0
\(431\) 280.014i 0.649685i −0.945768 0.324843i \(-0.894689\pi\)
0.945768 0.324843i \(-0.105311\pi\)
\(432\) 0 0
\(433\) 119.000i 0.274827i 0.990514 + 0.137413i \(0.0438789\pi\)
−0.990514 + 0.137413i \(0.956121\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 178.191 0.407760
\(438\) 0 0
\(439\) −727.000 −1.65604 −0.828018 0.560701i \(-0.810532\pi\)
−0.828018 + 0.560701i \(0.810532\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 526.087 1.18756 0.593778 0.804629i \(-0.297636\pi\)
0.593778 + 0.804629i \(0.297636\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −152.735 + 432.000i −0.341689 + 0.966443i
\(448\) 0 0
\(449\) 254.558i 0.566945i 0.958980 + 0.283473i \(0.0914865\pi\)
−0.958980 + 0.283473i \(0.908513\pi\)
\(450\) 0 0
\(451\) −72.0000 −0.159645
\(452\) 0 0
\(453\) 562.857 + 199.000i 1.24251 + 0.439294i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 310.000i 0.678337i 0.940726 + 0.339168i \(0.110146\pi\)
−0.940726 + 0.339168i \(0.889854\pi\)
\(458\) 0 0
\(459\) 360.000 + 585.484i 0.784314 + 1.27557i
\(460\) 0 0
\(461\) 721.249i 1.56453i −0.622945 0.782266i \(-0.714064\pi\)
0.622945 0.782266i \(-0.285936\pi\)
\(462\) 0 0
\(463\) 730.000i 1.57667i 0.615244 + 0.788337i \(0.289058\pi\)
−0.615244 + 0.788337i \(0.710942\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −195.161 −0.417905 −0.208952 0.977926i \(-0.567005\pi\)
−0.208952 + 0.977926i \(0.567005\pi\)
\(468\) 0 0
\(469\) 119.000 0.253731
\(470\) 0 0
\(471\) −145.000 + 410.122i −0.307856 + 0.870747i
\(472\) 0 0
\(473\) 347.897 0.735511
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 415.779 + 336.000i 0.871654 + 0.704403i
\(478\) 0 0
\(479\) 390.323i 0.814870i 0.913234 + 0.407435i \(0.133577\pi\)
−0.913234 + 0.407435i \(0.866423\pi\)
\(480\) 0 0
\(481\) −50.0000 −0.103950
\(482\) 0 0
\(483\) −178.191 + 504.000i −0.368925 + 1.04348i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 473.000i 0.971253i 0.874167 + 0.485626i \(0.161408\pi\)
−0.874167 + 0.485626i \(0.838592\pi\)
\(488\) 0 0
\(489\) 161.000 455.377i 0.329243 0.931241i
\(490\) 0 0
\(491\) 814.587i 1.65904i 0.558479 + 0.829518i \(0.311385\pi\)
−0.558479 + 0.829518i \(0.688615\pi\)
\(492\) 0 0
\(493\) 1080.00i 2.19067i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −296.985 −0.597555
\(498\) 0 0
\(499\) −175.000 −0.350701 −0.175351 0.984506i \(-0.556106\pi\)
−0.175351 + 0.984506i \(0.556106\pi\)
\(500\) 0 0
\(501\) −312.000 110.309i −0.622754 0.220177i
\(502\) 0 0
\(503\) 347.897 0.691643 0.345822 0.938300i \(-0.387600\pi\)
0.345822 + 0.938300i \(0.387600\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1289.76 456.000i −2.54391 0.899408i
\(508\) 0 0
\(509\) 729.734i 1.43366i 0.697247 + 0.716831i \(0.254408\pi\)
−0.697247 + 0.716831i \(0.745592\pi\)
\(510\) 0 0
\(511\) −490.000 −0.958904
\(512\) 0 0
\(513\) −98.9949 161.000i −0.192973 0.313840i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 504.000 + 178.191i 0.971098 + 0.343335i
\(520\) 0 0
\(521\) 568.514i 1.09120i 0.838047 + 0.545599i \(0.183698\pi\)
−0.838047 + 0.545599i \(0.816302\pi\)
\(522\) 0 0
\(523\) 175.000i 0.334608i 0.985905 + 0.167304i \(0.0535061\pi\)
−0.985905 + 0.167304i \(0.946494\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 178.191 0.338123
\(528\) 0 0
\(529\) 119.000 0.224953
\(530\) 0 0
\(531\) 192.000 237.588i 0.361582 0.447435i
\(532\) 0 0
\(533\) 212.132 0.397996
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 118.794 336.000i 0.221218 0.625698i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 863.000 1.59519 0.797597 0.603191i \(-0.206104\pi\)
0.797597 + 0.603191i \(0.206104\pi\)
\(542\) 0 0
\(543\) −613.769 217.000i −1.13033 0.399632i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 778.000i 1.42230i −0.703039 0.711152i \(-0.748174\pi\)
0.703039 0.711152i \(-0.251826\pi\)
\(548\) 0 0
\(549\) −7.00000 5.65685i −0.0127505 0.0103039i
\(550\) 0 0
\(551\) 296.985i 0.538992i
\(552\) 0 0
\(553\) 406.000i 0.734177i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 424.264 0.761695 0.380847 0.924638i \(-0.375632\pi\)
0.380847 + 0.924638i \(0.375632\pi\)
\(558\) 0 0
\(559\) −1025.00 −1.83363
\(560\) 0 0
\(561\) −216.000 + 610.940i −0.385027 + 1.08902i
\(562\) 0 0
\(563\) 381.838 0.678220 0.339110 0.940747i \(-0.389874\pi\)
0.339110 + 0.940747i \(0.389874\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 554.372 119.000i 0.977728 0.209877i
\(568\) 0 0
\(569\) 381.838i 0.671068i 0.942028 + 0.335534i \(0.108917\pi\)
−0.942028 + 0.335534i \(0.891083\pi\)
\(570\) 0 0
\(571\) 535.000 0.936953 0.468476 0.883476i \(-0.344803\pi\)
0.468476 + 0.883476i \(0.344803\pi\)
\(572\) 0 0
\(573\) −59.3970 + 168.000i −0.103660 + 0.293194i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 49.0000i 0.0849220i 0.999098 + 0.0424610i \(0.0135198\pi\)
−0.999098 + 0.0424610i \(0.986480\pi\)
\(578\) 0 0
\(579\) 25.0000 70.7107i 0.0431779 0.122126i
\(580\) 0 0
\(581\) 831.558i 1.43125i
\(582\) 0 0
\(583\) 504.000i 0.864494i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 415.779 0.708311 0.354156 0.935186i \(-0.384768\pi\)
0.354156 + 0.935186i \(0.384768\pi\)
\(588\) 0 0
\(589\) −49.0000 −0.0831919
\(590\) 0 0
\(591\) 384.000 + 135.765i 0.649746 + 0.229720i
\(592\) 0 0
\(593\) −772.161 −1.30213 −0.651063 0.759024i \(-0.725677\pi\)
−0.651063 + 0.759024i \(0.725677\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −291.328 103.000i −0.487987 0.172529i
\(598\) 0 0
\(599\) 644.881i 1.07660i −0.842754 0.538298i \(-0.819067\pi\)
0.842754 0.538298i \(-0.180933\pi\)
\(600\) 0 0
\(601\) 455.000 0.757072 0.378536 0.925587i \(-0.376428\pi\)
0.378536 + 0.925587i \(0.376428\pi\)
\(602\) 0 0
\(603\) −96.1665 + 119.000i −0.159480 + 0.197347i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 566.000i 0.932455i 0.884665 + 0.466227i \(0.154387\pi\)
−0.884665 + 0.466227i \(0.845613\pi\)
\(608\) 0 0
\(609\) −840.000 296.985i −1.37931 0.487660i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 578.000i 0.942904i 0.881892 + 0.471452i \(0.156270\pi\)
−0.881892 + 0.471452i \(0.843730\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 636.396 1.03144 0.515718 0.856758i \(-0.327525\pi\)
0.515718 + 0.856758i \(0.327525\pi\)
\(618\) 0 0
\(619\) 593.000 0.957997 0.478998 0.877816i \(-0.341000\pi\)
0.478998 + 0.877816i \(0.341000\pi\)
\(620\) 0 0
\(621\) −360.000 585.484i −0.579710 0.942809i
\(622\) 0 0
\(623\) −950.352 −1.52544
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 59.3970 168.000i 0.0947320 0.267943i
\(628\) 0 0
\(629\) 50.9117i 0.0809407i
\(630\) 0 0
\(631\) 559.000 0.885895 0.442948 0.896547i \(-0.353933\pi\)
0.442948 + 0.896547i \(0.353933\pi\)
\(632\) 0 0
\(633\) 19.7990 + 7.00000i 0.0312780 + 0.0110585i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 240.000 296.985i 0.375587 0.464765i
\(640\) 0 0
\(641\) 543.058i 0.847204i 0.905848 + 0.423602i \(0.139235\pi\)
−0.905848 + 0.423602i \(0.860765\pi\)
\(642\) 0 0
\(643\) 854.000i 1.32815i −0.747666 0.664075i \(-0.768826\pi\)
0.747666 0.664075i \(-0.231174\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 322.441 0.498363 0.249181 0.968457i \(-0.419838\pi\)
0.249181 + 0.968457i \(0.419838\pi\)
\(648\) 0 0
\(649\) 288.000 0.443760
\(650\) 0 0
\(651\) 49.0000 138.593i 0.0752688 0.212892i
\(652\) 0 0
\(653\) 356.382 0.545761 0.272880 0.962048i \(-0.412024\pi\)
0.272880 + 0.962048i \(0.412024\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 395.980 490.000i 0.602709 0.745814i
\(658\) 0 0
\(659\) 890.955i 1.35198i −0.736911 0.675990i \(-0.763716\pi\)
0.736911 0.675990i \(-0.236284\pi\)
\(660\) 0 0
\(661\) −910.000 −1.37670 −0.688351 0.725378i \(-0.741665\pi\)
−0.688351 + 0.725378i \(0.741665\pi\)
\(662\) 0 0
\(663\) 636.396 1800.00i 0.959873 2.71493i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1080.00i 1.61919i
\(668\) 0 0
\(669\) 161.000 455.377i 0.240658 0.680683i
\(670\) 0 0
\(671\) 8.48528i 0.0126457i
\(672\) 0 0
\(673\) 742.000i 1.10253i −0.834332 0.551263i \(-0.814146\pi\)
0.834332 0.551263i \(-0.185854\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −432.749 −0.639216 −0.319608 0.947550i \(-0.603551\pi\)
−0.319608 + 0.947550i \(0.603551\pi\)
\(678\) 0 0
\(679\) 343.000 0.505155
\(680\) 0 0
\(681\) 168.000 + 59.3970i 0.246696 + 0.0872202i
\(682\) 0 0
\(683\) −661.852 −0.969037 −0.484518 0.874781i \(-0.661005\pi\)
−0.484518 + 0.874781i \(0.661005\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 274.357 + 97.0000i 0.399356 + 0.141194i
\(688\) 0 0
\(689\) 1484.92i 2.15519i
\(690\) 0 0
\(691\) −302.000 −0.437048 −0.218524 0.975832i \(-0.570124\pi\)
−0.218524 + 0.975832i \(0.570124\pi\)
\(692\) 0 0
\(693\) 415.779 + 336.000i 0.599969 + 0.484848i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 216.000i 0.309900i
\(698\) 0 0
\(699\) −744.000 263.044i −1.06438 0.376314i
\(700\) 0 0
\(701\) 178.191i 0.254195i 0.991890 + 0.127098i \(0.0405662\pi\)
−0.991890 + 0.127098i \(0.959434\pi\)
\(702\) 0 0
\(703\) 14.0000i 0.0199147i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 415.779 0.588089
\(708\) 0 0
\(709\) −95.0000 −0.133992 −0.0669958 0.997753i \(-0.521341\pi\)
−0.0669958 + 0.997753i \(0.521341\pi\)
\(710\) 0 0
\(711\) −406.000 328.098i −0.571027 0.461459i
\(712\) 0 0
\(713\) −178.191 −0.249917
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −59.3970 + 168.000i −0.0828410 + 0.234310i
\(718\) 0 0
\(719\) 873.984i 1.21555i 0.794107 + 0.607777i \(0.207939\pi\)
−0.794107 + 0.607777i \(0.792061\pi\)
\(720\) 0 0
\(721\) 1078.00 1.49515
\(722\) 0 0
\(723\) 336.583 + 119.000i 0.465536 + 0.164592i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 871.000i 1.19807i −0.800721 0.599037i \(-0.795550\pi\)
0.800721 0.599037i \(-0.204450\pi\)
\(728\) 0 0
\(729\) −329.000 + 650.538i −0.451303 + 0.892371i
\(730\) 0 0
\(731\) 1043.69i 1.42776i
\(732\) 0 0
\(733\) 406.000i 0.553888i −0.960886 0.276944i \(-0.910678\pi\)
0.960886 0.276944i \(-0.0893217\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −144.250 −0.195726
\(738\) 0 0
\(739\) 830.000 1.12314 0.561570 0.827429i \(-0.310198\pi\)
0.561570 + 0.827429i \(0.310198\pi\)
\(740\) 0 0
\(741\) −175.000 + 494.975i −0.236167 + 0.667982i
\(742\) 0 0
\(743\) 1306.73 1.75873 0.879363 0.476152i \(-0.157969\pi\)
0.879363 + 0.476152i \(0.157969\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 831.558 + 672.000i 1.11320 + 0.899598i
\(748\) 0 0
\(749\) 1247.34i 1.66534i
\(750\) 0 0
\(751\) −350.000 −0.466045 −0.233023 0.972471i \(-0.574862\pi\)
−0.233023 + 0.972471i \(0.574862\pi\)
\(752\) 0 0
\(753\) 288.500 816.000i 0.383134 1.08367i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 265.000i 0.350066i 0.984563 + 0.175033i \(0.0560032\pi\)
−0.984563 + 0.175033i \(0.943997\pi\)
\(758\) 0 0
\(759\) 216.000 610.940i 0.284585 0.804928i
\(760\) 0 0
\(761\) 1069.15i 1.40492i 0.711722 + 0.702461i \(0.247915\pi\)
−0.711722 + 0.702461i \(0.752085\pi\)
\(762\) 0 0
\(763\) 175.000i 0.229358i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −848.528 −1.10629
\(768\) 0 0
\(769\) 529.000 0.687906 0.343953 0.938987i \(-0.388234\pi\)
0.343953 + 0.938987i \(0.388234\pi\)
\(770\) 0 0
\(771\) 336.000 + 118.794i 0.435798 + 0.154078i
\(772\) 0 0
\(773\) 1009.75 1.30627 0.653136 0.757240i \(-0.273453\pi\)
0.653136 + 0.757240i \(0.273453\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −39.5980 14.0000i −0.0509627 0.0180180i
\(778\) 0 0
\(779\) 59.3970i 0.0762477i
\(780\) 0 0
\(781\) 360.000 0.460948
\(782\) 0 0
\(783\) 975.807 600.000i 1.24624 0.766284i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1519.00i 1.93011i −0.262039 0.965057i \(-0.584395\pi\)
0.262039 0.965057i \(-0.415605\pi\)
\(788\) 0 0
\(789\) −24.0000 8.48528i −0.0304183 0.0107545i
\(790\) 0 0
\(791\) 118.794i 0.150182i
\(792\) 0 0
\(793\) 25.0000i 0.0315259i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 526.087 0.660085 0.330042 0.943966i \(-0.392937\pi\)
0.330042 + 0.943966i \(0.392937\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 768.000 950.352i 0.958801 1.18646i
\(802\) 0 0
\(803\) 593.970 0.739688
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 59.3970 168.000i 0.0736022 0.208178i
\(808\) 0 0
\(809\) 823.072i 1.01739i −0.860945 0.508697i \(-0.830127\pi\)
0.860945 0.508697i \(-0.169873\pi\)
\(810\) 0 0
\(811\) 319.000 0.393342 0.196671 0.980470i \(-0.436987\pi\)
0.196671 + 0.980470i \(0.436987\pi\)
\(812\) 0 0
\(813\) −1329.36 470.000i −1.63513 0.578106i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 287.000i 0.351285i
\(818\) 0 0
\(819\) −1225.00 989.949i −1.49573 1.20873i
\(820\) 0 0
\(821\) 118.794i 0.144694i 0.997380 + 0.0723471i \(0.0230489\pi\)
−0.997380 + 0.0723471i \(0.976951\pi\)
\(822\) 0 0
\(823\) 1375.00i 1.67072i 0.549706 + 0.835358i \(0.314740\pi\)
−0.549706 + 0.835358i \(0.685260\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −280.014 −0.338590 −0.169295 0.985565i \(-0.554149\pi\)
−0.169295 + 0.985565i \(0.554149\pi\)
\(828\) 0 0
\(829\) 142.000 0.171291 0.0856454 0.996326i \(-0.472705\pi\)
0.0856454 + 0.996326i \(0.472705\pi\)
\(830\) 0 0
\(831\) −217.000 + 613.769i −0.261131 + 0.738590i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 98.9949 + 161.000i 0.118274 + 0.192354i
\(838\) 0 0
\(839\) 1247.34i 1.48669i −0.668906 0.743347i \(-0.733237\pi\)
0.668906 0.743347i \(-0.266763\pi\)
\(840\) 0 0
\(841\) −959.000 −1.14031
\(842\) 0 0
\(843\) −517.602 + 1464.00i −0.614000 + 1.73665i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 343.000i 0.404959i
\(848\) 0 0
\(849\) 65.0000 183.848i 0.0765607 0.216546i
\(850\) 0 0
\(851\) 50.9117i 0.0598257i
\(852\) 0 0
\(853\) 1057.00i 1.23916i −0.784935 0.619578i \(-0.787304\pi\)
0.784935 0.619578i \(-0.212696\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 627.911 0.732685 0.366342 0.930480i \(-0.380610\pi\)
0.366342 + 0.930480i \(0.380610\pi\)
\(858\) 0 0
\(859\) −946.000 −1.10128 −0.550640 0.834743i \(-0.685616\pi\)
−0.550640 + 0.834743i \(0.685616\pi\)
\(860\) 0 0
\(861\) 168.000 + 59.3970i 0.195122 + 0.0689860i
\(862\) 0 0
\(863\) 390.323 0.452286 0.226143 0.974094i \(-0.427388\pi\)
0.226143 + 0.974094i \(0.427388\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1015.41 + 359.000i 1.17117 + 0.414072i
\(868\) 0 0
\(869\) 492.146i 0.566336i
\(870\) 0 0
\(871\) 425.000 0.487945
\(872\) 0 0
\(873\) −277.186 + 343.000i −0.317510 + 0.392898i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1463.00i 1.66819i −0.551623 0.834094i \(-0.685991\pi\)
0.551623 0.834094i \(-0.314009\pi\)
\(878\) 0 0
\(879\) 480.000 + 169.706i 0.546075 + 0.193067i
\(880\) 0 0
\(881\) 1069.15i 1.21356i −0.794870 0.606779i \(-0.792461\pi\)
0.794870 0.606779i \(-0.207539\pi\)
\(882\) 0 0
\(883\) 1289.00i 1.45980i −0.683556 0.729898i \(-0.739568\pi\)
0.683556 0.729898i \(-0.260432\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1289.76 1.45407 0.727037 0.686599i \(-0.240897\pi\)
0.727037 + 0.686599i \(0.240897\pi\)
\(888\) 0 0
\(889\) −238.000 −0.267717
\(890\) 0 0
\(891\) −672.000 + 144.250i −0.754209 + 0.161897i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −636.396 + 1800.00i −0.709472 + 2.00669i
\(898\) 0 0
\(899\) 296.985i 0.330350i
\(900\) 0 0
\(901\) 1512.00 1.67814
\(902\) 0 0
\(903\) −811.759 287.000i −0.898957 0.317829i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 14.0000i 0.0154355i 0.999970 + 0.00771775i \(0.00245666\pi\)
−0.999970 + 0.00771775i \(0.997543\pi\)
\(908\) 0 0
\(909\) −336.000 + 415.779i −0.369637 + 0.457402i
\(910\) 0 0
\(911\) 695.793i 0.763768i 0.924210 + 0.381884i \(0.124725\pi\)
−0.924210 + 0.381884i \(0.875275\pi\)
\(912\) 0 0
\(913\) 1008.00i 1.10405i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1366.13 1.48978
\(918\) 0 0
\(919\) −1423.00 −1.54842 −0.774211 0.632927i \(-0.781853\pi\)
−0.774211 + 0.632927i \(0.781853\pi\)
\(920\) 0 0
\(921\) −521.000 + 1473.61i −0.565689 + 1.60001i
\(922\) 0 0
\(923\) −1060.66 −1.14914
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −871.156 + 1078.00i −0.939758 + 1.16289i
\(928\) 0 0
\(929\) 415.779i 0.447555i −0.974640 0.223778i \(-0.928161\pi\)
0.974640 0.223778i \(-0.0718389\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −33.9411 + 96.0000i −0.0363785 + 0.102894i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1655.00i 1.76628i −0.469114 0.883138i \(-0.655427\pi\)
0.469114 0.883138i \(-0.344573\pi\)
\(938\) 0 0
\(939\) −119.000 + 336.583i −0.126731 + 0.358448i
\(940\) 0 0
\(941\) 1663.12i 1.76739i 0.468062 + 0.883696i \(0.344952\pi\)
−0.468062 + 0.883696i \(0.655048\pi\)
\(942\) 0 0
\(943\) 216.000i 0.229056i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 415.779 0.439048 0.219524 0.975607i \(-0.429550\pi\)
0.219524 + 0.975607i \(0.429550\pi\)
\(948\) 0 0
\(949\) −1750.00 −1.84405
\(950\) 0 0
\(951\) 432.000 + 152.735i 0.454259 + 0.160605i
\(952\) 0 0
\(953\) 1680.09 1.76294 0.881472 0.472236i \(-0.156553\pi\)
0.881472 + 0.472236i \(0.156553\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1018.23 + 360.000i 1.06399 + 0.376176i
\(958\) 0 0
\(959\) 831.558i 0.867109i
\(960\) 0 0
\(961\) −912.000 −0.949011
\(962\) 0 0
\(963\) 1247.34 + 1008.00i 1.29526 + 1.04673i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1162.00i 1.20165i −0.799379 0.600827i \(-0.794838\pi\)
0.799379 0.600827i \(-0.205162\pi\)
\(968\) 0 0
\(969\) −504.000 178.191i −0.520124 0.183892i
\(970\) 0 0
\(971\) 712.764i 0.734051i 0.930211 + 0.367026i \(0.119624\pi\)
−0.930211 + 0.367026i \(0.880376\pi\)
\(972\) 0 0
\(973\) 1078.00i 1.10791i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −899.440 −0.920614 −0.460307 0.887760i \(-0.652261\pi\)
−0.460307 + 0.887760i \(0.652261\pi\)
\(978\) 0 0
\(979\) 1152.00 1.17671
\(980\) 0 0
\(981\) 175.000 + 141.421i 0.178389 + 0.144160i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1043.69i 1.05530i
\(990\) 0 0
\(991\) 535.000 0.539859 0.269929 0.962880i \(-0.413000\pi\)
0.269929 + 0.962880i \(0.413000\pi\)
\(992\) 0 0
\(993\) 1182.28 + 418.000i 1.19062 + 0.420947i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1274.00i 1.27783i −0.769276 0.638917i \(-0.779383\pi\)
0.769276 0.638917i \(-0.220617\pi\)
\(998\) 0 0
\(999\) 46.0000 28.2843i 0.0460460 0.0283126i
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 1200.3.c.h.449.4 4
3.2 odd 2 inner 1200.3.c.h.449.2 4
4.3 odd 2 150.3.b.a.149.1 4
5.2 odd 4 1200.3.l.p.401.1 2
5.3 odd 4 1200.3.l.i.401.2 2
5.4 even 2 inner 1200.3.c.h.449.1 4
12.11 even 2 150.3.b.a.149.3 4
15.2 even 4 1200.3.l.p.401.2 2
15.8 even 4 1200.3.l.i.401.1 2
15.14 odd 2 inner 1200.3.c.h.449.3 4
20.3 even 4 150.3.d.b.101.2 yes 2
20.7 even 4 150.3.d.a.101.1 2
20.19 odd 2 150.3.b.a.149.4 4
60.23 odd 4 150.3.d.b.101.1 yes 2
60.47 odd 4 150.3.d.a.101.2 yes 2
60.59 even 2 150.3.b.a.149.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.3.b.a.149.1 4 4.3 odd 2
150.3.b.a.149.2 4 60.59 even 2
150.3.b.a.149.3 4 12.11 even 2
150.3.b.a.149.4 4 20.19 odd 2
150.3.d.a.101.1 2 20.7 even 4
150.3.d.a.101.2 yes 2 60.47 odd 4
150.3.d.b.101.1 yes 2 60.23 odd 4
150.3.d.b.101.2 yes 2 20.3 even 4
1200.3.c.h.449.1 4 5.4 even 2 inner
1200.3.c.h.449.2 4 3.2 odd 2 inner
1200.3.c.h.449.3 4 15.14 odd 2 inner
1200.3.c.h.449.4 4 1.1 even 1 trivial
1200.3.l.i.401.1 2 15.8 even 4
1200.3.l.i.401.2 2 5.3 odd 4
1200.3.l.p.401.1 2 5.2 odd 4
1200.3.l.p.401.2 2 15.2 even 4