Properties

Label 1344.4.p.b.223.1
Level $1344$
Weight $4$
Character 1344.223
Analytic conductor $79.299$
Analytic rank $0$
Dimension $16$
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] = [1344,4,Mod(223,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.223");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} + 58 x^{13} + 178264 x^{12} - 331354 x^{11} + 307862 x^{10} - 610 x^{9} + 8375926786 x^{8} - 15937543350 x^{7} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.1
Root \(11.6544 + 11.6544i\) of defining polynomial
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.b.223.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -10.6345 q^{5} +(-18.0158 - 4.29303i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -10.6345 q^{5} +(-18.0158 - 4.29303i) q^{7} -9.00000 q^{9} +5.02265 q^{11} +80.6582 q^{13} +31.9036i q^{15} -31.3183i q^{17} +74.0722i q^{19} +(-12.8791 + 54.0475i) q^{21} -37.3407i q^{23} -11.9070 q^{25} +27.0000i q^{27} -242.998i q^{29} +284.926 q^{31} -15.0679i q^{33} +(191.590 + 45.6543i) q^{35} +167.404i q^{37} -241.975i q^{39} +270.989i q^{41} -169.804 q^{43} +95.7107 q^{45} +639.550 q^{47} +(306.140 + 154.685i) q^{49} -93.9548 q^{51} -534.235i q^{53} -53.4134 q^{55} +222.217 q^{57} +56.9436i q^{59} -217.266 q^{61} +(162.142 + 38.6372i) q^{63} -857.761 q^{65} -684.081 q^{67} -112.022 q^{69} +966.305i q^{71} -325.451i q^{73} +35.7211i q^{75} +(-90.4871 - 21.5624i) q^{77} -856.194i q^{79} +81.0000 q^{81} -459.647i q^{83} +333.055i q^{85} -728.995 q^{87} -1077.84i q^{89} +(-1453.12 - 346.268i) q^{91} -854.778i q^{93} -787.722i q^{95} -1225.31i q^{97} -45.2038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 144 q^{9} + 56 q^{13} + 36 q^{21} + 80 q^{25} + 392 q^{49} + 336 q^{57} + 184 q^{61} - 1536 q^{65} + 864 q^{69} - 240 q^{77} + 1296 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) −10.6345 −0.951180 −0.475590 0.879667i \(-0.657765\pi\)
−0.475590 + 0.879667i \(0.657765\pi\)
\(6\) 0 0
\(7\) −18.0158 4.29303i −0.972763 0.231802i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 5.02265 0.137671 0.0688357 0.997628i \(-0.478072\pi\)
0.0688357 + 0.997628i \(0.478072\pi\)
\(12\) 0 0
\(13\) 80.6582 1.72081 0.860406 0.509608i \(-0.170210\pi\)
0.860406 + 0.509608i \(0.170210\pi\)
\(14\) 0 0
\(15\) 31.9036i 0.549164i
\(16\) 0 0
\(17\) 31.3183i 0.446811i −0.974726 0.223406i \(-0.928283\pi\)
0.974726 0.223406i \(-0.0717175\pi\)
\(18\) 0 0
\(19\) 74.0722i 0.894385i 0.894438 + 0.447193i \(0.147576\pi\)
−0.894438 + 0.447193i \(0.852424\pi\)
\(20\) 0 0
\(21\) −12.8791 + 54.0475i −0.133831 + 0.561625i
\(22\) 0 0
\(23\) 37.3407i 0.338525i −0.985571 0.169263i \(-0.945861\pi\)
0.985571 0.169263i \(-0.0541386\pi\)
\(24\) 0 0
\(25\) −11.9070 −0.0952563
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 242.998i 1.55599i −0.628271 0.777994i \(-0.716237\pi\)
0.628271 0.777994i \(-0.283763\pi\)
\(30\) 0 0
\(31\) 284.926 1.65078 0.825390 0.564563i \(-0.190955\pi\)
0.825390 + 0.564563i \(0.190955\pi\)
\(32\) 0 0
\(33\) 15.0679i 0.0794846i
\(34\) 0 0
\(35\) 191.590 + 45.6543i 0.925273 + 0.220485i
\(36\) 0 0
\(37\) 167.404i 0.743813i 0.928270 + 0.371906i \(0.121296\pi\)
−0.928270 + 0.371906i \(0.878704\pi\)
\(38\) 0 0
\(39\) 241.975i 0.993512i
\(40\) 0 0
\(41\) 270.989i 1.03223i 0.856519 + 0.516115i \(0.172622\pi\)
−0.856519 + 0.516115i \(0.827378\pi\)
\(42\) 0 0
\(43\) −169.804 −0.602205 −0.301103 0.953592i \(-0.597355\pi\)
−0.301103 + 0.953592i \(0.597355\pi\)
\(44\) 0 0
\(45\) 95.7107 0.317060
\(46\) 0 0
\(47\) 639.550 1.98485 0.992425 0.122848i \(-0.0392028\pi\)
0.992425 + 0.122848i \(0.0392028\pi\)
\(48\) 0 0
\(49\) 306.140 + 154.685i 0.892536 + 0.450976i
\(50\) 0 0
\(51\) −93.9548 −0.257967
\(52\) 0 0
\(53\) 534.235i 1.38458i −0.721619 0.692291i \(-0.756602\pi\)
0.721619 0.692291i \(-0.243398\pi\)
\(54\) 0 0
\(55\) −53.4134 −0.130950
\(56\) 0 0
\(57\) 222.217 0.516374
\(58\) 0 0
\(59\) 56.9436i 0.125651i 0.998025 + 0.0628257i \(0.0200112\pi\)
−0.998025 + 0.0628257i \(0.979989\pi\)
\(60\) 0 0
\(61\) −217.266 −0.456033 −0.228016 0.973657i \(-0.573224\pi\)
−0.228016 + 0.973657i \(0.573224\pi\)
\(62\) 0 0
\(63\) 162.142 + 38.6372i 0.324254 + 0.0772672i
\(64\) 0 0
\(65\) −857.761 −1.63680
\(66\) 0 0
\(67\) −684.081 −1.24737 −0.623685 0.781676i \(-0.714365\pi\)
−0.623685 + 0.781676i \(0.714365\pi\)
\(68\) 0 0
\(69\) −112.022 −0.195448
\(70\) 0 0
\(71\) 966.305i 1.61520i 0.589730 + 0.807601i \(0.299234\pi\)
−0.589730 + 0.807601i \(0.700766\pi\)
\(72\) 0 0
\(73\) 325.451i 0.521797i −0.965366 0.260898i \(-0.915981\pi\)
0.965366 0.260898i \(-0.0840188\pi\)
\(74\) 0 0
\(75\) 35.7211i 0.0549962i
\(76\) 0 0
\(77\) −90.4871 21.5624i −0.133922 0.0319125i
\(78\) 0 0
\(79\) 856.194i 1.21936i −0.792648 0.609679i \(-0.791298\pi\)
0.792648 0.609679i \(-0.208702\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 459.647i 0.607865i −0.952694 0.303932i \(-0.901700\pi\)
0.952694 0.303932i \(-0.0982997\pi\)
\(84\) 0 0
\(85\) 333.055i 0.424998i
\(86\) 0 0
\(87\) −728.995 −0.898351
\(88\) 0 0
\(89\) 1077.84i 1.28372i −0.766822 0.641860i \(-0.778163\pi\)
0.766822 0.641860i \(-0.221837\pi\)
\(90\) 0 0
\(91\) −1453.12 346.268i −1.67394 0.398887i
\(92\) 0 0
\(93\) 854.778i 0.953079i
\(94\) 0 0
\(95\) 787.722i 0.850722i
\(96\) 0 0
\(97\) 1225.31i 1.28259i −0.767293 0.641297i \(-0.778397\pi\)
0.767293 0.641297i \(-0.221603\pi\)
\(98\) 0 0
\(99\) −45.2038 −0.0458905
\(100\) 0 0
\(101\) −1513.47 −1.49105 −0.745523 0.666480i \(-0.767800\pi\)
−0.745523 + 0.666480i \(0.767800\pi\)
\(102\) 0 0
\(103\) −997.456 −0.954197 −0.477098 0.878850i \(-0.658311\pi\)
−0.477098 + 0.878850i \(0.658311\pi\)
\(104\) 0 0
\(105\) 136.963 574.769i 0.127297 0.534207i
\(106\) 0 0
\(107\) −959.164 −0.866597 −0.433299 0.901250i \(-0.642650\pi\)
−0.433299 + 0.901250i \(0.642650\pi\)
\(108\) 0 0
\(109\) 717.352i 0.630365i −0.949031 0.315183i \(-0.897934\pi\)
0.949031 0.315183i \(-0.102066\pi\)
\(110\) 0 0
\(111\) 502.212 0.429440
\(112\) 0 0
\(113\) 940.776 0.783193 0.391596 0.920137i \(-0.371923\pi\)
0.391596 + 0.920137i \(0.371923\pi\)
\(114\) 0 0
\(115\) 397.101i 0.321999i
\(116\) 0 0
\(117\) −725.924 −0.573604
\(118\) 0 0
\(119\) −134.450 + 564.224i −0.103572 + 0.434642i
\(120\) 0 0
\(121\) −1305.77 −0.981047
\(122\) 0 0
\(123\) 812.968 0.595958
\(124\) 0 0
\(125\) 1455.94 1.04179
\(126\) 0 0
\(127\) 2089.41i 1.45988i 0.683509 + 0.729942i \(0.260453\pi\)
−0.683509 + 0.729942i \(0.739547\pi\)
\(128\) 0 0
\(129\) 509.411i 0.347683i
\(130\) 0 0
\(131\) 892.922i 0.595534i 0.954639 + 0.297767i \(0.0962418\pi\)
−0.954639 + 0.297767i \(0.903758\pi\)
\(132\) 0 0
\(133\) 317.994 1334.47i 0.207320 0.870025i
\(134\) 0 0
\(135\) 287.132i 0.183055i
\(136\) 0 0
\(137\) −1440.52 −0.898338 −0.449169 0.893447i \(-0.648280\pi\)
−0.449169 + 0.893447i \(0.648280\pi\)
\(138\) 0 0
\(139\) 2001.02i 1.22104i 0.792001 + 0.610520i \(0.209040\pi\)
−0.792001 + 0.610520i \(0.790960\pi\)
\(140\) 0 0
\(141\) 1918.65i 1.14595i
\(142\) 0 0
\(143\) 405.118 0.236907
\(144\) 0 0
\(145\) 2584.17i 1.48003i
\(146\) 0 0
\(147\) 464.055 918.420i 0.260371 0.515306i
\(148\) 0 0
\(149\) 1978.85i 1.08801i −0.839081 0.544006i \(-0.816907\pi\)
0.839081 0.544006i \(-0.183093\pi\)
\(150\) 0 0
\(151\) 1906.09i 1.02725i −0.858014 0.513626i \(-0.828302\pi\)
0.858014 0.513626i \(-0.171698\pi\)
\(152\) 0 0
\(153\) 281.864i 0.148937i
\(154\) 0 0
\(155\) −3030.05 −1.57019
\(156\) 0 0
\(157\) 280.991 0.142838 0.0714188 0.997446i \(-0.477247\pi\)
0.0714188 + 0.997446i \(0.477247\pi\)
\(158\) 0 0
\(159\) −1602.70 −0.799388
\(160\) 0 0
\(161\) −160.305 + 672.724i −0.0784707 + 0.329305i
\(162\) 0 0
\(163\) −2809.63 −1.35011 −0.675053 0.737770i \(-0.735879\pi\)
−0.675053 + 0.737770i \(0.735879\pi\)
\(164\) 0 0
\(165\) 160.240i 0.0756042i
\(166\) 0 0
\(167\) 101.283 0.0469310 0.0234655 0.999725i \(-0.492530\pi\)
0.0234655 + 0.999725i \(0.492530\pi\)
\(168\) 0 0
\(169\) 4308.75 1.96120
\(170\) 0 0
\(171\) 666.650i 0.298128i
\(172\) 0 0
\(173\) 408.564 0.179552 0.0897761 0.995962i \(-0.471385\pi\)
0.0897761 + 0.995962i \(0.471385\pi\)
\(174\) 0 0
\(175\) 214.515 + 51.1172i 0.0926618 + 0.0220806i
\(176\) 0 0
\(177\) 170.831 0.0725449
\(178\) 0 0
\(179\) −1624.36 −0.678272 −0.339136 0.940737i \(-0.610135\pi\)
−0.339136 + 0.940737i \(0.610135\pi\)
\(180\) 0 0
\(181\) −1303.28 −0.535203 −0.267601 0.963530i \(-0.586231\pi\)
−0.267601 + 0.963530i \(0.586231\pi\)
\(182\) 0 0
\(183\) 651.797i 0.263291i
\(184\) 0 0
\(185\) 1780.26i 0.707500i
\(186\) 0 0
\(187\) 157.301i 0.0615131i
\(188\) 0 0
\(189\) 115.912 486.427i 0.0446103 0.187208i
\(190\) 0 0
\(191\) 117.261i 0.0444226i −0.999753 0.0222113i \(-0.992929\pi\)
0.999753 0.0222113i \(-0.00707066\pi\)
\(192\) 0 0
\(193\) 292.337 0.109030 0.0545151 0.998513i \(-0.482639\pi\)
0.0545151 + 0.998513i \(0.482639\pi\)
\(194\) 0 0
\(195\) 2573.28i 0.945009i
\(196\) 0 0
\(197\) 4190.92i 1.51569i −0.652436 0.757843i \(-0.726253\pi\)
0.652436 0.757843i \(-0.273747\pi\)
\(198\) 0 0
\(199\) −1747.17 −0.622381 −0.311191 0.950347i \(-0.600728\pi\)
−0.311191 + 0.950347i \(0.600728\pi\)
\(200\) 0 0
\(201\) 2052.24i 0.720169i
\(202\) 0 0
\(203\) −1043.20 + 4377.82i −0.360681 + 1.51361i
\(204\) 0 0
\(205\) 2881.84i 0.981837i
\(206\) 0 0
\(207\) 336.067i 0.112842i
\(208\) 0 0
\(209\) 372.038i 0.123131i
\(210\) 0 0
\(211\) −628.802 −0.205159 −0.102579 0.994725i \(-0.532710\pi\)
−0.102579 + 0.994725i \(0.532710\pi\)
\(212\) 0 0
\(213\) 2898.91 0.932537
\(214\) 0 0
\(215\) 1805.78 0.572806
\(216\) 0 0
\(217\) −5133.17 1223.19i −1.60582 0.382654i
\(218\) 0 0
\(219\) −976.353 −0.301260
\(220\) 0 0
\(221\) 2526.08i 0.768879i
\(222\) 0 0
\(223\) −2081.12 −0.624942 −0.312471 0.949927i \(-0.601157\pi\)
−0.312471 + 0.949927i \(0.601157\pi\)
\(224\) 0 0
\(225\) 107.163 0.0317521
\(226\) 0 0
\(227\) 806.786i 0.235896i −0.993020 0.117948i \(-0.962368\pi\)
0.993020 0.117948i \(-0.0376315\pi\)
\(228\) 0 0
\(229\) −1726.29 −0.498152 −0.249076 0.968484i \(-0.580127\pi\)
−0.249076 + 0.968484i \(0.580127\pi\)
\(230\) 0 0
\(231\) −64.6871 + 271.461i −0.0184247 + 0.0773197i
\(232\) 0 0
\(233\) 2208.17 0.620868 0.310434 0.950595i \(-0.399526\pi\)
0.310434 + 0.950595i \(0.399526\pi\)
\(234\) 0 0
\(235\) −6801.31 −1.88795
\(236\) 0 0
\(237\) −2568.58 −0.703997
\(238\) 0 0
\(239\) 1379.52i 0.373362i −0.982421 0.186681i \(-0.940227\pi\)
0.982421 0.186681i \(-0.0597731\pi\)
\(240\) 0 0
\(241\) 5254.27i 1.40439i 0.711986 + 0.702194i \(0.247796\pi\)
−0.711986 + 0.702194i \(0.752204\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) −3255.65 1645.00i −0.848963 0.428960i
\(246\) 0 0
\(247\) 5974.53i 1.53907i
\(248\) 0 0
\(249\) −1378.94 −0.350951
\(250\) 0 0
\(251\) 4235.01i 1.06499i −0.846434 0.532493i \(-0.821255\pi\)
0.846434 0.532493i \(-0.178745\pi\)
\(252\) 0 0
\(253\) 187.549i 0.0466052i
\(254\) 0 0
\(255\) 999.164 0.245373
\(256\) 0 0
\(257\) 3842.32i 0.932596i −0.884628 0.466298i \(-0.845587\pi\)
0.884628 0.466298i \(-0.154413\pi\)
\(258\) 0 0
\(259\) 718.670 3015.92i 0.172417 0.723554i
\(260\) 0 0
\(261\) 2186.99i 0.518663i
\(262\) 0 0
\(263\) 5327.10i 1.24898i −0.781031 0.624492i \(-0.785306\pi\)
0.781031 0.624492i \(-0.214694\pi\)
\(264\) 0 0
\(265\) 5681.33i 1.31699i
\(266\) 0 0
\(267\) −3233.53 −0.741156
\(268\) 0 0
\(269\) −7196.26 −1.63109 −0.815546 0.578692i \(-0.803563\pi\)
−0.815546 + 0.578692i \(0.803563\pi\)
\(270\) 0 0
\(271\) 3998.83 0.896352 0.448176 0.893945i \(-0.352074\pi\)
0.448176 + 0.893945i \(0.352074\pi\)
\(272\) 0 0
\(273\) −1038.80 + 4359.37i −0.230298 + 0.966452i
\(274\) 0 0
\(275\) −59.8048 −0.0131141
\(276\) 0 0
\(277\) 6116.25i 1.32668i −0.748319 0.663339i \(-0.769139\pi\)
0.748319 0.663339i \(-0.230861\pi\)
\(278\) 0 0
\(279\) −2564.33 −0.550260
\(280\) 0 0
\(281\) 6280.80 1.33338 0.666692 0.745333i \(-0.267710\pi\)
0.666692 + 0.745333i \(0.267710\pi\)
\(282\) 0 0
\(283\) 894.355i 0.187858i 0.995579 + 0.0939292i \(0.0299427\pi\)
−0.995579 + 0.0939292i \(0.970057\pi\)
\(284\) 0 0
\(285\) −2363.17 −0.491164
\(286\) 0 0
\(287\) 1163.36 4882.10i 0.239273 1.00412i
\(288\) 0 0
\(289\) 3932.17 0.800360
\(290\) 0 0
\(291\) −3675.94 −0.740506
\(292\) 0 0
\(293\) 6903.17 1.37641 0.688204 0.725517i \(-0.258399\pi\)
0.688204 + 0.725517i \(0.258399\pi\)
\(294\) 0 0
\(295\) 605.568i 0.119517i
\(296\) 0 0
\(297\) 135.611i 0.0264949i
\(298\) 0 0
\(299\) 3011.84i 0.582539i
\(300\) 0 0
\(301\) 3059.15 + 728.972i 0.585803 + 0.139592i
\(302\) 0 0
\(303\) 4540.40i 0.860855i
\(304\) 0 0
\(305\) 2310.51 0.433769
\(306\) 0 0
\(307\) 7933.97i 1.47497i 0.675364 + 0.737485i \(0.263987\pi\)
−0.675364 + 0.737485i \(0.736013\pi\)
\(308\) 0 0
\(309\) 2992.37i 0.550906i
\(310\) 0 0
\(311\) 6215.65 1.13330 0.566652 0.823957i \(-0.308238\pi\)
0.566652 + 0.823957i \(0.308238\pi\)
\(312\) 0 0
\(313\) 3466.13i 0.625933i −0.949764 0.312966i \(-0.898677\pi\)
0.949764 0.312966i \(-0.101323\pi\)
\(314\) 0 0
\(315\) −1724.31 410.888i −0.308424 0.0734950i
\(316\) 0 0
\(317\) 1277.86i 0.226409i −0.993572 0.113204i \(-0.963888\pi\)
0.993572 0.113204i \(-0.0361115\pi\)
\(318\) 0 0
\(319\) 1220.50i 0.214215i
\(320\) 0 0
\(321\) 2877.49i 0.500330i
\(322\) 0 0
\(323\) 2319.81 0.399622
\(324\) 0 0
\(325\) −960.400 −0.163918
\(326\) 0 0
\(327\) −2152.05 −0.363942
\(328\) 0 0
\(329\) −11522.0 2745.61i −1.93079 0.460092i
\(330\) 0 0
\(331\) 3351.13 0.556480 0.278240 0.960512i \(-0.410249\pi\)
0.278240 + 0.960512i \(0.410249\pi\)
\(332\) 0 0
\(333\) 1506.64i 0.247938i
\(334\) 0 0
\(335\) 7274.87 1.18647
\(336\) 0 0
\(337\) −5362.98 −0.866885 −0.433442 0.901181i \(-0.642701\pi\)
−0.433442 + 0.901181i \(0.642701\pi\)
\(338\) 0 0
\(339\) 2822.33i 0.452177i
\(340\) 0 0
\(341\) 1431.08 0.227265
\(342\) 0 0
\(343\) −4851.30 4101.04i −0.763689 0.645584i
\(344\) 0 0
\(345\) 1191.30 0.185906
\(346\) 0 0
\(347\) −3355.31 −0.519084 −0.259542 0.965732i \(-0.583572\pi\)
−0.259542 + 0.965732i \(0.583572\pi\)
\(348\) 0 0
\(349\) −5141.54 −0.788597 −0.394298 0.918983i \(-0.629012\pi\)
−0.394298 + 0.918983i \(0.629012\pi\)
\(350\) 0 0
\(351\) 2177.77i 0.331171i
\(352\) 0 0
\(353\) 8307.43i 1.25258i −0.779591 0.626289i \(-0.784573\pi\)
0.779591 0.626289i \(-0.215427\pi\)
\(354\) 0 0
\(355\) 10276.2i 1.53635i
\(356\) 0 0
\(357\) 1692.67 + 403.350i 0.250940 + 0.0597971i
\(358\) 0 0
\(359\) 10202.5i 1.49991i 0.661489 + 0.749955i \(0.269925\pi\)
−0.661489 + 0.749955i \(0.730075\pi\)
\(360\) 0 0
\(361\) 1372.31 0.200075
\(362\) 0 0
\(363\) 3917.32i 0.566408i
\(364\) 0 0
\(365\) 3461.02i 0.496323i
\(366\) 0 0
\(367\) −5765.85 −0.820095 −0.410047 0.912064i \(-0.634488\pi\)
−0.410047 + 0.912064i \(0.634488\pi\)
\(368\) 0 0
\(369\) 2438.90i 0.344077i
\(370\) 0 0
\(371\) −2293.48 + 9624.68i −0.320948 + 1.34687i
\(372\) 0 0
\(373\) 5679.15i 0.788352i −0.919035 0.394176i \(-0.871030\pi\)
0.919035 0.394176i \(-0.128970\pi\)
\(374\) 0 0
\(375\) 4367.82i 0.601475i
\(376\) 0 0
\(377\) 19599.8i 2.67757i
\(378\) 0 0
\(379\) −6030.20 −0.817284 −0.408642 0.912695i \(-0.633997\pi\)
−0.408642 + 0.912695i \(0.633997\pi\)
\(380\) 0 0
\(381\) 6268.23 0.842864
\(382\) 0 0
\(383\) −1734.63 −0.231425 −0.115712 0.993283i \(-0.536915\pi\)
−0.115712 + 0.993283i \(0.536915\pi\)
\(384\) 0 0
\(385\) 962.287 + 229.305i 0.127384 + 0.0303545i
\(386\) 0 0
\(387\) 1528.23 0.200735
\(388\) 0 0
\(389\) 9051.11i 1.17972i 0.807507 + 0.589858i \(0.200816\pi\)
−0.807507 + 0.589858i \(0.799184\pi\)
\(390\) 0 0
\(391\) −1169.45 −0.151257
\(392\) 0 0
\(393\) 2678.77 0.343832
\(394\) 0 0
\(395\) 9105.21i 1.15983i
\(396\) 0 0
\(397\) 422.728 0.0534411 0.0267206 0.999643i \(-0.491494\pi\)
0.0267206 + 0.999643i \(0.491494\pi\)
\(398\) 0 0
\(399\) −4003.41 953.982i −0.502309 0.119696i
\(400\) 0 0
\(401\) −2221.19 −0.276610 −0.138305 0.990390i \(-0.544165\pi\)
−0.138305 + 0.990390i \(0.544165\pi\)
\(402\) 0 0
\(403\) 22981.6 2.84068
\(404\) 0 0
\(405\) −861.396 −0.105687
\(406\) 0 0
\(407\) 840.812i 0.102402i
\(408\) 0 0
\(409\) 5178.51i 0.626066i −0.949742 0.313033i \(-0.898655\pi\)
0.949742 0.313033i \(-0.101345\pi\)
\(410\) 0 0
\(411\) 4321.57i 0.518655i
\(412\) 0 0
\(413\) 244.461 1025.89i 0.0291262 0.122229i
\(414\) 0 0
\(415\) 4888.12i 0.578189i
\(416\) 0 0
\(417\) 6003.07 0.704968
\(418\) 0 0
\(419\) 8420.46i 0.981782i −0.871221 0.490891i \(-0.836671\pi\)
0.871221 0.490891i \(-0.163329\pi\)
\(420\) 0 0
\(421\) 14115.9i 1.63413i −0.576548 0.817063i \(-0.695601\pi\)
0.576548 0.817063i \(-0.304399\pi\)
\(422\) 0 0
\(423\) −5755.95 −0.661617
\(424\) 0 0
\(425\) 372.908i 0.0425616i
\(426\) 0 0
\(427\) 3914.22 + 932.727i 0.443612 + 0.105709i
\(428\) 0 0
\(429\) 1215.35i 0.136778i
\(430\) 0 0
\(431\) 9628.09i 1.07603i −0.842935 0.538015i \(-0.819174\pi\)
0.842935 0.538015i \(-0.180826\pi\)
\(432\) 0 0
\(433\) 1048.69i 0.116390i −0.998305 0.0581952i \(-0.981465\pi\)
0.998305 0.0581952i \(-0.0185346\pi\)
\(434\) 0 0
\(435\) 7752.51 0.854493
\(436\) 0 0
\(437\) 2765.91 0.302772
\(438\) 0 0
\(439\) −5992.62 −0.651508 −0.325754 0.945455i \(-0.605618\pi\)
−0.325754 + 0.945455i \(0.605618\pi\)
\(440\) 0 0
\(441\) −2755.26 1392.16i −0.297512 0.150325i
\(442\) 0 0
\(443\) −14401.7 −1.54457 −0.772286 0.635275i \(-0.780887\pi\)
−0.772286 + 0.635275i \(0.780887\pi\)
\(444\) 0 0
\(445\) 11462.3i 1.22105i
\(446\) 0 0
\(447\) −5936.56 −0.628164
\(448\) 0 0
\(449\) −12503.5 −1.31420 −0.657100 0.753803i \(-0.728217\pi\)
−0.657100 + 0.753803i \(0.728217\pi\)
\(450\) 0 0
\(451\) 1361.08i 0.142109i
\(452\) 0 0
\(453\) −5718.26 −0.593084
\(454\) 0 0
\(455\) 15453.3 + 3682.39i 1.59222 + 0.379414i
\(456\) 0 0
\(457\) −2149.64 −0.220035 −0.110018 0.993930i \(-0.535091\pi\)
−0.110018 + 0.993930i \(0.535091\pi\)
\(458\) 0 0
\(459\) 845.593 0.0859889
\(460\) 0 0
\(461\) 14484.2 1.46334 0.731668 0.681661i \(-0.238742\pi\)
0.731668 + 0.681661i \(0.238742\pi\)
\(462\) 0 0
\(463\) 6792.36i 0.681788i 0.940102 + 0.340894i \(0.110730\pi\)
−0.940102 + 0.340894i \(0.889270\pi\)
\(464\) 0 0
\(465\) 9090.15i 0.906549i
\(466\) 0 0
\(467\) 3760.27i 0.372600i −0.982493 0.186300i \(-0.940350\pi\)
0.982493 0.186300i \(-0.0596497\pi\)
\(468\) 0 0
\(469\) 12324.3 + 2936.78i 1.21340 + 0.289142i
\(470\) 0 0
\(471\) 842.973i 0.0824674i
\(472\) 0 0
\(473\) −852.864 −0.0829064
\(474\) 0 0
\(475\) 881.980i 0.0851958i
\(476\) 0 0
\(477\) 4808.11i 0.461527i
\(478\) 0 0
\(479\) 8749.41 0.834594 0.417297 0.908770i \(-0.362977\pi\)
0.417297 + 0.908770i \(0.362977\pi\)
\(480\) 0 0
\(481\) 13502.5i 1.27996i
\(482\) 0 0
\(483\) 2018.17 + 480.914i 0.190124 + 0.0453051i
\(484\) 0 0
\(485\) 13030.6i 1.21998i
\(486\) 0 0
\(487\) 13362.5i 1.24335i −0.783274 0.621677i \(-0.786452\pi\)
0.783274 0.621677i \(-0.213548\pi\)
\(488\) 0 0
\(489\) 8428.89i 0.779483i
\(490\) 0 0
\(491\) 7703.62 0.708064 0.354032 0.935233i \(-0.384810\pi\)
0.354032 + 0.935233i \(0.384810\pi\)
\(492\) 0 0
\(493\) −7610.29 −0.695234
\(494\) 0 0
\(495\) 480.721 0.0436501
\(496\) 0 0
\(497\) 4148.37 17408.8i 0.374406 1.57121i
\(498\) 0 0
\(499\) 10522.8 0.944019 0.472009 0.881594i \(-0.343529\pi\)
0.472009 + 0.881594i \(0.343529\pi\)
\(500\) 0 0
\(501\) 303.848i 0.0270956i
\(502\) 0 0
\(503\) 20002.0 1.77305 0.886527 0.462678i \(-0.153111\pi\)
0.886527 + 0.462678i \(0.153111\pi\)
\(504\) 0 0
\(505\) 16095.0 1.41825
\(506\) 0 0
\(507\) 12926.3i 1.13230i
\(508\) 0 0
\(509\) −18691.1 −1.62764 −0.813819 0.581118i \(-0.802615\pi\)
−0.813819 + 0.581118i \(0.802615\pi\)
\(510\) 0 0
\(511\) −1397.17 + 5863.27i −0.120953 + 0.507585i
\(512\) 0 0
\(513\) −1999.95 −0.172125
\(514\) 0 0
\(515\) 10607.5 0.907613
\(516\) 0 0
\(517\) 3212.24 0.273257
\(518\) 0 0
\(519\) 1225.69i 0.103665i
\(520\) 0 0
\(521\) 7044.44i 0.592365i 0.955131 + 0.296183i \(0.0957137\pi\)
−0.955131 + 0.296183i \(0.904286\pi\)
\(522\) 0 0
\(523\) 5379.79i 0.449793i −0.974383 0.224897i \(-0.927796\pi\)
0.974383 0.224897i \(-0.0722045\pi\)
\(524\) 0 0
\(525\) 153.352 643.545i 0.0127482 0.0534983i
\(526\) 0 0
\(527\) 8923.38i 0.737588i
\(528\) 0 0
\(529\) 10772.7 0.885401
\(530\) 0 0
\(531\) 512.493i 0.0418838i
\(532\) 0 0
\(533\) 21857.5i 1.77627i
\(534\) 0 0
\(535\) 10200.2 0.824290
\(536\) 0 0
\(537\) 4873.09i 0.391600i
\(538\) 0 0
\(539\) 1537.63 + 776.927i 0.122877 + 0.0620865i
\(540\) 0 0
\(541\) 20045.6i 1.59303i 0.604621 + 0.796513i \(0.293325\pi\)
−0.604621 + 0.796513i \(0.706675\pi\)
\(542\) 0 0
\(543\) 3909.83i 0.308999i
\(544\) 0 0
\(545\) 7628.69i 0.599591i
\(546\) 0 0
\(547\) 10939.2 0.855079 0.427539 0.903997i \(-0.359381\pi\)
0.427539 + 0.903997i \(0.359381\pi\)
\(548\) 0 0
\(549\) 1955.39 0.152011
\(550\) 0 0
\(551\) 17999.4 1.39165
\(552\) 0 0
\(553\) −3675.66 + 15425.0i −0.282649 + 1.18615i
\(554\) 0 0
\(555\) −5340.79 −0.408475
\(556\) 0 0
\(557\) 10198.6i 0.775817i −0.921698 0.387909i \(-0.873198\pi\)
0.921698 0.387909i \(-0.126802\pi\)
\(558\) 0 0
\(559\) −13696.1 −1.03628
\(560\) 0 0
\(561\) −471.902 −0.0355146
\(562\) 0 0
\(563\) 17772.4i 1.33040i −0.746663 0.665202i \(-0.768345\pi\)
0.746663 0.665202i \(-0.231655\pi\)
\(564\) 0 0
\(565\) −10004.7 −0.744957
\(566\) 0 0
\(567\) −1459.28 347.735i −0.108085 0.0257557i
\(568\) 0 0
\(569\) −22205.7 −1.63604 −0.818022 0.575187i \(-0.804929\pi\)
−0.818022 + 0.575187i \(0.804929\pi\)
\(570\) 0 0
\(571\) −12606.3 −0.923918 −0.461959 0.886901i \(-0.652853\pi\)
−0.461959 + 0.886901i \(0.652853\pi\)
\(572\) 0 0
\(573\) −351.784 −0.0256474
\(574\) 0 0
\(575\) 444.617i 0.0322467i
\(576\) 0 0
\(577\) 7952.14i 0.573747i 0.957968 + 0.286874i \(0.0926159\pi\)
−0.957968 + 0.286874i \(0.907384\pi\)
\(578\) 0 0
\(579\) 877.010i 0.0629487i
\(580\) 0 0
\(581\) −1973.28 + 8280.91i −0.140904 + 0.591308i
\(582\) 0 0
\(583\) 2683.27i 0.190617i
\(584\) 0 0
\(585\) 7719.85 0.545601
\(586\) 0 0
\(587\) 9107.77i 0.640405i 0.947349 + 0.320203i \(0.103751\pi\)
−0.947349 + 0.320203i \(0.896249\pi\)
\(588\) 0 0
\(589\) 21105.1i 1.47643i
\(590\) 0 0
\(591\) −12572.7 −0.875082
\(592\) 0 0
\(593\) 16773.3i 1.16155i −0.814066 0.580773i \(-0.802751\pi\)
0.814066 0.580773i \(-0.197249\pi\)
\(594\) 0 0
\(595\) 1429.81 6000.25i 0.0985153 0.413422i
\(596\) 0 0
\(597\) 5241.52i 0.359332i
\(598\) 0 0
\(599\) 6931.31i 0.472798i 0.971656 + 0.236399i \(0.0759672\pi\)
−0.971656 + 0.236399i \(0.924033\pi\)
\(600\) 0 0
\(601\) 7300.46i 0.495494i 0.968825 + 0.247747i \(0.0796902\pi\)
−0.968825 + 0.247747i \(0.920310\pi\)
\(602\) 0 0
\(603\) 6156.73 0.415790
\(604\) 0 0
\(605\) 13886.3 0.933152
\(606\) 0 0
\(607\) −20396.1 −1.36384 −0.681920 0.731427i \(-0.738855\pi\)
−0.681920 + 0.731427i \(0.738855\pi\)
\(608\) 0 0
\(609\) 13133.5 + 3129.60i 0.873882 + 0.208239i
\(610\) 0 0
\(611\) 51585.0 3.41556
\(612\) 0 0
\(613\) 11938.4i 0.786602i 0.919410 + 0.393301i \(0.128667\pi\)
−0.919410 + 0.393301i \(0.871333\pi\)
\(614\) 0 0
\(615\) −8645.52 −0.566864
\(616\) 0 0
\(617\) 18423.2 1.20209 0.601045 0.799215i \(-0.294751\pi\)
0.601045 + 0.799215i \(0.294751\pi\)
\(618\) 0 0
\(619\) 8807.50i 0.571896i 0.958245 + 0.285948i \(0.0923084\pi\)
−0.958245 + 0.285948i \(0.907692\pi\)
\(620\) 0 0
\(621\) 1008.20 0.0651492
\(622\) 0 0
\(623\) −4627.20 + 19418.2i −0.297568 + 1.24875i
\(624\) 0 0
\(625\) −13994.8 −0.895670
\(626\) 0 0
\(627\) 1116.12 0.0710899
\(628\) 0 0
\(629\) 5242.81 0.332344
\(630\) 0 0
\(631\) 1067.09i 0.0673223i −0.999433 0.0336611i \(-0.989283\pi\)
0.999433 0.0336611i \(-0.0107167\pi\)
\(632\) 0 0
\(633\) 1886.41i 0.118449i
\(634\) 0 0
\(635\) 22219.9i 1.38861i
\(636\) 0 0
\(637\) 24692.7 + 12476.6i 1.53589 + 0.776046i
\(638\) 0 0
\(639\) 8696.74i 0.538400i
\(640\) 0 0
\(641\) −7062.71 −0.435195 −0.217598 0.976039i \(-0.569822\pi\)
−0.217598 + 0.976039i \(0.569822\pi\)
\(642\) 0 0
\(643\) 12546.4i 0.769490i 0.923023 + 0.384745i \(0.125711\pi\)
−0.923023 + 0.384745i \(0.874289\pi\)
\(644\) 0 0
\(645\) 5417.34i 0.330709i
\(646\) 0 0
\(647\) 405.177 0.0246200 0.0123100 0.999924i \(-0.496082\pi\)
0.0123100 + 0.999924i \(0.496082\pi\)
\(648\) 0 0
\(649\) 286.008i 0.0172986i
\(650\) 0 0
\(651\) −3669.58 + 15399.5i −0.220925 + 0.927120i
\(652\) 0 0
\(653\) 2647.84i 0.158680i −0.996848 0.0793401i \(-0.974719\pi\)
0.996848 0.0793401i \(-0.0252813\pi\)
\(654\) 0 0
\(655\) 9495.79i 0.566460i
\(656\) 0 0
\(657\) 2929.06i 0.173932i
\(658\) 0 0
\(659\) 9246.93 0.546600 0.273300 0.961929i \(-0.411885\pi\)
0.273300 + 0.961929i \(0.411885\pi\)
\(660\) 0 0
\(661\) −22699.7 −1.33573 −0.667863 0.744284i \(-0.732791\pi\)
−0.667863 + 0.744284i \(0.732791\pi\)
\(662\) 0 0
\(663\) −7578.23 −0.443912
\(664\) 0 0
\(665\) −3381.71 + 14191.5i −0.197199 + 0.827551i
\(666\) 0 0
\(667\) −9073.74 −0.526742
\(668\) 0 0
\(669\) 6243.36i 0.360811i
\(670\) 0 0
\(671\) −1091.25 −0.0627827
\(672\) 0 0
\(673\) −3196.22 −0.183069 −0.0915343 0.995802i \(-0.529177\pi\)
−0.0915343 + 0.995802i \(0.529177\pi\)
\(674\) 0 0
\(675\) 321.490i 0.0183321i
\(676\) 0 0
\(677\) −21840.2 −1.23987 −0.619933 0.784655i \(-0.712840\pi\)
−0.619933 + 0.784655i \(0.712840\pi\)
\(678\) 0 0
\(679\) −5260.30 + 22075.0i −0.297307 + 1.24766i
\(680\) 0 0
\(681\) −2420.36 −0.136194
\(682\) 0 0
\(683\) 7500.96 0.420229 0.210114 0.977677i \(-0.432616\pi\)
0.210114 + 0.977677i \(0.432616\pi\)
\(684\) 0 0
\(685\) 15319.3 0.854481
\(686\) 0 0
\(687\) 5178.88i 0.287608i
\(688\) 0 0
\(689\) 43090.4i 2.38261i
\(690\) 0 0
\(691\) 32124.5i 1.76856i −0.466960 0.884278i \(-0.654651\pi\)
0.466960 0.884278i \(-0.345349\pi\)
\(692\) 0 0
\(693\) 814.384 + 194.061i 0.0446405 + 0.0106375i
\(694\) 0 0
\(695\) 21279.9i 1.16143i
\(696\) 0 0
\(697\) 8486.92 0.461212
\(698\) 0 0
\(699\) 6624.52i 0.358458i
\(700\) 0 0
\(701\) 22998.8i 1.23916i −0.784933 0.619581i \(-0.787303\pi\)
0.784933 0.619581i \(-0.212697\pi\)
\(702\) 0 0
\(703\) −12400.0 −0.665255
\(704\) 0 0
\(705\) 20403.9i 1.09001i
\(706\) 0 0
\(707\) 27266.4 + 6497.35i 1.45043 + 0.345627i
\(708\) 0 0
\(709\) 4735.32i 0.250831i 0.992104 + 0.125415i \(0.0400263\pi\)
−0.992104 + 0.125415i \(0.959974\pi\)
\(710\) 0 0
\(711\) 7705.75i 0.406453i
\(712\) 0 0
\(713\) 10639.3i 0.558831i
\(714\) 0 0
\(715\) −4308.23 −0.225341
\(716\) 0 0
\(717\) −4138.55 −0.215561
\(718\) 0 0
\(719\) 33697.1 1.74783 0.873914 0.486081i \(-0.161574\pi\)
0.873914 + 0.486081i \(0.161574\pi\)
\(720\) 0 0
\(721\) 17970.0 + 4282.11i 0.928207 + 0.221184i
\(722\) 0 0
\(723\) 15762.8 0.810823
\(724\) 0 0
\(725\) 2893.39i 0.148218i
\(726\) 0 0
\(727\) −2679.22 −0.136680 −0.0683402 0.997662i \(-0.521770\pi\)
−0.0683402 + 0.997662i \(0.521770\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 5317.96i 0.269072i
\(732\) 0 0
\(733\) 23133.2 1.16568 0.582841 0.812586i \(-0.301941\pi\)
0.582841 + 0.812586i \(0.301941\pi\)
\(734\) 0 0
\(735\) −4935.00 + 9766.95i −0.247660 + 0.490149i
\(736\) 0 0
\(737\) −3435.90 −0.171727
\(738\) 0 0
\(739\) 7748.32 0.385692 0.192846 0.981229i \(-0.438228\pi\)
0.192846 + 0.981229i \(0.438228\pi\)
\(740\) 0 0
\(741\) 17923.6 0.888583
\(742\) 0 0
\(743\) 34547.9i 1.70584i −0.522043 0.852919i \(-0.674830\pi\)
0.522043 0.852919i \(-0.325170\pi\)
\(744\) 0 0
\(745\) 21044.1i 1.03490i
\(746\) 0 0
\(747\) 4136.82i 0.202622i
\(748\) 0 0
\(749\) 17280.1 + 4117.72i 0.842994 + 0.200879i
\(750\) 0 0
\(751\) 363.938i 0.0176835i 0.999961 + 0.00884174i \(0.00281445\pi\)
−0.999961 + 0.00884174i \(0.997186\pi\)
\(752\) 0 0
\(753\) −12705.0 −0.614870
\(754\) 0 0
\(755\) 20270.3i 0.977102i
\(756\) 0 0
\(757\) 31211.6i 1.49855i 0.662258 + 0.749276i \(0.269598\pi\)
−0.662258 + 0.749276i \(0.730402\pi\)
\(758\) 0 0
\(759\) −562.648 −0.0269075
\(760\) 0 0
\(761\) 23313.8i 1.11055i −0.831668 0.555273i \(-0.812614\pi\)
0.831668 0.555273i \(-0.187386\pi\)
\(762\) 0 0
\(763\) −3079.61 + 12923.7i −0.146120 + 0.613196i
\(764\) 0 0
\(765\) 2997.49i 0.141666i
\(766\) 0 0
\(767\) 4592.97i 0.216223i
\(768\) 0 0
\(769\) 10415.1i 0.488400i 0.969725 + 0.244200i \(0.0785253\pi\)
−0.969725 + 0.244200i \(0.921475\pi\)
\(770\) 0 0
\(771\) −11527.0 −0.538435
\(772\) 0 0
\(773\) 26311.8 1.22428 0.612140 0.790749i \(-0.290309\pi\)
0.612140 + 0.790749i \(0.290309\pi\)
\(774\) 0 0
\(775\) −3392.62 −0.157247
\(776\) 0 0
\(777\) −9047.77 2156.01i −0.417744 0.0995450i
\(778\) 0 0
\(779\) −20072.8 −0.923212
\(780\) 0 0
\(781\) 4853.41i 0.222367i
\(782\) 0 0
\(783\) 6560.96 0.299450
\(784\) 0 0
\(785\) −2988.20 −0.135864
\(786\) 0 0
\(787\) 17169.8i 0.777684i 0.921304 + 0.388842i \(0.127125\pi\)
−0.921304 + 0.388842i \(0.872875\pi\)
\(788\) 0 0
\(789\) −15981.3 −0.721102
\(790\) 0 0
\(791\) −16948.9 4038.78i −0.761861 0.181545i
\(792\) 0 0
\(793\) −17524.3 −0.784747
\(794\) 0 0
\(795\) 17044.0 0.760362
\(796\) 0 0
\(797\) 874.370 0.0388604 0.0194302 0.999811i \(-0.493815\pi\)
0.0194302 + 0.999811i \(0.493815\pi\)
\(798\) 0 0
\(799\) 20029.6i 0.886854i
\(800\) 0 0
\(801\) 9700.58i 0.427906i
\(802\) 0 0
\(803\) 1634.63i 0.0718365i
\(804\) 0 0
\(805\) 1704.76 7154.10i 0.0746398 0.313228i
\(806\) 0 0
\(807\) 21588.8i 0.941712i
\(808\) 0 0
\(809\) 33010.4 1.43459 0.717296 0.696769i \(-0.245380\pi\)
0.717296 + 0.696769i \(0.245380\pi\)
\(810\) 0 0
\(811\) 20230.9i 0.875958i 0.898985 + 0.437979i \(0.144306\pi\)
−0.898985 + 0.437979i \(0.855694\pi\)
\(812\) 0 0
\(813\) 11996.5i 0.517509i
\(814\) 0 0
\(815\) 29879.0 1.28419
\(816\) 0 0
\(817\) 12577.7i 0.538603i
\(818\) 0 0
\(819\) 13078.1 + 3116.41i 0.557981 + 0.132962i
\(820\) 0 0
\(821\) 4732.53i 0.201177i 0.994928 + 0.100589i \(0.0320726\pi\)
−0.994928 + 0.100589i \(0.967927\pi\)
\(822\) 0 0
\(823\) 41290.9i 1.74886i 0.485152 + 0.874430i \(0.338764\pi\)
−0.485152 + 0.874430i \(0.661236\pi\)
\(824\) 0 0
\(825\) 179.415i 0.00757141i
\(826\) 0 0
\(827\) −21503.5 −0.904170 −0.452085 0.891975i \(-0.649320\pi\)
−0.452085 + 0.891975i \(0.649320\pi\)
\(828\) 0 0
\(829\) −36515.6 −1.52984 −0.764922 0.644123i \(-0.777223\pi\)
−0.764922 + 0.644123i \(0.777223\pi\)
\(830\) 0 0
\(831\) −18348.7 −0.765958
\(832\) 0 0
\(833\) 4844.46 9587.77i 0.201501 0.398795i
\(834\) 0 0
\(835\) −1077.09 −0.0446399
\(836\) 0 0
\(837\) 7693.00i 0.317693i
\(838\) 0 0
\(839\) −17897.1 −0.736443 −0.368222 0.929738i \(-0.620033\pi\)
−0.368222 + 0.929738i \(0.620033\pi\)
\(840\) 0 0
\(841\) −34659.2 −1.42110
\(842\) 0 0
\(843\) 18842.4i 0.769830i
\(844\) 0 0
\(845\) −45821.5 −1.86545
\(846\) 0 0
\(847\) 23524.6 + 5605.72i 0.954326 + 0.227408i
\(848\) 0 0
\(849\) 2683.07 0.108460
\(850\) 0 0
\(851\) 6250.99 0.251799
\(852\) 0 0
\(853\) 35335.7 1.41837 0.709187 0.705021i \(-0.249062\pi\)
0.709187 + 0.705021i \(0.249062\pi\)
\(854\) 0 0
\(855\) 7089.50i 0.283574i
\(856\) 0 0
\(857\) 2.07218i 8.25955e-5i −1.00000 4.12977e-5i \(-0.999987\pi\)
1.00000 4.12977e-5i \(-1.31455e-5\pi\)
\(858\) 0 0
\(859\) 38481.3i 1.52848i −0.644930 0.764241i \(-0.723114\pi\)
0.644930 0.764241i \(-0.276886\pi\)
\(860\) 0 0
\(861\) −14646.3 3490.09i −0.579726 0.138144i
\(862\) 0 0
\(863\) 2091.21i 0.0824861i −0.999149 0.0412431i \(-0.986868\pi\)
0.999149 0.0412431i \(-0.0131318\pi\)
\(864\) 0 0
\(865\) −4344.88 −0.170787
\(866\) 0 0
\(867\) 11796.5i 0.462088i
\(868\) 0 0
\(869\) 4300.36i 0.167871i
\(870\) 0 0
\(871\) −55176.8 −2.14649
\(872\) 0 0
\(873\) 11027.8i 0.427531i
\(874\) 0 0
\(875\) −26230.0 6250.39i −1.01341 0.241488i
\(876\) 0 0
\(877\) 15517.2i 0.597467i −0.954337 0.298734i \(-0.903436\pi\)
0.954337 0.298734i \(-0.0965642\pi\)
\(878\) 0 0
\(879\) 20709.5i 0.794670i
\(880\) 0 0
\(881\) 47831.0i 1.82913i −0.404434 0.914567i \(-0.632531\pi\)
0.404434 0.914567i \(-0.367469\pi\)
\(882\) 0 0
\(883\) 50738.9 1.93375 0.966874 0.255254i \(-0.0821590\pi\)
0.966874 + 0.255254i \(0.0821590\pi\)
\(884\) 0 0
\(885\) −1816.70 −0.0690032
\(886\) 0 0
\(887\) −6294.31 −0.238267 −0.119133 0.992878i \(-0.538012\pi\)
−0.119133 + 0.992878i \(0.538012\pi\)
\(888\) 0 0
\(889\) 8969.89 37642.5i 0.338403 1.42012i
\(890\) 0 0
\(891\) 406.834 0.0152968
\(892\) 0 0
\(893\) 47372.9i 1.77522i
\(894\) 0 0
\(895\) 17274.3 0.645159
\(896\) 0 0
\(897\) −9035.51 −0.336329
\(898\) 0 0
\(899\) 69236.5i 2.56860i
\(900\) 0 0
\(901\) −16731.3 −0.618647
\(902\) 0 0
\(903\) 2186.92 9177.46i 0.0805936 0.338213i
\(904\) 0 0
\(905\) 13859.7 0.509074
\(906\) 0 0
\(907\) −28639.8 −1.04848 −0.524239 0.851571i \(-0.675650\pi\)
−0.524239 + 0.851571i \(0.675650\pi\)
\(908\) 0 0
\(909\) 13621.2 0.497015
\(910\) 0 0
\(911\) 28088.6i 1.02153i −0.859719 0.510767i \(-0.829361\pi\)
0.859719 0.510767i \(-0.170639\pi\)
\(912\) 0 0
\(913\) 2308.64i 0.0836856i
\(914\) 0 0
\(915\) 6931.54i 0.250437i
\(916\) 0 0
\(917\) 3833.34 16086.7i 0.138046 0.579313i
\(918\) 0 0
\(919\) 143.830i 0.00516270i 0.999997 + 0.00258135i \(0.000821670\pi\)
−0.999997 + 0.00258135i \(0.999178\pi\)
\(920\) 0 0
\(921\) 23801.9 0.851574
\(922\) 0 0
\(923\) 77940.4i 2.77946i
\(924\) 0 0
\(925\) 1993.29i 0.0708528i
\(926\) 0 0
\(927\) 8977.10 0.318066
\(928\) 0 0
\(929\) 12743.0i 0.450037i −0.974355 0.225018i \(-0.927756\pi\)
0.974355 0.225018i \(-0.0722442\pi\)
\(930\) 0 0
\(931\) −11457.8 + 22676.4i −0.403347 + 0.798271i
\(932\) 0 0
\(933\) 18647.0i 0.654313i
\(934\) 0 0
\(935\) 1672.82i 0.0585101i
\(936\) 0 0
\(937\) 1125.79i 0.0392509i −0.999807 0.0196254i \(-0.993753\pi\)
0.999807 0.0196254i \(-0.00624737\pi\)
\(938\) 0 0
\(939\) −10398.4 −0.361382
\(940\) 0 0
\(941\) 8478.11 0.293707 0.146854 0.989158i \(-0.453085\pi\)
0.146854 + 0.989158i \(0.453085\pi\)
\(942\) 0 0
\(943\) 10118.9 0.349436
\(944\) 0 0
\(945\) −1232.67 + 5172.92i −0.0424324 + 0.178069i
\(946\) 0 0
\(947\) −40984.8 −1.40636 −0.703182 0.711010i \(-0.748238\pi\)
−0.703182 + 0.711010i \(0.748238\pi\)
\(948\) 0 0
\(949\) 26250.3i 0.897915i
\(950\) 0 0
\(951\) −3833.57 −0.130717
\(952\) 0 0
\(953\) −32319.7 −1.09857 −0.549285 0.835635i \(-0.685100\pi\)
−0.549285 + 0.835635i \(0.685100\pi\)
\(954\) 0 0
\(955\) 1247.02i 0.0422539i
\(956\) 0 0
\(957\) −3661.49 −0.123677
\(958\) 0 0
\(959\) 25952.2 + 6184.21i 0.873870 + 0.208236i
\(960\) 0 0
\(961\) 51391.7 1.72508
\(962\) 0 0
\(963\) 8632.48 0.288866
\(964\) 0 0
\(965\) −3108.86 −0.103707
\(966\) 0 0
\(967\) 186.698i 0.00620870i −0.999995 0.00310435i \(-0.999012\pi\)
0.999995 0.00310435i \(-0.000988147\pi\)
\(968\) 0 0
\(969\) 6959.44i 0.230722i
\(970\) 0 0
\(971\) 38237.7i 1.26375i −0.775068 0.631877i \(-0.782285\pi\)
0.775068 0.631877i \(-0.217715\pi\)
\(972\) 0 0
\(973\) 8590.45 36050.1i 0.283039 1.18778i
\(974\) 0 0
\(975\) 2881.20i 0.0946382i
\(976\) 0 0
\(977\) −39341.4 −1.28827 −0.644137 0.764910i \(-0.722783\pi\)
−0.644137 + 0.764910i \(0.722783\pi\)
\(978\) 0 0
\(979\) 5413.62i 0.176731i
\(980\) 0 0
\(981\) 6456.16i 0.210122i
\(982\) 0 0
\(983\) 33595.5 1.09006 0.545031 0.838416i \(-0.316518\pi\)
0.545031 + 0.838416i \(0.316518\pi\)
\(984\) 0 0
\(985\) 44568.4i 1.44169i
\(986\) 0 0
\(987\) −8236.82 + 34566.1i −0.265634 + 1.11474i
\(988\) 0 0
\(989\) 6340.59i 0.203862i
\(990\) 0 0
\(991\) 13906.6i 0.445770i 0.974845 + 0.222885i \(0.0715475\pi\)
−0.974845 + 0.222885i \(0.928453\pi\)
\(992\) 0 0
\(993\) 10053.4i 0.321284i
\(994\) 0 0
\(995\) 18580.4 0.591997
\(996\) 0 0
\(997\) −37389.4 −1.18770 −0.593848 0.804577i \(-0.702392\pi\)
−0.593848 + 0.804577i \(0.702392\pi\)
\(998\) 0 0
\(999\) −4519.91 −0.143147
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 1344.4.p.b.223.1 yes 16
4.3 odd 2 inner 1344.4.p.b.223.10 yes 16
7.6 odd 2 1344.4.p.a.223.16 yes 16
8.3 odd 2 1344.4.p.a.223.8 yes 16
8.5 even 2 1344.4.p.a.223.15 yes 16
28.27 even 2 1344.4.p.a.223.7 16
56.13 odd 2 inner 1344.4.p.b.223.2 yes 16
56.27 even 2 inner 1344.4.p.b.223.9 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1344.4.p.a.223.7 16 28.27 even 2
1344.4.p.a.223.8 yes 16 8.3 odd 2
1344.4.p.a.223.15 yes 16 8.5 even 2
1344.4.p.a.223.16 yes 16 7.6 odd 2
1344.4.p.b.223.1 yes 16 1.1 even 1 trivial
1344.4.p.b.223.2 yes 16 56.13 odd 2 inner
1344.4.p.b.223.9 yes 16 56.27 even 2 inner
1344.4.p.b.223.10 yes 16 4.3 odd 2 inner